Axion Quasiparticles for Axion Dark Matter Detection
Jan Schütte-Engel, David J. E. Marsh, Alexander J. Millar, Akihiko Sekine, Francesca Chadha-Day, Sebastian Hoof, Mazhar Ali, Kin-Chung Fong, Edward Hardy, Libor ?mejkal
PPrepared for submission to JCAP
Axion Quasiparticles for Axion DarkMatter Detection
Jan Sch¨utte-Engel, a,b,c
David J. E. Marsh, d Alexander J.Millar, e,f
Akihiko Sekine, g Francesca Chadha-Day, h SebastianHoof, d Mazhar Ali, i Kin-Chung Fong, j Edward Hardy, k andLibor ˇSmejkal l,m a Department of Physics, University of Illinois at Urbana-Champaign,Urbana, IL 61801, U.S.A. b Illinois Center for Advanced Studies of the Universe,University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A. c Hamburg University, 22761 Hamburg, Germany d Institut f¨ur Astrophysik, Georg-August-Universit¨at G¨ottingen, Friedrich-Hund-Platz 1,37077 G¨ottingen, Germany e The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, StockholmUniversity, AlbaNova, 10691 Stockholm, Sweden f Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstulls-backen 23, 10691 Stockholm, Sweden g RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan h Department of Physics, University of Durham, South Rd,Durham DH1 3LE, United Kingdom i Max Planck Institute of Microstructure Physics, Weinberg 2, 06120 Halle — Germany j Raytheon BBN Technologies, Quantum Engineering and Computing,Cambridge, Massachusetts 02138, USA k Mathematical Sciences, The University of Liverpool,Liverpool, L69 7ZL, United Kingdom l Institut f¨ur Physik, Johannes Gutenberg Universit¨at Mainz, 55128 Mainz, Germany m Institute of Physics, Czech Academy of Sciences, Cukrovarnick´a 10, 162 00,Praha 6, Czech Republic a r X i v : . [ h e p - ph ] F e b -mail: [email protected], [email protected] Abstract. ontents
Discussion and Conclusions 73
A Antiferromagnetic Resonance and Magnons for Particle Physicists 78
A.1 Effective Field Theory of AFMR 78A.1.1 AFMR in Cartesian Coordinates 79A.1.2 AFMR in Polar Coordinates 82A.1.3 Longitudinal Spin Waves in the Heisenberg Model 83A.2 The Landau-Lifshitz Equations 85
B Axion Dark Matter and the Millielectronvolt Range 87C Comparison to earlier results 89
The quantum chromodynamics (QCD) axion [1–3] solves the charge-parity ( CP ) problemof the strong nuclear force [4–6], and is a plausible candidate [7–9] to compose thedark matter (DM) in the cosmos [10]. The axion mass is bounded from above [11–13] and below [14, 15] by astrophysical constraints (for reviews, see Refs. [16–19], andAppendix B), placing it in the range1 peV (cid:46) m a (cid:46)
20 meV . (1.1)The local DM density is known from stellar motions in the Milky Way [20]. As-suming axions comprise all the (local) DM, the axion number density is given by n a = ρ loc /m a . Due to the very small axion mass, the number density is very large and axionscan be modelled as a coherent classical field, φ . The field value is: φ = Φ cos( m a t ) , (1.2)where Φ is Rayleigh-distributed [21, 22] with mean √ ρ loc /m a and linewidth ∆ ω/ω ∼ − given by the Maxwell-Boltzmann distribution of axion velocities around the localgalactic circular speed, v loc ≈
200 km / s (see e.g. refs. [21, 23]).Axions couple to electromagnetism via the interaction L = g aγ φ E · B . Thus, inthe presence of an applied magnetic field, B , the DM axion field in Eq. (1.2) acts as asource for the electric field, E . This is the inverse Primakoff process for axions, and leadsto axion-photon conversion in a magnetic field. The rate of axion-photon conversion de-pends on the unknown value of the coupling g aγ and happens at an unknown frequency ω = m a ± ∆ ω . For the QCD axion (as opposed to a generic “axion like particle” [16])the mass and coupling are linearly related, g aγ ∝ m a , although different models for thePeccei-Quinn [1] charges of fundamental fermions predict different values for the con-stant of proportionality. The two historical reference models of Kim-Shifman-Vainshtein-Zhakarov (KSVZ) [24, 25] and Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) [26, 27] span– 1 – narrow range, while more recent generalisations with non-minimal particle contentallow for more variation [19, 28, 29].The axion-photon coupling g aγ is constrained by a large number of null-results fromexperimental searches and astrophysical considerations [20]. For experimentally allowedvalues of ( m a , g aγ ), and accessible magnetic field strengths, the photon production ratein vacuum is unobservably small. The power can be increased in two basic ways. If theconversion happens along the surface of a magnetized mirror, then the produced photonscan be focused onto a detector [30]. This approach is broadband, and does not dependon the axion mass. Reaching sensitivity to the QCD axion requires very large mirrors,very sensitive detectors, and control over environmental noise. Alternatively, the signalcan be resonantly or coherently enhanced (e.g. Refs. [31, 31–40]). These approaches arenarrow band, and require tuning to the unknown DM axion frequency.Depending on the model of early Universe cosmology, and the evolution of the ax-ion field at high temperatures T (cid:29) − x Fe x ) Se , the quest to realiserelated materials in the lab has picked up incredible pace. A currently favoured candi-date Mn Bi Te [43], is, however, yet to be fabricated successfully. AQ materials allowthe possibility to explore aspects of axion physics in the laboratory [44]. The AQ res-onance hyrbidises with the electric field forming an axion-polariton [42]. The polaritonfrequency is of order the AF anisotropy field, with typical values O (1 meV), and is tune-able with applied static field B [41]. This proposal opens the possibility for large volumeTHz resonance, easily tuneable with an applied magnetic field, thus overcoming the firsthurdle to detection of meV axions. The proposal makes use of the current interest inmanufacture of low noise, high efficiency single photon detectors (SPDs) in THz [45].The development of such detectors has benefits for sub mm astronomy and cosmology,as well as application to other DM direct detection experiments [30].The present paper expands on the ideas outlined in Paper I with more in depthmodelling and calculations. A guide to the results is given below. Axion Quasiparticle Materials
We begin with a detailed treatment of the materials science, and outline a scheme toprove the existence of AQs in TMIs, and measure their parameters.• We introduce the basic model for the equations coupling the electric field and theAQ. There are two parameters that determine the model: the AQ mass, m Θ , andthe decay constant, f Θ , as summarised in Section 2.1.– 2 – Next in Section 2.2, we clarify the microscopic model for AQs in TMIs. We beginwith the symmetry criteria, followed by a microscopic model based on the DiracHamiltonian. The AQ is the longitudinal fluctuataion of the antiferromagneticorder parameter in the Hubbard model. The Appendix summarises the relatedphenomenon of antiferromagnetic resonance and transverse magnons in the effec-tive field theory of the Heisenberg model.• Both m Θ and f Θ can be estimated from known material properties. We consider(Bi − x Fe x ) Se , the candidate material from Paper I and Ref. [42], and also themore recent candidate material Mn Bi Te [43]. The results of this study are givenin Tables 4 and 5.• We next consider sources of loss. The largest sources of loss are identified to beconductive losses to the electric field, and crystal and magnetic domain inducedline broadening for the AQ. The loss model is summarised in Table 6.• Using the model thus developed, we present a computation of the transmissionspectrum of an AQ material. The spectrum shows two peaks due to the mixingof the electric field and the AQ, the locations of which can be used to measurethe parameters m Θ and f Θ . The width of the resonances provides a measurementof the loss parameters on resonance, which cannot otherwise be identified fromexisting measurements. Such a measurement can be performed using THz timedomain spectroscopy [46]. The procedure is shown schematically in Fig. 6 Axion Dark Matter Detection • Axion DM acts as a source to the AQ model developed in the previous sections.Axion-photon conversion in a magnetic field sources photons, which hybridize withthe AQ forming polaritons, and thus acquire an effective mass. It is shown thatthis model can be treated in the same way as a dielectric haloscope [47]. Theresonance in the polariton spectrum leads to an effective refractive index n < β ( ω ), for a range of plausible values forthe model parameters, losses, and material thickness. See, for example, Fig. 17.• The power enhancement is driven by the material thickness, d , which should ex-ceed the wavelength of emitted photons. When losses are included, we identifya maximum thickness above which the power enhancement decreases due to thefinite skin-depth. See Fig. 19.• We perform forecasts for the limits on axion DM parameter space, ( m a , g aγ ), thatcan be obtained for a range of plausible material and THz detector parameters. Weidentify pessimistic and optimistic possibilities for the discovery reach, summarisedin Fig. 23.We use units (cid:126) = c = k B = 1 throughout most of the text, in combination with SIwhere appropriate. – 3 – Axion Quasiparticle Materials
Axion quasiparticles (AQs) are defined, for our purposes, as a degree of freedom, denotedby δ Θ, coupled to the electromagnetic Chern-Simons term: S topo = απ Z d x ( δ Θ + Θ ) E · B , (2.1)where Θ is the constant electromagnetic Chern-Simons term, equal to zero in ordinaryinsulators and π in topological insulators (TIs). In these materials, surface currents areaccounted for by inclusion of a non-zero value for Θ (the topological magneto-electriceffect due to the Hall conductivity [48, 49]). In the presence of a dynamical AQ field, δ Θ, the static vacuum value Θ is allowed to take on a continuum of values between 0and π . The total axion field is denoted by Θ = δ Θ + Θ . We review these conceptsfurther below, for a detailed presentation see Ref. [50].The dynamics of the AQs are described by [42, 51] S Θ = f Z d x h ( ∂ t δ Θ) − ( v i ∂ i δ Θ) − m δ Θ i , (2.2)where f Θ , v i and m Θ are the stiffness, velocity and mass of the AQ. The velocities v i are of the order of the spin wave speed in typical antiferromagnets, v s ∼ − c [52]. Inthe coupled equations of motion for the electric field and the AQ (see Section 3.2), f Θ enters in the combination b = απ √ B e √ (cid:15)f Θ = 1 . (cid:18) (cid:15) (cid:19) / (cid:18) B e (cid:19) (cid:18)
70 eV f Θ (cid:19) . (2.3)In addition to the action for the AQ we consider electromagnetic fields governed byMaxwell’s equations in media, which depend on the complex valued dielectric function,˜ (cid:15) = (cid:15) + i(cid:15) = (cid:15) + iσ/ω (where σ is the conductivity), and magnetic susceptibility, χ m . Where there is no room for confusion we use (cid:15) = (cid:15) in some of the following. Thephenomenological model also requires the specification of a loss matrix, Γ . The idea to realise axion electrodynamics in solids was originally developed by Wilczek [44]who, however, could not identify a magnetic solid that breaks parity and time-reversalwhile preserving its combination: as we will see, necessary conditions for AQs. Recentdevelopments in nonmagnetic and magnetic electronic topological phases of matter, andstudy of the topological magnetoelectric effect associated with the Chern-Simons term inmagnetoelectrics [53, 54] have led to the identification of several routes to realise axionelectrodynamics in energy bands of magnetic topological insulators and Dirac quasipar-ticle antiferromagnets. The electronic, magnetic, topological energy bands can couple Note that we use the Lorentz-Heaviside convention, where 1 T ≈
195 eV . – 4 –o spin fluctuations, and thus generate a dynamical axion phase on the electromagneticChern-Simons term.In this section we discuss the Dirac quasiparticle model of AQs in electronic energybands. We compare the symmetry criteria for static and dynamical axion topologicalantiferromagnets, and discuss the most prominent material candidates. The topological Θ term is called also an axion angle as it can take any value between 0and 2 π . The operations of charge conjugation C , parity P (known as inversion symmetryin condensed matter, a terminology we adopt throughout this section to distinguish itfrom other types of parity operation in solids), and time-reversal T are the discretesymmetries constraining the values of Θ, and which define the properties of fundamentalforces in nature via the CPT theorem. CP breaking means that the physical laws are notinvariant under combination of interchanging particle with its antiparticle with invertingthe spatial coordinates. If Θ = 0 , π , then CP is violated. The combined CPT symmetryis believed to be preserved (i.e. the so called
CPT theorem) and thus the violation of CP implies the violation of T symmetry, i.e. the reversal of the time coordinate, andthus particle motion. Realisation of CP -broken theory and axion electrodynamics withnon-quantized axion angle can be achieved in materials with broken T symmetry [55–57]. In materials, magnetic ordering can break the T symmetry. In this section we willdiscuss the symmetries of magnetic axion insulators which exhibit nonzero pseudoscalaraxion quasiparticle Θ (we use capital letter to label the solid state quasiparticle axionto distinguish it from the DM axion).The nonzero axion response can be find in subgroup of conventional and topologicalmagnetoelectric materials. The conventional magneto-electric polarizability tensor isdefined as[58, 59]: α ij = ( ∂P i /∂B j ) E = ( ∂M j /∂E i ) B . (2.4)Here P i , B j , M j , and E i are electric polarisation, magnetic field, magnetization, andelectric field. The magnetoelectric polarizability tensor can be decomposed as [58]: α ij = ˜ α ij + Θ e πh δ ij , (2.5)where the first term is the non-diagonal part of the tensor arising from spin, orbitaland ionic contribution [60]. The second term is the diagonal pseudoscalar part of thecoupling related to the axion angle Θ.We will now review symmetry criteria for nonzero axion quasiparticle Θ. In solidstate potentials, discrete symmetries impose severe constraint on the existence and formof the topological axion angle [61], and provide robust insight into the topological char-acterisation of the energy bands [62–65]. The topological classification assigns two insu-lators into the same category as long as it is possible to connect the two correspondingHamiltonians by a continuous deformation without closing an energy gap and whilepreserving all symmetries [53, 54]. – 5 – Axion quasiparticle responseNon-quantised Quantised Magnetoelectric point groups (Tab. 1) (Antiferromagnetic) topological insulators Axion insulators Dynamical axion quasiparticleConventional diagonal magneto electric Bulk Dirac quasiparticles Axion odd symmetry (Tab. 2)Spin dynamics ( t ) T
32 ( m ) 422 4 m (4 /m m m ) Cr O [73]6 m m
622 6 m /m m m ) (cid:16) α xx − α xx (cid:17)
4, 42 m (4 /m ), 4 mm m /m mm ) (cid:16) α xx α xx α xx (cid:17)
23, ( m ), 432, 4 m ( m m ) Table 1 : Table of antiferromagnetic and ferromagnetic nonquantized axion magneto-electric symmetry groups and candidate material. In the first column we show only thediagonal part of magnetoelectric polarizability tensor α ij . The symbols 1 and 1 markspatial inversion and time-reversal symmetry, respectively. FM and AF MPG refers toferromagnetic and antiferromagnetic magnetic point group [75].mangetic point groups which allow for the nonzero diagonal magnetoelectric responseelements [73, 74]. We also list whether the material has allowed ferromagnetism (FM,12 magnetic point groups) or is enforced by the point group symmetry to be antiferro-magnetic (AF, 28 point groups) [75] together with several material examples. We seethat the magnetoelectric response can be anisotropic what was confirmed experimen-tally [76]. Note that the third row of the Tab. 1 gives zero trace. This analysis excludesfrom pseudoscalar magnetoelectric coupling materials which do exhibit only tracelessmagnetoelectric coupling. When the system breaks P and T but preserves its combi-nation, in can host also bulk Dirac quasiparticles [68]. We mark the PT symmetricmagnetoelectric pseudoscalar point groups by brackets in Tab. 1.In topological insulators, such as Bi Se (the nonmagnetic phase of crystal shown inFig. 2(a)) and Bi Te , the presence of T symmetry in combination with nontriivial bandinversion ensures the axion angle Θ to be π [77], requires zero surface Hall conductivity,and the topological magneto-electric effect [78]. The topological magneto-electric effectin topological insulators refers to a quantized magneto-electric response, and has beenobserved also by magneto-optical measurements [77]. In fact, the quantization of Θin non-magnetic topological insulators can be taken as defining property of topologicalinsulators [78]. Recently, also antiferromagnetic topological insulator [79] was foundin MnBi Te [80]. Antiferromagnetic topological insulator state is protected by time-reversal symmetry coupled with partial unit cell translation t as we show in Fig. 2(b).The static axion insulators are magnetic topological insulators, such as MnBi Te [81,82], which break T symmetry via the presence of a magnetic ion (in this case, Mn). How-ever, they exhibit axion response with Θ = π , protected by the presence of axion oddsymmetries such as inversions, see inversion centre in Fig. 2(b), or crystalline symmetries.The axion-odd symmetries are the symmetries which reverse the sign of Θ and supportthe so called Z classification [62, 63, 83]. Among the additional axion-odd symmetriesare improper rotations, and antiunitary proper rotations (for instance rotation combined– 7 – a) (b) TeBi T t
Figure 2 : (a) Crystal structure of topological insulator Bi Se consist from quintuplelayers forming rhombohedral unit cell. Antiferromagnetism of the magnetically doped(Fe,Bi) Se breaks spatial inversion P and time-reversal T symmetry, but preserve com-bined PT symmetry. (b) Crystal structure of intrinsic antiferromagnetic axion insulatorMnBi Te . The quantized value of axion angle is protected by the inversion symmetry P (we maek two inversion symmetry points in the lattice). The system exhibits alsopartial unit cell translation t combined with time-reversal symmetry T .with time-reversal). In the Table 2, we list axion angle quantizing symmetry operations, g . We decompose the symmetry operation g = g k ◦ g ⊥ into the parts g k and g ⊥ which areparallel and perpendicular (in the surface plane) to the given surface normal ˆ z [61]. Weremark that we list the point group operation, but in general we need to pay attentionto the nonsymmoprhic partial translations of the group, for details see [61].Finally, the dynamical axion insulator allows for nonquantized dynamical axionangle. The dynamics of the axion angle was suggested to be induced by chiral magneticeffect, antiferromangetic resonance [72], longitudinal spin fluctuations [71] in an anti-ferromagnet or spin fluctuations in paramagnetic state [84]. In Fig. 2(a), we show anexample of lattice with dynamical axion insulator state - Fe-doped (Bi − x Fe x ) Se with PT symmetric crystal. Here, the antiferromagnetism breaks the inversion and time-reversal symmetries of the Bi Se crystal. The symmetry breaking is desribed by massterm M which corresponds to the band-gap in surface state. The combined PT sym-metry is in the (Bi − x Fe x ) Se crystal preseved and enforces Kramers degenerate bands.This can be seen by acting PT symmetry on the Bloch state to show that these twostates have the same energy and are orthoghonal [68, 69, 85–87]. The presence of PT – 8 – g k g ⊥ Operations reversing ˆ z M z M z EP C S , , C , , ¯ C T M d T Operations preserving ˆ z ET E ETC , , , T C , , TM d M d Table 2 : Axion angle quantizing symmetries. E , P , M z , M d , C , , , , S , , , and T markunitary symmmetry operations of identity, inversion, mirror parallel and perpendicularto surface normal ˆ z , rotational axis, improper rotations, and time-reversal, respectively.Overbar marks inversion. Adapted after [61].allows for antiferromagnetic Dirac quasipaticles [68, 69] with plethora of unconventionaland practically useful response such as large anisotropic magnetoresistance [68, 88]. Wediscuss the material physics requirements for dynamical axion insulator in the sectionfollowed by section on minimal effective model of a dynamical axion insulator. In this section we list requirements for a dynamical axion insulator which is also suitablefor dark axion detection [89]. In addition to constraints comming from requiring dynam-ical axion quasiparticles, we need to ensure strong coupling of the Θ magneto-electricresponse to the fluctuations of the magnetic order parameter. The concept was originallydeveloped for the longitudinal fluctuations in the N´eel order parameter in magneticallydoped topological insulators ((Bi − x Fe x ) Se in Ref. [42]) and recently extended intothe instrinsic antiferromagnet Mn Bi Te [72]. This dynamical axion field is quite weakdue to the low magnetoelectric coupling and trivial electronic structure in conventionalmaterials such as Cr O [90] and BiFeO [91] with Θ = 10 − and 10 − , respectively,see Tab. 3. The dynamical axion effect (i.e. a large Θ response to external perturba-tions) can be enhanced in the proximity of the topological phase transitions[72]. Wenow summarise the material criteria for a dynamical axion quasiparticles for detectingdark matter axion:• Nonzero dynamical axion angle. The material symmetry allows for dynamicalaxion insulator state and axion spin density wave [71, 72, 84] with mass in therange of meV. This is one of the main advantage of using axion quasiparticles inantiferromagnets for detecting light and weakly interacting DM axions [92].• Large bulk band-gap [56]. The material is in bulk semicondcuting or insulatingwith a large bulk band-gap, without disturbing bulk metallic states. In turn itslow energy physics is governed solely by the axion coupling.– 9 – Topological mass term M should be of order of dynamical axion fluctuation mass m a to ensure resonance with DM axion [71, 92].• Large fluctiation in axion angle. This can be achieved close to the magnetic andtopological phase transition as δ Θ( r , t ) = δM /g , and 1 /g ∼ / M (0) [72]. Thetopological phase transition should be approached from the topological side. Prac-tically, one can tune this mass term by alloying. The alloying can effectively tunethe strength of the spin orbit interaction. However, the proximity close to themagnetic transition can compromise narrow linewidth, see next point.• Trade-off among narrow linewidth and sensitivity. Narrow linewidth of the axionrespones where the thermal fluction and scattering are supressed. This imposistmeperature constraints (i.e. T (cid:28) T N , T (cid:28) m ). In contrast, enhanced responseclose the mangetic phase transition could enhance sensitivity.• Robust magnetic ordering with elevated critical (N´eel) temperature.• As we will see in chapter about power output we need large spin-flop fields ( > PT sym-metric antiferromagnetism seems to be favourable over ferromagnetism as it naturaly– 10 – hase Material class T C (K) ∆ [meV] Θ Magnetoelectric BiFeO
643 950 0.9e-4 [91]Cr O
343 1300 1.3 e-3 [90]Magnet/TIs CrI /Bi Se /MnBi Se >
10 5.6 π [97]Intrinsic P AFs MnBi Te < < π EuIn(Sn) As(P) < π Doped TIs Cr(Fe)-Bi Se ∼ ∼
30 nonquantizedIntrinsic PT AFs Mn(Eu) Bi Te ∼
50 0 . π [97] Table 3 : Table of magnetoelectric insulating material classes and candidates. FM (AF)marks (anti)ferromagnetism. T C critical temperature. ∆ bulk band gap.provides for Dirac quasiparticles with tunable axion quasiparticles masses, longitudinalspin waves, larger spin-flop fields, elevated N´eel temperatures, possibility to combinechemistry required for magnetism and spin-orbit coupling in single material platform.We list some of the promissing building block materials and systems for dynamical axionquasiparticles in the Tab. 3. Besides listing materials which are directly dynamical axioninsulators we added also materials which can be used as starting configurations to buildthe dynamical axion insulator, for instance, by alloying of the static axion insulators.We emphasize that the bulk energy bands encode the information about the dynam-ical axion insulator response, and its surface states [71]. We can see this on expressionfor the intrinsic magnetoelectric susceptibility, axion coupling, can be calculated in theBloch representation as [71]:Θ = − π Z BZ d k(cid:15) αβγ Tr (cid:20) A α ∂ β A γ − i A α A β A γ (cid:21) . (2.8)Here we explictily see the axion angle relation to the non-abelian Berry connection A α,nm ( k ) = h u n k | i∂ k α | u m i constructed from the Bloch functions | u n k i . The trace isover occupied valence bands.The first-principle calculations of the axion angle is reserch topic on its own [57, 61].For the sake of brevity we will adopt here simpler approach. We can use first-prinicplecalculations and symmetry analysis to identify and parametrize low energy effectiveHamiltonian for which the calculation of axion angle and its dynamical response isnumerically less demanding. We will now describe dynamical axion quasiparticle modelwhich is applicable to Fe-doped Bi Se [71] and intrinsic antiferromagnet Mn Bi Te [72]and also heterostructures [97]. We can derive the minimal model of dynamical axion insulator starting from the Diracquasiparticle model for the bulk states of topological insulator Bi Se [99]. The lowenergy physics can be captured by four-band Hamiltonian in the basis of bonding and– 11 –ntibonding Bi p z states | P − z , ↑ ( ↓ ) i and (cid:12)(cid:12) P + z , ↑ ( ↓ ) (cid:11) [71, 72, 99]: H Dirac = (cid:15) ( k ) + X a =1 d a ( k )Γ a . (2.9)Here Γ a refer to the Dirac matrices representation:Γ (1 , , , , = ( σ x ⊗ s x , σ x , ⊗ s y , σ y ⊗ I × , σ z ⊗ I × , σ x ⊗ s z ) (2.10)in the basis ( | P + z , ↑i , | P + z , ↓i , | P − z , ↑i , | P − z , ↓i ). σ and s are orbital and spin Paulimatrices. The 4 × a satisfy the Clifford algebra { Γ a , Γ b } = 2 δ ab with Γ =Γ Γ Γ Γ . This model can be tuned to the trivial (Θ = 0) or topological insulator state(Θ = π ). To induce nonzero δ Θ and dynamical axion state we need to add P and T symmetry breaking terms due to the antiferromagnetism.The crystal momentum dependent coefficients take the form: d , , ( k ) = A , , ( k ) + m x,y,z , (2.11) (cid:15) ( k ) = C + 2 D + 4 D − D cos k z − D (cos k x + cos k y ) (2.12) d ( k ) = M ( k ) = M − B − B + 2 B cos k z + 2 B (cos k x + cos k y ) (2.13) d ( k ) = M . (2.14)Here the fourth term M ( k ) controls the topological phase transition from the trivial totopological insulator, is invariant under T , and we denote M ( k = 0) = M . The topo-logical insulating phase is achieved when M, B , B > m x,y,z and M (a CP -odd chiral mass term). We see that the spatialinversion P = σ z ⊗ I x and time-reversal operators T = iI x ⊗ s y K do not commutewith the Hamiltonian, while their combination does. Here K is complex conjugation.Only the last mass term M induces linear perturbations to Θ as we will showfurther, and without loss of generality one can set m x,y,z = 0. In turn, the M term opensa surface band gap in the surface states Dirac Hamiltonian as we show in Fig. 3. The A, B, C , D , and masses M, M constants are material dependent and can be determinedby fitting the electronic structure calculated from the first-principles [71, 97, 99, 100].We also remark, that for calculating the complete response of the material we need toknow the full periodic Hamiltonian Eq. (2.14).When its sufficient to study small wavector excitations we can use continuum vari-ant, k , p -expansion, around momentum points X f , where q = k − X f : H f ( q ) = q x α + q y α + q z α + M α + M f α . (2.15)Here we use the standard Dirac equation basis: β = α = I − I ! , α i =1 , , = σ i − σ i ! ⇒ α = II ! . (2.16)Furthermore, the subscript f denotes the valley degree of freedom in the low-energyelectronic band of the system, and can be understood as the Dirac quasiparticle flavour .– 12 – latexit sha1_base64="OX8Q0Mq6XWofB2fYhTzlpzTU9ok=">AAAB/HicbVDLSgMxFM34rPU12qWbYBFcDRmx6EYounFZwT6gHYZMmmlDk8yYZIQy1F9x40IRt36IO//GtJ2Fth64cDjnXu69J0o50wahb2dldW19Y7O0Vd7e2d3bdw8OWzrJFKFNkvBEdSKsKWeSNg0znHZSRbGIOG1Ho5up336kSrNE3ptxSgOBB5LFjGBjpdCtiDCvTa4g8moC9iR9gAiGbhV5aAa4TPyCVEGBRuh+9foJyQSVhnCsdddHqQlyrAwjnE7KvUzTFJMRHtCupRILqoN8dvwEnlilD+NE2ZIGztTfEzkWWo9FZDsFNkO96E3F/7xuZuLLIGcyzQyVZL4ozjg0CZwmAftMUWL42BJMFLO3QjLEChNj8yrbEPzFl5dJ68zzkeffnVfr10UcJXAEjsEp8MEFqINb0ABNQMAYPINX8OY8OS/Ou/Mxb11xipkK+APn8wfXWpLv
2, and note that the spin wavespeed appears in the spatial derivatives by choice of units.
