Axion/Hidden-Photon Dark Matter Conversion into Condensed Matter Axion
KKEK-TH-2299
Axion/Hidden-Photon Dark Matter Conversion intoCondensed Matter Axion
So Chigusa ( a,b,c ) , Takeo Moroi ( d,e ) and Kazunori Nakayama ( d,e ) ( a ) Berkeley Center for Theoretical Physics, Department of Physics,University of California, Berkeley, CA 94720, USA ( b ) Theoretical Physics Group, Lawrence Berkeley National Laboratory,Berkeley, CA 94720, USA ( c ) KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, Japan ( d ) Department of Physics, Faculty of Science,The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan ( e ) Kavli IPMU (WPI), The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
Abstract
The QCD axion or axion-like particles are candidates of dark matter of the universe.On the other hand, axion-like excitations exist in certain condensed matter systems,which implies that there can be interactions of dark matter particles with condensedmatter axions. We discuss the relationship between the condensed matter axion and acollective spin-wave excitation in an anti-ferromagnetic insulator at the quantum level.The conversion rate of the light dark matter, such as the elementary particle axion orhidden photon, into the condensed matter axion is estimated for the discovery of thedark matter signals. a r X i v : . [ h e p - ph ] F e b ontents A.1 Tight-binding model with spin-orbit interaction . . . . . . . . . . . . . . . . 19A.2 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21A.3 Model independence of the spin-orbit interaction term . . . . . . . . . . . . . 22
B Transformation of α matrix 24C Berry connection and topological term 25 C.1 Dimensional reduction of (4 + 1)-dimensional quantum Hall insulator . . . . 25C.2 Hamiltonian expression of θ . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Introduction
The QCD axion is a hypothetical elementary particle that solves the strong CP problem [1–3]and is a candidate of dark matter (DM) of the universe [4–6] (see Refs. [7–9] for reviews).Recently people often consider axion-like particles (ALPs) in a broad sense, partly motivatedby the developments in string theory [10–12]. ALPs do not necessarily address the strong CPproblem, but they are also good DM candidates and may be experimentally probed through,e.g., the axion-photon coupling of the form
L ∝ a (cid:126)E · (cid:126)B where a denotes the ALP field and (cid:126)E ( (cid:126)B ) denotes the electric (magnetic) field respectively. There are many experimental ideas tosearch for ALPs including the QCD axion, although still it is not discovered yet [13–33].On the other hand, the axion-like excitation also appears in the condensed matterphysics [34, 35] (see Refs. [36, 37] for reviews). To distinguish it from the elementary par-ticle axion or ALP, we call such an axion-like excitation in condensed matter context as“condensed matter axion (CM axion)”. The CM axion δθ has an interaction with the elec-tromagnetic field as L ∝ δθ (cid:126)E · (cid:126)B , similar to the ALP. We call such an insulator an axionicinsulator.Let us briefly mention a relation between the topological insulator and axionic insula-tor. In general, topological electromagnetic responses of a three-dimensional insulator aredescribed by the topological term in the Lagrangian: L = θ α e π F µν (cid:101) F µν = θ α e π (cid:126)E · (cid:126)B. (1.1)For example, it implies that there appears a magnetization (electric polarization) propor-tional to the applied electric (magnetic) field: (cid:126)M ∝ θ (cid:126)E ( (cid:126)P ∝ θ (cid:126)B ). If the Hamiltonian of thesystem is invariant under the time-reversal symmetry, the coefficient θ can only take a valueeither 0 or 1 /
2: i.e., such an insulator is classified by a discrete Z index [38–43]. The caseof θ = 1 / θ does not have to be quantized but can take arbitraryvalues possibly with a space-time dependence: θ = θ ( (cid:126)x, t ). If θ is a dynamical field, it iscalled the CM axion. Although it is often helpful to start from the topological insulatorfor understanding the origin of CM axion, the existence of CM axion does not necessarilyrequire that the insulator is topological. One can generally write θ ( (cid:126)x, t ) = θ + δθ ( (cid:126)x, t ) sothat δθ ( (cid:126)x, t ) expresses the CM axion while θ is the background value. The value of θ depends on the properties of the material and can be zero. It has been known that in a classof magnetically doped topological insulators, the fluctuation of the anti-ferromagnetic orderparameter (the so-called Neel field) plays a role of CM axion [35].In this paper, we consider a process like the light DM conversion into the CM axion andestimate the conversion rate. Such a process has been considered in Ref. [33] for the detection In the following we use the terminology “ALP” for general elementary axion-like particles including theQCD axion.
Time-reversal invariant topological insulators have been first considered in two-dimensional systems [44,45].
2f axion-like DM. One of the main purposes of this paper is to discuss the origin of CM axionin a comprehensive and self-consistent manner for particle physicists. We will explicitly showthe relationship between the CM axion and the spin-wave fluctuation (magnon) based on amodel presented in Ref. [46]. Another purpose is to provide a useful method to calculate theDM conversion rate into the CM axion in a quantum mechanical way. As an illustration, wewill consider the case of ALP DM and hidden-photon DM.This paper is organized as follows. In Sec. 2 we review the (anti-ferromagnetic) Heisen-berg model of the localized electron spin system on the lattice. It gives a basis of thecollective spin-wave excitation (magnon) and its dispersion relation, which will turn out tobe identified with the CM axion in a certain setup. In Sec. 3 the so-called (half-filling) Hub-bard model is briefly introduced. Electrons in solids are often modeled by a tight-bindingHamiltonian plus the Coulomb repulsive force between electrons on the same lattice point(Hubbard interaction). It is shown that the limit of large Hubbard interaction reduces to the(anti-ferromagnetic) Heisenberg model. Therefore, the Hubbard model on a certain latticemay describe both the electron energy band structure as well as the anti-ferromagnetic orderand magnon excitation around it. In Sec. 4 we introduce the Fu-Kane-Mele-Hubbard modelas a concrete setup and show that it contains an excitation that is regarded as the CM axionalong the line of Ref. [46]. It will become clear that the CM axion is described by the useof anti-ferromagnetic magnon and its dispersion can be estimated as explained in Sec. 2. InSec. 5 we estimate the conversion rate of light bosonic DM into the CM axion. We considertwo DM models: ALP and hidden photon. We conclude in Sec. 6.
Let us start with the Heisenberg anti-ferromagnet model [47–49].
