Axion-matter coupling in multiferroics
Henrik S. Røising, Benjo Fraser, Sinéad M. Griffin, Sumanta Bandyopadhyay, Aditi Mahabir, Sang-Wook Cheong, Alexander V. Balatsky
AAxion-matter coupling in multiferroics
Henrik S. Røising, Benjo Fraser, Sin´ead M. Griffin,
2, 3
Sumanta Bandyopadhyay, Aditi Mahabir, Sang-Wook Cheong, and Alexander V. Balatsky
1, 4, ∗ Nordita, KTH Royal Institute of Technology and Stockholm University,Roslagstullbacken 23, SE-106 91 Stockholm, Sweden Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Department of Physics, University of Connecticut, Storrs, CT 06269, USA Rutgers Center for Emergent Materials and Department of Physics and Astronomy,Rutgers University, Piscataway, NJ 08854, USA (Dated: February 23, 2021)Multiferroics (MFs) are materials with two or more ferroic orders, like spontaneous ferroelec-tric and ferromagnetic polarizations. Such materials can exhibit a magnetoelectric effect wherebymagnetic and ferroelectric polarizations couple linearly, reminiscent of, but not identical to the elec-tromagnetic E · B axion coupling. Here we point out a possible mechanism in which an externaldark matter axion field couples linearly to ferroic orders in these materials without external appliedfields. We find the magnetic response to be linear in the axion-electron coupling. At temperaturesclose to the ferromagnetic transition fluctuations can lead to an enhancement of the axion-inducedmagnetic response. Relevant material candidates such as the Lu-Sc hexaferrite family are discussed. I. INTRODUCTION
Dark matter (DM) comprises over three quarters ofthe universe’s mass-density, yet so far it eludes detec-tion in spite of the intensive search using numerous de-tection schemes . One possible explanation is that theDM particles are too light to strongly scatter off nucleiand electron-based detection schemes currently proposed,motivating the need for new ideas to explore the sub-GeVrange of DM masses. In this regard, the O (meV) axion, apseudoscalar boson introduced to solve the strong charge-parity problem in quantum chromodynamics , offers aparticularly well-motivated DM candidate .A new avenue to DM detection is offered by quan-tum materials, where the energy scales of the excitationscoinicide with the requirements for lower-mass DM de-tection. Moreoever, quantum materials possess entan-gled collective degrees of freedom that allow for an en-hanced coupling to axions. Traditionally, the majority ofaxion detection proposals have focused on exploiting theaxion-photon coupling in cavities or plasma haloscopesusing strong external fields . More recently, axionphysics has been increasingly discussed in the quantumcondensed matter context primarily in analogy toaxion electrodynamics in topological insulators , butalso in terms of particle axion detection via resonant cou-pling to quasiparticles . In most of the existing propos-als, the detection scheme uses DM particle absorption toinduce single particle excitations in the sensor material,e.g. the particle-hole excitations in a small gap material.An alternative approach is to detect DM by their cou-pling to collective modes of the quantum material. Withthe rapid progression of the field of axion electrodynam-ics, we anticipate more such proposals for axion sensingschemes will be explored.Here we explore multiferroics (MFs) as a platform forDM detection using materials with parallel ferroic orders FIG. 1. An incident axion, a , couples to multiferroic ordersvia the electron spin and momentum below the Curie temper-ature. Close to the onset of ferromagnetism, softness in themagnetization makes the system susceptible to the axion cou-pling. The magnetic material response is imagined sensed bya sensitive magnetometer, such as a superconducting quan-tum interference device (SQUID). The inset shows a sketchof the A phase of multiferroic Lu − x Sc x FeO ( h -LSFO) . P (cid:107) M as an “impedance matched” medium to interactwith axion DM. Here, P ( M ) is the electric (magnetic)polarization vector. Multiferroics with P (cid:107) M act asthe “condensate” axion field seen by an arriving externalaxion. The external axion field linearly couples to the po-larization fields and both changes net magnetization andexcites the collective modes of the MF. The mechanism a r X i v : . [ h e p - ph ] F e b for DM detection proposed here is qualitatively differentfrom existing schemes in the sense that the axion DM af-fects the magnetization – a macroscopic observable – bymodifying the parameters that control the macroscopicstate. As such, the proposed method of detection is sim-ilar to the transition-edge sensors, except that our sensor(the multiferroic sample) remains in a macroscopicallyordered state before and after interaction with DM.To place the current discussion in a broader context:one finds two main developing threads for axion DM sen-sors. One approach is relying on the sensor device beingplaced in external electric ( E ), magnetic ( B ) or bothfields simultaneously, whereby one breaks time-reversal( T ) and parity ( P ) to induce coupling to the axion field.Another approach is to use systems that take advantageof spontaneous symmetry breaking (no external fields ap-plied) in quantum materials. In the phase with broken P T symmetry coupling of the axion DM field to matterfields occurs naturally. The coupling induces collectiveexcitations that are subsequently read out by appropri-ate sensing means. Here we propose a new realization ofthe latter platform where