For concreteness, let us consider the AF insulator phase of (Bi − x Fe x ) Se and Mn Bi Te such that there is a single degree of freedom with M , = M , = 0, and M , =– 15 – (2 / U n k , where n k is parallel to the easy-axis anisotropy. In terms of the AF or-der parameters the AQ is given by expanding eq. (2.26), leading to δ Θ ≈ U M δn k . (2.28)Thus, we see that the AQ δ Θ is the longitudinal fluctuation in the AF order.The EFT of transverse magnons is presented in Appendix A.1, and is based on theHeisenberg model. The Heisenberg model is the strong coupling limit of the Hubbardmodel used to describe the AQ, but nonetheless it provides some insight into the physics,which we discuss briefly. The EFT describes the AF order parameter, n . Let us denotethe components of n as n k along the easy-axis and n ⊥ , , orthogonal to it. In the EFTwe have that: δn k ≈ − δn ⊥ , − δn ⊥ , . (2.29)Thus the AQ is related non-linearly to the transverse magnons of the Heisenberg EFT.In the Dirac model for the AQ, the interaction between δ Θ and electromagnetismis given entirely by the chiral anomaly, i.e. the interaction δ Θ E · B . On the other handthe Heisenberg EFT contains the spin interaction L em = µ B s · H , with s = ˙ n × n atleading order. As we have just established, however, the Heisenberg model fields are notlinearly related to the AQ in the Hubbard model with t/U (cid:28)
1. We therefore neglect theinteraction L em in our subsequent calculations based on the effective action Eq. (2.27).If only the axion, δ Θ ∝ δn k , is present, then indeed ˙ n × n = 0.However, if the AFMR fields δn ⊥ are also excited, then L em mixes the fields andleads to the Kittel shift in the frequencies of these fields ω = µ B H + p m s + v k (seeAppendix A.1). The Kittel shift would also mix the AFMR fields with the axion. Itis not clear to us how to model these two effects, the AQ and AFMR with an appliedfield, at the same time because the two descriptions are valid in opposite regimes of theHubbard model parameters. The splitting µ B H (cid:28) m s for fields H ∼ Q .In this case, one arrives at a quadratic Lagrangian for the transverse and longitudinalmagnons with coupling to external sources [93]. However, in addition to these desiredingredients there are also spinor degrees of freedom, the “holons” describing the spin-charge separation. Another possibility, which we suggest, is to generalise the N´eel orderparameter to an SU (2) doublet with the AQ a Goldstone boson associated to a Chiral U (1) subgroup under which the Dirac quasiparticles are charged. Other approaches to the longitudinal mode include Refs. [105, 106]. – 16 – able 4 : Material parameters for AQ materials. Mn Bi Te has not been experimentallyrealised, and parameters in the references are calculated ab initio , rather than measured.Values in parentheses were assumed in Ref. [42]. We assume the dielectric constants forboth materials are equal to the undoped Bi Se extrapolated to low energy [108, 109].Symbol Name (Bi − x Fe x ) Se Mn Bi Te µ B H E Exchange 1 meV [110] 0.8 meV [111] µ B H A Anisotropy 16 meV [107] 0.1 meV [111] V u.c. Unit cell volume 440 ˚A
270 ˚A U Hubbard term 3 eV [107] 3 eV [111] M Bulk band gap 0 .
03 eV (0 . t Nearest neighbour hopping a S Magnetic moment 4.99 [107] 4.59 [111] T N N´eel temperature 10 K [98] 6 K b (cid:15) Dielectric constant 25 (100) 25 a The hopping parameters t are derived from H E assuming half-filling. b Estimated from the Liechtenstein magnetic force theorem, T N = 3 µ B H E / k B [112]. Three unknown quantities determine the AQ model: the mass m Θ , decay constant f Θ ,and speed v s (from the spatial derivatives, giving the wave speed). We generally workin the limit v s (cid:28) c and ignore the magnon dispersion relative to the E -field. This leavestwo parameters, m Θ and f Θ . We show in detail in section 3.1 how both m Θ and f Θ canbe determined experimentally from the polariton resonances and gap via transmissionspectroscopy (related to the total reflectance measurement proposed by Ref. [42]). In thissection, however, we wish to estimate these parameters from known material properties.We consider two candidate materials, firstly the magnetically doped TI (Bi − x Fe x ) Se of Ref. [42]. Reference [107] considered a number of different TIs doped with differentmagnetic ions, and found that only (Bi − x Fe x ) Se is both antiferromagnetic and in-sulating. (Bi − x Fe x ) Se has been successfully fabricated. However, the magentism isfragile due to the doping (required around 3.5%), and the region of the phase diagramexhibiting the AQ is small. Therefore, we also consider the new class of intrinsicallymagnetic TIs, Mn x Bi y Te z , of which only Mn Bi Te is thought to contain an AQ, buthas yet to be fabricated. Material properties for both cases are listed in Table 4, whilethe derived parameters are given in Table 5. Our estimates for the derived parametersare discussed in the following.The microscopic model for the AQ is derived from the Hubbard model in the weakcoupling limit. In the Hubbard model, one allows hopping of spins between lattice sites.– 17 – able 5 : Derived AQ parameters. “Material 1” is our best approximation to(Bi − x Fe x ) Se . We report the results of Ref. [42], who assumed a cubic lattice toevaluate the band integrals, but rescaled by our values of M . We use a combinationof normalisation to the cubic lattice result, and the material properties in Table 4 toestimate the parameters for “Material 2”, our best approximation to Mn Bi Te .Symbol Name Equations “Material 1” “Material 2” m Θ AQ mass (2.35), (2.38) 2 meV 1.8 meV f Θ AQ decay constant (2.34), (2.37) 30 eV 70 eVThe Hubbard Hamiltonian is: H = − t X h ij i ,σ a † iσ a jσ + U X i n i ↑ n i ↓ , (2.30)where a † iσ and a iσ are the creation and annihilation operators for a spin σ at latticesite i and the first sum is over nearest neighbour sites. n i ↑ and n i ↓ are the spin upand spin down density operators for the i th lattice site. The first term describes thekinetic energy of the system, whose scale is given by the hopping parameter t . Thesecond term describes the interaction between spins on the same site, with scale givenby the Hubbard term U . In the limit of half filling and U (cid:29) t , the Hubbard model isequivalent to a Heisenberg model with J H ∼ t /U [113]. The exchange field is relatedto the Heisenberg Hamiltonian via Eq. (A.15) as H E = 2 SJ H gµ B , (2.31)where S is the ion spin and g is the spectroscopic splitting factor [114] (see Appendix Afor more details). This relation was used in Table 4 to set the hopping parameter t given U , S and µ B H E and taking g = 2.The electron band energies d i , Eqs. (2.23), (2.24), appearing in the microscopicmodel are normalized with respect to t . The Brillouin zone (BZ) momentum, k , on theother hand, is normalised with respect to the unit cell. This suggests normalizing theintegrals Eqs. (2.23,2.24) as (we consider only the case with a single Dirac fermion fromnow on and drop the subscript f ): J = Z BZ d k (2 π ) P i d i | d | = I V u.c. t , (2.32)(note that this J is not Heisenberg J H , in fact J H ∝ /J ) and m J = M Z BZ d k (2 π ) | d | = M I V u.c. t , (2.33)– 18 –here V u.c. is the volume of the unit cell. It then follows that the AQ mass is: m Θ = M s I I = 2 SU s I I . (2.34)Notice that for an exact Dirac dispersion for d , the integrals over the BZ vanish if theDirac mass, M , vanishes, as we expect from the Gell–Mann-Oakes-Renner relation [115].However, these integrals should be evaluated for d ’s computed in the full theory, i.e. abinitio density functional theory for the Hubbard model.In the full theory, the normalized integrals I depend on the ratio t/U . In terms ofthe Hubbard model parameters we have M = (2 / U n z , where n z = S is the normalisedAF order. The decay constant is: f Θ2 = 2 M I V u.c. t , (2.35)Using a cubic lattice model, Ref. [42] computed the BZ integrals for (Bi − x Fe x ) Se .The integrals depend on the ratio t/U , so we can also use this result for Mn Bi Te ifwe extract the values of the normalised integrals.Ref. [42] report b = 0 . m Θ = 2 meV. Ref. [42] assumed values forthe dielectric constant (taken at the gap instead of near the spin wave resonance) andbulk band gap (taken from the model without doping [116]) of (Bi − x Fe x ) Se , whichwe wish to update (in Table 4, the values assumed by Ref. [42] are given in parentheses).Fortunately, both of these quantities can be factored out of the relevant expressions toarrive simply with the normalised integrals. We find: I = 4 × − , I = 4 I × − . (2.36)Leading to the derived model parameters: f Θ = 30 eV (cid:18) M .
03 eV (cid:19) . (cid:18) V u.c. (cid:19) − . (cid:18) t .
04 eV (cid:19) − . (cid:18) I × − (cid:19) . (2.37) m Θ = 2 meV (cid:18) S . (cid:19) (cid:18) U (cid:19) (cid:18) I / I × − (cid:19) . . (2.38)The derived parameters are presented in Table 5, where we adopt the less committalnames “Material 1” and “Material 2” for (Bi − x Fe x ) Se and Mn Bi Te respectively,to acknowledge the limitations of our estimates.Note that in Table 4 we quote the anisotropy field µH A , but that this plays norole in our estimation of the AQ parameters. The anisotropy field in fact determinesthe transverse magnon masses (see Appendix A.1), and not the mass of the longituninalAQ. In Paper I we mistakenly assumed to use the transverse magnon mass for the AQ(along with a doping fudge factor). The transverse and longitudinal modes turn out tohave similar masses. While we do not know of a fundamental reason for this coincidence,they are both clearly governed by the same O (meV) magnetic energy scales.– 19 – able 6 : Summary of the loss model. Elements of Γ are specified before diagonalisingthe kinetic term (see Eq. 3.28). The only E -field loss is the conductance. The total AQloss is given by the sum of the remaining terms. Loss channels deemed negligible includeAQ decay to photons, AQ-photon scattering, and off-diagonal losses. Type of losses Symbol Parameterisation Reference values CommentsConductance Γ ρ (cid:15) ω − ω Extrapolated fromoptical wavelengthsGilbert damping Γ lin α G (1 + χ − m ) ω − ω χ m given in Ref. [119] † Magnon scattering Γ m n/a Boltzmann-suppressedfor T < m
Impurities& domains Γ cryst . ( δL/L ) ω [10 − , − ] ω Typical impurity scale L ∼ † We thank Chang Liu for providing this result.
Finally, we mention the important spin flop transition (for a detailed descriptionand bibliography, see ref. [117]). Large magnetic fields cause spins to align and inducenet magnetization. The magnetization increases linearly for fields larger than the spin-flop field, H SF , eventually destroying the AF order. The spin flop field for MnBi Te is 3.5 T [118]. In easy axis systems, the AF order is destroyed completely when themagnetization saturates. This occurs at the spin flip transition for fields larger than theexchange field, H E . Large applied fields that destroy AF order will also destroy the AQ.For the exchange fields given in table 4 we expect these transitions to happen in themany Tesla regime. In the following we consider fields up to 10 T for illustration. As discussed below, the magnon and photon losses are crucial in determining how effec-tive an AQ material is for detection of DM. In order to detect the AQ and measure itsproperties, it is essential that any experiment is carried out at temperatures below theN´eel temperature. Fortunately both candidate materials have T N > T (cid:28) ω a dilution refrigeratortemperatures. Material conductance (inverse resistivity) appears in the E -field equations of motion asa damping term Γ ρ = 1 /ρ = 0 . ρ/ (Ω cm)] − , from which we see that a resonancenear 1 meV requires ρ (cid:29)
1Ω cm) for Q = ω/ Γ (cid:29)
1. For a resonance involving theelectric field, one requires large resistance, i.e. low conductance.Ref. [120] measure ρ in the optical ( ω ∼ T ≈ ρ = 2 × − Ωcm forundoped Bi Se , lowering to ρ = 5 × − Ωcm with doping. However, it is shown that– 20 – . . . . . (cid:15) MeasurementFit, p = − . λ [nm] − . . . R e s . Figure 4 : Bi Se dielectric function, (cid:15) , as a function of wavelength λ in the opticalregime ( ω ≈ (cid:15) ∝ λ p with p = − . T can increase ρ to be as large as 1 Ω cm. For MnBi Te thesituation is similar, with two different measurements giving a longitudinal ρ ≈ − Ωcmat T ∼ O (few) K [121]. In the case of MnBi Te , resistivity can be raised by doping withantimony (Sb) [122]. Even so, topological insulators are actually very poor insulators attypical electronic frequencies.The measurements of bulk ρ for both Bi Se and MnBi Te are taken at high energynear the band gap around 1 eV, and far from the spin wave resonance frequency at lowenergies. References [108, 109] studied the dielectric function of Bi Se as a functionof probe wavelength for the trigonal and orthorhombic phases. The complex dielectricfunction is ˜ (cid:15) ( ω ) = (cid:15) − i(cid:15) . For energies below the gap, E (cid:46) (cid:15) has value around25 at the longest wavelenths measured and is only slowly decreasing, while (cid:15) tends tozero rapidly at large wavelengths in the trigonal case (which is thus more favourable forour purposes). The value of (cid:15) is considerably smaller than the (cid:15) = 100 estimate usedin Paper I and assumed in Ref. [42]. As we show below, smaller values of (cid:15) are highlydesirable for DM detection.The resistivity is given by ρ ( ω ) = 1 / [ ω(cid:15) ( ω )]. A narrow linewidth on resonancerequires to (cid:15) ( ω + ) (cid:28)
1. Measurements in Ref. [109] extend to a maximum wavelength2800 nm where (cid:15) ∼
1. A simple power law extrapolation to THz wavelengths gives (cid:15) (1 meV) = 9 . × − (see fig. 4). Thus, the resistivity on the polariton resonanceat wavelengths of order 1 mm is significantly higher than the bulk measurements inthe optical. The value of (cid:15) is different for different crystal structures of Bi Se , and weconsider only the most favourable case with the highest resistivity. We take the value (cid:15) =– 21 –0 − as a reference scale, however, we do not include any further frequency dependence,which would certainly be different for different materials, such as Mn Bi Te . Theresistivity on resonance can be determined from the linewidth as measured by THztransmission spectroscopy, as we demonstrate in Section 3.2. As we have discussed, the AQ is not described by the same EFT as ordinary AF-magnons.However, due to the relation between the AQ and the magnon fluctuation, we use thewell-studied magnon case as a means to assess the possible magnitude of the axionlinewidth, and the qualitative possibilities. Furthermore, as we will see, the dominantcontribution is estimated to be due to material impurities, which do not depend on themicroscopic model for the AQ. We split the magnon losses into different contributions:Γ m = X i Γ i , (2.39)where the index i sums over terms defined in the following subsections.Ref. [123] gives a comprehensive account of non-linear wave dynamics relevant to themagnon linewidth. Early works on magnon scattering and linewidth include Ref. [124].The recent pioneering work of Refs. [125, 126] showed how neutron diffraction with en-ergy resolution down to 1 µ eV can be used to confirm the theoretical predictions for theAF-magnon linewidth, and the dependence on temperature and momentum across thewhole Brillouin zone, including many of the contributions discussed in the following. Wefocus on a few channels for losses, by means of example, closely following Ref. [126]. Scat-tering channels that we have not considered include AF-magnon-ferromagnetic magnonscattering, and magnon-phonon scattering: these are discussed in e.g. Ref. [123].In the present work, we are only concerned with the q ≈ T (cid:28) T N , wheremany contributions can be neglected. In this regime, as we show in Section 3.2, the totalAQ contribution to the linewidth can be measured using THz transmission spectroscopy. “Linear” Losses and Gilbert Damping, Γ lin Losses are historically incorporated for spin waves by the introduction of the phenomeno-logical Gilbert damping term into the Landau-Lifshitz equation, making the Landau-Lifshitz-Gilbert (LLG) equation. Gilbert damping is a linear loss, since it simply rep-resents decay of spin waves due to torque. There is not a universally accepted firstprinciples model of Gilbert damping. One possible model is presented in Ref. [127],where Gilbert damping is shown to arise due to spin orbit coupling in the Dirac equa-tion (other models include Refs. [128, 129]). In this case, the damping term is writtenas: Γ lin = α G (1 + χ − m ) ω . α G = eµ Σ s m e , (2.40)where m e is the electron mass and χ m is the dimensionless magnetic susceptibility (vol-ume susceptibility in SI units), and Σ s = S/V u.c. . The dimensionless prefactor α G isof order 10 − for (Bi − x Fe x ) Se and Mn Bi Te . The value of χ m was measured for– 22 – f ✓ ( k ) ✓ ( k ) ✓ ( k ) ✓ ( k ) ✓ ( k ) ✓ ( k ) ( k ) ( k ) ↵f ⇥ ↵f ⇥ Figure 5 : Left : Four magnon scattering. In EFT, the amplitude can be calculated asshown in Ref. [130]. As shown in Ref. [126], it is the leading contribution to the magnonlinewidth for T ∼ T N . For temperatures far below the spin wave mass, this termis Boltzmann suppressed. Right : Feynman diagram for the s -channel of the processEq. (2.49) mediated by the axion term in the Lagrangian. The parametric dependenceof the vertex factors is shown in red. This process is suppressed by two powers of α compared to the four magnon amplitude, Eq. (2.45).MnBi Te in Ref. [119] and found to be of order χ m ≈ − for T < T N (see Table 6).Thus the relative width, Γ /ω , is of order 10 − , which is negligible compared to the othersources of loss in the following. Furthermore, χ m is small enough to be neglected in themagnetic permeability (with c = 1), µ m = 1 + χ m , which we fix to unity. Magnon-Magnon Scattering, Γ m Reference [126] showed that two-to-two magnon scattering is the dominant contributionto the linewidth above ∼
10 K in the antiferromagnets Rb MnF and MnF as measuredby neutron scattering. The linewidth at 10 K due to this process is Γ m ≈ µ eV, fallingrapidly at lower temperatures. We will show how this behaviour arises below. Indeed,as noted in [125], for q → T →
0, all scattering contributions to the magnonlinewidth vanish. Ref [125] also find that this is true for scattering between the magnonand longitudinal spin fluctuations such as the axion. We find it useful to derive in somedetail the scattering contribution to the linewidth, and demonstrate why it vanishes atlow temperature, since this is the most well understood part of our loss model.The magnon modes obey a Boltzmann equation. Mode coupling via non-linearitiesinduces an effective lifetime for any initial configuration. Mode coupling arises from thefour-magnon amplitude: δθ ( k ) + δθ ( k ) ←→ δθ ( k ) + δθ ( k ) , (2.41)which has matrix element M ( k , k , k , k ), and is shown in Fig. 5. The state with mo-mentum k is the mode in the condensate of interest, k is a thermal magnon. Magnons k and k are modes scattered out of the condensate, and thus losses. This matrix ele-ment appears in the collisional Boltzmann equation for the magnon distribution function– 23 – ≡ f ( k ) as (see e.g. Ref. [131]):d f d t = Z Y i =2 dΦ i (2 π ) δ (3) ( k + k − k − k ) δ ( ω + ω − ω − ω ) |M| [ f f (1 + f )(1 + f ) − f f (1 + f )(1 + f )] , (2.42)= − Z d k (2 π ) f f | v − v | σ , (2.43)where dΦ i = d k i (2 π ) is the non-relativistic phase space element for state with momen-tum k i , the Dirac delta’s enforce energy-momentum conservation, k i represents the3-momentum of the ith particle, and the f i factors assume the particles are bosons.Ref. [126] caution that when such integrals are evaluated numerically, one should becareful to include the Umklapp processes, related to conservation of crystal momen-tum.vWe formulate the integral non-relativistically as the material picks out a preferredframe for the magnons. In the second line, the first term in the square brackets representsproduction of states k , k (the inverse process in Eq. 2.41), while the second termrepresents losses. In the last line we have assumed f = f = 0 for unoccupied finalstates, and used the definition of the differential cross section (this structure is familiarfrom particle physics scattering theory [132]). Ref. [123] derives an equivalent equationbeginning from the LLG equation, which also shows this non-linear loss term explicitlyin terms of the four-magnon amplitude.Eq. (2.43) is the collisional Boltzmann equation, ∂ t f = C [ f ], where C [ f ] is thescattering integral. Factoring out f for the condensate, the scattering integral takes theform C [ f ] ∼ /τ and we identify the relaxation time τ for the distribution function tochange significantly from its initial state. This gives the result that:Γ m = 1 /τ ∼ h σv i , (2.44)where the angle brackets denote the thermal average, i.e. phase space integral with thethermal distribution f .Magnons can be described by EFT, as discussed in Appendix A.1. The four magnonamplitude is given by the equivalent of the QCD pion amplitude evaluated around non-zero quark masses [130]. M = 14 √ ω ω ω ω v m F { δ ab δ cd (cid:18) v m ω ω − k · k + m m (cid:19) + δ ac δ bd (cid:18) − v m ω ω + 2 k · k + m m (cid:19) + δ ad δ bc (cid:18) − v m ω ω + 2 k · k + m m (cid:19) } , (2.45)where a, b, c, d = 1 , v m is the magnon velocity and m m is the magnon mass.This is the amplitude appropriate to a non-relativistic normalization, with 1 par-ticle per unit volume rather than the usual 2 ω particles per unit volume in relativisticquantum mechanics. – 24 –n this case, the cross section is related to the T-matrix above as [130]: dσ = |M| v (2 π ) δ ( k + k − k − k ) d k (2 π ) d k (2 π ) , (2.46)where v = | v − v | is the relative velocity of the incoming particles.We integrate over k and k to obtain the total cross section for given incomingmomenta k and k : σ ( k , k ) = Z d k (2 π ) d k (2 π ) dσ = Z dk (2 π ) d Ω k |M| v (2 π ) δ ( ω + ω − ω − ω )) (cid:12)(cid:12)(cid:12)(cid:12) ( k + k − k − k =0) , (2.47)where ω = q k v m + m m and ω = p ( k + k − k ) v m + m m . At this point in thecalculation, we might be tempted to move to the centre of mass frame. However, thiswould change the magnon velocity v m , with the new magnon velocity depending on Ω,leading to a magnon dispersion relation that depends on Ω. Therefore, it is in our bestinterests to remain in the rest frame of the material. We thus obtain the differentialcross section: dσd Ω = 1(2 π ) v |M| k k ω + k ω , (2.48)where k , ω , k , ω are defined by conservation of energy and momentum for a given Ω.Now let us consider the scaling of Γ m with temperature T . We note first that thefactor of v in Γ m is cancelled by the factor of v in dσd Ω . We will focus first on the scalingof the line widths measured in [126] at temperatures from 3 K to 0 . T N for magnons withmomentum k = 0 to k = q Z B at the edge of the zone boundary. The contributions of T to Γ m are as follows:• Thermal magnons have an energy set by T . We assume that T (cid:38) m m , such thatthermal magnons can be excited. We therefore take ω ∼ k ∼ T .• The scaling of the outgoing momenta with T depends on the relative sizes of T and ω . The energy at the zone boundary in [126] is 6 . MnF and6 . , while the temperature ranges from 3 K = 0 .