Suppose a bipartitelattice consisting of sublattices A and B, and on each lattice point (cid:96) ∈ A or (cid:96) (cid:48) ∈ B there isan electron spin (cid:126)S . Applying an external magnetic field B along the z direction, the modelHamiltonian is given by H = − J (cid:88) (cid:104) (cid:96),(cid:96) (cid:48) (cid:105) (cid:126)S (cid:96) · (cid:126)S (cid:96) (cid:48) − gµ B ( B A + B ) (cid:88) (cid:96) ∈ A S z(cid:96) + gµ B ( B A − B ) (cid:88) (cid:96) (cid:48) ∈ B S z(cid:96) (cid:48) , (2.1)where J < g = 2 and µ B = e/ (2 m e ) is the Bohr magneton,and B A is the anisotropy field. The collective excitation of the spin-wave around the groundstate, called magnon, is analyzed through the Holstein-Primakoff transformation, S + (cid:96) = (cid:113) s − a † (cid:96) a (cid:96) a (cid:96) , S − (cid:96) = a † (cid:96) (cid:113) s − a † (cid:96) a (cid:96) , S z(cid:96) = s − a † (cid:96) a (cid:96) , (2.2) S + (cid:96) (cid:48) = b † (cid:96) (cid:48) (cid:113) s − b † (cid:96) (cid:48) b (cid:96) (cid:48) , S − (cid:96) (cid:48) = (cid:113) s − b † (cid:96) (cid:48) b (cid:96) (cid:48) b (cid:96) (cid:48) , S z(cid:96) (cid:48) = − s + b † (cid:96) (cid:48) b (cid:96) (cid:48) , (2.3) As explained in Sec. 3, the Heisenberg anti-ferromagnet model may be understood from the Hubbardmodel in the limit of strong electron self-interaction at each site. S ± (cid:96) = S x(cid:96) ± iS y(cid:96) and S ± (cid:96) (cid:48) = S x(cid:96) (cid:48) ± iS y(cid:96) (cid:48) , and the creation-annihilationoperators satisfy the commutation relation (cid:2) a (cid:96) , a † m (cid:3) = δ (cid:96)m , (cid:104) b (cid:96) (cid:48) , b † m (cid:48) (cid:105) = δ (cid:96) (cid:48) m (cid:48) . (2.4)In addition, s is the spin quantum number; the eigenvalue of (cid:126)S (cid:96) · (cid:126)S (cid:96) is given by s ( s + 1). TheHamiltonian is rewritten in terms of the creation-annihilation operators as H = 2 N zs J − N sω A − J s (cid:88) (cid:104) (cid:96),(cid:96) (cid:48) (cid:105) (cid:16) a † (cid:96) a (cid:96) + b † (cid:96) (cid:48) b (cid:96) (cid:48) + a † (cid:96) b † (cid:96) (cid:48) + a (cid:96) b (cid:96) (cid:48) (cid:17) + ( ω A + ω L ) (cid:88) (cid:96) a † (cid:96) a (cid:96) + ( ω A − ω L ) (cid:88) (cid:96) (cid:48) b † (cid:96) (cid:48) b (cid:96) (cid:48) , (2.5)where N is the total number of sites in a sublattice, z denotes the number of adjacent latticepoints (e.g. z = 6 for simple bipartite cubic lattice), and ω L ≡ gµ B B , ω A ≡ gµ B B A . (2.6)Now let us move to the Fourier space. We define the Fourier component as a (cid:96) = 1 √ N (cid:88) (cid:126)k e − i(cid:126)k · (cid:126)x (cid:96) a (cid:126)k , b (cid:96) (cid:48) = 1 √ N (cid:88) (cid:126)k e i(cid:126)k · (cid:126)x (cid:96) (cid:48) b (cid:126)k . (2.7)Substituting this into the Hamiltonian, we find H = (cid:88) (cid:126)k (cid:104) ( ω J + ω A + ω L ) a † (cid:126)k a (cid:126)k + ( ω J + ω A − ω L ) b † (cid:126)k b (cid:126)k + ω J γ (cid:126)k ( a (cid:126)k b (cid:126)k + a † (cid:126)k b † (cid:126)k ) (cid:105) , (2.8)where ω J ≡ − zsJ and γ (cid:126)k = 1 z (cid:88) (cid:126)δ e i(cid:126)k · (cid:126)δ , (2.9)with (cid:126)δ being the vector connecting the adjacent lattice points. Finally, it is diagonalizedthrough the Bogoliubov transformation: α (cid:126)k = u (cid:126)k a (cid:126)k − v (cid:126)k b † (cid:126)k , β † (cid:126)k = u (cid:126)k b † (cid:126)k − v (cid:126)k a (cid:126)k . (2.10)One can check that the canonical commutation relation is maintained if | u (cid:126)k | − | v (cid:126)k | = 1.The concrete expression is given by | u (cid:126)k | = 12 (cid:32) ω J + ω A (cid:112) ( ω J + ω A ) − | γ (cid:126)k | ω J (cid:33) , | v (cid:126)k | = 12 (cid:32) − ω J + ω A (cid:112) ( ω J + ω A ) − | γ (cid:126)k | ω J (cid:33) , (2.11)4ith arg( γ (cid:126)k ) = 2 arg( u (cid:126)k ) = − − v (cid:126)k ). (Thus, u (cid:126)k v (cid:126)k is real and negative.) Note that, when ω A (cid:28) ω J , we have large Bogoliubov coefficients | u (cid:126)k | ∼ | v (cid:126)k | (cid:29) | (cid:126)k · (cid:126)δ | (cid:28)
1. Then, onefinds the diagonal Hamiltonian: H = (cid:88) (cid:126)k (cid:104) ( ω (cid:126)k + ω L ) α † (cid:126)k α (cid:126)k + ( ω (cid:126)k − ω L ) β † (cid:126)k β (cid:126)k (cid:105) . (2.12)Here, ω (cid:126)k represents the magnon dispersion relation (besides the overall offset coming fromthe Larmor frequency ω L ), ω (cid:126)k = ω J (1 − | γ (cid:126)k | ) + ω A ( ω A + 2 ω J ) . (2.13)In the low frequency limit | (cid:126)k · (cid:126)δ | (cid:28)
1, we obtain γ (cid:126)k (cid:39) i (cid:80) (cid:126)δ ( (cid:126)k · (cid:126)δ ) /z − (cid:80) (cid:126)δ ( (cid:126)k · (cid:126)δ ) /z . Itimplies the linear dispersion relation, ω (cid:126)k ∝ | (cid:126)k | for large | (cid:126)k | (but still it satisfies | (cid:126)k · (cid:126)δ | (cid:28) ω (cid:126)k ∝ k . Theyare related to the so-called type-I and type-II Nambu-Goldstone boson dispersion relationas generally classified in Refs. [50, 51]. A tight-binding model is one of the approaches to estimate the electron energy band structurein solids. In this approach, one starts with the picture that each electron is rather tightlybounded by each atom and then takes into account the overlap between the nearest electronwave function.Let us consider only one electron orbital at each site and neglect the interaction amongdifferent orbits, spin-orbit coupling, electron self-interaction, etc.
In the second quantiza-tion picture, the tight-binding Hamiltonian is given by H = − t (cid:88) (cid:104) i,j (cid:105) ,σ c † iσ c jσ , (3.1)where c † iσ and c iσ denote the electron creation and annihilation operators at the site i withspin σ ( ↑ or ↓ ) and the summation is taken over the combination of adjacent sites (cid:104) i, j (cid:105) . Thecreation and annihilation operators satisfy the anti-commutation relation (cid:110) c iσ , c † jσ (cid:48) (cid:111) = δ ij δ σσ (cid:48) . (3.2) Effects of the interaction among different orbitals and spin-orbit coupling are important for the topo-logical insulator. The electron self-interaction will be taken into account in the next subsection. c iσ = 1 √ N (cid:88) (cid:126)k e − i(cid:126)k · (cid:126)x i c (cid:126)k,σ . (3.3)The Hamiltonian is rewritten in a diagonal form as H = (cid:88) (cid:126)k,σ (cid:15) (cid:126)k c † (cid:126)k,σ c (cid:126)k,σ , (cid:15) (cid:126)k = − t ( γ (cid:126)k + γ ∗ (cid:126)k ) . (3.4)This (cid:15) (cid:126)k denotes the electron energy band. In a simple cubic lattice, for example, we obtain (cid:15) k = 2 t (cid:16) − (cid:80) i = x,y,z cos( k i a ) (cid:17) .The conductivity of this model is determined by the number of electrons in the system.If each orbital is filled, i.e., there are two electrons with opposite spins at each site, theenergy band is filled and this becomes an insulator as far as there is an energy gap to thenext energy band. If there is only one electron at each orbital, the energy band is not filledand it becomes a metal. Let us add the effect of interaction between electrons at the same site i to the tight-bindingHamiltonian. The resulting Hamiltonian is called the Hubbard model: H = H t + H U = − t (cid:88) (cid:104) i,j (cid:105) ,σ c † iσ c jσ + U (cid:88) i n i ↑ n i ↓ , (3.5)where U > n i ↑ = c † i ↑ c i ↑ and n i ↓ = c † i ↓ c i ↓ .The Hubbard model is characterized by several parameters: the relative interactionstrength U/t and the number of electrons per site, N e /N s . The case of N e /N s = 1 iscalled the half-filling (it is “half” because of the spin degree of freedom) and its propertiesare well understood. Below, we consider the half-filling case. Naively, one may consider thatthe half-filling Hubbard model describes a metal since electrons are in a conducting band.It is true in the limit U = 0, but it is not necessarily true for sizable interaction strength.The interaction term can split the energy band and make a gap, which would result in aninsulator. Such an insulator is called the Mott insulator.Now we consider the large interaction limit: U/t (cid:29)