P T breaking, and hence axionelectrodynamics, occurs naturally.Our axion-matter sensing proposal is sketched inFig. 1. Our central scheme is that in MFs with parallel ferroic orders in zero external fields one has a mediumthat linearly couples to an axion field, a , through an a P · M term in the Hamiltonian. This term allows for anovel axion DM detection scheme: Impinging axions willhybridize and directly couple with ferroic orders in MFswithout requiring an intermediate photon coupling. Formacroscopic domains (microns and larger) with P (cid:107) M each unit cell of the MF couples linearly to a . Thiscoupling between the axion field and matter fields (with P (cid:107) M ) is a bulk effect. The electrons in the MF producea P T breaking term that can be expressed in terms of P and M but which microscopically results in the bulkexpectation value of the axion-electron coupling as de-scribed in Sec. III A. The macroscopically ordered stateprovides a set of sensors where each “atomic” unit inter-acts with the same axion field, and their contributionsadd coherently – the signal scales with a macroscopicnumber of atoms. Phrased differently, we obtain Avo-gadro scaling due to the P (cid:107) M coherent state of thematerial. The proposed scheme thus offers a quantumadvantage at the macroscopic level. This “Avogadro ad-vantage” is a unique feature of macroscopically coherentquantum matter where each unit cell, or each part of thecoherent condensate, adds to increased detection sensi-tivity. We see this advantage operational for the numer-ous detection schemes relying on the quantum macro-scopically coherent states ranging from ferromagnets tosuperfluid/superconducting condensates and MFs. Todiscuss multiferroics as a proposed platform in this con-text we will use the acronym MERMAID (MultifERroicMatter AxIon Detector).We also present a known material possessing our re-quired coupling to give materials-specific estimates of our MERMAID scheme. We focus on the recently discov-ered hexagonal LuFeO thin films and the Lu − x Sc x FeO crystal family, which have a ferromagnetic ground statefor x ≈ . T C M ≈
160 K . Sample candi-dates may be synthesized and characterized using THzmagneto-optical spectroscopy to determine the nature ofthe collective excitations, including electromagnons, andthe resulting P (cid:107) M coupling strength.This paper is organized as follows. We provide back-ground on axions in Sec. II. An effective theory of theaxion-matter coupling in multiferroics is presented inSec. III. In Sec. III A we discuss the effective axion-electron coupling in multiferroics, and we pursue an ef-fective Ginzburg–Landau theory to describe the dynam-ical response in Sec. III B. Critical slowing down andthe axion-induced magnetic response are discussed inSec. III C and III D, respectively. In Sec. III E we dis-cuss the advantages of Avogadro scaling in quantum ma-terials. In Sec. IV we discuss material candidates andestimates from ab initio density functional theory calcu-lations for a specific compound. We provide conclusionsand an outlook in Sec. V. II. BACKGROUND: AXIONS
The axion is an hypothesized pseudoscalar boson orig-inally proposed to solve the strong charge-parity problemvia the Peccei–Quinn mechanism in quantum chromody-namics (QCD) . It was realized that the axion couldbe a viable DM candidate, which has significantly fuelledinterest in it . The axion couples to electromagnetism(in units where c = (cid:126) = 1) via L a = 12 ( ∂ µ a )( ∂ µ a ) − m a a , (1) L γγa = − θ F µν ˜ F µν = θ E · B , (2) θ ≡ ag γγa , (3)where F µν is the electromagnetic tensor with ˜ F µν = ε µναβ F αβ its dual ( ε µναβ is the Levi–Civita symbol),and where a ( m a ) is the axion field (mass), and g γγa is adimensionful coupling strength.Although axion-like particles with a Lagrangian likeEqs. (1) and (2) within a vast mass range could ac-count for a partial DM abundance, QCD axions havethe particular coupling strength g γγa = C a m a GeV , where C a is a model dependent number that ranges between[ − . , .
15] for various theories . Despite an in princi-ple still vast ( m a , g γγa ) parameter space, the axion massis restricted by m a (cid:46) . and by m a (cid:38) − eV from cosmological bounds .Moreover, the coupling strength is roughly limited to | g γγa | < − GeV − by helioscope experiments . In-terest in the mass range around meVs has been fuelledby extensive lattice QCD computations and searches atthe XENON experiment .Moreover the axion couples to Dirac fermions via L af = − g af ∂ µ a m f ¯ ψγ γ µ ψ, (4)where ψ is the Dirac spinor and g af is a dimension-less coupling strength. The coupling to electrons is lim-ited by | g ae | < · − from solar neutrino experi-ments . The relevant low-energy Hamiltonian result-ing from Eq. (4) is derived in Appendix A. We note thatthis Hamiltonian contains an effective term of the form θ P · M which is present even in the absence of externalfields ( A = 0), see Eq. (A4).Since the axion is expected to be light, we can treat a as a classical field with a long de Broglie wavelengthcompared to the typical experimental probe length scaleto give an estimate of θ in Eq. (3) for QCD axions. Weassume that the the axion field is oscillating with a fre-quency set by its mass, a = a exp( im a t ). Assumingthat the axion makes up the local DM density , ρ DM ≈
300 MeVcm − , the amplitude a is fixed by ρ DM = m a | a | , yielding e.g. | θ | ≈ | a g γγa | ∼ − . III. EFFECTIVE THEORY
In this section we develop a framework to study themagnetic response in MFs mediated by an axion-fermioncoupling. In Sec. III A we discuss how an effective matter-analogue of the axion E · B coupling emerges in MFsystems, with details of the derivation listed in Appen-dices A, B, and C. A. Multiferroics and effective matter coupling
With a matter-analogue of Eq. (2) in mind, multi-ferroics are materials in which spontaneous electric andmagnetic polarizations can develop . In solid-statecrystals, ferroelectricity is normally driven by lone-pair,geometric, charge ordering, or spin-driven mechanisms ,which result in a macroscopic breaking of inversion sym-metry. As a result of spontaneous symmetry breaking in the MF bulk of both time-reversal and inversion symme-try, there exist internal electric and magnetic fields (evenin the absence of external fields) resulting from thesesymmetry-broken order parameters. Based on symme-try considerations one should expect an effective χθ P · M coupling to be present in the MF phase, despite this termbeing commonly overlooked in the high-energy literature.Microscopically, an effective P · M axion term can bederived from the coupling to electrons, as per Eq. (4),in the MF ground state. For details we refer to Appen-dices A and B. The key result