26 meV to 0 . T N ,corresponding to 2 . . T > ω and cases where ω > T are measured. When T (cid:29) ω , the temperatureprovides most of the energy in the scattering process and we have k , ω , k , ω ∼ T .When T (cid:28) ω , the energy of the damped magnon provides most of the energy inthe scattering process and we have k , ω , k , ω ∼ ω .• The number of thermal magnons also scales with T . Assuming that there is nosignificant mass gap at k = 0 for the magnons considered in [126], we have R d k f ∼ T , as for a black body.• We have also R d k d k ∼ T when T (cid:29) ω from the factor of k in the phasespace integral. – 25 – As T (cid:38) m m , the ω and k terms in M dominate. Using the scalings above, thisgives M ∼ T − / .Putting these elements together, we find Γ m ∼ T T T − = T for ω (cid:28) T andΓ m ∼ T T − = T for ω (cid:29) T . We can compare this prediction with the measuredresult in Figure 4 in [126]. For low magnon wavenumber q (corresponding to low ω ,we have Γ m ∼ T as expected. As q is increased, the scaling with T decreases towardsΓ m ∼ T as predicted. However, the measured Γ m ∼ T when q = 0 case is notexplained by this analysis.We would also expect that for temperatures much lower than the magnon mass,very few thermal magnons would be excited, and Γ m would be exponentially suppressed.For a magnon mass m m ∼ T <
10 K.Starting from the Boltzman equation, we have argued that magnon-magnon scat-tering decays with T , reproducing the experimentally observed trends in [126], and isthen exponentially suppressed at temperatures below the magnon mass. The scatteringcontribution to the antiferromagnetic magnon linewidth is calculated analytically forseveral low T regimes in [124]. This yields a power law fall off with T in each case.We therefore conclude that, at low T , and particularly for temperatures below themagnon mass, the magnon scattering contribution to the linewidth is negligible. Axion-Photon Scattering, Γ γm Scattering of magnons from thermal photons contributes to the magnon line-width Γ m .This process is induced by the four particle amplitude δθ ( k ) + γ ( k ) ←→ δθ ( k ) + γ ( k ) , (2.49)i.e. magnon/AQ-photon scattering mediated by the Chern-Simons interaction, Eq. (2.1).Inspecting the Feynman diagram, Fig. 5 (right panel), this amplitude is suppressed bytwo powers of the fine structure constant α with respect to the four magnon amplitude,and so we do not expect magnon-photon scattering to be significant compared withmagnon-magnon scattering. The inverse process, scattering thermal magnons from theelectric field, is similarly suppressed, and thus likely to be subdominant to conductivelosses to E . Axion Lifetime, Γ mγγ The Chern-Simons interaction leads to direct decay of an AQ into two photons. Thecontribution to the width is:Γ mγγ = α π m s f = 6 . × − eV (cid:18) m s meV (cid:19) (cid:18)
100 eV f Θ (cid:19) , (2.50)corresponding to a lifetime on the order of months. This process can be safely neglectedcompared to all other scales in the problem.– 26 – ff-Diagonal Losses The off-diagonal terms in the loss correspond to loss terms of the form dφ k dt ∼ h A k i and dA k dt ∼ h φ k i . As we generically expect dφ k dt ∼ h φ k i and dA k dt ∼ h A k i , these will all be ofthe form: dφ k dt ∼ h A k ih φ k i , (2.51) dA k dt ∼ h A k ih φ k i . (2.52)These off-diagonal loss terms are therefore not present at linear order in the per-turbations A k and φ k . Impurities and Domains, Γ cryst . At low temperatures, the dominant contribution contribution to the magnon linewidthin Ref. [126] is attributed to scattering of magnons off magnetic domains and crystalimpurities, which is T -independent .In the simplest picture, scattering from magnetic domains leads to a lifetime: τ ∼ L mag . v ( q ) , (2.53)where v ( q ) is the velocity of the mode with momentum q , and L mag . is the size ofthe domain. The Ref. [119] crystals of MnBi Te have estimated magnetic domainsize L mag . ∼ µ m. We require the axion-polariton to propagate at least through thethickness, d , of the sample, and thus magnetic domains appear to strongly affect theskin depth and resonance width of axion-quasiparticle dominated polaritons in the limit d (cid:29) L mag . .However, in the q → v s = 0 and ignorethe magnon propagation compared to the electric field. It is currently unknown howscattering from domains will affect such long wavelength mixed modes. On one hand,it may be that the domain walls appear as small scale fluctuations that decouple fromlarge wavelength modes. Conversely, given that the domain walls disrupt the short rangeinteractions that support the small q magnons it is possible that they have non-trivialeffects despite the scale separation.A second T -independent contribution to the linewidth, which is expected to remainin the q → cryst . = (cid:18) δLL (cid:19) ω ( k ) (2.54)where δL is the lattice constant, and L is the spacing between impurities, thus δL/L isthe average number of lattice sites between impurities. The model Eq. (2.54) accounts in– 27 –he same manner for magnetic and crystal impurities. In Ref. [119] the crystal impuritiesoccur on the same scale as the magnetic domains, L cryst . ∼ µ m, while δL ∼ ( V u . c . ) / ∼ cryst . = 7 × − ω (cid:18) δL (cid:19) µ m L cryst . ! . (2.55)We estimate that crystal impurity scattering is the dominant contribution to theAQ linewidth in the regime of interest. Given the lack of conclusive calculations ormeasurements in the literature (or even, as far as we can tell, a detailed model), weregard this as a question best resolved by experimental studies. Indeed, an understandingof the dynamics of small q magnons and axion-polaritons is an interesting off-shoot ofthe studies proposed in Section 3. However, given the importance of this linewidthcontribution to our proposed dark matter search, we must adopt a reference value. Weadopt the range given in Table 6, Γ cryst . ∈ [10 − , − ] meV, corresponding to impurityseparations of order 1 µ m. One of the methods proposed by Ref. [42] to detect the presence of AQs in TMIs wastotal-reflectance measurement, and idea we explore further here. In the following weshow to compute the transmission function of TMIs using axion electrodynamics. Thetransmission function is shown to display a gap, leading to total reflectance. Further-more, by using a wideband THz source, such a measurement can also determine theaxion-polariton resonant frequencies, and loss parameters. The concept of this
THztransmission spectroscopy measurement is shown in Fig. 6. Similar measurements havebeen performed on antiferromagents (e.g. Ref. [46]), which demonstrate AFMR anddetermine the magnon linewidth (losses on resonance) for an electromagnetic source. Such a measurement has not to date been performed on any AQ candidate material.
In this section, we review the axion-Maxwell equations for TMIs. We then derive a one-dimensional model as well as the correct interface conditions for all fields involved. Basedon the one-dimensional model, we compute the reflection and transmission coefficientsfor incoming THz radiation. Crucially, for our purposes, such a measurement uses precisely the same physics (oscillating E -fieldsource) as occurs for dark axion detection. This is in contrast to neutron scattering of antiferromagnets(e.g. Ref. [126]), which determines the linewidth for a different excitation mechanism. – 28 – o u r c e P o w e r f [THz] d B e yxz | T y | f [THz] EB EB E THz
N S B Wideband THz SourceDetectorAQ Material | T y | f [THz] T r a n s m i ss i o n F un c t i o n f [THz] m ⇥ ! LO res Figure 6 : Proposed transmission experiment to detect the axion-polariton.
Left:
THzsource power spectrum.
Centre:
Transmission experiment concept. A source field, whichpropagates along the negative z -direction is incident on a TMI. An external B -field B e is applied parallel to the TMI surface. If AQs exist in the material, the dispersionrelation has a gap where no propagating modes exist, thus altering the spectrum ofthe transmitted radiation. Right:
Theoretical transmission spectrum. The green linecorresponds to the case where a dynamical AQ is present. The gap is indicated by thevertical green dotted lines. The width on resonance, Γ res , serves to measure the polaritonlosses.
The macroscopic axion-Maxwell equations for a three-dimensional TMI are [42] ∇ · D = ρ f − απ ∇ ( δ Θ + Θ ) · B , (3.1) ∇ × H − ∂ t D = J f + απ ( B ∂ t ( δ Θ + Θ ) − E × ∇ ( δ Θ + Θ )) , (3.2) ∇ · B = 0 , (3.3) ∇ × E + ∂ t B = 0 , (3.4) ∂ t δ Θ − v i ∂ i δ Θ + m δ Θ = Λ E · B , (3.5)where δ Θ is the pseudoscalar axion quasiparticle (AQ) field, Θ ∈ [0 , π ] a constant, f the AQ decay constant, v i (with i = x, y, z ) is the spin wave velocity, m Θ the spin wavemass, E is the electric field, B the magnetic flux density, D the displacement field, H the magnetic field strength, ρ f the free charge density, and J f the free current density,– 29 –hich fulfill the continuity equation ∇ · J f + ˙ ρ f = 0 as in usual electrodynamics. Inwhat follows we often use the linear constitutive relations D = (cid:15) E and H = µ − B , (3.6)where (cid:15) and µ are the scalar permittivity and permeability, respectively. Note that it isimportant to include the Θ term in the equations above: while Θ is some constant inthe TMI, it is always zero in vacuum. Applying the nabla operator can therefore give adelta function at the boundaries of the TMI, i.e. a boundary charge term.Equations (3.1) and (3.2) can be written such that the terms including the dy-namical AQ field δ Θ can be interpreted as additional contributions to polarization andmagnetization, i.e. ∇ · D Θ = ρ f , (3.7) ∇ × H Θ − ∂ t D Θ = J f , (3.8) ∇ · B = 0 , (3.9) ∇ × E + ∂ t B = 0 , (3.10)where we define D Θ = D + απ (Θ + δ Θ) B , (3.11) H Θ = H − απ (Θ + δ Θ) E . (3.12)To derive interface conditions for the electromagnetic fields, we consider two domainslabeled 1 and 2. Both domains have different (cid:15) , µ , and Θ . Transforming Eqs. (3.7)–(3.10) into their integral representation, and applying Gauss’s (Stokes’) theorem to aninfinitesimal volume (surface) element, leads to the following interface conditions for theelectromagnetic fields: n × ( E − E ) = 0 , (3.13) n · ( D Θ , − D Θ , ) = σ S , (3.14) n · ( B − B ) = 0 , (3.15) n × ( H Θ , − H Θ , ) = J S , (3.16)where σ S and J S are free surface charge and current densities (both assumed to be zero inwhat follows) and n is a unit vector pointing from domain 1 to domain 2. It is importantto stress that Eqs. (3.13)–(3.16) are interface conditions, not boundary conditions.As described above, interface conditions follow from the differential equation intheir integral form. In contrast, boundary conditions can be applied at the boundarydomains for which a partial differential equation is solved, and do not follow from theintegral representation of the differential equation. This is also the reason why theinterface conditions are specified only for the electromagnetic fields, and not for thedynamical axion field δ Θ. In this section we only consider the case of a TMI surrounded– 30 –y a non-topological material/vacuum with Θ = 0. We then only need to imposeinterface conditions for the electromagnetic fields; while they exist in both the TMI andthe adjacent region, the dynamical AQ only exists in the TMI. The AQ is therefore onlysubject to boundary conditions. We revisit and deepen this discussion in Section 3.2,in the context of calculating reflection and transmission coefficients for a layer of TMIsurrounded by vacuum. To develop a one-dimensional model, we assume that all fields only depend on the z -coordinate and time. Furthermore, all fields are taken to be transverse fields, i.e. B z = H z = D z = E z = 0. Then, in a domain with constant Θ , Eqs. (3.1)–(3.5) reduceto: ∂ z − H y H x ! − ∂ t D x D y ! − J f = απ " B x B y ! ∂ t δ Θ + − E y E x ! ∂ z δ Θ , (3.17) ∂ z − E y E x ! + ∂ t B x B y ! = 0 , (3.18) ∂ t δ Θ − v z ∂ z δ Θ + m δ Θ = Λ( E x B x + E y B y ) , (3.19)where we assumed that no free static charges exist, i.e. ρ f = 0. The interface con-ditions (3.14) and (3.15) are trivially fulfilled in the one-dimensional model since the z -components of all electromagnetic fields vanish, and n = ˆ e z . The sources in Eqs. (3.17) and (3.19) are non-linear and, therefore, finding analyticsolutions is in general not possible. However, we are interested in the special caseof solving the equations in presence of a strong, static external B -field B e = B e ˆ e y .We may therefore separate the total B -field into a static and a dynamical part, i.e. B → B e + B ( x , t ). Similarly, the free current J f can be split into a part which sources B e , and an additional reaction current, i.e. J f → J f + J f . Physically, the reactioncurrent describes losses of the electromagnetic fields in the materials. Note that B e fulfils ∇ × H e = J f , and J f satisfies the continuity equation ∇ · J f = 0. With theseassumptions the resulting equations are: ∂ z − H y H x ! − ∂ t D x D y ! − σ E x E y ! = απ " B x B e ! ∂ t δ Θ + − E y E x ! ∂ z δ Θ , (3.20) ∂ z − E y E x ! + ∂ t B x B y ! = 0 , (3.21) ∂ t δ Θ − v z ∂ z δ Θ + m δ Θ = Λ( E x B x + E y B e ) , (3.22) This does not mean that J f vanishes. Since ρ f and J f are connected via a continuity equation, J f only has to fulfil ∇ · J f = 0 if ρ f = 0. – 31 –here we substitute the reaction current J f with the loss term σ E (Ohm’s law). Whenderiving Eqs. (3.20) and (3.22), we used that the external field B e is much larger thanthe y -component of the reaction B -field, B y . Note that it is straightforward to includean external source field in E and B .Let us now justify why the non-linear terms on the right-hand side in Eqs. (3.20)and (3.22) can be linearized. Consider the two distinct cases where a strong externallaser field is parallel or orthogonal to the static external B -field: first, assume that theexternal laser field is parallel to B e = B e ˆ e y . Note that B e ∂ t δ Θ ∂ z δ Θ ≈ × Vm (cid:18) B e (cid:19) , (3.23)where we approximated ∂ t δ Θ ∂ z δ Θ with a typical spin wave velocity, which is on the order of v s = 10 − [133]. Typical THz sources have a power around P = 10 − W, which leadsto E y = 27 Vm for a beam surface area of 10 mm . Equation (3.23) is therefore fulfilledfor sufficiently large external B -fields. With these considerations we see directly that B e ∂ t δ Θ (cid:29) ∂ z δ Θ E x since E x is even smaller than E y . It follows that the non-linear termin the second component on the right-hand side in Eq. (3.20) can be neglected.Next, we consider the two source terms in the first equation in the reft-hand sideof Eq. (3.20). The term E y ∂ z δ Θ dominates over the term B x ∂ t δ Θ since E y contains theexternal laser source. However, the large source term B e ∂ t δ Θ in the term in Eq. (3.20)is larger than the dominating source in the first term: B e ∂ t δ Θ (cid:29) E y ∂ z δ Θ, cf. Eq. (3.23).From Eq. (3.21) it is clear that ∂ t B y = − ∂ z E x and therefore due to H y ∼ B y the sourceof the first component in (3.20) sources the E y -component. Therefore we can ignore thenon-linear sources in the first equation in (3.20) and focus only on the E y -component,e.g. the large linear source in the second equation in (3.20). The non-linear term E x B x in Eq. (3.22) can also be neglected since it is much smaller than the term E y B e , whichincludes two external fields.Second, in the case that the external laser field is orthogonal to B e = B e ˆ e y , thedominating source of the Klein-Gordon equation, cf. Eq. (3.22) is the linear term E y B e .Note that the fields B x and E y can only be induced by polarization rotation and are bothon the order of απ . However, since 3 × (cid:16) B e (cid:17) (cid:29) E x , we can linearize the sourceterm of the Klein-Gordon, cf. Eq. (3.22), i.e. E x B x (cid:28) E y B e . The second component ofEq. (3.20) can be linearized because any available THz lasers has an amplitude that isbelow the limit in Eq. (3.23). The first component of Eq. (3.20) can also be linearized,i.e. the source terms are neglected since both source terms include electromagnetic fieldsthat are only generated via polarization rotation.In summary, whether an external laser E -field is parallel or orthogonal to B e , theequations can be linearized, and they reduce to: ∂ z E x − n ∂ t E x − µσ∂ t E x = 0 , (3.24) ∂ z E y − n ∂ t E y − µσ∂ t E y = απ µB e ∂ t δ Θ , (3.25) ∂ t δ Θ − v z ∂ z δ Θ + m δ Θ = Λ E y B e , (3.26)– 32 –here we explicitly use the linear constitutive relations, cf. Eq. (3.6). Furthermore therefractive index is given by n = (cid:15) µ . (3.27)The material properties µ , (cid:15) , m Θ , σ , Θ , v z , and Λ are constants in the equations of mo-tion. Regions with different material properties are linked by using interface conditionsfor the fields.The corresponding interface conditions are given in Eqs. (3.13) and (3.16) with n = ˆ e z . Equation (3.13) remains unchanged after linearization, while the definition of H Θ in Eq. (3.16) changes due to the linearization to H Θ = H + απ Θ E . Losses can appear in the linearized equations of motion (3.24)–(3.26) in case of a finiteconductivity σ . However, magnon losses, and losses that mix between magnons andphotons, are not included. We now generalize Eqs. (3.24)–(3.26) to include all possibletypes of losses. The equations then read: K ∂ t X − Γ ∂ t X + M X = 0 , (3.28)where we define X = E x E y δ Θ , K = απ B e (cid:15) , Γ = Γ ρ ρ Γ × , × , Γ m , M = k n k n − Λ B e v z k + m , (3.29)and where Γ ρ = σ/(cid:15) is the photon loss, Γ m is the equivalent loss for magnons, and Γ × , / are mixed losses that can arise when photons and magnons interact. We retain these forthe most general treatment, and set them to zero later. Note that not all Γs have the samemass dimension since [Γ ρ ] = [Γ m ] = 1, while [Γ × , ] = 3 and [Γ × , ] = −
1. The approachalso gives the possibility to define different refractive indices n and photon losses Γ ρ for the E x and E y components. However, these effects can only become importantwhen polarization rotation effects are discussed in detail. In the following, polarizationrotation effect are computed, however they are not discussed at a level of detail, suchthat including different refractive indices for different polarizations would not changethe results significantly.The interface conditions (3.13)–(3.16) remain the same in the presence of losses,because it is assumed that all losses are bulk losses. The presence of an AQ leads to a gap in the dispersion relation, which does not includeany propagating modes. Based on this, Li et al. [42] proposed a transmission measure-ment (cf. Fig. 6) to determine the band gap in a TMI polariton spectrum, opened by– 33 –he presence of the AQ (cf. Fig. 7). We now compute the transmission and reflectioncoefficients, and we demonstrate how to experimentally determine the parameters ofinterest – in particular the relevant terms of the loss matrix Γ . Our strategy for solving the linearized equations is as follows: we solve the equationsfor each spatial domain of constant material properties. We then apply the appropriateinterface conditions to match the solutions in the different domains.
Lossless case (Γ = 0).
The dispersion relation for the E x -component, see Eq. (3.24),is the usual photon dispersion relation: k = n ω ≡ k p . (3.30)The E y -component mixes with the AQ and, in the v z = 0 case, we find a typical polaritondispersion [42, 134]: ω ± = 12 h ω LO + k n i ± h(cid:0) ω LO − k n (cid:1) + 4 b k n i / , (3.31)where we have defined b ≡ απ Λ B e (cid:15) , (3.32) ω LO ≡ b + m . (3.33)The case v z = 0 is discussed later since v z is on the order of the spin wave velocity 10 − and therefore the expected effect is small.We show ω ± as a function of the wave number k in the left panel of Fig. 7. Thehorizontal black lines indicate the gap between m Θ and ω LO , where total reflection isexpected. The resulting frequencies for m Θ and ω LO are in the THz regime what makesclear why THz sources are needed to probe the gap in the dispersion relation. ω + converges for large k to a photon dispersion (dashed blue line). ω − has for small k analmost photon-like dispersion ω − = kn m √ b + m (dashed red line).Inverting Eq. (3.31) gives: k = n ω h − b ω − m i ≡ k ≡ n ω . (3.34)We show k as a function of ω in the right panel of Fig. 7. In the limit of b →
0, Eq. (3.34)becomes the usual photon dispersion relation. For ω we have two solutions, while thesolution for k can be described by a single function. Inside the bandgap, k is negative,thus k is purely imaginary, and no propagating mode is present. In the following sectionit is explicitly shown that this leads to total reflection and zero transmission.– 34 –
10 20 30 40 50 k [meV] . . . . ω [ m e V ] ω + ω − knkn m Θ √ b + m k [ ] f [ T H z ] m Θ ω LO . . . . ω [meV] k [ m e V ] Re( k )Im( k )Re( k ) f [THz] k [ mm ] m Θ ω LO Figure 7 : Polariton dispersion relation, arising from the mixing of AQs and photonsfor a spin wave velocity v z = 0. We use typical material values for a Mn Bi Te TMI,cf. table 5 and eq. (2.3), and n = 5 and µ = 1. The external B -field is B e = 2 T. Theleft panel shows the ω ± mode, which has a bandgap between m Θ and ω LO (horizontallines). The right panel illustrates the inverse of the dispersion relation for k . Inside thebandgap (vertical lines), k is only imaginary, and hence no propagating modes exist.The most general ansatz for the field evolution in a TMI medium are E x ( z ) = ˆ E + x e ik p z + ˆ E − x e − ik p z , (3.35) E y ( z ) = ˆ E + y e ik Θ z + ˆ E − y e − ik Θ z , (3.36) δ Θ( z ) = δ ˆΘ + e ik Θ z + δ ˆΘ − e − ik Θ z , (3.37)where we omitted the time dependence e − iωt in each line. After plugging the solutionsinto the equations of motion, cf. Eq. (3.28) the following relations are obtained: δ ˆΘ ± = Θ E ˆ E ± y , Θ E = Λ B e m − ω , (3.38)or, equivalently, ˆ E ± y = E Θ δ ˆΘ ± , E Θ = − απ µω B e k p − k . (3.39)In the following, the relations in Eq. (3.38) are used to reduce the number of unknownsin the ansatz (3.37): δ Θ( z ) = Θ E ˆ E + y e ik Θ z + Θ E ˆ E − y e − ik Θ z . (3.40)The remaining constants ˆ E ± y can be determined by using the interface conditions (explic-itly shown in Section 3.2.2). The AQ field δ Θ is completely determined, cf. Eq. (3.40),and no boundary conditions for δ Θ have to be applied when, for example, a layer of TMIsurrounded by vacuum is considered. It will become clear in the following that this is aconsequence of the v z = 0 limit. – 35 –
20 40 60 80 100 k [meV] . . . . ω [ m e V ] ω + ω − knkn m Θ √ b + m k [ ] f [ T H z ] m Θ ω LO . . . . ω [meV] k [ m e V ] k + Re ( k − ) Im ( k − ) f [THz] k [ mm ] m Θ ω LO Figure 8 : Dispersion relation for a non-zero spin wave velocity of v z = 0 .