1. In this limit, the tight-bindingpart is regarded as a perturbation. In the ground state, one electron is localized at eachsite to minimize the Hubbard interaction energy (hence it is expected that it behaves as aninsulator rather than metal). Thus, the ground state is expressed as | (cid:101) σ (cid:105) = (cid:32)(cid:89) i c † iσ i (cid:33) | (cid:105) , (3.6)6here (cid:101) σ schematically represents the array of spin, e.g., (cid:101) σ = ( . . . , ↑ , ↑ , ↓ , . . . ) and so on.There are 2 N e degenerate ground states corresponding to the spin degree of freedom at eachsite.We want to consider an effective Hamiltonian regarding H t as a perturbation. Noting (cid:104) (cid:101) σ | H t | (cid:101) σ (cid:105) = 0, the nontrivial effect appears at the second-order in H t . The effective Hamil-tonian is given by H eff = −P H t H U H t P = − t U P (cid:88) (cid:104) i,j (cid:105) σσ (cid:48) (cid:16) c † iσ c jσ c † jσ (cid:48) c iσ (cid:48) + c † jσ c iσ c † iσ (cid:48) c jσ (cid:48) (cid:17) P , (3.7)where P denotes the projection operator to the Hilbert space spanned by the ground state(3.6). The physical meaning is that, for σ (cid:54) = σ (cid:48) , it exchanges the spin at the adjacent sites i and j for a given ground state. This is rewritten in terms of the spin operator as H eff = 4 t U (cid:88) (cid:104) i,j (cid:105) (cid:126)S i · (cid:126)S j , (3.8)where we have defined S zi = 12 ( c † i ↑ c i ↑ − c † i ↓ c i ↓ ) , S + i ≡ S xi + iS y = c † i ↑ c i ↓ , S − i ≡ S x − iS y = c † i ↓ c i ↑ . (3.9)Since the coefficient t /U is positive, it represents the Heisenberg anti-ferromagnet modelwith J = − t /U . Thus, the half-filling Hubbard model may describe both the metal phasein the limit U → U limit. A three-dimensional topological insulator has been proposed in Refs. [39, 40]. An exampleis the diamond lattice with a strong spin-orbit coupling. On the other hand, taking accountof the Hubbard on-site interaction between electrons may lead to the anti-ferromagneticphase, leading to the topological anti-ferromagnet. Such a model is called the Fu-Kane-Mele-Hubbard model and studied in Ref. [46]. Actually, it is found in Ref. [46] that there isa topological anti-ferromagnetic phase depending on the interaction strength, in which thespin-wave excitation (magnon) has an axionic coupling to the electromagnetic field.Now, we briefly review the Fu-Kane-Mele-Hubbard model on the diamond lattice. Weassume the half-filling case, i.e., there is only one electron at the electron orbitals of ourinterest at each site. The model Hamiltonian is given by H = H + H U : H = (cid:88) (cid:104) i,j (cid:105) σ t ij c † iσ c jσ + i λa (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) c † i (cid:126)σ · ( (cid:126)d ij × (cid:126)d ij ) c j , (4.1) H U = U (cid:88) i n i ↑ n i ↓ , (4.2)7here c i ≡ ( c i ↑ , c i ↓ ) T . Here, (cid:126)d ij and (cid:126)d ij are the two vectors that connect two adjacent sites: a (1 , , a (1 , − , − a ( − , , − a ( − , − , a being the lattice constant and λ represents the strength of the spin-orbit coupling. Note that the diamond lattice consistsof two sublattices (which we call A and B) both of which are face-centered cubic. (cid:104)(cid:104) i, j (cid:105)(cid:105) denotes a set of the next-nearest neighbor sites, and hence sites i and j belong to the samesublattice. (For more detail about the interaction of electrons in next-nearest neighbor sites,see App. A.)Let us study the energy bands of this model neglecting the Hubbard interaction term [39,40]. In the Fourier space, the Hamiltonian is expressed as the matrix form in the basis c (cid:126)k ≡ ( c (cid:126)k ↑ ,A , c (cid:126)k ↓ ,A , c (cid:126)k ↑ ,B , c (cid:126)k ↓ ,B ) T as H = (cid:88) (cid:126)k c † (cid:126)k H c (cid:126)k , H = (cid:88) µ =1 R µ ( (cid:126)k ) α µ , (4.3)where R ( (cid:126)k ) = λ (cid:104) sin( (cid:126)k · (cid:126)a ) − sin( (cid:126)k · (cid:126)a ) − sin( (cid:126)k · ( (cid:126)a − (cid:126)a )) − sin( (cid:126)k · ( (cid:126)a − (cid:126)a )) (cid:105) , (4.4) R ( (cid:126)k ) = λ (cid:104) sin( (cid:126)k · (cid:126)a ) − sin( (cid:126)k · (cid:126)a ) − sin( (cid:126)k · ( (cid:126)a − (cid:126)a )) − sin( (cid:126)k · ( (cid:126)a − (cid:126)a )) (cid:105) , (4.5) R ( (cid:126)k ) = λ (cid:104) sin( (cid:126)k · (cid:126)a ) − sin( (cid:126)k · (cid:126)a ) − sin( (cid:126)k · ( (cid:126)a − (cid:126)a )) − sin( (cid:126)k · ( (cid:126)a − (cid:126)a )) (cid:105) , (4.6) R ( (cid:126)k ) = t (cid:104) (cid:126)k · (cid:126)a ) + cos( (cid:126)k · (cid:126)a ) + cos( (cid:126)k · (cid:126)a ) (cid:105) + δt, (4.7) R ( (cid:126)k ) = t (cid:104) sin( (cid:126)k · (cid:126)a ) + sin( (cid:126)k · (cid:126)a ) + sin( (cid:126)k · (cid:126)a ) (cid:105) , (4.8)with (cid:126)a = a (0 , , , (cid:126)a = a (1 , , , (cid:126)a = a (1 , ,
0) and α i = (cid:18) σ i − σ i (cid:19) , α = (cid:18) (cid:19) , α = (cid:18) i − i (cid:19) . (4.9)These α matrices are Hermite and satisfy the anti-commutation relation { α µ , α ν } = 2 δ µν .Then, it is easy to show that the energy eigenvalues are given by E ± = ± (cid:115)(cid:88) µ (cid:16) R µ ( (cid:126)k ) (cid:17) . (4.10)This gives the dispersion relation of the bulk electron. It is found that, at the so-called X r points ( r = 1 , ,
3) of the momentum space, (cid:126)k X = πa (1 , , , (cid:126)k X = πa (0 , , , (cid:126)k X = πa (0 , , E ± = 0 in thelimit of δt = 0. Thus, this material is regarded as a semimetal in this limit. For example,the dispersion relation around (cid:126)k = (cid:126)k X is given by E ± ( (cid:126)q ) = ± (cid:113) ( tq x ) + 4 λ ( q y + q z ) + ( δt ) , (4.11)where we have taken (cid:126)k = (cid:126)k X + (cid:126)q . Thus, nonzero δt gives the energy gap between two energybands, which makes the material the bulk insulator (topological insulator, actually).8 .2 Axionic excitation in anti-ferromagnetic phase It is expected that the inclusion of the Hubbard interaction H U may lead to the anti-ferromagnetic ordering. Actually, it is found that the anti-ferromagnetic phase appears forsizable U/t in the mean field approximation [46]. Under this approximation, the Hubbardinteraction term can be rewritten as H U (cid:39) U (cid:88) i (cid:16) (cid:104) n i ↑ (cid:105) n i ↓ + (cid:104) n i ↓ (cid:105) n i ↑ − (cid:104) n i ↑ (cid:105) (cid:104) n i ↓ (cid:105)− (cid:68) c † i ↑ c i ↓ (cid:69) c † i ↓ c i ↑ − (cid:68) c † i ↓ c i ↑ (cid:69) c † i ↑ c i ↓ + (cid:68) c † i ↑ c i ↓ (cid:69) (cid:68) c † i ↓ c i ↑ (cid:69) (cid:17) , (4.12)with (cid:104)O(cid:105) being the ensemble average of the operator O . We use the operator equations n i ↑ ( ↓ ) = ± S (cid:48) zi + 12 ( n i ↑ + n i ↓ ) , (4.13) c † i ↑ c i ↓ = S (cid:48) xi + iS (cid:48) yi , (4.14) c † i ↓ c i ↑ = S (cid:48) xi − iS (cid:48) yi , (4.15)with (cid:126)S (cid:48) i being spin operators in the coordinate system used in the previous subsection, withwhich three Dirac points are defined. Note that, in the U → ∞ limit of a half-filling model,we can safely restrict ourselves to states with (cid:104) n i ↑ + n i ↓ (cid:105) = 1. Then, neglecting constantterms, the Hubbard interaction becomes H U (cid:51) (cid:88) (cid:126)k c † (cid:126)k H U c (cid:126)k , H U = − U (cid:88) r =1 m r α r , (4.16)with m r are defined through (cid:68) (cid:126)S i,A (cid:69) = − (cid:68) (cid:126)S i,B (cid:69) ≡ (cid:126)m, (4.17)which characterizes the anti-ferromagnetic ordering.Under this background and assuming U | (cid:126)m | (cid:28) λ , the X r points ( r = 1 , ,