is that the axion-electroncoupling at low energy contains the term δE ae ≈ − i (cid:126) g ae ∂ t a m e c (cid:104) Ψ | σ · ∇ | Ψ (cid:105) , (5)where | Ψ (cid:105) is the MF ground state. In the MF phaseEq. (5) is dictated by the distortions of electronic orbits(densities) such that one obtains a spontaneous magneticand electric dipole moment throughout the whole sam-ple. In a (rare) subclass of MFs these moments are par-allel, as discussed in Sec. IV. Despite the m a /m e sup-pression, coming from the µ = 0 contribution to Eq. (4),we argue in Sec. III E that the energy of Eq. (5) canadd coherently over a macroscopic volume limited by theMF domain size, which can result in an appreciable ef-fect. To be clear, the interaction in Eq. (5), evaluated onelectronic states, yields a bulk contribution that is eval-uated by model calculations (Sec. III E) and ab initio means (Sec. IV). In Appendix A we discuss the relationto other couplings which are considered in existing de-tection schemes . B. Ginzburg–Landau theory
Using an isotropic sample bulk model, we proceed withan effective description of the longitudinal dynamics ofthe internal MF degrees of freedom. This approach isvalid close to the transition temperature where the freeenergy can be expanded in powers of the respective or-der parameters. In terms of the free energy density F , F [ M , P ] = (cid:82) d r F [ M , P ], in the absence of externalelectric and magnetic fields, we have (in SI units) F [ M , P ] = µ F M [ M ] + 1 ε F P [ P ] − (cid:114) µ ε ( α ij M i P j + χθ P · M ) , F M [ M ] = α M | ∂ t M | − β M | ∇ · M | − γ M ( T − T C M ) | M | − λ M | M | + . . . , F P [ P ] = α P | ∂ t P | − β P | ∇ · P | − γ P ( T − T C P ) | P | − λ P | P | + . . . , (6)where ε ( µ ) is the vacuum permittivity (permeability),and where α ’s, β ’s, γ ’s, and λ ’s are phenomenological co-efficients, α ij is the material dependent ME tensor, and χ is a material dependent susceptibility. The transition temperature T C M ( T C P ) denotes the magnetic (electric)Curie temperature above which the ferromagnetic (ferro-electric) order melts. In analogy to the P T -odd axionfield, where P ( T ) is the inversion (time-reversal) sym-metry operator, the ME tensor is P T -odd if the system(crystal) otherwise respects inversion and time-reversalsymmetry. The magnitude of the elements α ij are lim-ited by thermodynamic stability of the theory .We consider the case T ≈ T C M (cid:28) T C P (as applicableto the majority of MF materials). This makes P act asa “hard” and M as a “soft” field; for the latter fluctua-tions are greater. Hence we take the electric polarizationto be close to its minimum energy value, and considerfluctuations, γ P ( T − T C P ) | P | + λ P | P | → m P | δ P | ,where δ P are the ferroelectric fluctuations, P = P + δ P ,and where m P is the effective mass. Below we relabel δ P → P and bear in mind that P is a fluctuation field.When associating F in Eq. (6) with a Lagrangian densityit yields the classical equations of motion for M and P : α M ∂ t M − β M ∇ M + γ M ( T − T C M ) M + c α + χθ )( P + P ) + 2 λ M | M | M = 0 , (7) α P ∂ t P − β P ∇ P + 12 m P P + 12 c ( α + χθ ) M = 0 , (8)where c is the speed of light and α is the ME tensor withelements α ij . Turning off both α and θ reduces Eqs. (7)and (8) to two uncoupled Klein–Gordon equations whenignoring the cubic (Gross–Pitaevskii) term, which is jus-tified when T > T C M .We emphasize that the above effective theory for themagnetic order is a classical and minimal description ofthe longitudinal magnon dynamics. In situations wherethe transverse modes are involved, dynamics from theLandau–Lifshitz–Gilbert equation is expected to play arole . Finally, we note that in a crystal the Ginzburg–Landau theory away from low filling should respect sym-metries of the crystal point group and partition the solu-tion into irreducible representations. This layer of com-plexity is omitted in the isotropic bulk theory consideredhere and would constitute a natural extension of the the-ory. C. Critical slowing down
On general grounds we expect to have two regimes ofthe coupling following the above equations. The axionfield provides a periodic perturbation set by the timescale τ a ∼ /ω a , while the characteristic time of magne-tization is set by the mass term τ ∼ | T − T C M | − / , andmore generally, the magnetization dynamics experiencesa critical slowing down near T C M : τ ∼ | T − T C M | − νz , (9)for some dynamical critical exponent zν ∼ . The (i) adiabaticregime of the axion-magnetization dynamics occurs at τ a > τ . In this case the effective axion field seen by mag-netization has a similar effect to a steady external field,and the response accumulates in time. In the opposite FIG. 2. Sketch of critical slowing down for the magnetizationdynamics time scale τ (blue curve). The horizontal dashedline indicates the steady axion dynamics time scale τ a . Tem-perature tuning can be used to match the magnetic time scalewith the axion time scale. The adiabatic regime away fromthe critical point, where axion dynamics is slow compared tomagnetization time scale τ < τ a , is shown with a green dashedlines on the temperature axis. case of (ii) antiadiabatic evolution, τ a < τ , the magne-tization “sees” a rapidly oscillating θ ( t ) term that effec-tively averages to zero. The most direct and strongesteffects of the axion field will be seen in the temperatureregime where the axion field dynamics is slow comparedto the magnetization time, i.e. regime (i). This require-ment places a constraint on how close one can approachcriticality, as illustrated in Fig. 2.To give a numerical estimate of this constraint we as-sume that τ = τ (1 − T /T c ) − νz for T < T c , where τ is material dependent. The crossover between the adi-abatic and the antiadiabatic regime occurs at T /T c =1 − ( τ ω a ) / ( νz ) , given that τ ω a <
1. For m a = 1 meVthis would require τ (cid:46) . havefound τ in the range of τ ∼ , which if appliedto our context would limit the crossover to occur for lightaxions m a (cid:46) µ eV. For instance, for m a = 0 . µ eV and νz = 1 we find T /T c ≈ .