01. Thisexaggerated value was chosen because for realistic value of 10 − the effect of band crossingof the + mode is not visible. We use typical material values for a Mn Bi Te TMI, cf.table 5 and eq. (2.3), and n = 5 and µ = 1. The external B -field is B e = 2 T.Note that the relations in Eq. (3.39) could have also been used to reduce theconstants in Eq. (3.36). However, a short calculation reveals that this would result inthe same outcome, regardless whether the relations in Eq. (3.38) or (3.39) was used toreduce the constants.A finite spin wave velocity, v z = 0, leads a slightly modified dispersion relation: ω ± = 12 h ω LO + k ( v z + 1 n ) i ± h(cid:0) ω LO + k ( v z − n ) (cid:1) + 4 b k n i / . (3.41)Equation (3.41) is not a typical polariton dispersion relation, since the sign of v z underthe square root is positive, not negative. The dispersion relation for ω ± from Eq. 3.41 isshown in the left panel of Fig. 8, where we used an unrealistically large vale of v z = 0 . v z are on the order of 10 − . A non-zerovalue of v z leads to a gap-crossing of the ω − mode. However, due to the smallness ofthe spin wave velocity compared to the speed of light, the gap crossing happens at largevalues of the wave number k .Inverting Eq. (3.41) yields two modes for k , k ± = 12 v z (cid:20) ω − m + n ω v z ± (cid:16)(cid:0) m + ω ( n v z − (cid:1) + 4 ω n b v z (cid:17) / (cid:21) , (3.42)whereas we only obtained one mode for k in the v z = 0 case, cf. Eq. (3.34). Thefunctional dependence of Eq. (3.42) is shown in the right panel of Fig. 8. The imaginarypart of the k − mode, which for v z = 0 was only present inside the gap, now keeps risingoutside of the gap for frequencies ω < m Θ . The k + mode crosses the gap such thatfor ω > ω LO two propagating modes exist. However the wavelength of the k + mode isalways much shorter than the wavelength of the k − mode.– 36 –he most general ansatz in the case of non-vanishing spin wave velocity is: E x ( z ) = ˆ E + x e ik p z + ˆ E − x e − ik p z , (3.43) E y ( z ) = ˆ E ++ y e ik + z + ˆ E + − y e − ik + z + ˆ E − + y e ik − z + ˆ E −− y e − ik − z , (3.44) δ Θ( z ) = δ ˆΘ ++ e ik + z + δ ˆΘ + − e − ik + z + δ ˆΘ − + e ik − z + δ ˆΘ −− e − ik − z . (3.45)Relations for the unknown constants in the ansatz (3.43)–(3.45) can be derived incomplete analogy to the v z = 0 case. However, we would now need to specify boundaryconditions for the dynamical axion in order to determine all constants. We do notperform the explicit calculation here since we expect the difference to the v z = 0 case tobe minimal, thanks to the smallness of the spin wave velocity. To see this, consider thefollowing argument:Let an incoming electromagnetic wave in vacuum be described by A e ik p z . In theTMI material with v z = 0, two modes are present. Around ω ∼ ω LO , the first mode k s has a wavelength that is much shorter than k p , while the second mode k l has a muchlonger wavelength than k p , i.e. | k s | (cid:29) | k p | (cid:29) | k l | . This is exactly the situation that weface (cf. Fig. 8), where k s = k + and k l = k − . Neglecting reflections, the fraction ofthe amplitudes of the two modes in medium 1 are (cid:12)(cid:12)(cid:12)(cid:12) A l A s (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) k a − k p k p − k l (cid:12)(cid:12)(cid:12) ≈ (cid:12)(cid:12)(cid:12) k s k p (cid:12)(cid:12)(cid:12) (cid:29)
1, wherethe index 1 refers to medium 1. Therefore the amplitude of long wavelength mode A l is much larger than the amplitude of the short wavelength mode A s . Based on thesearguments, the contribution of the k + mode can therefore be neglected – even though itis in principle present. In what follows, we will consequently assume that v z = 0. Case with losses (Γ = 0). If material losses are included, the dispersion relations(3.30) and (3.34) are modified. The dispersion relation of the E x -component is k = n ω (cid:18) i Γ ρ ω (cid:19) =: k p (3.46)and the dispersion relation for the mixed system of E y and δ Θ is k ≡ k = n ω b − i Γ m ω + m − ω + i Γ ρ ω ! + n ω iB e (cid:16) απ Γ × , ω (cid:15) + ΛΓ × , (cid:17) − ω Γ × , Γ × , − i Γ m ω + m − ω . (3.47)The first part of the dispersion relation in Eq. (3.47) only includes the diagonal lossesΓ m and Γ ρ , while the second part also includes mixed losses. We argued in Section 2.5that mixed losses are smaller than the diagonal losses Γ ρ and Γ m . We therefore neglectmixed losses in what follows. Note that due to the fact that we plot k + only up to 100 meV, the much larger values of k + around ω LO are not visible in the right panel of Fig. 8. – 37 – . . . . ω [meV] k [ m e V ] m Θ ω LO Γ ρ =0.4 meV, Γ m =0 meVRe( k )Im( k ) . . . . ω [meV] k [ m e V ] m Θ ω LO Γ ρ =0 meV, Γ m =0.05 meVRe( k )Im( k ) . . . . ω [meV] k [ m e V ] m Θ ω LO Γ ρ =0 meV, Γ m =0.2 meVRe( k )Im( k ) Figure 9 : Dispersion relation of the axion-polariton for magnon and photon losses, Γ m and Γ ρ . Mixed losses are neglected. Left: Γ ρ = 0 . m = 0. Photon lossesintroduce an almost constant imaginary part to the dispersion relation if the chosenfrequency interval is not too large. Middle: Γ ρ = 0 and Γ m = 0 .
05 meV, right: Γ ρ = 0and Γ m = 0 . Bi Te TMI, cf.table 5 and eq. (2.3), and n = 5 and µ = 1. The external B -field is B e = 2 T.Rewriting the dispersion relation (3.47) without mixed losses gives: k ≡ k = n ω m − ω ) ωb Γ m ω + ( m − ω ) + i Γ m ωb Γ m ω + ( m − ω ) + i Γ ρ ω ! . (3.48)Equation (3.48) shows that the Γ ρ contribution is unaffected by any other materialproperties, and it stays approximately constant when ω does not vary too much. Weshow an example for Γ m = 0 in the left panel of Fig. 9. While the peak of the resonanceis not affected much by the losses, Γ ρ introduces an almost constant imaginary for allfrequencies. In contrast, magnon losses Γ m are dominant around m Θ . This can be seenfrom the third term in Eq. (3.48) which represents a Lorentzian curve that peaks around ω = m Θ and has a full width at half maximum (FWHM) of Γ m . In the middle and rightpanels of Fig. 9 we show examples for Γ ρ = 0. The larger Γ m , the larger the FWHM of theimaginary part in the dispersion relation. In other words, frequencies away from the gapare damped more strongly when Γ m is large. Furthermore, the resonance becomes lesspronounced for large Γ m . As a consequence, it will be difficult to confirm the existenceof the gap in the spectrum, and the presence of a dynamical AQ, when large losses arepresent. We investigate this more quantitatively in Section 3.2.3, where we calculate thereflection and transmission coefficients for a single TMI layer.In the presence of losses the most general solution, cf. Eq. (3.35)–(3.37), is stillvalid. However, the relations in the Eqs. (3.38) and (3.39) are modified: δ ˆΘ ± = Θ E ˆ E ± , Θ E = Λ B e + iω Γ × , − ω + m − iω Γ m , (3.49)or, equivalently, ˆ E ± y = E Θ δ ˆΘ ± , E Θ = απ ω µB e + in ω Γ × , k − k p . (3.50)– 38 – =z z z z r z N Medium 0 11 2 r-1 r N ... ...
Figure 10 : Multilayer system of different materials. Each medium is characterized by (cid:15) r , µ r , Γ r , and Θ r . The external B -field has the same strength and polarization in eachmedium.It can be checked that Eqs. (3.49) and (3.50) reduce to Eqs. (3.38) and (3.39) in thelimit of Γ →
0. In complete analogy to the case without losses, Eq. (3.49) determinesthe dynamical AQ field, cf. Eq. (3.40).
In the previous section, we discussed the solutions of the one-dimensional axion-Maxwellequations in a homogeneous TMI. Here, we consider N + 1 media, separated by N interfaces, as shown in Fig. 10. Let the first interface be located at z = z , and thelast interface at z N . We label each medium with an index r , i.e. r = 0 , . . . , N . Forexample, the permittivity and permeability of medium r are thus denoted by µ r and (cid:15) r ,respectively. Recall that, in all media, we set v z = 0 and define the constant external B -field to be B e = B e ˆ e y .We now develop a matrix formalism to link the solutions in different materials toeach other. This makes it possible to compute the scattering of incoming electromagneticradiation from a multilayer system. The simplest application is the computation of thereflection and transmission coefficients for THz radiation that hits a layer of TMI; wediscuss this case at the end of this section.The most general ansatz in medium r is given by: E rx = ˆ E + x,r e ik rp ( z − z r ) + ˆ E − x,r e − ik rp ( z − z r ) ,E ry = ˆ E + y,r e ik r Θ ( z − z r ) + ˆ E − y,r e − ik i Θ ( z − z r ) ,δ Θ r = Θ rE ˆ E + y,r e ik r Θ ( z − z r ) + Θ rE ˆ E − y,r e − ik r Θ ( z − z r ) , (3.51)where, compared to Eqs. (3.35)–(3.37), we introduce different phase shifts z r for eachmedium. The expressions for k p , k Θ , and E Θ were derived already in Eq. (3.30), (3.34),and (3.38) for the case Γ = 0 and in Eq. (3.46), (3.47), and (3.49), for the case Γ = 0.Applying the interface conditions (3.13) and (3.16) for the electromagnetic fields at z r yields the following system of equations: t r = M − r M r − P r − t r − , (3.52)– 39 –ith t r = ˆ E + x,r ˆ E − x,r ˆ E + y,r ˆ E − y,r , M r = k rp ωµ r − k rp ωµ r − απ Θ r − απ Θ r − απ Θ r − απ Θ r − k r Θ ωµ r k r Θ ωµ r , (3.53)and P r = diag (cid:16) e i ∆ pr , e − i ∆ pr , e i ∆ Θ r , e − i ∆ Θ r (cid:17) . (3.54)The phases are defined as: ∆ Θ r ≡ k r Θ ( z r +1 − z r ) and ∆ pr ≡ k rp ( z r +1 − z r ).Let us define the matrix S to relate the incoming field amplitude from medium 0to the outgoing field amplitude in medium N : t N = S t . (3.55)For instance, for a single interface, S is given by S = M − M P (3.56)and, for two interfaces, S is given by S = M − M P M − M P . (3.57)Finally, for N interfaces, we find S to be given by S = M − N M N − P N − M − N − M N − P N − M N − · · · M − M P M − M P . (3.58)For electromagnetic radiation coming into the system from medium 0, ˆ E + x and ˆ E + y, are known and ˆ E − x,N = ˆ E − y,N = 0. The other unknown field values can be determinedfrom the elements of S , i.e. S ij , via ˆ E + x,N ˆ E − x, ˆ E + y,N ˆ E − y, = − S S S S S − S S S − · − S ˆ E + x − S ˆ E + y − S ˆ E + x − S ˆ E + y − S ˆ E + x − S ˆ E + y − S ˆ E + x − S ˆ E + y . (3.59) Let us now apply the matrix formalism to a system with N = 2. However, note that thematrix approach developed here is able to describe more complicated systems, consistingof many layers. One particular example could be a layered system of different topologicalinsulators with different material properties. The matrix formalism with N > N = 2 boundaries, and hence three– 40 –edia. Media 0 and 2 are vacuum while medium 1 is a TMI, hosting a dynamical AQ.The THz laser radiation is coming from medium 0 and hits the layer of TMI. In whatfollows, we omit the subscripts r that label the materials because the only non-vacuummedium is the TMI, i.e. medium 1.We assume that the laser polarization is oriented in the y -direction, parallel tothe external B -field. In this case, we obtain – to lowest order in απ Θ – the followingreflection and transmission coefficients: T y = 2 i ˜ k (cid:16) ˜ k + 1 (cid:17) sin ∆ + 2 i ˜ k cos ∆ + O (cid:18) απ Θ (cid:19) ! , (3.60) R y = − (cid:16) ˜ k − (cid:17) sin ∆ (cid:16) ˜ k + 1 (cid:17) sin ∆ + 2 i ˜ k cos ∆ + O (cid:18) απ Θ (cid:19) ! , (3.61)where ˜ k = k Θ ωµ , ∆ ≡ dk Θ and d = z − z is the thickness of the layer. Note that,although ˜ k depends on the expansion parameter we did not expand ˜ k because otherwisethe expansion for the transmission and reflection coefficients would not be valid aroundthe resonance. The calculated transmission and reflection coefficients are valid for boththe case with and without losses since we assume all losses to be bulk losses. T y and R y agree with the normal transmission and reflection coefficients of a dielectric disk [47] ifthe coupling b of the AQ to the photon is set to zero, i.e. k Θ → nω , corresponding to f Θ → ∞ .We show the full functions for the reflection and transmission coefficients withoutlosses ( Γ = 0) in Fig. 11. The coefficients are shown for different values of the laserfrequency ω and sample thickness d . The left column assumes the presence of a dynamicalAQ, while the figures in the right column show the case when no dynamical AQ is present.Note that in both cases Θ = 0 . π is assumed although the shown results for T y and R y do not depend on Θ to lowest order, cf. equations (3.60) and (3.61).First, we discuss the figures in the top and middle row, which show the reflectionand transmission coefficients for the E y -components. If a dynamical AQ is present,the dispersion relation k Θ becomes imaginary between m Θ and ω LO . The gap betweenthese two frequencies is marked with the two vertical lines. For large thicknesses d ,all frequencies in the gap are reflected, and none are transmitted. This is a directconsequence of the purely imaginary k Θ in the gap. For small values of the thickness,the gap size is reduced. This happens around ω LO , i.e. the upper part of the gap, since theimaginary part gets reduced (the skin depth becomes larger) the more ω LO is approachedfrom smaller frequencies, cf. right panel of Fig. 7. When going away from the gap, thefigures in the left and right columns agree more and more. This is as expected since thedispersion relation k Θ differs only significantly from a normal photon dispersion aroundthe gap. In the case of no dynamical AQ (left panel) we notice a clear non-zero reflectionand transmission inside the gap. Comparing the figures on the left- and right-hand side,it is clear that the AQ causes an O (1) modification of the T y and R y coefficients comparedto the spectrum when no dynamical AQ is present.– 41 –ext, we discuss the bottom row of Fig. 11, which shows the transmission coef-ficient for the E x -component. If no dynamical AQ field is present (right panel) butwe have a topological material with Θ = 0 . π , the transmission T x vanishes. Thismay be surprising at fist glance because there is mixing at the interface of ordinary TIsand, hence, also a polarization rotation. However, the transmission in the x -componentvanishes since the polarization rotations at the two interfaces cancel each other. If inaddition to the static Θ = 0 . π a dynamical AQ is present (left panel), we get a smallnon-zero transmission T x . The signal is much smaller than in the case of the T y co-efficient. This is because the incoming laser is polarized in the E y component and anon-zero E x -component can only be induced due to a nonzero Θ , i.e. mixing at theinterfaces, which is proportional to the small parameter απ Θ . In conclusion, we shouldfirst look for the AQ by studying the E y -component because the AQ modification of thiscomponent is much larger than for the E x -component.However, once the AQ is found, one can also use the E x -component to determine,for example, Θ of the material by reflection and transmission measurements. Is is alsopossible to study the influence of non-linear effects with the x -components. In Eq. (3.20)it was shown that the laser sources the x -component in a non-linear fashion. This effectis neglected here because the equations are linearized.Figure 12 shows the transmission coefficient T y for different losses. The figures areproduced for our benchmark material Mn Bi Te with n = 5 , µ = 1 and Θ = 0 . π . Inthe top row we illustrate the influence of photon losses Γ ρ for a TMI with a dynamicalAQ (left panel) and for a normal TI (right panel). The transmission at large layerthicknesses becomes smaller independently of the resonance. This is due to the fact thatΓ ρ appears in the dispersion relation (3.48) as an additional term which is approximatelyconstant in the small shown frequency interval. The skin depth is of the order of Γ − ρ . Itis therefore advantageous to have thin material samples for distinguishing between thecase of a DA (left panel) and no DA (right panel). However, should not be too thin. Forvery small thicknesses the frequencies inside the gap lead to a transmission coefficient T y that is not very small anymore. Note that the effect of losses for large d becomesmore pronounced for larger refractive indices n .In the bottom row of Fig. 12 photon losses are zero and the effect of magnon lossesΓ m is illustrated. We do not show the case without an AQ because without AQ thereare no magnon losses and one should compare to the already existing Fig. 11 (middlerow, right). The larger the magnon losses, the more pronounced is the widening of thegap. This can be understood by looking at the dispersion relation in Eq. (3.48). Magnonlosses Γ m introduce a Lorentzian shaped imaginary part to the dispersion relation. Thewidth of the Lorentzian is proportional to Γ m . Due to the Lorentzian shape of thedamping imaginary part in the dispersion relation also frequencies that are not directlyin the gap – but close to the gap – can become highly damped. This effect becomesmore pronounced the thicker the sample is.From the previous discussion it becomes clear that finding the AQ will dependvery sensitively on the losses and thickness of the material. The losses that we show inFig. 12 are exaggerated and in reality we expect them to be much smaller. Therefore– 42 –rom Fig. 12 we find that with a layer thickness on the order of 0 .
03 mm and 0 . ρ are varied between Γ ρ = 3 × − meV and 0 . m = 3 × − meV and 0 . d = 0 .
03 mm. In each figure, we show three different external B -field values. Thevalues of m Θ and ω LO are marked with a vertical black and coloured lines, respectively.The weaker the external B -field, the smaller the gap.Figure 13 makes again clear that the larger the losses the harder it is to distinguishthe case where a dynamical AQ is present (solid coloured curves) from the case that notdynamical AQ (dashed black line) is present. For relatively small losses, the distinctionbetween the curves is very clear. We therefore conclude that comparing these results tofuture measurements will make it possible to explicitly determine the material param-eters, i.e. losses, refractive index, Θ , and the parameters that enter the AQ mass m Θ and the gap size parameter b .We now investigate further the resonance around ω LO , cf. Fig. 13. When the losses inFig. 13 are small the resonance frequency f res = ω res π corresponds to ω LO . However, withhigher losses, the resonance frequency ω res moves to higher frequencies, i.e. f res > ω LO π .With increasing losses, the resonance smears out until it vanishes completely. Figure (13)allows us to directly read off the amount of losses that would still be acceptable AQdetection (for a sample of thickness d = 0 .
03 mm). The resonance peaks to the rightof ω LO in Fig. 13 are not symmetric. We therefore define the width of the resonancepeak, Γ res , as two times the frequency interval that ranges from the frequency at thetransmission maximum down to the smaller frequency at half the transmission maximum.The ratio f res Γ res is called the Q -factor. It describes the quality of the resonance in thesense that large Q -factors give rise to a well-defined resonance, whereas low Q -factorsshow that the resonance is highly damped. In Fig. 14, the Q -factor is shown with respectto the applied external B -field for different losses. We consider the case of dominantconductive losses (red) and dominant magnon losses (blue). The largest Q -factor isobserved at small external B -field, and the low magnon losses lead to the largest Q .This is consistent with the intuition that at low B -field the polariton is largely magnon-like. For larger external B -fields the difference between the two cases becomes small, asboth sources of loss contribute almost equally.– 43 – . . . . ω [meV]0 . . . . . . d [ mm ] . . . . . . | R y | f [THz] 1 . . . . ω [meV]0 . . . . . . d [ mm ] . . . . . . | R y | f [THz]1 . . . . ω [meV]0 . . . . . . d [ mm ] . . . . . . | T y | f [THz] 1 . . . . ω [meV]0 . . . . . . d [ mm ] . . . . . . | T y | f [THz]1 . . . . ω [meV]0 . . . . . . d [ mm ] . . . . . . | T x | f [THz] 1 . . . . ω [meV]0 . . . . . . d [ mm ] . . . . . . | T x | f [THz] Figure 11 : Reflection and transmission coefficients for a laser that hits a materialwith ( left ) and without ( right ) a dynamical AQ. The laser polarization is in the y -direction, parallel to the external B -field. The materials have Θ = 0 . π , n = 5, and µ = 1. Typical material values for a Mn Bi Te TMI with an external B -field B e of 2 Tare chosen, cf. Table 5 and Eq. (2.3). m Θ and ω LO are marked with the black verticallines. The R y ( T y ) coefficient is always close to one (zero) inside the gap if a dynamicalAQ is present. The T x coefficient is only non-zero is a AQ is present. However the effectof an AQ in the E x -component is much smaller than in the E y -component since the E x -component can only be induced via polarization rotation with non-zero Θ .– 44 – . . . . ω [meV]0 . . . . . . d [ mm ] Γ ρ =10 − meV . . . . . . | T y | f [THz] 1 . . . . ω [meV]0 . . . . . . d [ mm ] Γ ρ =10 − meV . . . . . . | T y | f [THz]1 . . . . ω [meV]0 . . . . . . d [ mm ] Γ m =10 − meV . . . . . . | T y | f [THz] Figure 12 : Transmission coefficients for the E y -component (parallel to the external B -field) for exaggerated photon and magnon losses, Γ ρ and Γ m . We show the results forwhen a dynamical AQ field ( left ) and if no dynamical AQ is present; ( right ). In bothcases we have Θ = 0 . π . We use typical material values for a Mn Bi Te TMI, cf.table 5 and eq. (2.3), and n = 5 and µ = 1. The external B -field is B e = 2 T. m Θ and ω LO are marked with the black vertical lines.– 45 – | T y | Γ ρ [meV] → Γ m [meV] ↓ − − − − − | T y | − | T y | − .
50 0 . f [THz]01 | T y | − .
50 0 . f [THz] 0 .
50 0 . f [THz] 0 .
50 0 . f [THz] B e =1.0 T B e =1.5 T B e =2.0 T δ Θ = 0
Figure 13 : Transmission coefficient T y for a layer of TMI with thickness d = 0 .