3) are slightlyshifted as (cid:126)k (cid:101) X = (cid:18) πa , U m λa , − U m λa (cid:19) , (cid:126)k (cid:101) X = (cid:18) − U m λa , πa , U m λa (cid:19) , (cid:126)k (cid:101) X = (cid:18) U m λa , − U m λa , πa (cid:19) . (4.18)For example, the energy dispersion around the (cid:101) X point is given by E ± ( (cid:126)q ) = ± (cid:113) ( tq x ) + 4 λ ( q y + q z ) + ( δt ) + ( U m ) , (4.19)where we have taken (cid:126)k = (cid:126)k (cid:101) X + (cid:126)q . It is seen that there is an additional gap due to theanti-ferromagnetic order. 9he Hamiltonian around the (cid:101) X point is expressed as H (cid:101) X ( (cid:126)q ) = 1 a ( (cid:101) q x α + (cid:101) q y α + (cid:101) q z α ) + δt α + U m α , (4.20)where we have rescaled the momentum as tq x → (cid:101) q x /a , 2 λq y → (cid:101) q y /a , 2 λq z → (cid:101) q z /a . Inderiving Eq. (4.20), we have performed an appropriate change of the basis of the α matricesthrough a unitary transformation, with which α ↔ α (see App. B). The Hamiltonianaround the (cid:101) X and (cid:101) X points can also be reduced to the same form except for the last term,which becomes U m α and U m α , respectively. From this Hamiltonian, we can infer theeffective action for the electron which mimics the action of the relativistic Dirac fermion as S = (cid:90) d x (cid:88) r =1 , , ψ r [ iγ µ ( ∂ µ − ieA µ ) − δt − iγ U m r ] ψ r . (4.21)One can make a chiral rotation of the fermion to eliminate the γ dependent term, ψ r → e iγ θ r / ψ r . Then, there appears a topological term: S = (cid:90) d x θ α e π F µν (cid:101) F µν , θ ≡ θ + (cid:88) r θ r = θ + (cid:88) r tan − (cid:18) U m r δt (cid:19) , (4.22)where θ is either 0 or 1 / δt . (See App. C for another derivationof θ .) Note that the background magnetization (cid:126)m can fluctuate: it is a spin-wave or magnonexcitation, (cid:126)m ( (cid:126)x ). Then, θ ( (cid:126)x ) is not a constant but a dynamical field and it has an ax-ionic coupling to the electromagnetic field. Therefore, in this model, the magnon effectivelybehaves as an axion-like field (CM axion). To relate the axionic excitation (or the CM axion) θ to the conventional magnons defined inSec. 2, we repeat the analysis in the previous subsection, taking into account the fluctuationof the background magnetization in terms of magnon operators. We focus only on thespatially homogeneous spin fluctuations and consider their interaction with electrons ataround a Dirac point (cid:126)k ∼ (cid:126)k (cid:101) X r . Then, the relevant part of the Hubbard interaction term isschematically expressed as H U (cid:51) U (cid:88) r =1 , , (cid:88) (cid:126)k ∼ (cid:126)k (cid:101) Xr (cid:88) L = A,B (cid:104) (cid:101) F L ( n i ↑ ; (cid:126) c † (cid:126)k ↓ ,L c (cid:126)k ↓ ,L ) + (cid:101) F L ( n i ↓ ; (cid:126) c † (cid:126)k ↑ ,L c (cid:126)k ↑ ,L ) − (cid:101) F L ( c † i ↑ c i ↓ ; (cid:126) c † (cid:126)k ↓ ,L c (cid:126)k ↑ ,L ) − (cid:101) F L ( c † i ↓ c i ↑ ; (cid:126) c † (cid:126)k ↑ ,L c (cid:126)k ↓ ,L ) (cid:105) , (4.23)where the Fourier transform of operators O i is defined as (cid:101) F L ( O i ; (cid:126)q ) ≡ N (cid:88) i ∈ L O i e i(cid:126)q · (cid:126)x i . (4.24) Eq. (4.22) may not be applicable when
U m r /δt (cid:29) F L in Eq. (4.23) is determined by the magnetization, which may fluctuate around the averagevalue. We again use the operator equations Eqs. (4.13)–(4.15) to rewrite ˜ F L in terms of spinoperators (cid:126)S (cid:48) i . The relationship between (cid:126)S (cid:48) i and (cid:126)S i , which are defined in Sec. 2 and directlyrelated to magnon operators, is given by (cid:126)S (cid:48) A ( B ) i = O (cid:126)S A ( B ) i , (4.25)with O ≡ ( (cid:126)o (cid:126)o (cid:126)o ) being a 3 × (cid:126)m (cid:107) (cid:126)o . Taking everything into consideration, the magnon-Dirac electron interaction term is, upto some constant and quadratic terms of magnons, expressed as H U (cid:51) (cid:88) (cid:126)k c † (cid:126)k (cid:101) H U c (cid:126)k , (cid:101) H U = (cid:88) µ =1 (cid:101) R µ α µ + (cid:101) R α + (cid:101) R α + (cid:101) R α , (4.26)with α rr (cid:48) ≡ − iα r α r (cid:48) . Coefficients are given by (cid:101) R r = − U (cid:20) m r + (cid:114) s N (cid:16) ( O r − iO r )( u (cid:126) − v (cid:126) )( α (cid:126) − β † (cid:126) ) + h . c . (cid:17)(cid:21) ( r = 1 , , , (4.27) (cid:101) R = (cid:101) R = 0 , (4.28)where O rr (cid:48) is the ( r, r (cid:48) ) component of the rotation matrix O , while m r ≡ O r ( s − N (cid:80) (cid:126)q v (cid:126)q )is the r -th component of the sublattice magnetization in the ground state. In addition,here and hereafter, s = 1 /
2. Note that the expectation value of (cid:101) R r is proportional to the r -th component of the order parameter ( (cid:104) S i,A (cid:105) − (cid:104) S i,B (cid:105) ) /
2, while that of (cid:101) R rr (cid:48) to the averagemagnetization ( (cid:104) S i,A (cid:105) + (cid:104) S i,B (cid:105) ) /
2. The (cid:101) R rr (cid:48) terms induce interactions between magnon andelectron/hole. It may cause, for example, the decay of a magnon into an electron-holepair when the gap is small. Because we are interested in the magnon interaction withelectromagnetic fields, which is not induced by the (cid:101) R rr (cid:48) terms, we neglect them from nowon. Repeating the same procedure as Sec. 4.2, we obtain the relationship between the axionicexcitation and magnons. Finally, the electromagnetic interaction of magnons is described by H int = − α e π (cid:114) s N ( u (cid:126) − v (cid:126) ) (cid:104) D ∗ α † (cid:126) − Dβ † (cid:126) + h . c . (cid:105) (cid:90) d x (cid:126)E · (cid:126)B, (4.29)with D = (cid:88) r U/δt U m r /δt ( O r − iO r ) , (4.30)being an O (1) factor, assuming only a moderate hierarchy between U and δt . Note that( u (cid:126) − v (cid:126) ) is real because γ (cid:126) = 1. The interaction Hamiltonian shows that a linear combinationof magnon states is excited by a non-zero value of (cid:126)E · (cid:126)B . There is an ambiguity in the choice of (cid:126)o and (cid:126)o related to the SO (2) rotation around (cid:126)o . However,since (4.29) is unchanged under the SO (2) up to an overall phase factor, it does not affect the interactionstrength. From Eq. (4.29), one can read off the “decay constant” of the CM axion as f CM ∼ Dark matter conversion into condensed matter axion
Now we discuss the detection of the elementary-particle DM axion (or ALPs) and hiddenphoton through the interaction with CM axion. (To avoid confusion between the DM andCM axions, hereafter, the DM axion and ALPs are both called ALPs.)