85. For these estimates, whichare merely indicative, one would thus want to maintainthe system close to the temperature T = 0 . T c . In sub-sequent work we plan to undertake a more systematicanalysis of relaxation times. D. Magnetic response
The coupled dynamical equations (7) and (8) can betreated with linear response theory with θ = θ exp( iω a t )acting as a weak time-dependent driving force. Here weconsider the result of such a treatment in one spatialdimension with α ij = αδ ij assumed for simplicity, withthe details listed in Appendix D.When T < T C M (cid:28) T C P and the magnetization is softcompared to the electric polarization, the classical mag-netic response resembles that of a driven harmonic oscil-lator. In frequency space the magnetization has a linearresponse function of the form (cid:12)(cid:12)(cid:12) δMθ (cid:12)(cid:12)(cid:12) = 1 (cid:113) ( ω a − ω M ) + η M ω a χ α M (cid:12)(cid:12)(cid:12) αM m P − cP (cid:12)(cid:12)(cid:12) . (10)with M = M + δM , P ≈ P , where M and P are thestatic polarizations, and δM are magnetic fluctuations. ω M is the magnetic (magnon) frequency, and η M is adamping coefficient of a term η M ∂ t M added to simulateenergy dissipation in the system. Addressing how low η M can be pushed and whether it is a limiting factor experi-mentally is outside the scope of this paper. However, forspin waves to be resolved we require η M (cid:46) . J , where J ∼ O (1 meV) is the typical spin-exchange interaction.The expression Eq. (10) sheds light on the key factorsin the response. First, there is the overall coefficient,i.e. the factor inside the absolute value. This shows thatthere are two contributions to the driving: the parallelmagnetoelectric effect, α , and the static electric polariza-tion, P . Since the P term can reasonably be expectedto dominate in many cases , the ferroelectricity will bea controlling factor in the driving.Secondly, Eq. (10) has the characteristic frequency de-pendence of a driven oscillator. Considering the responseas a function of axion frequency, there are three regimes,each of which can be clearly seen in Fig. 3: (cid:63) ω a (cid:29) ω M : The antiadiabatic regime discussedabove. The response dies off quickly as the axionfrequency becomes much larger than the magnonfrequency, so this limit is not optimal. (cid:63) ω a (cid:39) ω M : In this regime the response is controlledby the magnon resonance. The response will go asthe size of the peak, which will be controlled by thedamping coefficient η M . (cid:63) ω a (cid:28) ω M : The adiabatic regime. The axion os-cillates much more slowly than the magnetic re-sponse of the material, therefore it can be approx-imated as a static external field. Setting ω a = 0 inEq. (10), the left hand side behaves as ω − M . The re-sponse will increase as the temperature approachesthe critical magnetic temperature because ω M be-comes small there.As discussed in III C, avoiding the first regime in-troduces a limitation on what range of axion frequen-cies can be spanned for a given material and temper-ature (approximately ω a (cid:46) . ω M = ω M (Γ) .In the second regime, damping will be a critical factor.The third regime is the one most likely to be accessiblein experiment, and the one on which the authors wouldlike to concentrate. It has an attractive feature: thetemperature can be used as a tuning dial to adjust boththe range of accessible axion frequencies and the size ofthe response. − − − log ( ω a [meV]) | δ M / θ | [ O e ] × δT = 1 µ K δT = 100 mK FIG. 3. The magnetic linear response for axions coupling toferroic orders, for two values of δT = T C M − T . Dynamical fre-quency response from Eq. (10) assuming that α − M = (1 µ eV) and η M = 5 µ eV. Here we used the parameters χ = 1, m P = 4 . · − , α = 10 − , b = 0 . γ M = 10 − K − , λ M = 9 .
25 kOe − , P = 0 . µ C/cm , and T C M = 160 K, asroughly motivated by the compound h -LSFO, see the maintext and Sec. IV. In Fig. 3 the frequency response Eq. (10) is shown forparameters motivated by h -LSFO. Both the static re-sponse enhancement near criticality and the resonancevariation with temperature are clearly visible.Focusing on the static regime, we would like to studythe behaviour of the magnetic response near the criticaltemperature. As already mentioned, the static responseis controlled by ω − M , which we must therefore write interms of the Ginzburg-Landau parameters in Eqs. (7)and (8). To proceed we need the full non-linear solutionfor the magnetization as presented in Appendix E – herewe summarize the key points. The static magnetizationsatisfies the cubic equation γ M ( T − T C M ) M + 2 λ M M + c α + χθ ) P = 12 m P ( α + χθ ) M . (11)The solution can generically be expanded to linear orderin θ as M ( θ ) = M (0) + f ( T ) χθ + O ( θ ) , (12)where f ( T ), which is just the static limit of the linearmagnetic response, depends on λ M , m P , α , P , and γ M .There is one extra subtlety: owing to restored time-reversal symmetry in the paramagnetic phase, α = 0 for T > T C M , and so α itself has a temperature de-pendence near criticality. We model this using α ( T ) = α (1 − T /T C M ) b for some positive exponent b . This extrascaling makes the T → T C M limit somewhat subtle, butwe find that f ( T ) ∼ ( T C M − T ) − b/ . Therefore the mag-netic response grows in principal without bound as thecritical temperature is approached. In practice, disorderwill pose a challenge for temperature tuning, which moti-vates consideration of a broader range of magnetoelectricmaterials for dark matter detection. E. Avogadro scaling