03 mmfor different B -fields (colours) are shown. Panels vary the losses Γ ρ (from left to right)and Γ m (from top to bottom). The vertical black line indicates the value of m Θ = 2 meV,while the other vertical lines indicate the value of ω LO for different values of the external B -field. The larger the external B -field, the larger the gap between m Θ and ω LO . Weuse typical material values for a Mn Bi Te TMI, cf. table 5 and eq. (2.3), and n = 5and µ = 1. The dashed black line shows the result when no dynamical AQ δ Θ is present,while the solid coloured lines are for the case with a dynamical AQ.– 46 –
Axion Dark Matter and Axion Quasiparticles
Paper I proposed using dynamical AQs in TMIs to detect DAs. This is possible sinceDAs can mix resonantly with axion polaritons. Compared to Paper I, we work outa more detailed calculation for the emitted photon signal by taking into account thecorrect interface conditions and material losses. This, in turn, allows us to present amore rigorous calculation of the sensitivity reach for DA searches using TMIs.As a starting point, we use the three-dimensional equations of motion, Eqs. (3.1)–(3.5). We linearize these and derive a one-dimensional model in analogy to Section 3.1.3.In what follows, the one-dimensional model is used to derive the photon signal generatedby DAs passing through a magnetized TMI that hosts dynamical AQs.
To describe the threefold mixing between AQs, DAs, and photons, we need to addthe Klein-Gordon equation for DAs, which is sourced by the electromagnetic fields, toEqs. (3.1)–(3.5). Additional source terms, arising due to the presence of DAs, havetherefore to be added to Eqs. (3.1) and (3.2). Doing so results in the following equations B e [T]50100150200 f r e s Γ r e s d=0.03 mm Γ ρ [meV], Γ m [meV]10 − , 10 − − , 10 − Figure 14 : TMI Quality factor, Q = f res / Γ res , for different losses, with respect to theexternal B -fields. f res is the frequency of the maximal transmission peaks around ω LO ,cf. Fig. 13. The TMI layer has a thickness of d = 0 .
03 mm.– 47 –f motion: ∇ · D = ρ f − απ ∇ ( δ Θ + Θ ) · B − g aγ ∇ a · B , (4.1) ∇ × H − ∂ t D = J f + απ h B ∂ t ( δ Θ + Θ ) − E × ∇ ( δ Θ + Θ ) i + g aγ ( B ∂ t a − E × ∇ a ) , (4.2) ∇ · B = 0 , (4.3) ∇ × E + ∂ t B = 0 , (4.4) ∂ t δ Θ − v i ∂ i δ Θ + m δ Θ = Λ E · B , (4.5)( ∂ t − ∇ + m a ) a = g aγ E · B . (4.6)where a is the pseudoscalar DA field, g aγ is the DA-photon coupling, and m a is the DAmass, in addition to the other variables already defined in Eqs. (3.1)–(3.5).In Section 3.1.1 we already noted that one cannot obtain interface conditions fromthe Klein-Gordon equation for an interface between media with and without AQs. How-ever, DAs are expected to permeate any medium due to the necessarily feeble interactionsof dark matter, and their presence in the Galaxy. Therefore, for two media that bothcontain DAs, Eq. (4.6) can be used to derive an interface condition for the DA field.Consider an infinitesimal volume element between two media, say, between medium 1and medium 2. We integrate over this infinitesimal volume element and apply the di-vergence theorem. It follows that the normal derivative of the DA field between twointerfaces has to be continuous, n · ( ∇ a − ∇ a ) = 0 . (4.7)Furthermore, we require that the DA field be continuous over the interface: a − a = 0 . (4.8)We stress that the continuity of the axion field in Eq. (4.8) does not follow from theaxion-Maxwell equations, but is a reasonable approximation. In other words, as DAsonly interacts with matter through very small couplings, and we are interested in theconversion of axions to photons again by a very small coupling, any modification due tothe axion interacting with the interface is at higher order, and thus negligible. Let us again assume the presence of a strong and static external B -field, B e . Withoutloss of generality, let B e be polarized in the y -direction, i.e. B e = B e ˆ e y . Then, similarsteps as in Sections 3.1.2 and 3.1.3 lead to the following linearized equations of motion: (cid:16) ∂ z − n ∂ t − σµ∂ t (cid:17) E y = µB e ∂ t (cid:18) απ δ Θ + g aγ a (cid:19) , (4.9) (cid:16) v z ∂ z − ∂ t − m (cid:17) δ Θ = − Λ B e E y , (4.10) (cid:16) ∂ z − ∂ t − m a (cid:17) a = − g aγ B e E y . (4.11)– 48 –he photon signal in the E x -component, induced by DAs, is always an order απ Θ smaller than the E y -component. This is due to the fact that only the E y -componentmixes with DAs and AQs. The E x -component can only be generated due to the mixing atthe interface, which is proportional to απ Θ . The main photon signal is therefore polarizedparallel to the external B -field in experimental DA searches, i.e. in the E y -component.Due to the suppression of the E x -component, it will be even more challenging to detect asignal in the E x -component. This justifies neglecting the E x -component in what follows.In addition to the linearization assumptions made in Section 3.1.3, we further as-sume that the non-linear terms, which include the DA, can also be linearized. This as-sumption is justified because of the small coupling and non-relativistic nature of GalacticDAs, for which ∂ t a/∂ z a ≈ − .The interface conditions for the electromagnetic fields after linearization are ob-tained with the linearized fields D Θ = D + απ (Θ + δ Θ) B e and H Θ = H − απ Θ E . Inthe one-dimensional model, the conditions n · ( D Θ2 − D Θ1 ) = 0 and n · ( B − B ) = 0 arealways fulfilled, since transverse waves are assumed to vanish, i.e. B z = 0 and E z = 0.The only non-trivial interface conditions are: n × ( H Θ , − H Θ , ) = 0 , (4.12) n × ( E − E ) = 0 , (4.13)where n = ˆ e z and it is assumed again that no free surface charges and currents arepresent.Including bulk losses in the one-dimensional model does not change the interfaceconditions. The magnon losses Γ m , photon losses Γ ρ , and mixed losses Γ × , and Γ × , areincluded in complete analogy to Section 3.1.4. The resulting equations of motion are: K ∂ t X − Γ ∂ t X + M X = 0 , (4.14)where we define X = E y δ Θ a , K = απ B e (cid:15) g aγ B e (cid:15) , Γ = Γ ρ Γ × , × , Γ m
00 0 0 , M = k n − Λ B e v z k + m − g aγ B e k + m a . (4.15)No losses for the DA are included since a valid DM candidate must, by necessity, havean astronomically long lifetime (indeed, the QCD axion in the mass range of interestsatisfies this constraint by many orders of magnitude). In this section, we solve the linearized equations of motion (4.14). We first consider thelossless case and then generalize the solutions to include losses. Material properties are– 49 –lways considered piecewise homogeneous, We introduce a matrix formalism to calculateemitted photon and axion power from an experimental setup with multiple TMI layers.We apply the matrix formalism to our benchmark setup, a single-TMI layer surroundedby vacuum. Using a multi-layer might be able to boost the signal, similar to multi-layerproposals for the MADMAX haloscope [47], although the higher frequencies consideredhere would lead to significant mechanical challenges if any tuning was required. Notethat we set v z = 0 in our calculations, cf. Section 3 for an explanation. We first focus on the case without losses. The dispersionrelation implied by Eq. (4.14) is: k − k a + k ! = b a k p + k a − k ! , (4.16)where b a = g aγ B e (cid:15) was defined in analogy to to b = απ Λ B e (cid:15) . The dispersion relations, upto leading order in the DA-photon coupling, therefore are k + = k Θ + O ( g aγ ) , (4.17) k − = k a + O ( g aγ ) . (4.18)The most general ansatz for the fields is: E = ˆ E ++ e ik Θ z + ˆ E + − e − ik + z + ˆ E − + e ik − z + ˆ E −− e − ik − z ,δ Θ = δ ˆΘ ++ e ik + z + δ ˆΘ + − e − ik + z + δ ˆΘ − + e ik − z + δ ˆΘ −− e − ik − z ,a = s ˆ a ++ e ik + z + ˆ a + − e − ik + z + ˆ a − + e ik − z + ˆ a −− e − ik − z . (4.19)In the following, we focus on the DA zero-velocity limit, i.e. k a = 0. This is an appro-priate approximation for dark matter and the most general ansatz in Eq. (4.19) reducesto: E = ˆ E ++ e ik Θ z + ˆ E + − e − ik + z + ˆ E − ,δ Θ = δ ˆΘ ++ e ik + z + δ ˆΘ + − e − ik + z + δ ˆΘ − ,a = ˆ a ++ e ik + z + ˆ a + − e − ik + z + ˆ a − , (4.20)where we omit the y index of the E -field since we ignore the E x -component. The caseof finite axion velocity was explored in Ref. [135].Plugging Eq. (4.20) into the equations of motion, Eqs. (4.9)–(4.11), we obtainrelations between the constants in the general ansatz. After plugging these relationsback into the ansatz (4.20), we obtain: Eδ Θ a = ˆ E ++ + E a + E e ik + z + ˆ E + − + E a + E e − ik + z + ˆ a − E − a Θ − a , (4.21)– 50 –here the following variables were defined:Θ E = Λ B e m − ω , a E = g aγ B e k , (4.22) E a = ω µg aγ B e k − k , Θ a = Θ E E a . (4.23)From Eq. (4.21) it becomes clear that the dynamical AQ is completely determined byfixing the variables ˆ E ++ , ˆ E + − , and ˆ a − . In the next section we show that these variablescan be fully determined by using the interface conditions for the electromagnetic andDA fields. Therefore no boundary conditions for the AQ need to be applied. The case with losses (Γ = 0). When losses are included, the full dispersion relation k ± takes on a more complicated form. However, in the limit g aγ → k − → k a and k → k , where k is given by Eq. (3.47). In what follows, a E and E a are needed also in the case of losses. a E in Eq. (4.22) does not get modified in the caseof losses, and E a has the same form as in Eq. (4.23). However, we now require the fullform of k Θ from Eq. (3.47). In the previous section, we described the solution of the linearized equations in a homo-geneous medium. Here, we discuss solutions for the fields in a multilayer system thatconsists of N + 1 media, cf. Fig. 10. We use the same labels for the media as as in Sec-tion 3.2.2. There are N interfaces, which we label by r = 0 , . . . , N . The first interface isat z = z and the last interface is at z N . The material properties in Eq. (4.21) of eachmedium are labeled with the corresponding index r as a subscript. The constant Θ does not influence the emitted photon signal at lowest order and is therefore neglectedin the following. We further introduce a phase similar to the case of AQ-photon mixingis introduced in the ansatz, cf. Eq. (3.51). The external B -field B e is the same in allmedia and is polarized in the y -direction. Recall that we consider the DA zero-velocitylimit with a zero spin wave velocity.Applying the interface conditions for the electromagnetic fields, cf. Eq. (4.12), (4.13),and (4.13) and for the DA, cf. Eq. (4.7), at z r between medium r − r we obtainthe following system of equations: t r = M − r M r − P r − t r − , (4.24)with M r = E − a,rk r + µ r − k r + µ r a + E,r a + E,r , t r = ˆ E ++ r ˆ E + − r ˆ a − r (4.25) If we were to consider a finite spin wave velocity, we would obtain three modes k , , . In this case,the most general ansatz would have six unknowns per field, and we would have to specify boundaryconditions for the AQ. – 51 –nd, defining ∆ + r ≡ k r + ( z r +1 − z r ) , P r = diag(e i ∆ + r , e − i ∆ + r , . (4.26)In complete analogy to Section 3.2.2 the S -matrix, which relates the states in media0 and N to each other, is defined via t N = S t . (4.27)The expressions for one, two, . . . , N interfaces are the same as in Eqs. (3.56)–(3.58).The unknown fields can be calculated from the S -matrix as follows: ˆ E ++ N ˆ E + − ˆ a − N = − ˆ a − − S S S − − · S S S , (4.28)where the amplitude of the DAs is known and, has to lowest order the same magnitudein each medium | ˆ a − | = | ˆ a − r | for all r = 1 , . . . , N . The emitted E -field in medium N thatpropagates in the positive z -direction is called ˆ E ++ N . The emitted E -field that propagatesin the negative z -direction is called ˆ E + − . Let us now consider the case of a single TMI layer (hosting a dynamical AQ) surroundedby vacuum. Dark axions are present in the form of an background field that oscillates intime with a frequency that is determined by the DA mass, m a . The DAs mix with theAQs and photons. In terms of the matrix formalism, there are two interfaces ( N = 2),with media 0 and 2 are vacuum, and medium 1 is a TMI of thickness d . The TMI hasconstant refractive index n = (cid:15) and losses Γ . The external B -field is present in allmedia. The DAs have the same magnitude in each medium, which is determined by theaxion dark matter density ρ a : | ˆ a − | = | ˆ a − r | = 2 ρ a /m a .The three-way mixing between DAs, AQs, and photons produces a photon at theboundary, which propagates away from the TMI layer. Note that, since we neglectthe spin wave velocity, the system behaves essentially as a two-level system of massivephotons and DAs. The emitted E -fields in media 0 and 2 are denoted by ˆ E + − andˆ E ++2 , respectively. Recall that ˆ E + − is the E -field amplitude that is emitted in negative z -direction in medium 0 and ˆ E ++2 is the emitted photon signal emitted in the positive z -direction in medium 2. We assume that the DA particles are effectively at rest. In thislimit there is no preferred direction and the magnitudes of ˆ E ++2 and ˆ E + − are the same. A nontrivial permeability, µ = 1, can be incorporated straight-forwardly into the matrix formalism,described in the previous sections. However, we set µ = 1 for simplicity and because this is a goodapproximation for the TMI materials discussed in Section 2.4. – 52 – ossless case (Γ = 0). The full formula for ˆ E ++2 from the matrix formalism is im-practical. We therefore quote the result first order in the DA-photon coupling, which,assuming that g aγ is sufficiently small, should be a good approximation:ˆ E ++2 = ˆ a − sin(∆ / (cid:0) n − (cid:1) n Θ ( n Θ sin(∆ /
2) + i cos(∆ / g aγ B e + O (cid:16) ( g aγ B e ) (cid:17) , (4.29)where we define the phase depth ∆ = dk Θ = dωn Θ (where k Θ is the lossless solution tothe dispersion relation, eq. (3.34)) and the effective refractive index is n = n − b ω − m ! . (4.30)Furthermore, we used in the language of the matrix formalism, such that a + E, = g aγ B e ω = a + E, , E − a, = − g aγ B e = E a, , and a + E, = g aγ B e n ω , E − a, = − g aγ B e n . From now on, terms ofthe order O (cid:0) ( g aγ B e ) (cid:1) are omitted to simplify the expressions. We also normalize thefield amplitude ˆ E ++2 to the DA-induced field in vacuum, E = g aγ B e a − ,ˆ E ++2 E = − sin(∆ / (cid:0) − n (cid:1) n Θ ( n Θ sin(∆ /
2) + i cos(∆ / . (4.31)Note that Eq. (4.31) has the same form as in the case of fields emitted from as a dielectricdisk [47], with the effective refractive index n Θ , which is equivalent to introducing aphoton mass.From analysing Eq. (4.31), it becomes clear that a resonance occurs if the condition∆ = ∆ j = n Θ ( ω j ) ω j d = (2 j + 1) π , j ∈ N , (4.32)is fulfilled. Here, ω j are the resonance frequencies, which are are located at ω j = ω LO s ω LO j b n d = ω LO + δω j + O j b n d ω LO ! , (4.33)where we have defined δω j ≡ ∆ j b n d ω LO . (4.34)From Eq. (4.33), it is evident that – in the lossless limit – the resonance frequenciesare always larger than ω LO , i.e. ω LO < ω < ω < . . . . As the thickness of the TMIincreases, d → ∞ , the resonant frequencies converge to the limiting value, i.e. ω j → ω LO . Furthermore, Eq. (4.33) implies that the resonance frequency ω j can be tuned via the external B -field, since b ∝ B . In Section 5.1 we investigate the frequency and,equivalently, DA mass range that can be scanned with our benchmark materials andrealistic external B -fields. – 53 –o understand why the frequencies defined via Eq. (4.32) are indeed resonancefrequencies, consider the following: Equation (4.32) implies that cos ∆ j = 0 and, hence,the emitted field in Eq. (4.31) is ˆ E ++2 ∼ /n − ∼ /n for small n Θ . In fact, thesmaller n Θ the more pronounced the resonance and, from Fig. 7, we can see that this isthe case when ω j ∼ ω LO . Consequently, resonances that are further away from ω LO (i.e. j >
0) have less pronounced peak values, such that the maximal value of ˆ E ++2 is alwaysobtained for j = 0. Furthermore, Eq. (4.33) reveals that resonances are more pronouncedfor larger sample thickness d of a given TMI. We now investigate the resonances in moredetail and provide analytical expressions for their widths and maximum values.Around the resonances we have | n Θ | (cid:28)
1, and Eq. (4.31) can be approximated as:ˆ E ++2 E = − n + i n Θ cot (cid:16) ∆2 (cid:17) . (4.35)Expanding n around ω j yields, to lowest order, n = n δω j b , (4.36)where we require that b > δω j . The expansion of cot(∆ /
2) leads, to lowest order, tocot (cid:18) ∆2 (cid:19) = − ω j dn n Θ ( ω j ) b (cid:16) ω − ω j (cid:17) . (4.37)The emitted fields in Eq. (4.35) can then be approximated about the resonances asfollows: ˆ E ++2 E = − iA j iγ j ω j + ( ω − ω j ) , (4.38)with γ j = 4 d ω j − ω LO ω j ≈ j b n d ω LO , (4.39) A j = 4 b n ω j d ≈ b n ω LO d , (4.40)where we used that in a resonant case ω j is close to ω LO .The power output on resonance is: P = | E | β A , (4.41)where A is the surface area of the TMI layer and where we used that the Poynting vectorin z -direction has magnitude | ˆ E ++2 | . The power boost factor β is defined as β = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ E ++2 E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4.42)– 54 – × - × - × - × - × - × - × - × - × - × - × - × - × - × - × - × - × - × - × - Figure 15 : Effect of material thickness, d , on the boost factor. In the losslesslimit for one layer β is given by Eq. (4.31). We assume zero DA velocity ( v DM = 0, validwhen the resonance is wide compared to the DA linewidth). Typical material values forMn Bi Te TMI with an external B -field B e of 2 T are chosen, cf. Table 5 and Eq. (2.3).Vertical lines mark the frequencies ω LO and the resonance frequencies ω j . The resonanceboost increases, and bandwidth decreases, as the thickness d increases.Following Ref. [47], we refer to the unsquared β as the boost factor.The full width at half the maximum value (FWHM) of β about the resonance ω j is given by γ j . The highest value at the resonance, the peak amplitude, ω j is given by β ( ω j ) = A j γ j ω j ≈ dω LO ∆ j ! ≈ n Θ ( ω j ) . (4.43)With Eq. (4.43) it now becomes explicitly clear that the higher modes have a lowermaximum resonance value, since ∆ j < ∆ j +1 . Also large layer thicknesses d increasesthe maximal emitted E -field on resonance. Therefore to achieve a certain amount ofsignal boost from one layer of TMI a relatively large layer thickness d > /ω is needed.Equation (4.39) tells us the necessary information about the width of the resonance. Firstnote, that going to larger modes j or larger b will increase the FWHM γ j . Relativelythick TMI layers, i.e. large d , yield a very narrow resonance. Therefore a good balancefor d has to be found because d should be relatively large to reach a high resonance– 55 –alue. The refractive index n does not affect the maximum value of the resonance, cf.Eq. (4.43). However large n makes the FWHM γ j very small. We therefore concludethat it is advantageous to have low n materials fro broadband response.In Fig. 15 the boost amplitude β is shown for our benchmark material Mn Bi Te for four different layer thicknesses. For the the thinnest case, d = 0 . ω (dashed vertical line) that is too far away from ω LO (solid vertical line) such that n Θ ( ω ) is not much smaller than unity. For d = 2 mm wehave a resonance at ω . A larger width, but a lower maximum value, is realized at thesecond resonance peak ω . According to Eq. (4.39), the width of the resonance around ω should be broader by a factor of about (∆ / ∆ ) = 9 than the resonance widtharound ω .For thicker samples still, d = 5 mm, d = 10 mm the resonant boost at ω increasesfurther. In particular from Eq. (4.43) we find that the peak heights scale as ( d /d ) .Furthermore, the width of the peak around ω shrinks as d increases. From Eq. (4.39)we can directly read off that the width shrinks by a factor ( d /d ) . For d = 10 mmthe linewidth of β is on the order of the DA linewidth, 10 − m a . Without taking intoaccount material losses the linewidth of the power boost factor is larger that the axionlinewidth if γ j > − ω j ≈ − ω LO . (4.44)Equation (4.44) tells us the requirements for the material parameters such that the powerboost factor bandwidth is larger than the axion linewidth.Before we discuss the influence of material losses we want to give a clearer physicalpicture of the observed resonances. In Fig. 16 we consider three domains. The middledomain is a TMI layer with thickness d and with an effective refractive index n Θ = .The two outer domains are vacuum with n = 1. The axion induced field, which is shownin blue, is one in vaccum and enhanced inside the TMI, cf. Eq. (4.22). The enhancementof the axion induced field in the TMI layer is proportional to n . Consider now theinterface between the TMI and the left vacuum. To fulfil the continuity requirementof the total electric field, propagating modes (red) are emitted to both sides. One cancheck that this is indeed the case by adding the red and blue amplitudes at the interface.The emitted amplitude in vacuum is one, while the propagating fields inside the TMIare enhanced, since they are proportional to n . The outlined scenario happens at bothinterfaces of the TMI. Let us first consider the emitted radiation that propagates insidethe TMI from left to right. The radiation hits the right interface. The transmissionand reflection coefficients determine the fraction of the radiation, which is transmittedto the outside or reflected. For plane waves we have T = n Θ n Θ and R = n Θ − n Θ . Theimportant point is that the transmitted radiation is added in phase ( T >
0) to theradiation which is emitted from the right interface to the outside. The part of radiationwhich is reflected at the right interface receives a phase shift since
R <
0. Thereforethe reflected radiation is coherently added to the radiation which is emitted from theright surface to the left in the TMI. A similar scenario happens at the left interface.Now since n Θ (cid:28) Θ =1/2 n=1 E[E ] ~1/n Θ ~1/n Θ T= >02n Θ Θ n Θ -11+n Θ R= <0d β =4 n=1 z~1/n Θ Figure 16 : Physical understanding of the resonant enhancement of the emitted electro-magnetic fields from the mixing between DAs, AQs and photons. The TMI layer (gray)of thickness d has an effective refractive index n Θ = < E -fields are given in units of the axion-induced field in vacuum, E . Due to the smallness of n Θ around the resonance the transmission coefficient for thefields which propagate inside the TMI is small, while the reflection coefficient is large.Therefore effectively the TMI works as a cavity. After bouncing many times betweenthe two interfaces the effective emitted field is proportinal to β (cid:29)
1. In the specific ex-ample of n Θ the emitted field is four times larger than the axion induced field in vacuum( β = 4). A more detailed description can be found in the text.large. Therefore the radiation bounces many times between both interfaces. After eachbounce a small fraction of the radiation is transmitted to the outside. This is exactlyhow a cavity works and due to the fact that the transmitted fields to the outsides are alladded coherently the total emitted field is enhanced by the boost factor β = 1 /n . Thetotal emitted field is shown in green in Fig. 15. With this physical picture in mind wecan also understand why larger thicknesses d lead to larger β ’s on resonance. To fulfillthe resonance condition n Θ ( ω j ) ω j d = π we need a smaller n Θ ( ω j ) the larger we make d .However making n Θ ( ω j ) smaller leads to a larger axion induced field and therefore alsoto a larger total emitted field. Case with losses (Γ = 0). In the case of losses we can use that g aγ is a relativelysmall coupling and therefore k + → k Θ , where k Θ now includes losses, cf. Eq. (3.47). k − → k a = 0 in the axion zero velocity limit. Furthermore we have shown in Section 4.2.1that the relations for a E and E a still hold when we include the losses into the effectiverefractive index. In conclusion we can use Eq. (4.31) also if losses are present. The only– 57 – - - - - - - - - × - × - × - × - × - × - × - × - Figure 17 : Effect of losses on the boost factor.