The dynamics of the ALP DM a and the photon in a material is described by L = 12 ( ∂ µ a ) − m a a + 12 (cid:32) (cid:15) | (cid:126)E | − | (cid:126)B | µ (cid:33) + g aγγ a (cid:126)E · (cid:126)B, (5.1)where (cid:15) and µ are the permittivity and permeability of the material. Hereafter, we treat theALP field as a classical background a ( (cid:126)x, t ) = a cos( m a t − m a (cid:126)v a · (cid:126)x + δ ) , (5.2)with | (cid:126)v a | ∼ O (10 − ). When the ALP explains the total amount of the dark matter ρ DM ∼ . / cm , we obtain m a a / ρ DM . We consider applying a constant magnetic field (cid:126)B = B ˆ z to the system, where ˆ z is a unit vector along the z -axis. This magnetic field,combined with the ALP background, generates an oscillating electric field (cid:126)E ( (cid:126)x, t ) = E ˆ z cos( m a t − m a (cid:126)v a · (cid:126)x + δ ) , (5.3)with E = − (cid:15) g aγγ a B . (5.4)The target mass range of this set up is m a ∼ O (10 − ) eV, which has a de-Broglie length (cid:96) ∼ /m a | (cid:126)v a | ∼ O (10) cm. We assume that (cid:96) is larger than the material size and neglect the (cid:126)x dependence of the ALP background inside the material. Since (cid:126)E · (cid:126)B is uniform in this case,only the magnon zero-modes may be excited, which are considered in Sec. 4.3. Substitutingthe value of (cid:126)E · (cid:126)B generated by the ALP background, the interaction Hamiltonian is rewrittenas H int = ( C ∗ a α † (cid:126) − C a β † (cid:126) + h.c.) cos( m a t + δ ) , (5.5)where C a ≡ − α e E B V π (cid:114) s N ( u (cid:126) − v (cid:126) ) D, (5.6) (cid:0) ( u (cid:126) − v (cid:126) ) | D | (cid:112) ω (cid:126) V unit (cid:1) − with ω (cid:126) being the magnon frequency at (cid:126)k = 0 and V unit = V /N the volumeof the magnetic unit cell. V being the material volume. H int describes the generation of both α - and β -modesof the magnon. However, as we will see below, one of them is highly enhanced when thecorresponding excitation energy matches with the ALP mass; in such a case, we may expectan observable signal rate at the laboratory. Accordingly, we will estimate a signal rate ofthe magnon excitation assuming that a single mode is selectively excited. We start from the α -mode, while the discussion for the β -mode is parallel, as we willcomment later. We define the ground and the one-magnon states of the material through α (cid:126)q | (cid:105) = β (cid:126)q | (cid:105) = 0 for any (cid:126)q and | (cid:105) ≡ α † (cid:126) | (cid:105) , respectively. Also, we express the state of thematerial at the time t as | ψ ( t ) (cid:105) ≡ a ( t ) | (cid:105) + a ( t ) | (cid:105) , (5.7)and consider its time evolution described by i ∂∂t | ψ ( t ) (cid:105) = ( H + H int ) | ψ ( t ) (cid:105) , (5.8)where H and H int are given in Eqs. (2.12) and (5.5), respectively. We treat H int as aperturbation and evaluate the time evolution perturbatively. Expressing the time derivativewith a dot, the evolution of coefficients a ( t ) and a ( t ) is described as i ˙ a = C ∗ a cos( m a t + δ ) a , (5.9) i ˙ a = m m a + C a cos( m a t + δ ) a , (5.10)where the magnon mass is defined as m m ≡ ω (cid:126) + ω L . By solving these equations, we obtain a ( t ) (cid:39) − C a e iδ ( m a − m m )( e im a t − e − im m t ) − e − iδ ( m a + m m )( e − im a t − e − im m t ) m a − m m . (5.11)The probability that we find a one-magnon state | (cid:105) at the time t is given by P ( t ) ≡ | a ( t ) | . P ( t ) is highly enhanced when m m (cid:39) m a , with which we obtain P ( t ) (cid:39) | C a | t . (5.12)For the β -mode, we can repeat the discussion by defining | (cid:105) ≡ β † (cid:126) | (cid:105) , and all the calculationsare the same but replacements C a → C ∗ a and ω L → − ω L . P ( t ) can not become infinitely large because there is an upper limit on t for severalreasons; one of them is the ALP coherence time τ a ∼ /m a v a and another is the magnondissipation time τ m . Neglecting other possible sources of limitation for simplicity, we define Precisely speaking, the α - and β − modes are not mass eigenstates since they mix with a photon, formingthe so-called axionic polariton [35]. However, since the mixing is expected to be small for a small momentum,we neglect it in our analysis. The occupation number can be larger than 1. In the present case, however, the expectation value of theoccupation number is much smaller than 1, and the states with higher occupation numbers are irrelevant. τ ≡ min( τ a , τ m ). Then, the average magnon excitation rate isevaluated as dN signal dt = P ( τ ) τ = | C a | τ . (5.13)Numerically, the signal rate is evaluated as dN signal dt ∼ .
002 s − (cid:18) B (cid:19) ( u (cid:126) − v (cid:126) ) (cid:18) V unit (0 . − (cid:19) (cid:18) V (10 cm) (cid:19) × | D | (cid:15) (cid:16) g aγγ − GeV − (cid:17) (cid:18) − eV m a (cid:19) (cid:18) τ . µ s (cid:19) , (5.14)where V /N = V unit with V unit being the volume of the magnetic unit cell. Note that, fromEq. (2.11), a straightforward calculation shows( u (cid:126) − v (cid:126) ) = (cid:114) ω J + ω A ω A , (5.15)and hence the signal rate is enhanced if ω J (cid:29) ω A .In Fig. 1, we show the sensitivity on the ALP parameter space taking ( u (cid:126) − v (cid:126) ) = 1and 10, V unit = (0 . − , and | D | = (cid:15) = 1 as the material properties and postulating V = (10 cm) . We also assume τ a < τ m and use τ = 1 m a v a ∼ . µ s (cid:18) − eV m a (cid:19) . (5.16)As for the magnon dispersion relation, we use typical values m m = 1 . ± . (cid:18) B (cid:19) meV , (5.17)where the plus (minus) sign is selected for the α - ( β -)mode. The magnetic field is assumedto be scanned within the range 1 T < B <
10 T. The β -mode is used for our analysisonly when B < m a ∼ /τ ∼ − eVand we use ∆ t scan ∼ s for an observation. Accordingly, in order to cover all the accessibleALP mass, it takes ∼ dN noise /dt ∼ − s − as is adopted in [33], which is an already demonstrated value14 log ( m a /eV) -16-14-12-10-8-6 l o g ( | g a |/ G e V ) Helioscopes (CAST)LSW PVLASKSVZDFSZ u v = 1-mode-mode Figure 1: Sensitivity of the magnon to the ALP DM in the m a vs. g aγγ plane. The orange(green) region corresponds to the sensitivity of the β -mode ( α -mode) with u (cid:126) − v (cid:126) = 10,while the dot-dashed line in each region shows the sensitivity of the corresponding modewith u (cid:126) − v (cid:126) = 1. We postulate the target volume V = (10 cm) and the magnetic fieldscanned over 1 T < B < < B <
10 T) for the β -mode ( α -mode). For each stepof the scan, we use ∆ t = 10 s for an observation, which requires ∼ T = 0 .