Quantum matter sensors constitute examples of en-tangled sensors . In the case we consider the groundstate of the sensor is a coherent MF state with a wave-function given by | Ψ (cid:105) = (cid:81) (cid:96) | P (cid:96) (cid:107) ˆ x, M (cid:96) (cid:107) ˆ x (cid:105) with M (cid:96) (cid:107) P (cid:96) (cid:107) ˆ x in each unit cell (cid:96) (more precisely, each oper-ational unit of the coherent state). The coherent stateof the sensor occurs as a result of spontaneous symmetrybreaking of both parity and time reversal symmetry with-out external electric or magnetic fields. This fact alonemight provide an added advantage in designing robustDM sensing schemes.Within a macroscopic coherent state scheme each unitcell is functioning as a sensor for DM detection, e.g. as il-lustrated above. Macroscopic coherence implies the syn-chronized response of the material. Each unit cell mayhave a very low sensitivity due to small coupling and sizemismatch between the DM field and the unit cell fields.In a macroscopically coherent quantum state, however,the unit cell sensors add coherently and provide an Avo-gadro scaling enhancement of the signal-to-noise ratio, atleast in principle.In the case of MFs as a putative sensor the magneticdipole moments, i.e. electron spins, are aligned with theelectric dipole moment. The ground state of the MF isdescribed by a coherent product state with a finite mag-netic moment and polarization in each unit cell. Thescalar contributions of Eq. (5) add constructively over amacroscopic volume ideally limited by the length scalemin { λ a , L domain } , where L domain is the linear ferroic do-main size and λ a is the axion de Broglie wavelength.Since this wavelength is on the order of meters or more ,we anticipate that coupling to a macroscopic P (cid:107) M do-main can still be ensured on appreciable sensor lengthscales in clean systems. The coherent nature of the re-sponse is sketched in Fig. 4. As an order-of-magnitudeestimate (see Appendix B) we expect effective couplingto the axion to be χθ ae ∼ . · − g ae , (13)so that the total deposited energy becomes δE tot ae = L χθ ae (cid:114) µ ε P · M , (14)where L domain is the linear size of the MF domains, andwhere we have assumed that the axion makes up the localDM density.Importantly, though the axion-induced magnetizationwill be on the order of δM ∼ O (1 aT) (in vacuum) dueto the small coupling of Eq. (13), the generated magneticflux scales with the area covered by the magnetometer,Φ ae ∼ L δM , which can be enhanced experimen-tally. In practice, the size of the ferroic domains is likelydictated by Kibble–Zurek scaling as we elaborate on inthe next Section. In Appendix F we provide estimatesof the signal-to-noise ratio for SQUID magnetometers, FIG. 4. Atomic sites in a multiferroic where the electronspins S (cid:96) are aligned with the electric dipole moments P (cid:96) .The de Broglie wavelength of the axion a extends to the scaleof meters . The coherent (scalar) addition of couplings overatomic sites, given in Eq. (5), ideally results in Avogadroscaling of the effective axion coupling. indicating that the Avogadro scaling makes a successfulmeasurement of the response under ideal conditions con-ceivable. Finally, we note that in reality decoherence andloss from various sources, such as acoustic phonons and(magnetic) impurities, may make a naive scaling difficultto achieve. IV. MATERIAL CANDIDATES
Since the discovery of multiferroicity in antiferro-magnetic BiFeO , the list of established MFs con-tinues to grow and garner interest for their potentialin next-generation microelectronic. However, ferromag-netic order in these materials, which is much sought-after, is rare. Established ferromagnetic cases includeCoCr O , DyFeO , GdFeO , Mn GeO , andmore recently hexagonally grown Lu − x Sc x FeO ( h -LSFO) with x ≈ . . Among these, the subset with P (cid:107) M is practically limited to the latter four, thoughthe polarizations are not spontaneous in DyFeO .The high-temperature paraelectric phase of the hexag-onal ferrite h -LSFO adopts the P /mmc space groupwhich has inversion symmetry. At the ferroelectric Curietemperature of T C P ≈ K phonon mode manifesting in a trimer-ization of the Fe-O polyhedra, and a polar Γ − phononmode causing a rigid shift in the Lu ions. The resultingbroken-inversion-symmetry crystal structure adopts the P cm space group, as shown in Fig. 5 (a), in which thearrows indicate atomic displacement in the ferroelectricphase .In contrast with conventional ferroelectrics, the hexag-onal manganites and ferrites form topological defects(vortices) at their ferroelectric phase transition owing tothe emergent U (1) symmetry of the structural trimeriza-tion at the onset of the ferroelectric phase transition .This U (1) symmetry further breaks into a Z groundstate corresponding to the three trimerization directionsof both the “up” and “down” polarization directions, thelatter of which can be directly imaged using piezore-sponse force microscopy . Interestingly for our case, LuFeO ca b P o l a r i z a t i on ( μ C / c m ) % distortionP6 /mmc P6 cm z = 0.5z = 0z = 0z = 0.5a b ab cd FIG. 5. (a) h -LuFeO adopting the polar P cm space group with the two primary structural changes causing polarizationmarked – a trimerization and tilting of the Fe-O polyhedra, and a corrugation of the Lu ions – resulting in inversion symmetrybreaking. (b) Calculated change in total polarization of h -LuFeO along a path connecting the paraelectric P /mmc structureto the ferroelectric P cm phase. (c) and (d) Magnetic order with A symmetry showing a net out-of-plane magnetization in(c), and compensated frustrated triangular antiferromagnetism in (d). the domain sizes are determined by the Kibble–Zurekdescription of nonequilibrium dynamics during a drivenphase transition: in this case the typical standard do-main size of 10 µ m can be controlled by the quench ratethrough the ferroelectric phase transition .The spin structure of the trimerized Fe ions allowsfor a net magnetization in the c direction (in the so-called A configuration) due to spin canting as illustrated withpurple and blue arrows in Fig. 5 (c). Ferromagnetic orderdevelops below T C M ≈