TMI layer of thickness d = 1 mm.Other parameters are as Fig. 15. Left : Varying magnon losses Γ m , with Γ ρ = 0. Right :Varying photon losses Γ ρ with Γ m = 0.thing that we have to do is to use the effective refractive index which includes losses: n = n b m − ω − iω Γ m + i Γ ρ ω ! . (4.45)We subsequently neglect mixed losses.We begin by expanding the boost factor around the resonance frequencies ω j , whichremain unmodified by the losses, cf. Eq. (4.32). Then an expansion can be done incomplete analogy to the lossless case. For | − n ( ω j ) | ≈ n Θ is given now by Eq. (4.45).We now expand the two terms, n and cot (cid:16) ∆2 (cid:17) , that appear in the denominator inEq. (4.35). We find n ( ω j ) = n δω j b + i ˜Γ j ! (4.46)with ˜Γ j ≡ Γ ρ ω j + ω j Γ m b . In deriving Eq. (4.46) we have assumed that b > δω j and b > ω j Γ m . If such a condition is not fulfilled, the material is likely too lossy to be usefulin DA detection. Next we consider the nearly lossless limit, i.e. δω j b > ˜Γ j . In this casethe relevant expressions for us are n Θ ( ω j ) = n δω j b (cid:18) i
12 ˜Γ j δω j b (cid:19) and n ( ω j ) = n δω j b ,where we have written down only the important leading order terms. Next we expandthe cot (cid:16) ∆2 (cid:17) term:cot (cid:18) ∆2 (cid:19) = cot (cid:18) ∆2 (cid:19) ω = ω j + (cid:20) ∂∂ω cot (cid:18) ∆2 (cid:19)(cid:21) ω = ω j (cid:16) ω − ω j (cid:17) + · · · (4.47)– 58 –here we approximate ω ≈ ω j for the linear dependencies and the dots representhigher order terms, which we do not have to consider for a reasonable expansion. Theexpansion in Eq. (4.47) can be simplified in the small thickness limit, Im [∆( ω j )] < (cid:18) ∆2 (cid:19) = − i Im [∆( ω j )]2 −
14 ∆ j δω j (cid:16) ω − ω j (cid:17) (4.48)Putting everything together we obtain – as in the lossless case, cf. Eq. (4.38) – aLorentzian shaped functional dependence around the the resonance frequencies. Thewidth of the curve receives an additional term in the presence of losses: γ j = 4 b ∆ j n ω LO d + Γ m + b ω LO Γ ρ ! . (4.49)and the amplitude A j remains unchanged with respect to the lossless case.In Fig. 17 we show the boost factor β around the first resonance ω for a layerthicknesses of d = 1 mm for different values of the loss parameters Γ ρ and Γ m . Weobserve that each loss parameter has a similar quantitative effect on redicing the boostfactor peak, with magnon losses being only slightly more important. This is due to thefact that Γ m directly enters the resonance, cf. Eq. (4.45), while photon losses Γ ρ onlyenter via an additional term that is added to the other terms of the dispersion relation.Next let us discuss the effect of losses on the higher resonance frequencies. In Fig. 18we show the first two resonance peaks at ω and ω for d = 5 mm. In the lossless casewe have β ( ω ) ≈ ω resonance is therefore moresevere than the resonance at ω . This is simply the case because the system is moreresonant at ω and losses lead to a larger reduction. We conclude that losses may leadto a scanning strategy in the end that uses a higher resonance mode j >
0. However, thefinal scanning strategy can only be given when the losses are determined experimentally.We can see from Eq. (4.49) that the point where losses dominate is a function ofthe refractive index, thickness of the material and intrinsic losses. While, in the losslesscase, increasing the thickness of the layer increases the resonance, we can see that thisis limited by the losses, which give a width independent of d . Looking at the the heightof the resonance in the loss dominated limit is quite revealing β ( ω j ) = A j γ j ω j = 4 b n ω LO d b ∆ j n ω d + b ω LO (cid:16) ω LO Γ m b + Γ ρ ω LO (cid:17) loss dom. ≈ dn (cid:18) ω b Γ m + Γ ρ (cid:19) . (4.50)Unlike the lossless case, increasing d now hinders the resonance height, if not its width.Similarly, while n does not effect γ j , again the height is significantly reduced on reso-nance, further discouraging high n materials. Once the loss term of a material is known,the optimal thickness can be found by requiring that losses do not dominate.To get an idea of the scale of the maximum losses that still allow for useful DAdetection, we plot the maximum of β at the first resonance ω as a function of d in– 59 – - - - - - - - - × - × - × - × - - × - × - × - × - - Figure 18 : Effect of losses on the higher resonance peaks.
TMI layer of thickness d = 5 mm. Other parameters are as Fig. 17. The value of β at ω without losses is around β ≈ ω than at ω . - - - Figure 19 : Maximal boost factor β on resonance at ω with respect to the TMI layerof thickness d for different levels of loss. Material parameters correspond to Mn Bi Te with B = 2 T. The bands show variation of the refractive index from n = 3 (upper curveof the bands) to n = 7 (lower curve of each band).Fig. 19. The different colours indicate different losses. We also vary the refractive indexaround our best guess value, n = 5: the upper band for each colour corresponds to n = 3,while the lower to n = 7. Lower values of nn lead to greater boost factor maxima.From Fig. 19 we can read off that for given material parameters there is an optimal– 60 –hickness, which maximizes β on resonance. Analytically one can show d opt = 2 ω LO (cid:18) ∆ j n (cid:19) Γ ρ ω LO + Γ m ω LO b . (4.51)Note that the otimal thickness d opt gives a thickness that is consistent with the small d limit that we have used in the expansion only if 2 (cid:16) n ˜Γ j ∆ j (cid:17) <
1. This inequalityis fulfilled for the cases that we are interested in. In the zero loss limit the optimalthickness d opt diverges . However it is important to stress that in this limit γ j → β onresonance. Which thickness will be the optimal one with respect to a scanning strategyand sensitivity reach will be discussed in the next section. If the losses are finite theyenter with the third root. Also note that scanning different frequencies changes ω LO andtherefore in principle for each scanning frequency a different optimal layer thickness isneeded. It is not possible to change the layer thickness for each scanning frequency, andtherefore the true optimal thickness will depend on the details of the frequencies to bescanned and the scan strategy.Note that in Fig. 19 we are approximating the boost factor as a resonance. However,as β → δ j = P i and d = d opt . By requiring that β (cid:29) ρ ω LO + Γ m ω LO b (cid:28) n . (4.52)Thus the highest allowable losses are actually set by the refractive index of the material,at least in order to ensure a resonance occurs. In this section, we review the suitability of TMIs hosting AQs for DA detection. We sys-tematically investigate the discovery reach of the proposed single-TMI-layer benchmarkexperiment, and the necessary requirements for THz detectors. Astrophysical limits onthe axion mass and coupling, and motivation for axion DM in the milliectronvolt range,is reviewed in Appendix B.
Before considering THz detection technology and the reach of the proposed experimentin terms of the DA coupling, g aγ , we first determine the range of DA masses thatcan in principle be accessed using TMIs. Recall from Section 4.2.3 that the resonancefrequencies of the experiment, ω j , can be tuned by changing the external magnetic field,– 61 – e . To estimate the resulting range, we look at the first resonance ω since | β ( ω ) | > | β ( ω j ) | for all j >
0. Doing so, we find that ω ( B e ) ≈ ω LO ( B e ) = q m + b ( B e ) , (5.1)where m Θ is the AQ mass and b is given by Eq. (2.3). In the limit of B e → ω → m Θ , while for very strong B -fields of B e = 10 T and our our benchmarkparameters in Table 8 (our best approximation to Mn Bi Te ), we find that1 . m Θ < ω < . . (5.2)As of now, the DA mass is unknown, so it is desirable to cover a wide range of axionmasses with a given TMI crystal. Since typical magnetic fields in the lab are restrictedto the order of a few tesla, we cannot arbitrarily increase B e and hence need to maximisethe relative response of the AQ to B e , viz.1 m Θ d b d B e = απ √ √ (cid:15)f Θ m Θ ≈ . (cid:18) (cid:15) (cid:19) / (cid:18)
70 eV f Θ (cid:19) (cid:18) . m Θ (cid:19) . (5.3)This means that smaller f Θ , m Θ , or (cid:15) are beneficial for a TMI in order to cover a largerrange of frequencies for a given maximum possible value of the applied B -field.A large relative AQ response in Eq. (5.3) is only beneficial if the applied B -fieldvalue can be controlled to sufficiently high accuracy over the course of the measurement.This is because fluctuations in B e will translate into fluctuations in ω , which might resultin the resonance around ω fluctuating in and out of the bandwidth of the detector. Themagnet design for TOORAD will thus require relatively precise control of the B -field,and could be a limitation in cost, field strength, or total volume. Searching for dark DAs is challenging because the resultant photon signal is very weakand can be hidden in wide range of frequencies, since the DA mass in unknown. Toimprove our chance of success, we need to understand the intrinsic and extrinsic back-ground noise of our photon detection system, coupling efficiency of photon detectors toour proposed experimental setup, and scalability in collecting more photons from thematerial that hosts the AQ. In earlier sections, we have discussed the generation of elec-tromagnetic radiation in the THz (millimeter wave) regime using AQs for DA detection.We will focus here to the available technology to detect these photons with energies from0.01 to a few THz.Detectors that have high sensitivity for the search of DAs in our frequency rangeof interest include amplifiers, heterodyne mixers, bolometers, and single-photon detec-tors (SPDs). We shall consider experiments performing at temperatures much lower thanthe frequency, i.e. T (cid:28) ω to avoid the thermal photons from the blackbody radiation andto focus on the fundamental limit of photon detection [136–139]. Since amplifiers andheterodyne mixers, e.g. superconductor-insulator-superconductor (SIS) and hot-electronbolometric mixers, are sensitive to the voltage or the electric field of the signal, we can– 62 – able 7 : Comparison of detector technologies for searching dark matter using quasi-particle axions. See the main text for explanations of the symbols.Detector type Fundamental noise limit MetricAmplifiers &heterodyne mixers quantum noise (cid:126) ω [136, 138, 139] T Q Bolometers thermal fluctuations √ G th k B T [137, 140] NEPCalorimetric SPDs energy resolutionfor finite bandwidth q C th k B T [141, 142] dark count rateput them into one category, while bolometers and SPDs go into another. We present acomparison of all detector types discussed in what follows in Table 7.State-of-the-art amplifiers and mixers can detect a very weak signal from as littleas a few photons by parametric amplification or non-linear mixing processes. As thesignal-to-noise ratio is given by the ratio of the number of photons in the signal to thatin the amplifier noise, the amplifier noise can be quantified naturally in units of photonquanta, i.e. (cid:126) ω , or amplifier noise temperature, i.e. T Q = (cid:126) ω/k B . For the linear, phase-preserving amplification, the minimum amplifier noise is one quanta [136, 138, 139]. Halfof this comes from the quantum fluctuation from the parametric pumping port used inmodulation for the amplification gain, whereas another half from the quantum noisein the signal port. At lower microwave frequencies, quantum noise-limited amplifiershave been achieved based on parametric effects [143–145] and, at higher frequencies,in SIS detectors [138, 146], hot electron bolometric mixers [147, 148], and plasmonicmixers [149]. However, the insertion loss and insufficient first amplification gain maydegrade the overall performance, resulting in a higher system noise temperature T sys ≥ T Q . For a total measurement time, t meas , and measurement bandwidth, BW, the averagenoise is given by the Dicke radiometer formula,Noise = T sys √ BW t meas (5.4)Instead of amplifying the voltage, bolometers are high-sensitivity, square-law de-tectors that measure the power of microwave and far-infrared radiation. They operateby first absorbing the incident radiation and subsequently inferring the radiation powerfrom the temperature rise due to the increase of its internal energy. The bolometersensitivity is quantified by noise equivalent power (NEP), measured in units of W / √ Hz,i.e. the power fluctuations of the bolometer in absence of any incident power dur-ing a 1-second averaging window. Previous experiments project NEP values as low as10 − W / √ Hz [150]. The sensitivity of this technique is not limited by quantum fluctua-tions, but rather by the fundamental thermal fluctuations [137, 140]. This fundamental-fluctuation-limited NEP is given by √ G th k B T with G th being the thermal conductanceof the bolometric material to the thermal bath, and T is the bolometer (bath) temper-ature. Therefore, bolometers for DA detection will require to operate at the lowestachievable temperatures with the least thermal conductance to its surrounding.– 63 –n addition to power detection by bolometer, single-photon detectors (SPD) is an-other viable option to capture the photons generated from DAs. Efficient DA searcheswill require the SPD to have simultaneously a high quantum efficiency to register everyprecious photon, and a low dark count rate to minimize the false positive signal. Natu-rally, these two requirements are competing against each other because a higher quantumefficiency also means the detector can be triggered by noises to produce a count in theabsence of photons. Fortunately at cryogenic temperatures, we can employ supercon-ductors to detect photons efficiently and accurately. When the photon energy is largerthan the superconducting gap energy, ∆ S , the incident photons can break Cooper pairsand produce a sizable number, η d (cid:126) ω/ ∆ S , of quasiparticles, with η d < .
12 THz [150].Recently, quantum capacitance detector [158] has demonstrated the detection of 1 . (cid:15) = q C th k B T with C th being the thermal heatcapacity of the calorimeter, is still a good benchmark for calorimeter SPDs. When theNEP is white-noise limited, we can use [162]∆ (cid:15) = NEP s C th G th (5.5)to compare the sensitivity with bolometers. For DA searches in wide frequency band-width, graphene-based single photon detection also has an advantage in wide bandwidthphoton coupling by impedance matching the input to the photon absorber with an an-– 64 –enna. Spiral, log-periodic, and bow-tie antennas have been implemented for graphenedetectors [163, 164]. As graphene-based bolometers have been demonstrated recently inthe microwave regime [140, 165] with energy resolution projections to a few 10 GHz, itcan potentially complement SPDs by operating at millimeter wave frequencies.In addition to superconductor-based and calorimeter SPD, superconducting qubitsand quantum dots can also detect single photons [166, 167]. These nano fabricateddevices have discrete energy states and can serve effectively as artificial atoms. Whenincident photons promote the qubit or quantum dot to an excited state, they can bedetected by measuring the state of the artificial atoms. Detection of single photonshas been demonstrated using superconducting qubits at microwave frequencies [168,169] and using quantum dots as low as 1 . λ d ∼ η = 0 .
01 [167, 171]. Overall, it is desirable to obtain a detector with theoptimal combination of low dark count rate and high efficiency, as this will ultimatelydetermine the sensitivity of TOORAD. Detectors that feature a better efficiency typicallyhave a worse dark count rate than the detector from Ref. [167] considered above. Wewill therefore define a pessimistic (optimistic) scenario by setting λ d ∼ η = 0 .
01 ( η = 1).Last but not least, we shall consider how to put the photon detector togetherwith the material that hosts the AQs. The efficiency of the dark matter search relieson this system integration. The goal is to maximize the photon coupling as the axionquasiparticle material scales up. Therefore we will need to design an antenna that cancollect the photons emitted from the AQs to the detector with the least inert loss.This will be an important factor to select a potential detector technology to develop.Ultimately, to detect a small signal from DA, the detector metric should be the totalexperimental averaging time for an experiment to reach a statistical significance andwill depend on both efficiency and sensitivity. To improve our chance of detecting darkmatter, we need more research on detector technologies, which are also be useful inother applications including radio astronomy, spectroscopy, and medical imaging [158,172, 173]. As discussed above, the signal from the DA-polariton-photon conversion may be detectedusing an SPD, which is superior compared to heterodyne power detection in THz. In thissection we quantify the sensitivity and discovery reach for a photon-counting experiment.The detection of individual photon events is governed by Poisson statistics i.e. thelikelihood of detecting N photons given model parameters x (the set of DA and material– 65 –roperties) is given by p ( N | x ) = ( η n s + n d ) N N ! e − η n s − n d , (5.6)where n s = λ s τ and n d = λ d τ are the number of expected signal and dark count events,respectively, as calculated from their respective rates, λ s and λ d , and total observationtime τ . The parameter η describes the total detector efficiency i.e. takes into account theintrinsic efficiency of the detector as well as any other imperfections in the experimentalsetup. Note that Eq. (5.6) assumes that there are no external backgrounds present.While we do not use a likelihood approach based on Eq. (5.6) directly for our esti-mates, it should be noted that the form above is a better approach than the asymptotic,approximate equations used in what follows. For the case of a single-bin Poisson distri-bution without any nuisance parameters – i.e. assuming that the material and detectorproperties are perfectly known – we performed a Monte Carlo simulation to check thevalidity of the asymptotic formulae that we employ. We found them to be conservativeand, hence, suitable for the purpose of estimating TOORAD’s sensitivity. For an actualanalysis of experimental data, however, a likelihood-based approach should be used. In order to compute the sensitivity, we assume that no significant signal over backgroundis found. The significance is S = 2( √ n s + n d − √ n d ), where n s is the number of signalevents and n d the number of dark count events [174–176]. Then the exclusion limit at95% C.L. for photon counting based on Poisson statistics (Eq. (5.6)) is obtained from S <
2, i.e. λ s < τ + 2 q λ d τ , where λ d is the dark count rate, λ s the signal rate and τ the measurement time. For a discovery one would require S >
5. In Section 5.2 weargued that λ d = 1 mHz is reasonable. In the following we estimate the sensitivity intwo scenarios. The case τ < q λ d τ can be achieved for sufficiently long measurementtimes and is called the background dominated scenario, i.e. τ > λ d = 250 s. If themeasurement time is short τ < λ d = 250 s then it is not background dominated.First, we investigate the case that the measurement is not background dominated.The number of signal photons per measurement time is λ s = η | E | ω A β where A isthe surface area of the TMI, η the photon counting efficiency and ω = ω j ≈ ω LO isthe resonance frequency where the power boost factor peaks. The power boost factoris the emitted electromagnetic field normalized to the axion induced field E , which isdetermined by the local axion dark matter density ρ a , the axion photon coupling g aγ andthe strength of the external B -field: E = g aγ B e a − ’ g aγ √ ρ a m a B e . Putting everythingtogether we obtain the sensitivity estimate: g aγ > . × − GeV − (cid:18) . η (cid:19) (cid:18) B e (cid:19) (cid:18) β (cid:19) (0 . A ! (cid:18) τ (cid:19) ×× . GeVcm ρ a ! (cid:18) m a .
83 meV (cid:19) , (negligible backgrounds, τ < λ − d ) (5.7)– 66 –here an axion mass m a = 2 .
83 meV corresponds to the scanned axion mass with a 2 Texternal B -field under the assumption of the benchmark material ( n = 5 , f Θ = 64 eVand m Θ = 2 meV). The reference area in Eq. (5.7), A = (0 . . v = 10 − c andmass 2 .
83 meV. Single crystals of MnBi Te grown in Ref. [119] are on the order ofcm . Reaching large surface area will thus require tiling and machining many crystalstogether. Tiling is known to introduce significant complications for dielectric haloscopeslike MADMAX [177, 178]. Further, as the axion gives an opening angle of v ∼ − thecollecting area of the THz detector must be large. We anticipate that this problem canbe overcome with the correct antenna.When there are finite losses, we can use the peak value from Eq. (4.50) to eliminate β and obtain: g aγ > . × − GeV − (cid:18) . η (cid:19) (cid:18) B e (cid:19) (0 . A ! (cid:18) τ (cid:19) . GeVcm ρ a ! ×× (cid:18) .
83 meV m a (cid:19) (cid:18) ∆ j π (cid:19) (cid:18) d (cid:19) × Σ . (5.8)where we have defined in the dimensionless quantity:Σ ≡ dd opt ! . (5.9)We did not plug in any specific value for Σ in the sensitivity estimate because when thethickness is chosen to be close or equal to the optimal thickness Σ is of the order one.The losses, AQ decay constant, f Θ , and refractive index, n, all appear implicitly via thedetermination of d opt , the optimal material thickness.Compare the the sensitivity Eq. (5.8) to that obtained with heterodyne detection.In this case we use the Dicke radiometer equation with noise temperature T . The signalover noise ratio is given by SNR = P s T sys q τ ∆ ν a , where ∆ ν a = 10 − m a is the DA linewidth,and P s . If the physical system temperature is low enough, cf. Section 5.2, T sys is limitedby the standard quantum limit (SQL) T sys = ω = m a . The resulting sensitivity is: g aγ > . × − GeV − (cid:18) SNR2 (cid:19) (cid:18) m a .
83 meV (cid:19) (cid:18) B e (cid:19) . GeVcm ρ a ! ×× (cid:18) β (cid:19) (0 . A ! (cid:18) τ (cid:19) (heterodyne SQL). (5.10)The sensitivity is worse than the SPD, cf. Eq. (5.7), by approximately an order ofmagnitude. This is as expected since for high frequencies the SQL pushes T to largevalues. The SQL can, however, be overcome by “squeezing” [37].Next we focus on the case that the measurement is background dominated ( λ s < q λ d τ ). For our benchmark dark count rate of λ d this gives τ >
250 s. Long measurement– 67 –imes on a fixed frequency could be adopted in a “hint” scenario where the axion massis thought to be known by some other means (for example, an astrophysical hint, orhighly accurate relic density prediction), and a resonant DM search is required to verifythe hint. To consider this scenario, we take the measurement time on each frequency tobe τ = 3 yr, i.e. an entire experimental campaign. The sensitivity in this case is: g aγ > . × − GeV − (cid:18) . η (cid:19) (cid:18) B e (cid:19) (cid:18) β (cid:19) (0 . A ! (cid:18) λ d − Hz (cid:19) (cid:18) τ (cid:19) ×× . GeVcm ρ a ! (cid:18) m a .
83 meV (cid:19) (background dominated) . (5.11)Using now again the maximum peak value from Eq. (4.43) to eliminate β in the previousequation we obtain the sensitivity estimate g aγ > . × − GeV − (cid:18) . η (cid:19) (cid:18) B e (cid:19) (0 . A ! (cid:18) λ d − Hz (cid:19) (cid:18) τ (cid:19) ×× . GeVcm ρ a ! (cid:18) .
83 meV m a (cid:19) (cid:18) ∆ j π (cid:19) (cid:18) d (cid:19) × Σ . (5.12)To complete our discussion we also estimate the sensitivity for bolometric detectorswhose performance is specified by the NEP. The minimal detectable signal power whichsuch a detector can detect is P s > NEP / √ τ . Evaluating this leads to the sensitivity: g aγ > . × − GeV − NEP10 − W / √ Hz ! (0 . A ! (cid:18) τ (cid:19) ×× (cid:18) m a .
83 meV (cid:19) . GeVcm ρ a ! (cid:18) β (cid:19) (5.13)The sensitivity estimate in Eq. (5.13) has a similar order of magnitude as the SPDsensitivity, cf. Eq. (5.11) but a slightly different scaling with the axion mass. We now compute forecasts for the baseline parameters of “Material 2” (best approx-imation to Mn Bi Te , with refractive index n = 5 and µ = 1), and consider threepossibilities for the losses, Γ m /ω = Γ ρ /ω = 10 − , − , − . Assuming a fixed ratioΓ /ω is consistent with our model for the impurity based losses, and assumes (cid:15) is ap-proximately constant in the relevant range. For fixed ratio Γ /ω , there are larger lossesat higher frequencies. We first assumed SPD efficiency of η = 0 .
01 and dark count rate λ d = 10 − Hz, which has been demonstrated. We also show a more optimistic sensitivityestimate with η = 1 (dotted line) for Γ ρ /ω = Γ m /ω = 10 − . The surface area of the TMIlayer is fixed to be A = (0 . , where 0 . j = 0 for the sensitivity estimate.We consider two different scanning scenarios, with B -field values from 1 T to 10 T:– 68 – m /m a = Γ ρ /m a − − − m a [meV] − − − − − − − g a γ [ G e V ] (a) Scanning scenario (I) m a [meV] − − − − − − − g a γ [ G e V ] d=2.0mmd=0.9mmd=0.4mm (b) Scanning scenario (II) m a [meV] − − − − − − − g a γ [ G e V ] (c) Scanning scenario (I) m a [meV] − − − − − − − g a γ [ G e V ] d=2.0mmd=0.9mmd=0.4mm (d) Scanning scenario (II)
Figure 20 : Sensitivity for “Material 2” baseline parameters (see text for details) forvarious loss values Γ. Top row: η = 0 .