05 K [52]. Weestimate the sensitivity by requiring the signal-to-noise ratio (SNR)(SNR) ≡ ( dN signal /dt ) ∆ t scan (cid:112) ( dN noise /dt ) ∆ t scan , (5.18)to be larger than 3 for each scan step.In the figure, the orange and green regions correspond to the sensitivity using β - and α -modes, respectively, with u (cid:126) − v (cid:126) = 10, while the dot-dashed line in each region showsthe sensitivity of the corresponding mode with u (cid:126) − v (cid:126) = 1. The other colored regionsshow existing constraints from the Light-Shining-through-Walls (LSW) experiments such asthe OSQAR [53] (yellow), the measurement of the vacuum magnetic birefringence at thePVLAS [54] (pink), and the observation of the ALP flux from the sun using the helioscopeCAST [55] (blue). We also show the predictions of the KSVZ and DFSZ axion models withblack solid and dashed lines, respectively. We can see that the use of both α - and β -modesgives a detectability over a broad mass range of 10 − –10 − eV and the sensitivity may reachboth the KSVZ and DFSZ model predictions for some mass range. It is also notable that15he sensitivity becomes much better for the lighter (heavier) mass region with the β -mode( α -mode), both of which correspond to larger B , due to the B dependence of the signalrate. We consider a hidden U (1) gauge field H µ , which has a kinetic mixing with the U (1) Y hypercharge gauge boson B µ . The relevant Lagrangian is L = − H µν H µν − B µν B µν + (cid:15) Y H µν B µν + 12 m H H µ H µ , (5.19)where m H is the hidden photon mass. Below, we use the convention that the expressionssuch as H µν and B µν denote the field strengths of the corresponding gauge fields H µ and B µ , respectively. After redefining fields as B (cid:48) µ ≡ B µ − (cid:15) Y H µ and H (cid:48) µ ≡ (cid:112) − (cid:15) Y H µ , we canrewrite the kinetic terms in the canonical form and obtain L = − H (cid:48) µν H (cid:48) µν − B (cid:48) µν B (cid:48) µν + 12 m H (cid:48) H (cid:48) µ H (cid:48) µ , (5.20)with m H (cid:48) ≡ m H / (cid:112) − (cid:15) Y . After the electroweak symmetry breaking, there appear addi-tional mass terms and further mixing occurs. The mass terms are given by L mass = m Z c W W µ − s W B µ ) + m H (cid:48) H (cid:48) µ H (cid:48) µ , (5.21)where m Z is the Z -boson mass, W µ is the third component of the SU (2) L gauge bosons,while c W ≡ cos θ W and s W ≡ sin θ W with θ W being the Weinberg angle. The mass terms areapproximately diagonalized by performing the unitary transformation W µ B (cid:48) µ H (cid:48) µ = c W − s W s W c W (cid:15) Y − s W c W s W (cid:15) Y − s W (cid:15) Y Z µ A µ H (cid:48)(cid:48) µ , (5.22)up to terms of O ( (cid:15) Y m (cid:48) H ) and O ( (cid:15) Y m Z ). The mass-squared eigenvalues are m Z , 0, and m (cid:48) H for Z µ , A µ , and H (cid:48)(cid:48) µ fields, respectively.According to the mixing among gauge bosons described above, the interaction between H (cid:48)(cid:48) µ and electrons is induced as L int = − (cid:15) H eH (cid:48)(cid:48) µ ¯ ψγ µ ψ, (5.23)where (cid:15) H ≡ (cid:15) Y c W and ψ is an electron field. Since the electromagnetic interaction of magnons(4.29) originates from the triangle diagram of Dirac electrons, a hidden photon field canreplace a photon field in the interaction at the cost of a factor (cid:15) H , leading to the magnon-hidden photon-photon interaction H int = − (cid:15) H α e π (cid:114) s N ( u (cid:126) − v (cid:126) ) (cid:104) D ∗ α † (cid:126) − Dβ † (cid:126) + h . c . (cid:105) (cid:90) d x (cid:126)E H · (cid:126)B, (5.24)16ith (cid:126)E H ≡ − (cid:126) ∇ H (cid:48)(cid:48) − ˙ (cid:126)H (cid:48)(cid:48) being the hidden electric field.From now on, let us resort to the abbreviation of H µ and m H for the mass eigenstateand eigenvalue of the hidden photon for notational simplicity. We consider the light hiddenphoton to explain the whole amount of the DM. Taking into account the equation ofmotion ( (cid:3) + m H ) H µ = 0 and ∂ µ H µ = 0, we can express each component of the hiddenphoton field as H ( t, (cid:126)x ) = − (cid:126)v H · (cid:126) ˜ H cos( m H t − m H (cid:126)v · (cid:126)x + δ ) , (5.25) (cid:126)H ( t, (cid:126)x ) = (cid:126) ˜ H cos( m H t − m H (cid:126)v · (cid:126)x + δ ) , (5.26)with ρ DM = m H ˜ H / H ≡ | (cid:126) ˜ H | . In this parametrization, the hidden electric field isexpressed as (cid:126)E H = (cid:126) ˜ Hm H sin( m H t + δ ) . (5.27)By repeating the same analysis as in the previous subsection, we can estimate the magnonexcitation rate from the existence of the hidden photon coherent oscillation. The rate is givenby dN signal /dt = | C H | τ / C H = − α e ˜ Hm H B V π cos θ (cid:114) s N ( u (cid:126) − v (cid:126) ) D, (5.28)and τ = min( τ H , τ m ) with τ H ∼ /m H v H . θ is defined as an angle between (cid:126) ˜ H and (cid:126)B .Numerically, we obtain the estimation dN signal dt ∼ .
02 s − (cid:18) B (cid:19) ( u (cid:126) − v (cid:126) ) (cid:18) V unit (0 . − (cid:19) (cid:18) V (10 cm) (cid:19) ×| D | (cid:16) (cid:15) H − (cid:17) (cid:18) cos θ / (cid:19) (cid:18) τ . µ s (cid:19) . (5.29)Note that the signal rate is proportional to a different power of the magnetic field and theDM mass compared with that for the ALP (5.14).In Fig. 2, we show the sensitivity in the hidden photon parameter space. The assumptionsfor the material properties are the same as those used in the previous subsection, while weassume τ = τ H ∼ /m H v H and cos θ = 1 / β - and α -modes, respectively. The gray region showsexisting constraints taken from [63], while the magenta region shows a sensitivity of theproposal with a polar material [61]. The purple and green lines correspond to the sensitivityof the Dirac material [62] with gap sizes ∆ = 2 . − –10 − eV of the hiddenphoton. The sensitivity has a smaller mass dependence compared with the result for theALP because of the smaller power of B in the expression of the signal rate. The correct relic abundance of hidden photon DM of meV mass range is reasonably explained by thegravitational production mechanism [56–59] or the production from cosmic strings [60]. log ( m H /eV) -19-17-15-13-11-9-7 l o g H Dirac,= 2.5 meVDirac, = 0 u v = 1-mode-modepolar material Figure 2: Sensitivity of the magnon to the DM hidden photon in the m H vs. (cid:15) plane. Thecolor and line style convention and the experimental set up are the same as those explainedin Fig. 1. The gray region is a combination of existing constraints, while the magenta regionshows a sensitivity of the polar material [61]. The purple and green lines correspond to thesensitivity of the Dirac material [62] with gap sizes ∆ = 2 . Motivated by recent developments in the axion electrodynamics in the context of condensedmatter physics, we considered a possibility of DM detection through DM conversion into thecondensed-matter (CM) axion. We formulated a way how the CM axion degree of freedomappears starting from the tight-binding model of the electrons on the lattice. In a particularexample, we have taken the model in [46], in which the CM axion may be interpreted as thespin wave or the (linear combination of) magnons in an anti-ferromagnetic insulator.