160 K, with much larger domainsof typical size 100 µ m .As a second promising material platform we mentionthe olivine compound Mn GeO . In this material theferroic orders originate from spiral spin-order, resultingfrom transverse conical chains that yield a net magnetiza-tion and electric polarization pointing along the c direc-tion. Moreover, magnetic order here sets in at 47 K. Low-ering the temperature leads to two additional first-ordertransitions, and below T C P ≈ . M (cid:107) P (cid:107) c arespontaneously generated. Ferroelectric domains on thescale of 500 µ m have been observed in this compound .To give materials-specific estimates of the expected ax-ion coupling to the matter P and M we use ab initio calculations based on Density Functional Theory (DFT)in the case of hexagonal LuFeO . The calculation detailsare given in Appendix G. Here we summarize the keyparameters used in the estimates of the strength of theeffective matter coupling, namely the expectation valueof the parallel polarization P and net ferromagnetism M .We first confirmed that the A magnetic orderingadopted by the P cm polar structure is a frustrated120 ° antiferromagnetic ordering with a slight out-of-planecanting. The calculated out-of-plane spontaneous mag-netization was found to be 0.02 µ B /Fe, consistent withprevious theoretical and experimental reports . Wefind this value is relatively insensitive to choice of U eff inour DFT+U calculations for modest values of U eff . How-ever, we note that an alternative magnetic order with A symmetry is essentially degenerate with the A magneticorder with a calculated energy of 0.06 meV/f.u. lowerin energy than A , which is also consistent with otherexperiments . Such a small energy barrier betweenthese two magnetic orders and the contradictory exper-imental measurements point to highly tunable magneticorders that can be sensitive to synthesis conditions andstrain, for example. We briefly note that this opens apossibility for increasing the observed out-of-plane netspontaneous magnetization through strain, chemical sub-stitution, or doping.Next we calculated the magnitude of the spontaneouspolarization of the P cm structure using the Berryphase approach. Debate over the definition of macro-scopic polarization in bulk periodic systems by summa-tion over local electric dipoles included whether it wastruly a bulk effect, or a result of surface terminationdetails, and how to resolve the multi-valuedness of thepolarisation resulting from unit cell choices . Theseissues were resolved with the introduction of the ModernTheory of Polarisation which defined the bulk polarisa-tion from the Berry phase of the constituent wavefunc-tions, naturally prescribing the gauge freedom arisingfrom the Berry phase . To resolve this latter point incalculations, we generate a set of interpolated structuresbetween the paraelectric P /mmc and the ferroelectric P cm to give a smooth interpolation of the Berry phasecalculated electronic polarization. Our final calculatedpolarization was 10 . µ C / cm in the out-of-plane direc-tion. V. CONCLUSION AND OUTLOOK
In this paper we have considered coupling between ax-ion dark matter and macroscopic orders in quantum ma-terials. The proposed scheme fits into the growing setof proposals to use quantum materials where the (bro-ken) symmetry in the ground state enables the couplingbetween the axion and quantum matter fields. Key ad-vantages of the proposed MERMAID platform for DMdetection include (i) reliance on the spontaneously bro-ken symmetry state rather than external fields, and (ii)Avogadro scaling that allows long-range crystalline orderin nature to build a coherent stack of sensors instead ofexperimenters.Using materials-specific quantities, we estimated theaxion-electron coupling in multiferroic materials withparallel magnetic and electric polarizations. The effec-tive axion coupling in these materials is expected to beof the form P · M , similar to a linear magnetoelectric ef-fect. We argued that the coupling is enhanced/resonantas the longitudinal magnon frequency matches the axionfrequency. While we use an established MF material forthese estimates, further studies should explore how toenhance the linear polarizations, and hence the couplingto axions, in real materials. Moreover, while here we fo-cus on the case of spontaneously broken symmetries toinduce electric- and magnetic-dipoles without using anyexternal magnetic or electric fields, one can enhance themagnetization and polarization in the material with ap-plied fields.Temperature may in principle be used as a tuning knobto scan axion frequencies and to boost the response nearcritically. The sensing is based on extreme sensitivityof the multiferroic magnetic response near the ferromag-netic transition to the parameter changes induced by theaxion field. In this respect the idea is similar to thetransition-edge sensor.We discussed Avogadro scaling as a result of the co- herent response of quantum materials with macroscopicquantum order. The quantum coherence of all unit cellsmeans that the amplitude for scattering at each cell inthe coherent state effectively enhances the signal-to-noiseratio. In the case of multiferroics this coherence emergesas a result of a macroscopic ferromagnetic and ferroelec-tric state.High sensitivity magnetometers such as SQUIDscombined with discoveries of P (cid:107) M multiferroics, suchas the ferrites Lu − x Sc x FeO and the olivine compoundMn GeO , may provide a realization of the axion-mattercoupling. In future work we would like to expand thesensitivity analysis and address the expected limitationsin greater detail. As identification of suited quantumand condensed matter systems for axion and dark matterdetection is in its infancy, exploitation of multiferroics forthis purpose has the potential of opening a new avenueof research. There is also a possibility of consideringdark matter detection schemes via the axion-mattercoupling in a wider class of magnetoelectric materialsthan in the fairly limited class considered here. ACKNOWLEDGMENTS
We thank Frank Wilczek, Alexander John Millar,Matthew Lawson, Yonathan Kahn, Ilya Sochnikov, Ste-fano Bonetti, Jan Conrad, Alfredo Ferella, AndrewGeraci, Johan Hellsvik, Rohit Prasankumar, KonstantinBeyer, Felix Flicker, and Nicola Spaldin for useful discus-sions.The work was supported by the research environmentgrant “Detecting Axion Dark Matter In The Sky And InThe Lab (AxionDM)” funded by the Swedish ResearchCouncil (VR) under Dnr 2019-02337. HSR, BF, SB, AM,and AVB were also supported by the University of Con-necticut, the European Research Council under the Euro-pean Unions Seventh Framework ERS-2018-SYG 810451HERO and VILLUM FONDEN via the Centre of Ex-cellence for Dirac Materials (Grant No. 11744). SMGwas supported by the Quantum Information Science En-abled Discovery (QuantISED) for High Energy Physics(KA2401032). This work used the Extreme Science andEngineering Discovery Environment (XSEDE), which issupported by National Science Foundation grant numberACI-1548562. 