01. Bottom row: η = 1. We fix the dark countrate λ d = 10 − Hz. The yellow band shows QCD axion models, and the dashed blue linethe CAST exclusion on g aγ . The scanning scenarios are defined in the text.• Scanning I. We begin at the highest frequency with the largest B -field where thebase power is largest and the QCD band is at the largest g aγ . We scan to the topof the QCD band. We then move by the width γ on to the next frequency atlower B , and repeat for a total scan time of 3 years. Fig. (20,a). We computethe optimal thickness with the largest axion mass within the scanned region.• Scanning II. We scan for a fixed time set equal on all frequencies and repeat for Note that we assume the peak power is achieved over the width γ . While this is less conservativethan assuming the minimal value (i.e., half the maximum), note that power also exists outside of γ ,which would still be integrated over during the scan. As this is an estimate, rather than a detailedexclusion limit of an experiment, the final limit would likely fall somewhere between the two. – 69 – m /m a = Γ ρ /m a − − − m a [meV] − − − − − γ m a Figure 21 : Linewidth of the boost parameter for “Material 2” baseline parameters (seetext for details) for various loss values Γ.a total scan time of 3 years. In each step we move in frequency by the width γ ,Fig. 21 (Eq. 4.49). We compute the optimal thickness with the axion mass that isin the middle of the scanned interval.In each case the limit is found for signal to noise equal to two, 95% C.L. exclusion.In the Scanning II case we assume that each individual scan takes the same amountof time τ . Then with the bandwidth from Eq. (4.49) (Fig. 21) we can calculate the totalnumber of scans. From this we then calculate the scan time for each individual scan suchthat the total scanning time for each case is t scan = 3 years. Depending on the individualscan time τ we calculate the sensitivity in the right limit, cf. Eq. (5.8) and (5.12).In the Scanning I scenario for η = 0 .
01, we find that a wide range of the QCDband can only be covered in the case with extremely small losses, Γ /ω = 10 − . Withthis scanning strategy, η = 1 detection efficiency allows a wide range of the top of QCDband to be scanned for all loss parameters. In the Scanning II scenario the QCD bandcannot be reached with η = 0 .
01. However, with η = 1 we find that a reasonable portionof the upper part of the QCD band can be scanned with Γ /ω = 10 − . With very lowlosses Γ /ω = 10 − and η = 1 the Scanning II scenario reaches almost KSVZ sensitivityacross a wide range of masses. We also considered the intermediate case η = 0 .
1, whichallows some sensitivity to the QCD axion band with Γ /ω = 10 − . We conclude that asuccessful QCD-sensitive experiment requires high efficiency SPDs.– 70 – able 8 : Parameter reference values and ranges. Our benchmark material is “Material2”, based on Mn Bi Te .Parameter name & symbol Range Benchmark TMI parameters
Decay constant f Θ [50, 200] eV 70 eVAQ mass m Θ ∼ O (meV) 1 . (cid:15) [9,49] 25Magnetic permeability µ ∼ O (1) 1Magnon losses Γ m [10 − , 10 − ] meVSpecific conductance Γ ρ [10 − , 10 − ] meVArea of crystal face A (0 . Thickness d d opt , cf. Eq. (4.51)
Experimental parameters
External B -field B e [1, 10] T 2 TDetection effciency η [0.01, 1] 0.01Dark count rate λ d (cid:38) We now wish to investigate how the sensitivity and scan range depend on the yet un-known material parameters of the TMIs. In this section we consider only the scanningII scenario. In Table 8 we list the unknown parameters, and reasonable ranges theymight take in different materials within our rough approximations to the theoreticaluncertainties. The ranges for the parameters have been motivated in Section 2.4.In Fig. 22 we study the effect of varying the AQ decay constant f Θ and the refractiveindex n on the scan range and sensitivity (we do not vary the AQ mass, since this hasthe trivial effect of changing the lower limit of the scan range). The sensitivity andother parameters are fixed as described in the previous subsection. Let us first discussthe scanning range. The smaller n and f Θ the larger is the axion mass range that canbe probed. This is because the upper range of the scanned axion mass is determined byEq. (5.1).To understand the effect of n and f Θ on the sensitivity, it is enlightening to studythe behaviour of the sensitivity estimates in the limit that the external B -field is verylarge, i.e. m a ≈ ω LO ≈ b ∼ B e nf Θ . Both sensitivity estimates in the background dominated,cf. Eq. (5.11), and in the non-background dominated limit, cf. Eq. (5.8), are proportionalto g aγ ∼ B e d , where we have assumed that Σ does not vary too much. Plugging in Remember that Σ would be exactly 3, if we would choose for each axion mass that is scanned theexact optimal thickness. However, in a scanning scenario this will for practical reasons not be possibleand we choose d to be the optimal thickness that corresponds to the axion mass that is in the center ofall axion masses that are scanned. As a consequence Σ can also be slightly larger than 3 in the wholeaxion mass that is being scanned. – 71 –he optimal thickness we obtain the scaling behaviour: g aγ ∼ (cid:18) B e (cid:19) (cid:18) f Θ (cid:19) √ n. (5.14)The strongest scaling is induced by the AQ decay constant f Θ . This view is also con-firmed by the plots in Fig. 22. However increasing f Θ also leads to a smaller scanninginterval such that the reached C aγ in the QCD band is almost constant. The refractiveindex n enters in the sensitivity only weakly with a square root dependence. Howeverfor fixed f Θ it is visible from the plots in Fig. 22 that decreasing n gives a slightly betterlimit on the DA-photon coupling. Furthermore, the scaling in eq. (5.14) only applies solong as the approximation m a ≈ ω LO ≈ b ∼ B e nf Θ holds. At large f Θ this approximationbreaks down for suitable values of B e (either the experimental maximum, or spin flopfield, whichever is lower).With these effects in mind, we revisit the candidate AQ material (Bi − x Fe x ) Se (“Material 1”), considered in Paper I. We estimate that this material has slightly smaller f Θ , and will thus have a slightly worse sensitivity to g aγ than the alternative Material2, although it will have a narrower possible scan range. To be more optimistic withMaterial 1, we adopt n = 3 for presentation (although this has a very small effect). Ourresults are collected in fig. 23. Appendix B gives more details about the QCD axion model assumptions indicated in this figure. – 72 – m /m a = Γ ρ /m a − − − − − − − g a γ [ G e V − ] f Θ [eV] → n ↓ − − − − g a γ [ G e V − ] m a [meV] − − − − g a γ [ G e V − ] m a [meV] m a [meV] Figure 22 : Sensitivity estimate for the DA-photon coupling g aγ varying the external B -field from 1 T to 10 T. The surface area is fixed to A = (0 . . The thickness d isset to the optimal thickness, cf. Eq. (4.51). We assume each frequency is scanned for thesame amount of time, and the total scanning time is t scan = 3 years. For the detector n b = 10 − Hz and efficiency η = 1. The yellow band represents the QCD band with C aγ = 12 . · · · .
25, cf. eq. (B.3) for the definition of C aγ . The dashed blue line showsthe CAST limit. The present work has developed the theory of axion quasiparticles in topological mag-netic insulators, and how such materials can be used to detect axion dark matter.– 73 – − .
01 0 . Dark axion mass m a [meV] − − − − − − − − − − D a r k a x i o n - ph o t o n c o up li n g g a γ [ G e V − ] H a l o s c o p e s ( f u t u r e ) H a l o s c o p e s ( c u rr e n t) CAST
IAXO (future) K S V Z Hot DMSN 1987AGorghetto+ ’20 realignment only
Material 1Material 2 realignment + strings
TOORAD (this work) η Γ / meV0 .
01 10 − − − Figure 23 : The projected TOORAD sensitivity for Material 1 [(Bi − x Fe x ) Se -inspired]and Material 2 (Mn Bi Te -inspired) for different losses and detector sensitivities.See Table 8 for all other benchmark parameter values. We show limits and fore-casts [179] for CAST [180, 181], IAXO [182], and various haloscopes [31, 183–192] (for ρ loc = 0 . / cm ) as well as the bounds from hot dark matter constraints [193], en-ergy loss arguments in SN1987A [13]. The preferred regions cold dark matter [194] inthe realignment scenario, and with the latest cosmic string decay calculations [195] arealso indicated as horizontal arrows. The QCD axion band encompasses all “preferred”KSVZ-type axion models as defined in ref. [29], in addition to the original KSVZ andDFSZ models. Model of Axion Quasiparticles:
We first presented in some detail the symmetrycriteria for the existence of axion quasiparticles, and the Dirac model for their realisationin topological magnetic insulators. While already known in the literature (e.g. refs. [42,43, 50, 51]), these have not been shown in detail in relation to axion DM, and provideimportant background to the subsequent results. We laid out carefully the symmetrycriteria necessary for a material to posses an AQ. Our exploration of the model shedslight on the nature of the AQ as a longitudinal magnon, i.e. a spatially and temporallyvarying AF spin fluctuation. It is non-linearly related to the transverse magnons ofordinary AFMR.In order to estimate the parameters f Θ and m Θ of the model, we used the resultof the ab initio calculation given in ref. [42] for (Bi − x Fe x ) Se on a cubic lattice. Werescaled the results to use updated values of the material parameters of (Bi − x Fe x ) Se ,and Mn Bi Te , for which there is not a result available in the literature. More accurate– 74 – b initio calculations of the parameters for both (Bi − x Fe x ) Se and Mn Bi Te arehighly desirable. We considered multiple possible sources of loss in these materials, andattempted to estimate the contributions to the polariton linewidth. This often involvedextrapolation of results obtained at different frequencies and only measured in relatedmaterials. Direct spectroscopic measurement of all these parameters is thus necessary. Axion Quasiparticle Detection:
We computed explicitly the transmission func-tion of AQ materials. This transmission function displays a magnetic field-dependentgap, and a series of resonances, which depend on the size of the loss terms. By mea-suring the frequency of the upper and lower ends of this gap, and the linewidths of theresonances, one could determine the parameters of the model directly. Furthermore,the gap in the polariton spectrum, and the scaling of the gap size with field strength,demonstrate directly the existence of the AQ and its coupling to the electromagneticfield via a Chern-Simons interaction. Thus, THz transmission spectroscopy can be usedto discover the AQ.The considered material candidates that can host an AQ are all antiferromagnets.Antiferromagnets exhibit an antiferromagnetic resonance (AFMR) with typical reso-nance frequencies in the THz regime. This raises the question how one can distinguishthe AFMR from the axion-polariton resonance in the transmission spectrum. It is wellknown how the AFMR frequency scales with a non-zero external B -field [134, 196, 197].This scaling is distinct from that of the axion-polariton resonance, which consists ofa fixed resonance at m Θ , and a second one near ω LO = q m + b ( B/B ) (where b = b ( B ) and B is a reference scale). We expect transmission spectra of the AFaxion insulator MnBi Te to show the single AFMR, while the AQ material Mn Bi Te will show both the axion polariton resonances and AFMR. Comparing results for bothmaterials and the B -field dependence will help isolate the effect of the AQ. Axion Dark Matter Detection:
We developed the computation of the poweroutput of an AQ material in the presence of axion DM. The system bears many simi-larities to dielectric and plasma haloscopes, and is characterised by a boost amplitude, β ( ω ). The boost amplitude increases with thicker sample sizes, and the height and widthof the boost are affected by magnon and photon losses. The power is amplified by β compared to a magnetized mirror, and for realistic models of the loss 10 (cid:46) β (cid:46) with a bandwidth of order 10 − to 10 − .Figure 23 shows our best estimates for the discovery potential of TOORAD com-pared to other constraints on axion dark matter, and proposals for future experiments.The present best estimate shows that TOORAD, using a material similar to Mn Bi Te could scan an O (1) range in the upper half of the QCD axion model band if the SPDefficiency is very good, η ≈
1. An extremely low loss material (Γ /ω ≤ − ) would beneeded to reach sensitivity to the KSVZ axion.The primary difference between the two material candidates considered lies in theestimated value of f Θ , with slightly higher values being favourable in the scan depth,but having a slightly narrower total range. If the spin flop transition of the materialis lower than the maximum 10 T field assumed, then the scans would begin at lowerfrequencies, and span a slightly smaller range of masses.– 75 – .2 DiscussionComparison to other axion detection proposals: We have considered detectingthe dark matter axion via the axion-photon coupling, g aγ , combined with the mixingbetween the photon and the AQ. It is interesting to note that if the dark matter axion alsopossess a coupling to electrons, g ae , then this can excite AFMR in the TMI via the “axionwind” derivative interaction [198] (this interaction has been successfully constrained withnuclear magnetic resonance [199] and ferromagnetic resonance [40, 200]). The AFMRaxion wind interaction opens the possibility that AQ materials could measure bothcouplings, g aγ and g ae , with the same material by tuning to different resonant modes.This could be used to perform model discrimination between the KSVZ model, withloop suppressed electron coupling, and the DFSZ model, with leading order electroncoupling. This would be an interesting line of future research.Similarly to dielectric and plasma haloscopes, TOORAD aims to avoid the Comp-ton wavelength limits imposed in traditional cylindrical cavities. Most experiments tryto avoid this limit through breaking translation invariance on roughly half Comptonwavelength scales. Examples include dielectric haloscopes [33] like MADMAX [201] andLAMPOST [202], multicavity arrays [203, 204] such as RADES [35, 205] and hybrid ap-proaches using dielectric loaded resonators [206, 207] such as Orpheus [208]. In contrast,TOORAD aims to give the photon an effective mass (in the low spin wave momentumlimit). In this sense, the most similar analogue in axion experimental design is a plasmahaloscope [209], which directly gives the photon a mass in the form of a plasma frequency.The THz regime represents a unique challenge for axion detection, as it representsan intermediate regime between scales and technologies. Dielectric haloscopes have beenproposed at lower [201] and higher [202] frequencies. THz represents a middle groundbetween the use of discrete, movable disks and and O (1000) layer deposited thin filmsimplying unique engineering challenges to cover the available parameter space.Dish antennas [30] are the simplest structure to target THz, due to their broadbandnature, however they lack resonant enhancement that could allow a more targeted searchat higher signal to noise. Currently the only proposed dish antenna in this range isBRASS [210].A more recent idea in the meV range is to use the axion’s coupling to phononpolaritons or magnons [211], however the resonance frequency in this proposal is noteasily tuned, which makes scanning axion masses difficult. To cover a range of axionmasses, different materials of high quality would need to be measured. Further, thesingle quanta measurement of such particles remains challenging [211].As the field of THz axion detection is still very young, and each approach hasdifferent material or engineering challenges, it is important to have a wide range of ideasin order have a chance to look in this well motivated, but very difficult, parameter space. Materials Science:
In terms of material research we have revealed there is a starkcontrast between conventional strong dynamical axion response in solids and dynamicalaxion quasiparticle response suitable for DM detection discussed here.• The axion quasiparticles for DM detection favour longitudinal spin waves withlinear coupling to photons. In contrast, the heterogeneous dynamical axion field– 76 –resent in the chiral magnetic effect or antiferromagnetic resonance of the stan-dard transversal spin modes does not provide within minimal models for such acoupling [72].• While conventional large axion response can be achieved close to the magneticphase transition [72], a DM search favours lower temperatures, ensuring sharperresonance linewidth free of thermal and scattering disorder.• The static quantised axion insulators are protected by axion odd symmetries suchas spatial inversion (parity). Our dynamical axion quasiparticles favour PT sym-metric systems: PT allows for Dirac quasiparticles enhancing the (dynamical)nonquantized AQ response by allowing tunability close to the topological phasetransition.Antiferromagnetism is favourable in many ways for axion DM detection. Reasons forthis include its compatibility with tunable axionic Dirac quasiparticles [68], availabilityof semiconducting band-structure with potentially large band-gaps, high critical tem-peratures, and large spin-flop fields. Furthermore, multi-sublattice systems can providefor a combination of separated heavy atomic elements with strong spin-orbit interactionand lighter magnetic elements. Materials wishlist:
We close with stating the desirable properties of an AQmaterial for axion DM detection.• Longitudinal spin wave mass, m Θ , in the meV range. The goal is to detect theQCD axion in this mass range. With much smaller m Θ there are already existingtechnologies, while for much larger values the QCD axion is already excluded.• Decay constant, f Θ , in the 10 to 100 eV range. For f Θ much larger than 100 eVthe AQ is not strongly coupled enough to the Θ term for efficient mixing. Anotherway to express this requirement is that the polariton gap for fields of order 1 Tshould be of order m Θ .• Low refractive index ( n (cid:46)
5) and high resistivity ( ρ > meV − ) in THz, prefer-ably measured from the axion-polariton spectrum resonance.• Low impurity density: impurity separation scale of microns or larger.• High spin flop field. This should definitely exceed 1 T for sufficiently large poweroutput. Larger spin flop fields permit a wider scan range.• High N´eel temperature. The experiment can be operated in a dilution refrigeratorwith T (cid:28) T (cid:28) T N .• Ability to manufacture samples with thickness in excess of 1 mm. Ultimately onemust also machine multiple samples together into a large surface area disk. Recall that in the Dirac model f = 2 M J where M is the bandgap and J is the spin wave stiffness. – 77 –e have shown that, with plausible assumptions, Mn Bi Te and (Bi − x Fe x ) Se bothsatisfy many of these requirements, although we expect the AQ phase of Mn Bi Te tobe more stable, since it does not require magnetic doping. If it can be proven that anymaterial satisfies the above requirements, then, in combination with existing detectorand magnet technology, such a material can be used to make an effective search for axiondark matter in the theoretically well-motivated mass range near 1 meV. Acknowledgments
We thank Bobby Acharya, Caterina Braggio, Nicol`o Crescini, Matthew Lawson, ErikLentz, Chang Liu, Eduardo Neto, Naomi Nimubona, Alireza Qaiumzadeh, AndreasRingwald, and David Tong for useful discussions. DJEM, SH, and MA are supportedby the Alexander von Humboldt Foundation and the German Federal Ministry of Ed-ucation and Research. JSE is supported through Germany’s ExcellenceStrategy - EXC2121 “Quantum Universe” - 390833306. KCF was supported in part by Army ResearchOffice under Cooperative Agreement Number W911NF-17-1-0574. FC-D is supportedby STFC grant ST/P001246/1, Stephen Hawking Fellowship EP/T01668X/1. EH issupported by STFC grant ST/T000988/1. AM is supported by the European ResearchCouncil under Grant No. 742104 and by the Swedish Research Council (VR) under Dnr2019-02337 “Detecting Axion Dark Matter In The Sky And In The Lab (AxionDM)”. ASis supported by the Special Postdoctoral Researcher Program of RIKEN. LS acknowl-edges the EU FET Open RIA Grant No. 766566, the Elasto-Q-Mat (DFG SFB TRR288), Czech Science Foundation Grant No. 19-28375X, and Sino-German DFG projectDISTOMAT.