Forthe convenience of readers of particle physics side, we have reviewed the Heisenberg modeland half-filling Hubbard model in a self-consistent and comprehensive manner. Based onthese basic ingredients, we can derive the CM axion dispersion relation and its interactionwith electromagnetic fields.As DM models, we considered two cases: the elementary particle axion (or ALP) andthe hidden photon. We calculated the DM conversion rate into the CM axion in a quantummechanical way and estimated the signal rate. It is possible to cover the parameter regionswhich have not been explored so far in the DM mass range of about meV. It may be possibleto reach the QCD axion. One should note, however, that our calculation is just based on
In the original proposal of dynamical axion in Fe-doped topological insulators such as Bi Se [35], theCM axion is interpreted as an amplitude mode of the anti-ferromagnetic order parameter and not expressedby a linear combination of magnons.
18n idealized theoretical model of the electron system in the anti-ferromagnetic insulator. Itis nontrivial how well such a description is when it is applied to a real material. We havenot provided a concrete way to detect the CM axion excitation. One possible way is to usethe photon emission through the CM axion-photon mixing (axionic polariton) and detect itby the dish antenna as discussed in Ref. [33]. In any case, it is important to understand theorigin of CM axion and its properties, and we believe our formulation gives a basis of theestimation of the CM axion production rate from background DM and is useful for futuredevelopments of this field.A final comment is that the physics of CM axion is very rich and the CM axion in adifferent material may have a different microphysical origin [36, 37]. It would be interestingto explore the physics of CM axion as a probe of DM in a broader class of materials.
Note added
While finalizing this manuscript, a related paper appeared on arXiv [64].
Acknowledgments
This work was supported by JSPS KAKENHI Grant (Nos. 20J00046 [SC], 16H06490 [TM],18K03608 [TM], 18K03609 [KN] and 17H06359 [KN]). SC was supported by the Director,Office of Science, Office of High Energy Physics of the U.S. Department of Energy under theContract No. DE-AC02-05CH1123.
A Note on spin-orbit interaction term
In this appendix, we see how to derive the spin-orbit interaction term given in Eq. (4.1).We will first discuss how the hamiltonian is expressed in terms of creation and annihilationoperators of the electron. Next, we derive the effective hamiltonian of graphene as anexample, which becomes the same form as (4.1), and then show that the result is modelindependent.
A.1 Tight-binding model with spin-orbit interaction
We consider a model in which atoms are attached to lattice points labeled by i with positionvectors (cid:126)r i . Each atom has its energy eigenstates generated by c † µi , where µ denotes anelectron orbital. The diagonal part of the tight-binding hamiltonian, H TB , is given by thesum of the hamiltonian of each atom. On the other hand, a small overlap between electronwave functions sit at different lattice sites induces relatively small off-diagonal elements. Weare particularly interested in the case where electrons in each atom are tightly bound on a19 \ ν s p x p z d zx s V ssσ n x V spσ n z V spσ √ n x n z V sdσ p x ∗ n x V ppσ + (1 − n x ) V ppπ n x n z V ppσ − n x n z V ppπ √ n x n z V pdσ + n z (1 − n x ) V pdπ p z ∗ ∗ n z V ppσ + (1 − n z ) V ppπ n x V pdπ d zx ∗ ∗ ∗ n x V ddπ + n y V ddδ Table 1: Table of off-diagonal elements of T ijµν [65]. (cid:126)n ≡ (cid:126)r j − (cid:126)r i denotes the lattice displace-ment vector. We omitted the principal quantum numbers associated with µ and ν since adifferent choice only results in different numerical values of V -factors such as V ssσ . The leftbottom elements with ∗ markers can be obtained by the relationship T ijµν = T jiνµ = T ijνµ (cid:12)(cid:12) (cid:126)n →− (cid:126)n . µ \ ν s p x p y p z s p x − is z is y p y is z − is x p z − is y is x µ \ ν d xy d x − y d zx d yz d z d xy is z − is x is y d x − y − is z is y is x d zx is x − is y − is z i √ s y d yz − is y − is x is z − i √ s x d z − i √ s y i √ s x (cid:68) (cid:126)L · (cid:126)S (cid:69) µν . It is implicitly assumed that theprincipal quantum numbers of µ and ν are the same. The left (right) panel shows the resultsfor s and p ( d ) orbitals. Note that the spin operators are related to the Pauli matrices as s f = σ f / f = x, y, z ).lattice point. In this case, we can neglect the overlap between two sites unless they are thenearest neighbors of each other. Accordingly, we obtain H TB = (cid:88) µ (cid:88) i (cid:15) µ c † µi c µi + (cid:88) µ,ν (cid:88) (cid:104) i,j (cid:105) T ijµν c † µi c νj , (A.1)where (cid:15) µ denotes the energy level of the electron orbital µ of a single atom. The off-diagonal elements T ijµν are calculated by Slater and Koster [65] as summarized in Table 1 forseveral important choices of electron orbitals. One of the important features of these resultsis the directional dependence (i.e., the existence of (cid:126)n ≡ (cid:126)r j − (cid:126)r i in the expressions), which issourced from the directional dependence of orbitals. Information of the shape of the latticecomes into the Hamiltonian due to this dependence.Next, we take into account the effects of the spin-orbit interaction. Due to the relativisticmotion of an electron inside an atom, it feels a magnetic field whose size and direction areproportional to its angular momentum (cid:126)L . As a result, we obtain the on-site spin-orbit In general, the energy level may change against the choice of the atom. However, we only focus on thecase where it is universal for all the atoms in this paper. H SO = 1 m e r dV ( r ) dr (cid:126)L · (cid:126)S, (A.2)where V ( r ) is the centrifugal potential in which the electron moves, while (cid:126)L and (cid:126)S are theelectron angular momentum and spin operators, respectively. Given that the operator (cid:126)L · (cid:126)S does not change the principal and azimuthal quantum numbers, this interaction induces theterm H SO = (cid:88) µ,ν (cid:88) i ξ n(cid:96) c † µi (cid:68) (cid:126)L · (cid:126)S (cid:69) µν c νi , (A.3)where n and (cid:96) are the common principal and azimuthal quantum numbers of µ and ν ,respectively, while ξ n(cid:96) denotes the radial average of the coefficient in Eq. (A.2). Some of thematrix elements (cid:68) (cid:126)L · (cid:126)S (cid:69) µν are shown in Table 2 as examples. A.2 Graphene