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The axion couples to fermions via (with c = (cid:126) = 1) L = ¯ ψ ( iγ µ D µ − m f ) ψ − g af ∂ µ a m f ¯ ψγ γ µ ψ, (A1)where D µ = ∂ µ − iqA µ is the covariant derivative, g af is a dimensionless coupling strength, and q and m f are thefermion charge and mass, respectively. Moreover, γ µ are Dirac matrices in a particle-hole spinor basis ψ = (cid:0) ψ f , ψ ¯ f (cid:1) .The Lagrangian of Eq. (A1) yields the equations of motion[ E + qϕ − m f + g ∇ a · σ ] ψ f = [ − g∂ t a + σ · ( p − q A )] ψ ¯ f , (A2)[ E + qϕ + m f + g ∇ a · σ ] ψ ¯ f = [ − g∂ t a + σ · ( p − q A )] ψ f , (A3)where g ≡ g af / (2 m f ), A = ( ϕ, A ), and i∂ ψ f = Eψ f , and σ is a vector of Pauli matrices. In the non-relativisticlimit, with E ≈ m f , and qϕ (cid:28) m f , these equations give the effective low-energy Hamiltonian (cf. Ref. 48) H af = 12 m f [ σ · ( p − q A )] − g af m f (cid:20) ∇ a · σ + ∂ t am f σ · ( p − q A ) (cid:21) + O ( g af , ∂ a ) , (A4)where the left out terms are either higher order in g af or in derivatives of a or in both. The ∂ t a term is expected tobe suppressed by m a / (2 m f ) (which is (cid:46) − for electrons) – nevertheless this is the term which we choose to focuson. The so-called “axion wind” coupling ∼ ∇ a term may be relevant some other settings . Appendix B: Axion-electron coupling in multiferroics
Here we consider the ∂ t a term of the coupling in Eq. (A4): H ae = − ia ˜ g p j · σ j , ˜ g ≡ g ae m a m e c , (B1)where the factor of m a appears when assuming an axion field a = a exp( im a t ), which is valid since the de Brogliewavelength is on the order of meters . As explained in Sec. II the magnitude of a is fixed by assuming that theaxion make up the local DM density , ρ DM = m a | a | ≈
300 MeVcm − .To quantify the size of the effect induced by Eq. (B1) one should ideally calculate δE ae = − ia ˜ g (cid:104) Ψ | σ · ∇ | Ψ (cid:105) , (B2)where | Ψ (cid:105) is the many-body electronic ground state (i.e. Ψ = A Ψ( r , s ; r , s ; . . . ; r N , s N ), A being the antisym-metrizing operator) with spin-orbit coupling included, which is a non-trivial problem. For the material estimates inSec. IV and App. G | Ψ (cid:105) is determined with spin-orbit coupling included self-consistently using density functional the-ory. Here we take a simplified approach to give an order-of-magnitude estimate for the non-relativistic outer electronorbitals. For the expectation value of Eq. (B1) one can generally insert a resolution of the identity such that δE ae = − ia ˜ g (cid:104) (cid:104) Ψ | p | Ψ (cid:105) · (cid:104) Ψ | σ | Ψ (cid:105) + (cid:88) n (cid:54) =Ψ (cid:104) Ψ | p | n (cid:105) · (cid:104) n | σ | Ψ (cid:105) (cid:105) . (B3)In the subsequent steps we ignore the off-diagonal terms in Eq. (B3). For an insulating state the excited statesare separated by the gap and their contributions to Eq. (B3) are expected to be small. A realistic estimate takingthem into account will have to be done using ab initio methods and will be a subject of subsequent work. We thusapproximate δE ae ≈ − ia ˜ g (cid:104) Ψ | p | Ψ (cid:105) · (cid:104) Ψ | σ | Ψ (cid:105) . (B4)To have a ferroelectric polarization, i.e. to account for the misalignment of charge centres, one needs a linear combi-nation of an even and odd orbital parity component, labelled by subscript ± : | Ψ (cid:105) = a | Ψ + (cid:105) + b | Ψ − (cid:105) , (B5)2 FIG. 6. Illustration of the ferroelectric polarization P generated by a mixed orbital state, consisting of Ψ + ( s -wave) and Ψ − ( p z -wave), with color representing charge density. as illustrated in Fig. 6, resulting in (cid:104) Ψ | p | Ψ (cid:105) = − i (cid:126) Re { a ∗ b } (cid:104) Ψ + | ∇ | Ψ − (cid:105) . (B6)The electric polarization takes the form P = ρ e (cid:82) d r r | Ψ( r ) | = ρ e (cid:104) Ψ | r | Ψ (cid:105) , where ρ e is the charge density. Thisnoticeably vanishes for any pure orbital state. However, for the mixed orbital state of Eq. (B5) we get P = 2 ρ e Re { a ∗ b } (cid:104) Ψ + | r | Ψ − (cid:105) ∝ ea B (cid:104) Ψ | p | Ψ (cid:105) , (B7)if the (atomic) spatial variations take place over the scale of the Bohr radius, a B . Hence Eq. (B7) shows that essentially P (cid:39) ea B Re { a ∗ b } (cid:104) Ψ + | ∇ | Ψ − (cid:105) . (B8)The spin expectation value defines a magnetic moment in the ferromagnetic phase, (cid:104) Ψ | σ | Ψ (cid:105) = − g s µ B µ , where g s isthe effective electron g -factor, µ B is the Bohr magneton, and µ is the total magnetic moment of the unit cell. Themicroscopic contributions are summed over individual spins in the unit cell (giving µ ). In the ferromagnetic state weget: δE ae ≈ a ˜ g (cid:126) g s αe P · µ = a ˜ g (cid:126) g s αe V uc P · M , (B9)where α is the fine structure constant and the subscript “uc” refers to the unit cell. This demonstrates how amacroscopic P · M coupling results from the axion-fermion coupling in a multiferroic. The above contributions addup coherently over a macroscopic number of unit cells, as dictated by the ferroic domain size. When comparing tothe definition of χθ ae in Eq. (6), we arrive at the effective dimensionless coupling: χθ ae ≡ g ae a m a (cid:126) m e c g s αe (cid:114) ε µ ∼ . · − g ae , (B10) δE tot ae ≈ L χθ ae (cid:114) µ ε P · M . (B11)Though the effective coupling is small, the total energy scales with volume and hence takes advantage of the coherentstate of the material. For representative parameters g ae = 10 − , M = 10 Oe, P = 0 . µ C/cm , and L domain = 1 mmthe deposited energy from the axion is δE ae = 0 . Appendix C: Relativistic corrections
The first relativistic correction to the Hamiltonian of Eq. (5) in natural units is H rel = − i g ae m e ( ∂ t a ) ∇ m e ( σ · ∇ ) . (C1)The scale of this correction with respect to the leading term we consider, Eq. (B1), is on the order of (cid:104) H rel /H ae (cid:105) ∼ Z e a B m e c ∼ − , (C2)which holds for the deepest electronic states of Lu ( Z = 71), and with a B being the Bohr radius, and H ae given byEq. (B1). Hence we conclude that for materials of interest the next-to-leading order term in the relativistic expansionof the axion-electron coupling is subdominant.3 Appendix D: Details of the dynamics