A Antiferromagnetic Resonance and Magnons for Particle Physicists
A.1 Effective Field Theory of AFMR
We follow Refs. [130, 212], and present the effective field theory of antiferromagneticresonance (EFT of AFMR), which we believe is illuminating, especially from a particlephysics perspective.The EFT of AFMR considers the dynamics of the AF magnetization n consideredas a field in the continuum limit of the Heisenberg model of the magnetic lattice, whichis equivalent to the Hubbard model in the half-filling limit, as discussed in section 2.4.The magnetic lattice consists of A sites and B sites, with spins S A and S B at eachsite, and n = ( S A − S B ) /
2. The symmetry group G = SO (3) is related to the internalrotations of n (not spatial rotations). This symmetry is broken by the groundstateAF order, h n i = ( h S A i − h S B i ) / H = SO (2) of rotations about the axis. Magnetic order implies thatthe groundstate breaks time translation invariance, T , which flips the spin orientations.However, the groundstate preserves an effective time translation invariance e T = T S ,where S swaps the A lattice sites for the B lattice sites. This leads, as we shall see, toa “relativistic” dispersion relation for AF spin waves. Spin-orbit effects (finite electron– 78 –ass corrections) lead to explicit breaking of SO (3), which can be considered as aperturbation, and leads to a preferred “easy axis” related to a direction in the crystallattice.The Lagrangian for fluctuations in n must be invariant under the coset space G/H ,which has the symmetry group of rotations on the surface of the two-sphere, S , andimposes the restriction n · n = 1. This restriction can be imposed as a constraint andexpanded for small perturbations in Cartesian coordinates for n , which is sufficient toderive the normal modes and dispersion relation. More generally, the constraint can beimposed by the correct choice of coordinates and metric, in this case the SO (3) invariantmetric on S , and leads to the full non-linear model in polar coordinates. We begin withthe first case, since we can align the coordinates with the spacetime directions and arriveat well known results quickly, while the second case is illuminating since it preserves thesymmetries manifestly, and leads to insights into the nature of the longitudinal mode. A.1.1 AFMR in Cartesian Coordinates
The Lagrangian at leading order in derivatives is: L = F n · ˙ n − F ∇ n · ∇ n , (A.1)where F is the spin wave stiffness, and F = v F with v the spin wave speed. Theexternal fields are the applied field, H , the probe photon with fields E γ , H γ andwavevector k γ , and the anisotropy field, H A , which defines the easy-axis in the material.In the simplest AFMR geometry we consider the applied field to be parallel to the z -axis, which is also parallel to the anisotropy field. We further consider the probe photon(RF-field) moving along the positive z -axis, polarised in the y -direction. The fields arethus: k γ = (0 , , k ) , H γ = ( H γ , , , E γ = (0 , E γ , , H = (0 , , H ) , H A = (0 , , H A ) . (A.2)For ordinary AFMR, the photon electric field is decoupled from the system.The applied field H and the photon magnetic field are coupled into the LagrangianEq. (A.1) by replacing the derivatives with SO (3) ∼ = SU (2) covariant derivatives: ∂ µ n a → D µ n a = ∂ µ n a + (cid:15) abc f µb n c , (A.3)where n a are the directions in the SO (3) group space, µ = 0 , , , µ B ), (cid:15) abc is theantisymmetric symbol in three dimensions with (cid:15) = 1 (i.e. the structure constants of SU (2)), and f µb is the applied field. For an applied magnetic field we have µ B H i = f i – 79 –hich allows us to relate the group space index a to the spacetime axis i = 1 , ,
3. Atlowest order in the applied fields, this result can be understood by appealing to theinteraction Lagrangian: L em = − µ B s · H , s = F ( ˙ n × n ) , ⇒ L em = µ B F (cid:15) ijk ˙ n i H j n k , (A.4)where the spin density s follows from the leading order term in the derivative expansionof the Noether current due to the SO (3) invariance.The anisotropy field is included in the Lagrangian via a perturbation of the form∆ L = O a n a and for our field geometry is given by:∆ L = µ B Σ s H A n (A.5)where Σ s = S/V u . c . is the “staggered magnetization”, ( S A − S B ) /
2, in the unit cell.In order to derive the dispersion relation (the propagator), we only require thequadratic Lagrangian. Anticipating the well-known Keffer-Kittel result for the AFMRpolarisations [196] we use n and n as coordinates, and Taylor expand for small n usingthe constraint, i.e. n = (1 − n − n ) / . Momentum conservation demands that k = k γ ,and with the given geometry this simplifies the problem to effectively one-dimensionalalong the z (3)-axis. After some basic algebra, the quadratic Lagrangian is found to be: L = F h ˙ n + ˙ n i − F h ( ∂ z n ) + ( ∂ z n ) i (A.6) − F µ B H γ [ ˙ n + µ B H n ] + F µ B H [ ˙ n n − ˙ n n + µ B H ( n + n )] − µ B Σ s H A n + n ) . The first line is the kinetic term, and the second line includes the effects of the externalfields. The photon field has been considered a perturbation, and thus couples linearlyto the fields n i in the Lagrangian. The photon field thus acts as an oscillating sourceterm in the equations of motion. On the other hand H and H A couple to quadraticcombinations of n i , and affect the dispersion relation.The equations of motion are:¨ n − µ B H ˙ n + (cid:18) v k + µ B Σ s H A F − µ B H (cid:19) n = µ B H γ H , (A.7)¨ n − µ B H ˙ n + (cid:18) v k + µ B Σ s H A F − µ B H (cid:19) n = µ B ˙ H γ . (A.8)To derive the dispersion relation, we consider the homogeneous equation with the righthand side set equal to zero, and move to frequency space by Fourier transforming t → ω .The system is diagonalised by the complex fields n ± = n ± in leading to the system: ω ± ∓ µ B H ω ± − (cid:18) v k + µ B Σ s H A F − µ B H (cid:19) = 0 , (A.9)– 80 – z xy Figure 24 : AFMR spin wave, with | n | = 1, indicating the higher order change in n associated with the spin precession.which is solved by ω + = µ B H ± r v k + µ B Σ s H A F ,ω − = − µ B H ± r v k + µ B Σ s H A F . (A.10)The dispersion relation, Eq. (A.10) for the fields n ± = n ± in displays all the well-known properties of AFMR. The two modes n ± = n ± in correspond to clockwise andanticlockwise precession of the N´eel vector [196]. The resulting spin wave is depictedin Fig. 24. The constraint | n | = 1 leads to an oscillation of n accompanying theprecession. As shown in Appendix A.2, n in this case oscillates with a frequency twicethat of the AFMR. However, if | n | = 1, then n in not an independent polarisation andits fluctuation does not change the length of the N´eel vector.In the absence of H A , the dispersion relation is linear in k . The application of H A induces a “mass term”, i.e. a term inducing a gap and leading order quadratic piece inthe dispersion relation near k = 0: m s = µ B Σ s H A F (A.11)Rearranging, we find m s F = µ B Σ s H A , (A.12)which has the form m s F = (spontaneous) × (explicit) symmetry breaking, and is theAF analogue of the Gell-Mann-Oakes-Renner relation [115] for pions [130] (and also theQCD axion). Furthermore, since F ∝ Σ s this fits with the microscopic interpretation– 81 –f F as arising from the staggered magnetization angular momentum per unit cellmentioned above.The applied field H , rather than leading to a mass term, instead induces a linearshift in the frequency, the “Kittel shift”, which arises from an effective (anti-)dampingterm and “negative mass squared” in the equations of motion for n ± .The exchange field, H E , is not incorporated directly in our treatment of EFT.However, as noted in Ref. [130], we should fix the EFT parameters with reference to amicroscopic theory. The microscopic theory (e.g. Ref. [213]) gives the spin wave massfrom the energy gap: m s = µ B H A (2 H E + H A ) , (A.13)where H E is the exchange (or Weiss) field. The second term of Eq. (A.13) is not presentin the EFT, which is linear in H A . Indeed, EFT is valid in the limit H A /H E (cid:28)
1, andbreaks down for large anisotropy fields [130]. Comparing Eq. (A.12) with the first termof Eq. (A.13) we identify H E = Σ s / µ B F leading to: F = Σ s µ B H E = Sµ B H E V u . c . . (A.14)The EFT of AFMR is based on the mean field Heisenberg model. The Heisenbergmodel is the strong coupling limit of the Hubbard model (the fundamental model onwhich our theory of the AQ is based), with different perturbative degrees of freedom. Inthe Heisenberg model with nearest neighbour interactions the Hamiltonian is: H = J H X
In the following we explicitly follow the treatment of Ref. [212], and use the field geom-etry: k γ = ( k, , , H γ = (0 , , H γ ) , E γ = (0 , E γ , , H = ( H , , , H A = ( H A , , . (A.16)In terms of polar coordinates, we have n = sin θ cos φ , n = sin θ sin φ , n = cos θ , (A.17)– 82 –F order breaks the SO (3) internal symmetry of the spins down to the coset space SO (3) /SO (2) which has the geometry of S . The dynamics of the Goldstone modescan be expressed using the polar coordinates. The easy axis has coordinates θ , φ , andwe normalise the order parameter to unity. The Goldstone mode Lagrangian at lowestorder in derivatives is: L = F γ ab ˙ ϑ a ˙ ϑ b − F γ ab ∇ ϑ a · ∇ ϑ b , (A.18)where a, b = θ, φ and the metric γ ab is the round metric on the sphere, γ ab = diag[1 , sin θ ].The dynamics is easiest to express choosing n to be the easy axis, θ = π/ φ = 0.We then find trivially that, at leading order in fluctuations: n = 1 − δθ − δφ , n = δφ , n = − δθ . (A.19)The longitudinal fluctuation, i.e. the change in n projected along the direction of travelof the spin wave, n in this case, is quadratic in the Goldstone modes, while the transversefluctuations are linear. Note, however, that in these coordinates we always have explicitlytwo polarizations and no change in the length of the N´eel vector. The anisotropy fieldperturbs the Lagrangian as above, ∆ L A = µ B Σ s H A n , and induces a mass term for δθ and δφ . The interaction with applied fields follows exactly as in the Cartesian case usingthe relations in Eq. (A.17). A.1.3 Longitudinal Spin Waves in the Heisenberg Model
The AQ is related to longitudinal fluctuations of the N´eel vector (i.e. those in thedirection of the anisotropy field), but this is not equivalent to a third longitudinal po-larisation that changes the length of the vector. Such a “true” longitudinal mode is themode that breaks SO (3) giving rise to AF order, i.e. the Higgs-like radial mode (see alsorefs. [214, 215]). When writing down the model, we need to be careful that it respects allthe symmetries. The field ~φ = ( h S A i − h S B i ) / ~φ = ρ ( x )( n ( x ) , n ( x ) , n ( x )) = ρ ( x ) n . (A.20)The field ρ is the longitudinal polarization, while n is the AFMR field introduced abovewith | n | = 1. There is a maximum magentization given by the spin density, and aminimum value pointing in the opposite direction. In our conventions, φ is dimensionlessand normalized to a maximum of unity, thus | φ | ≤ | φ | ≤ SO (3) invariant metric on S with unit radius. We use the field coordinates: ~ϕ = ( α, θ, φ ) , (A.21)where θ , φ are the AFMR variables in polar coordinates introduced above, and α is athird polar angle. The metric is ds = g AB dϕ A dϕ B = dα + sin αd Ω , (A.22)– 83 –here A, B = α, θ, φ , and d Ω = γ ab dϑ a dϑ b is the round metric on S , and ~ϑ = ( θ, φ ) asabove. We see that the polar angle α gives the radius of the S submanifold of S , asdesired and with the correct normalisation, ρ = sin α . We can interpret α as the anglebetween the spins in a “bending mode”.The Lagrangian in the absence of external fields is: L = F g AB ˙ ϕ A ˙ ϕ B − F g AB ∇ ϕ A ∇ ϕ B . (A.23)Specifying the anisotropy field allows us to identify the polar axis as n as above. Theanisotropy field introduces explicit symmetry breaking and a potential for α , V ( α ) ∝− n ∝ − sin α , which is minimized at α = π/
2. To consider the fluctuations, we write α = π/ − σ and σ is the angular field giving rise to the fluctuation in ρ , i.e. the thirdmagnon polarization. It has quadratic Lagrangian: L = F σ − F ∇ σ − µ B Σ s H A σ , . (A.24)The field σ couples to the other AFMR fields via the metric g AB : L = cos σ " F γ ab ˙ ϑ a ˙ ϑ b − F g ab ∇ ϑ a ∇ ϑ b . (A.25)Expanding cos σ = 1 − σ for the quadratic Lagrangian we see that at leading orderwe obtain the angular AFMR theory from above, and σ is decoupled. Similarly, σ isdecoupled from the external fields in the quadratic Lagrangian, since the spin density, s = F ˙ φ × φ , only contains σ at cubic order. Thus, in this S EFT of of the Heisenbergmodel, the σ degree of freedom corresponding to changes in the length of the N´eel vectoris stabilised by the anisotropy field, and is neither excited by external fields nor mixeswith the transverse AFMR polarisations. Could this mode be the AQ? We take thegeneral expression for δ Θ in eq. (2.26) and expand n A in the angular fields. Once again, δ Θ is quadratic in all the variables of this model, including σ . We have not been ableto obtain a quadratic kinetic term for δ Θ from an SO (3) invariant EFT including onlythe N´eel order parameter.The preceding discussion suggests a possible solution to the problem of the EFT ofthe AQ. We notice that S is in fact the spin group Sp (1) = Spin (3) = SU (2). Further-more SU (2) ∼ = SO (3) / Z , and for the AQ we are concerned with models that break thediscrete symmetries P and T . This suggests using a complex field φ in the fundamental2-dimensional representation of a chiral SU (2) to represent the AF order parameter,which now has four real degrees of freedom. Thus, after SSB this would give three goldstone modes: two “charged” goldstones, giving the transverse magnons, and one“neutral” goldstone, which we assume will be the longitudinal magnon. Each goldstonecorresponds to a U (1) subgroup of SU (2). The neutral goldstone is a pseudoscalar, andthus this U (1) group is itself chiral, i.e. a Peccei-Quinn symmetry. The Dirac fermionsin the band structure should be charged under this symmetry, such that they acquirechiral rotations (“ m ” mass) governed by the longitudinal mode. Just like the axionand the neutral pion, this new goldstone mode can now couple to E · B via the chiralanomaly. We have not, unfortunately, been able to work out this theory completely.– 84 – .2 The Landau-Lifshitz Equations In this appendix, following Refs. [216, 217], we describe an antiferromagnetic resonance(AFMR) state using the Landau-Lifshitz equation. We consider the action of the N´eelfield described by the non-linear sigma model [101], S AF = g J Z dtd r h ( ∂ µ n ) · ( ∂ µ n ) − ∆ n i . (A.26)In order to implement a little more realistic condition in Eq. (A.26), we take into accounta small net magnetization m satisfying the constraint n · m = 0 with | n | = 1 and | m | (cid:28)
1. Furthermore, we assume the case of AF insulators with easy-axis anisotropy.Then a modification of Eq. (A.26) gives the free energy of such AF insulators as [218, 219] F AF = Z d r a m + A X i = x,y,z ( ∂ i n ) − K n z − H · m , (A.27)where a and A are the homogeneous and inhomogeneous exchange constants, respec-tively, and K is the easy-axis anisotropy along the z direction. The fourth term is theZeeman coupling with H = gµ B B being an external magnetic field.In the case in which a dc magnetic field H and an ac magnetic field (i.e., RF field) h ( t ) are applied to the AF insulator, the total magnetic field in Eq. (A.27) is H = H + h ( t ) , (A.28)where H = gµ B B e z with B being much weaker than both the AF exchange couplingand easy-axial anisotropy and h ( t ) = h RF e − iω t . Here, e z is the unit vector parallel tothe easy axis of the AF order. Now we study the dynamics of m and n phenomenolog-ically, i.e., based on the Landau-Lifshitz-Gilbert (LLG) equation [216, 217, 219]. Fromthe free energy of the system F AF , the effective fields for n and m are given by f n = − δF AF δ n = A n × ( ∇ n × n ) + Kn z e z − ( n · H ) m , f m = − δF AF δ m = − a m + n × ( H × n ) , (A.29)The LLG equation is given by˙ n = ( γ f m − G ˙ m ) × n , ˙ m = ( γ f n − G ˙ n ) × n + ( γ f m − G ˙ m ) × m , (A.30)where γ = 1 / (cid:126) and G and G are dimensionless Gilbert damping constants. For thepurpose of deriving the AFMR state, we may neglect the Gilbert damping constants.Then, the LLG Eq. (A.30) is simplified as˙ n = γ ( − a m + H ) × n , (A.31a)˙ m = γKn z e z × n + γ H × m , (A.31b)– 85 –here we have assumed that n is spatially uniform, and we have used | n | = 1 and anidentity for matrices A × ( B × C ) = ( A · C ) B − ( A · B ) C . After some straightforwardmatrix algebra, we arrive at the following equation for the N´eel field: n × ¨ n + ω K ω a n z e z × n − γ ( n · H ) H × n + 2 γ ( n · H ) ˙ n + γ ( n · ˙ H ) n = γ ˙ H . (A.32)To obtain the AFMR state, where all the spins are fluctuating uniformly, we assumethe dynamics of the N´eel vector and the total magnetization around the easy axis as n ( t ) = e z + δ n ( t ) and m ( t ) = δ m ( t ) , (A.33)denoting that δ n ( t ) and δ m ( t ) are the small fluctuation components with | δ n | , | δ m | (cid:28) δ n ( t ) = R δ ˜ n ( ω ) e − iωt dω/ (2 π ), Eq. (A.32) reduces to [216, 217]2 iω H ωδ ˜ n /ω a + h(cid:16) ω + ω H (cid:17) /ω a − ω K i e z × δ ˜ n = D δ ( ω − ω ) , (A.34)where ω H = γgµ B B , ω a = γa , ω K = γK , and ω is the frequency of the RF field[ h ( t ) = h RF e − iω t ]. In Eq. (A.34), D = − iγω ( h x RF e x + h y RF e y ) is understood as the“driving force” vector causing the AFMR. Equation (A.34) is rewritten in the matrixform " iωω H − (cid:0) ω − ω a ω K + ω H (cid:1) ω − ω a ω K + ω H iωω H δ ˜ n x ( ω ) δ ˜ n y ( ω ) = ω a δ ( ω − ω ) " D x D y . (A.35)Multiplying the inverse matrix from the left hand side, we obtain " δ ˜ n x ( ω ) δ ˜ n y ( ω ) = " χ ( ω ) χ ( ω ) − χ ( ω ) χ ( ω ) D x D y , (A.36)where the susceptibility is defined as " χ ( ω ) χ ( ω ) − χ ( ω ) χ ( ω ) = ω a δ ( ω − ω )( ω − ω )( ω − ω − ) " iωω H ω − ω a ω K + ω H − (cid:0) ω − ω a ω K + ω H (cid:1) iωω H . (A.37)Here, ω ± = ω H ± √ ω a ω K (A.38)are the resonance frequencies. Note that these frequencies do not depend on the param-eters of the driving force D .Along with Eq. (A.34), the following equation is obtained from 2 γ ( n · H ) ˙ n + γ ( n · ˙ H ) n in Eq. (A.32), which describes the “longitudinal” AFMR state:2 ω H δ ˙ n z e z = iγω e − iω t ( h x RF δn x + h y RF δn y ) e z . (A.39)– 86 –ourier transforming this equation and substituting the solution for δn x and δn y [Eq. (A.36)]into it, we have δ ˜ n z ( ω ) ∝ h x RF δ ˜ n x ( ω − ω ) + h y RF δ ˜ n y ( ω − ω ) ∝ δ (2 ω − ω )[( ω − ω ) − ω ][( ω − ω ) − ω − ] , (A.40)which indicates that the resonance frequencies of the longitudinal AFMR are ω = 2 ω =2 ω ± . Eq. (A.40) reveals that the longitudinal mode is quadratic in the RF field, i.e., asecond-order response to the RF field, while the transverse mode [Eq. (A.36)] is a linearresponse to the RF field. B Axion Dark Matter and the Millielectronvolt Range
Since their initial proposal as a solution for the Strong CP problem more than 40 yearsago [1–3, 220], (QCD) axions have seen phases of growing interest due to a numberof breakthroughs. The first was the realisation that axions are excellent dark mattercandidates [7–9, 221], and that there are several ways to search for them experimen-tally [32, 222, 223]. Recently, there has been a huge growth of new ideas for axionsearches (see Ref. [18] for a review), which includes the present proposal (“Paper I”)using topological insulators [41].The QCD axion was originally proposed as the pseudo-Goldstone boson of a spon-taneously broken global U (1) symmetry, which couples to chiral fermions charged underthe strong nuclear force, SU (3) c gauge symmetry (i.e. quarks). Such a global symmetryis known as a Peccei-Quinn (PQ) symmetry, U (1) PQ . More generally, QCD axions can beregarded as pseudo-Goldstone bosons coupled to the QCD anomaly term, schematically G ˜ G , where G is the gluon field strength tensor, and ˜ G its dual.The PQ symmetry breaking scale, v PQ , is not predicted by the theory, although itis expected to be below the reduced Planck scale, M Pl = 2 . × GeV [224]. Thesymmetry breaking scale sets the axion mass, which arises due to the axion’s cou-pling to the QCD topological charge via SU (3) c instantons, and which reaches its zero-temperature value at temperatures lower than the QCD crossover temperature of around157 MeV [225, 226]. The axion mass at such temperatures is given by [227] m a = √ χ f a = 5 . GeV f a ! , (B.1)where χ is the zero-temperature QCD topological susceptibility, f a = v PQ / N , and N is the SU (3) c anomaly of U (1) PQ . The value of χ can be calculated from chiralperturbation theory [2, 227, 228] while, at higher temperatures, it can be calculated usinglattice quantum field theory (see e.g. Ref. [229]), or using instanton methods [230, 231].The QCD axion couples to the EM Chern-Simons term via two means. Firstly,by its model-independent mixing with pions, and secondly via the (model-dependent)– 87 –lectromagnetic anomaly ( E ) of fermions charged under U (1) PQ . The coupling is [e.g.228] ∆ L = g aγ a E · B , (B.2)where a ≡ f a θ is the canonically normalised axion field and g aγ is the axion-photoncoupling, which is given by g aγ = α πf a C aγ = α πf a (cid:20) EN − . (cid:21) , (B.3)where N is the SU (3) c anomaly of the PQ symmetry, and is equal to unity in theKSVZ model, while for the DFSZ model N = 6. The value of E / N depends on the PQcharges and gauge group representations of fermions. We define the QCD model bandaccording to the “preferred” models of ref. [29], which corresponds to 5 / < E / N < /
3. Experiments constrain | g aγ | , and so this band and encompasses the original KSVZ( E / N = 0) and DFSZ ( E / N = 8 /
3) models. For a generic “axion-like particle”, thecoupling g aγ is taken as a free parameter independent of m a .The QCD axion mass is bounded from above and below by astrophysical con-straints. The existence of BHs with with masses of order ten solar masses with highspins, stable over astrophysical timescales, would be impossible if the QCD axion ex-isted and m a (cid:46) − eV [14, 15, 232]. In such a case, the axion Compton wavelengthis resonant with the size of the BH ergoregion, causing axions to be abundantly createdfrom vacuum fluctuations, and rapidly draining the spin of the BH. On the other end ofthe mass scale, the QCD axion with m a (cid:38) .
02 eV is excluded by observations of neutri-nos coinciding with the galactic supernova SN1987A [11, 12]. The QCD axion couples tonuclei in the supernova, and axions are emitted by nuclear bremsstrahlung, cooling thesupernova more rapidly and shortening the neutrino burst if the axion-nucleon coupling(proportional to m a ) is too large. Since there is no statistically rigorous bound associ-ated with SN1987A, we also mention that a looser upper limit on m a can be derivedfrom constraints on the relativistic energy density in the early Universe (parameterisedas a hot DM component). Hot QCD axions are produced by their interaction withpions. The amount of hot axions produced is in conflict with the cosmic microwavebackground anisotropies as measured by the Planck satellite [10, 194] if m a (cid:38) . g aγ (cid:46) − GeV − . This is far above the coupling for typical QCDaxion models.Assuming that the QCD axion indeed composes the observed DM with cosmicdensity parameter Ω d h = 0 .
12 [194], it is possible to analyse the value of m a further(for a review, see Ref. [16]). If the maximum temperature of the Universe exceeds thePQ phase transition temperature, T PQ ∼ v PQ , or if the Hubble scale during inflation H I > πv PQ , then the PQ symmetry is unbroken at the end of inflation. When itsubsequently breaks, the Kibble mechanism leads to a network of topological strings– 88 –hat persists as the Universe expands and, when the axion mass becomes cosmologicallyrelevant m a ∼ H (where H is the Hubble parameter) domain walls form [235–237]. Suchdefects emit DM axions, and eventually decay when H (cid:28) m a . In this case, if N = 1(which imnolies the domain wall network is unstable, then the DM abundance is inprinciple calculable and depends only on m a . However the dynamics of the strings anddomain walls are complex and the axion abundance can only accurately be determinedby numerical simulations. This computation cannot be performed at the physical scaleseparation (between the string thickness and the Hubble length, which in turn sets thestring tension), and so the results must be extrapolated. In the case N >
1, the DMabundance depends on an additonal biasing parameter required such that the domainwall network decays [238].When N = 1, recent simulations [195] have placed a lower bound on the QCD ax-ion mass m a (cid:38) . Thiswork also estimates the bound m a (cid:38) . N >
1, in agreement with the gen-eral expectation that the axion mass should be larger in this scenario [238, 241, 242].One issue for axion direct detection in the post inflation scenario is the existence of ax-ion “miniclusters” [243–245]. Recent simulations of structure formation in this scenariosuggest that a large fraction of the DM (50% or more at the solar radius in the MilkyWay) is bound in dense, low mass objects [246–249]. These objects have a low collisioncross section with the Earth, and reduce the effective value of the local DM density fora direct detection experiment.In the alternative scenario for axion production, the PQ symmetry is broken in thevery early Universe during the hypothetical period of inflation [250–252], and axions aresubsequently produced when the initial vacuum state of the axion decays, in a processcalled “realignment”. This scenario has more free parameters than just m a , and it isnot possible to predict the axion mass based on the observed DM abundance. Thisscenario is incompatible with a large energy scale of inflation, and would be ruled outif primordial gravitational waves were observed [253]. If the initial vacuum value of θ is assumed to be of order 1 the axion mass in the pre-inflationary scenario is boundedto m a (cid:38) . C Comparison to earlier results
The forecasts shown in fig. 23 differ in many respects from those in Paper I, and weexplain briefly why, see fig. 25, which shows the same projection alongside those ofPaper I. For the detector, Paper I assumed a coupling factor equivalent to efficiency η = 0 .
01, and the same dark count rate as in the present work. The difference in the Other extrapolations to the physical scale separation lead to lower bounds on the mass [239, 240]. – 89 – . Dark axion mass m a [meV] − − − − − − − − − D a r k a x i o n - ph o t o n c o up li n g g a γ [ G e V − ] T OO R A D ( P a p e r I ) Material 1Material 2
TOORAD (this work) η Γ / meV0 .
01 10 − − − Figure 25 : Comparison between the forecasts made in Paper I (gray) and those in thepresent work (coloured lines).scan depth arises because the power output in Paper I was computed in analogy to aresonant cavity, and this formula is incorrect for a medium with a polariton resonance.Allowing for a few translations, however, the results can be compared. All in all, however,we stress that the power formula in Paper I was far too simplistic, and so should not beused.Our present corrected calculations have shown that the power does not scale withthe material volume as assumed in Paper I: rather it depends separately on the surfacearea and the material thickness. Losses lead to a maximum useful thickness, an effectwhich was not accounted for in Paper I. For the models presently considered, this leadsto a total useful material volume around 40 cm . Thus for comparison, we show thePaper I estimates for “Stage I”, which used a total volume of material 1 cm , and ‘StageII”, which used a total volume of material 100 cm , equivalent to d ≈ O ( µ eV) losses.Paper I included losses only by a rough estimate for the bulk quality factor, whichwas taken as Q = 10 , roughly Γ = 10 − meV (this assumed power law decreases in Γ m at low T as discussed in the present work, but neglected the impurity and conductancecontributions). On the other hand, Paper I assumed that the power was reduced by apolariton mode mixing factor, f + , which is absent in the present treatment (mode mixingmostly affects the width of the resonance). Together, these amount to an assumed β ≈ for some baseline parameters. Comparing to fig. 19, using the correct computationsfrom the present paper, such a large β could only be achieved with losses Γ = 10 − meV.These many considerations explain the different depth of the constraints in terms of g aγ – 90 –n the present work compared to Paper I.The difference in the scanned mass range in the present work compared to Paper Iis more useful and necessary to explain. It arises from the adopted values of m Θ and f Θ . In the present work we take our preferred material as Mn Bi Te , while Paper Iuses (Bi − x Fe x ) Se . Even so, there are differences to the (Bi − x Fe x ) Se parameterestimates in Paper I compared to the present work. In Paper I we erroneously assumedthat m Θ was equal to the AFMR frequency of the transverse magnon polarisations (witha reduction due to the doping required in (Bi − x Fe x ) Se ) leading to m Θ = 0 . m a in the Paper I treatment. Paper I also incorrectlyincluded the Kittel shift to m Θ . As discussed, the longitudinal magnon is not simplyrelated to the transverse modes and Paper I should have used the value of m Θ computedby Ref. [42], as we do in the present work. However, as we have noted, Ref. [42] useda square lattice approximation to compute m Θ . We do not know the error induced bythis assumption on the lower limit of the scannable mass range also in the present work.Furthermore, Paper I assumed f Θ for (Bi − x Fe x ) Se directly from Ref. [42]. In thepresent work we corrected this value using more up to date estimates of the bulk bandgap of (Bi − x Fe x ) Se including the effects of magnetic doping, leading to the “Material1” parameter estimates with lower f Θ . The lower value of f Θ in the present work allowsfor a wider range of masses to be scanned for the same range of B -field strengths. References [1] R. D. Peccei and H. R. Quinn,
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