We consider spatially homogeneous solutions of the longitudinal modes of Eqs. (7) and (8), and ignore the timedependence of P : P ( t, x ) = P ˆ e, M ( t, x ) = M ( t ) ˆ e, (D1)with ˆ e being a unit vector dictating the direction of the spontaneous magnetization/polarization. The dynamics isgoverned by the coupled equations α M ¨ M + γ M ( T − T C M ) M + 2 λ M M + c α + χθ ) P = 0 ,γ P ( T − T C P ) P + 2 λ P P + 12 c ( α + χθ ) M = 0 . (D2)Denote by M and P the values of the static polarizations in the absence of the axion field. These solve equations(D2) with the time derivative and θ set to zero. Now linearizing Eqs. (D2) around this static solution, M = M + δM , P = P + δP produces α M ¨ δM + m M δM + c α + χθ ) δP + P χθ ] = 0 , (D3) m P δP + 12 c [( α + χθ ) δM + M χθ ] = 0 , (D4)where we defined the (dimensionless) effective masses about equilibrium: m M ≡ γ M ( T − T C M ) + 6 λ M M , (D5) m P ≡ γ P ( T − T C P ) + 6 λ P P . (D6)Now we solve for δP , and also neglect terms ∝ δP θ, δM θ as second order quantities. Thus we arrive at an equationin the form of a standard driven harmonic oscillator¨ δM + η M ˙ δM + ω M δM = 12 α M (cid:18) αM m P − cP (cid:19) χθ, (D7) ω M ≡ α M (cid:20) m M − α m P (cid:21) . (D8)We have inserted by hand a phenomenological damping term ∼ η M into Eq. (D7), which does not follow from thepreceding equations but is expected on physical grounds, to model dissipation in the system. The driving term isproportional to the axion field. This is expected to oscillate, θ = θ exp( iω a t ), with the frequency associated to its restmass energy ω a = ( c / (cid:126) ) m a . We may therefore write down the linear response of the magnetization, as a function ofthe frequency: (cid:12)(cid:12)(cid:12) δMθ (cid:12)(cid:12)(cid:12) = 1 (cid:113) ( ω a − ω M ) + η M ω a χ α M (cid:12)(cid:12)(cid:12) αM m P − cP (cid:12)(cid:12)(cid:12) . (D9)The first term on the right hand side is the standard factor displaying the resonance between the magnetic responseand the axion field when ω a = ω M . A key question is how close we can tune the properties of our system to maximizethe response function of Eq. (D9).4 Appendix E: Static solution
The full expressions for the first terms in the series expansion of M ( θ ) as appearing in Eq. (12) are given by M ( θ ) = M (0) + χθf + O ( θ ) ,M (0) = 3 / α λ M + √ m P (cid:16)(cid:0) l − αcλ M P (cid:1) / + 2 √ γ M λ M ( T C M − T ) (cid:17) λ M m P ( l − αcλ M P ) / ,f = (cid:34) αl (cid:16) c P m P λ M − (cid:0) α + 2 m P γ M ( T C M − T ) (cid:1) (cid:17) λ M (cid:16) α c P m P λ M − ( α + 2 m P γ M ( T C M − T )) (cid:17) − √ (cid:16) α m P + γ M ( T C M − T ) (cid:17) (cid:18) αl (cid:16) c P m P λ M − ( α +2 m P γ M ( T CM − T )) (cid:17) α c P m P λ M − ( α +2 m P γ M ( T CM − T )) − cP λ M (cid:19) ( l − αcP λ M ) / + 2 √ α (cid:112) l − αcP λ M m P − cP λ M (cid:35)(cid:46)(cid:104) / (cid:0) l − αcP λ M (cid:1) / (cid:105) ,l = 1 m P (cid:104) α c P m M λ M − λ M (cid:0) α + 2 m P γ M ( T C M − T ) (cid:1) (cid:105) / . (E1) Appendix F: Signal-to-noise ratio
To give an estimate of the signal-to-noise ratio for the schematic setup of Fig. 1, we here compare the magnitudeof the axion induced magnetic field to presently achievable sensitivity levels for white noise in SQUIDs.From the response of Eq. (10), we obtain a (vacuum) magnetic response of magnitude δM ≈ .
35 aT ( δM ≈ .
31 aT)at ω a = ω M ( ω a (cid:28) ω M ), i.e. on (off) resonance, at T − T C M = 1 µ K and using the parameters of Fig. 3 which aremotivated by the compound h -LSFO. We reiterate, however, that at resonance the response is controlled by the lossrate δM ∝ /η M , which experimentally may well be the limiting factor.A key aspect of our proposal is that our signal will add coherently over the surface area of the sample, limitedonly by the maximum ferromagnetic domain size available. Given a linear domain size of 1 mm, we estimate thetotal magnetic flux measured over the whole area of the surface (1 mm ). This gives on the order of 10 − Φ , whereΦ ≡ ( hc ) / e is the fundamental flux quantum.On the other hand, presently available SQUID technologies can reach white noise sensitivities as low as12 nΦ / √ Hz (though 1 /f noise, of which origin is not fully understood, usually limits the sensitivity at frequencies f (cid:46) S/N ∼
1, we require a bandwidth of ∼ ∼ S/N likely are more restrictive in practice . Finally, we note thatimpurities may be another relevant and leading threat against the success of the scheme proposed here. Similarissues may, however, be even more pressing in doped topological insulators which tend to suffer from residual bulkconductance and charge puddles . Appendix G: Density functional theory calculation details
Our Density Functional Theory (DFT) calculations were performed with the Vienna Ab Initio Simulation Package(VASP) with projector augmented waves and the PBE exchange-correlation functional. We treated Lu (5s, 5p, 6s),Fe (3d, 4s, 4p), O (2s, 2p) electrons as valence. We used a planewave energy cutoff of 600 eV and a Γ-centered k-pointgrid of 6 × ×
2. Since semilocal density functionals suffer from the well-documented self-interaction error, we useDFT+U to effectively treat the Fe-d orbitals. In this case, we define an effective U eff ≡ U − J and select a value of U eff = 4 eV for the Fe-d orbitals, consistent with previous studies on h-LuFeO .We used the PBEsol functional for structural optimizations which is specifically tailored for accurate forces insolids . Structural optimizations of the unit cell shape and size in addition to the internal coordinates were performeduntil the Hellman–Feynman forces were less than 0.01 eV/˚A with the resulting lattice parameters given in the TableI. The spontaneous polarization was calculated using the Berry-phase approach with interpolated structures between5the nonpolar and polar hexagonal structures used to obtain consistent branch cuts for the calculated polarization.Spin-orbit coupling was included self-consistently for all calculations. TABLE I. Lattice parameters for the high-symmetry P /mmc and low-symmetry P cm phases of LuFeO calculated usingPBEsol. The last row are the measured lattice parameters from experiment taken from Magome et al. Space Group a (˚A) c (˚A) P /mmc P cm ) 11.523 (Th.)11.702 (Exp.94