Graphene is made of carbon atoms that are located on the two-dimensional honeycomblattice on the xy plane. Three out of four electrons of the outermost shell of each carbon in2 s , 2 p x , and 2 p y orbitals are shared among the nearest neighbor carbons to form the so-called σ bond. On the other hand, the other electron in the 2 p z orbital is also shared and calledthe π bond. The unit cell consists of two lattice sites, which we call A and B sublattices.Since we are particularly interested in the dynamics of electrons in p z orbitals of A and B sublattices, we construct an effective theory of electron states in p z orbitals by integratingout all the other states.Among the full hamiltonian H ≡ H TB + H SO , we treat the off-diagonal elements, i.e., thesecond term of Eq. (A.1) and H SO , as perturbations and name the corresponding part of H as V . Also, we call an effective theory hamiltonian H eff and its off-diagonal part V eff , bothof which are constructed only from c p z ,i and c † p z ,i . Then, the matching condition of the fulltheory to the effective theory is given by (cid:10) p z , i (cid:12)(cid:12) U † ( t, t ) (cid:12)(cid:12) p z , j (cid:11) = (cid:68) p z , i (cid:12)(cid:12)(cid:12) U † eff ( t, t ) (cid:12)(cid:12)(cid:12) p z , j (cid:69) , (A.4)where | µ, i (cid:105) ≡ c † µi | (cid:105) with | (cid:105) being the vacuum state, while U and U eff are the time evolutionoperators in the full and effective theories, respectively. Working in the interaction picture,they are given by U ( t, t ) = T (cid:26) exp (cid:20) i (cid:90) tt dt (cid:48) V I ( t (cid:48) ) (cid:21)(cid:27) , (A.5)with T being the time-ordering operator, and V I ( t ) ≡ e iH t V e − iH t , (A.6)21hile U eff can be obtained by substituting V with V eff .The left-handed side of Eq. (A.4) does not have a contribution from H SO at the first orderof V I since (cid:126)L · (cid:126)S does not have a non-zero matrix element. Also, there are contributions onlywith even numbers of (cid:126)L · (cid:126)S at the second order of perturbation. Such contributions justslightly modify (cid:15) µ and T ijµν and do not qualitatively change the physics, so we just neglect it.The third order contribution can be rewritten as (cid:90) tt dt (cid:48) (cid:90) tt (cid:48) dt (cid:48)(cid:48) (cid:90) tt (cid:48)(cid:48) dt (cid:48)(cid:48)(cid:48) (cid:88) µ,ν,k,p (cid:104) p z , i | iV I ( t (cid:48) ) | µ, k (cid:105) (cid:104) µ, k | iV I ( t (cid:48)(cid:48) ) | ν, p (cid:105) (cid:104) ν, p | iV I ( t (cid:48)(cid:48)(cid:48) ) | p z , j (cid:105) . (A.7)According to [66], it is known that the contributions from the spin-orbit interaction among3 d orbitals are numerically large in this model, so we may focus only on them. As a result,we deform (A.7) to obtain (cid:10) p z , i (cid:12)(cid:12) U † ( t, t ) (cid:12)(cid:12) p z , j (cid:11)(cid:12)(cid:12) V I (cid:39) − ( t − t ) ξ d V pdπ ( (cid:15) d − (cid:15) p ) (cid:126)s · ( (cid:126)d ij × (cid:126)d ij ) , (A.8)where (cid:126)d ij ≡ (cid:126)r k − (cid:126)r i and (cid:126)d ij ≡ (cid:126)r j − (cid:126)r k . The factor (cid:126)d ij × (cid:126)d ij forces the matrix element tobe zero when i = j and the only non-zero matrix elements are those with ( i, j ) being apair of next-nearest neighbors. Therefore, the subscript k in the definition of (cid:126)d , should beunderstood as the lattice site in between i and j . The corresponding matrix element in theright-handed side of Eq. (A.4) is given by (cid:68) p z , i (cid:12)(cid:12)(cid:12) U † eff ( t, t ) (cid:12)(cid:12)(cid:12) p z , j (cid:69) (cid:39) i ( t − t ) (cid:104) p z , i | V eff | p z , j (cid:105) , (A.9)so we conclude V eff (cid:51) i ξ d V pdπ ( (cid:15) d − (cid:15) p ) (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) c † p z ,i (cid:126)s · ( (cid:126)d ij × (cid:126)d ij ) c p z ,j . (A.10)This agrees with (4.1) when we set λ = a ξ d V pdπ / (cid:15) d − (cid:15) p ) . A.3 Model independence of the spin-orbit interaction term
So far, we have considered the spin-orbit interaction for a specific choice of the lattice struc-ture, i.e., the two-dimensional honeycomb lattice. Here, we argue that the structure of theinteraction, given in Eq. (4.1), can be understood by symmetries.Here, we consider the interaction between next-nearest neighbor sites induced by thespin-orbit interaction H SO . For this purpose, we consider a set of next-nearest neighbor sitesfrom the A -sublattice (called A ˆ i and A ˆ j ), which share only one nearest neighbor site (called B O ). The vectors pointing to A ˆ i and A ˆ j from B O are denoted as (cid:126)d i ˆ j and (cid:126)d i ˆ j , respectively.22 O xy AA i ^ j ^ d i ^ j ^ d i ^ j ^ Figure 3: The coordinate adopted in deriving the general form of the spin-orbit interaction.Here, we adopt a coordinate in which A ˆ i , A ˆ j , and B O are on the x vs. y plane; the positionof B O is set to be the origin and the y axis is chosen to be parallel to (cid:126)d i ˆ j − (cid:126)d i ˆ j (see Fig. 3).Hereafter, we assume that the whole lattice is invariant under the following transforma-tions and hence the Hamiltonian also is: • P : Parity, defined as the reflection with respect to the x vs. y plane: ( x, y, z ) P −→ ( x, y, − z ). With the P transformation, the angular momentum operator acting on theelectron on i -th site transforms as ( L ( i ) x , L ( i ) y , L ( i ) z ) P −→ ( − L ( P [ i ]) x , − L ( P [ i ]) y , L ( P [ i ]) z ), where i P −→ P [ i ]. (Thus, P [ˆ i ] = ˆ i .) In addition, the annihilation operator of the electrontransforms as c µ,i P −→ σ c P [ µ ] , P [ i ] , (A.11)where P [ µ ] denotes the P -transformed orbital of µ . (If µ is singlet under the P -transformation, P [ µ ] = µ .) • R : π rotation around the x axis: ( x, y, z ) R −→ ( x, − y, − z ). With this transformation,the lattice site i is moved to the position of R [ i ]. With R , the angular momentumoperator transforms as ( L ( i ) x , L ( i ) y , L ( i ) z ) R −→ ( L R [ j ] x , − L R [ j ] y , − L R [ j ] z ). In addition, c µ,i R −→ σ c R [ µ ] , R [ i ] , (A.12)where R [ µ ] denotes the R -transformed orbital of µ .For example, the diamond lattice used for the Fu-Kane-Mele-Hubbard model and the two-dimensional honeycomb lattice considered in the previous subsection are unchanged underthe P and R transformations. Then, one can find that the Hubbard model Hamiltonian given23n Eq. (3.5), tight-binding Hamiltonian given in Eq. (A.1), and the spin-orbit interactiongiven in Eq. (A.3) are invariant under the P and R transformations.Starting with the model that is invariant under the P and R transformations, the effectivetheory for the electrons in the orbitals of our interest should also respect these symmetries.In the effective theory, the interaction of the next-nearest neighbor sites can be expressed as H NNN = (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) (cid:16) (cid:96) a,ij c † i σ a c j + t ij c † i c j (cid:17) , (A.13)where (cid:104)(cid:104) i, j (cid:105)(cid:105) is a set of the next-nearest neighbor sites. (Here, we consider the effectivetheory containing only the electrons in the unique orbital of our interest, and the index forthe electron orbital is omitted for the notational simplicity.)Now, we discuss the properties of the coefficient (cid:96) a,ij and show that, with P and R symmetries, (cid:126)(cid:96) ij is proportional to (cid:126)d ij × (cid:126)d ij . To see this, we can use the following relations: (cid:96) a, ˆ i ˆ j c † ˆ i σ a c ˆ j P −→ − (cid:96) , ˆ i ˆ j c † ˆ i σ c ˆ j − (cid:96) , ˆ i ˆ j c † ˆ i σ c ˆ j + (cid:96) , ˆ i ˆ j c † ˆ i σ c ˆ j , (A.14) (cid:96) a, ˆ i ˆ j c † ˆ i σ a c ˆ j R −→ (cid:96) , ˆ i ˆ j c † ˆ j σ c ˆ i − (cid:96) , ˆ i ˆ j c † ˆ j σ c ˆ i − (cid:96) , ˆ i ˆ j c † ˆ j σ c ˆ i . (A.15)Eq. (A.14) results in (cid:96) ,ij = (cid:96) ,ij = 0 while Eq. (A.15) implies (cid:96) ,ji = − (cid:96) ,ij , and hence wecan find that (cid:126)(cid:96) ij ∝ (cid:126)d ij × (cid:126)d ij . B Transformation of α matrix The chiral representation of α matrices are defined as α i = (cid:18) σ i − σ i (cid:19) , α = (cid:18) − − (cid:19) , α = (cid:18) − ii (cid:19) , (B.1)where α = α α α α . They satisfy the anti-commutation relation { α µ , α ν } = 2 δ µν . Underthe unitary transformation α µ → (cid:101) α µ = U † α µ U , the anti-commutation relation remainsintact. For some choice of U , the α matrices are exchanged. Examples are summarized inTable. 3, where U = 1 √ − i − i − i − i , U = 1 √ −
10 1 1 00 − , U = 1 √ − i
00 1 0 i − i i . (B.2)Note that they have the form of U i = 1 √ (cid:18) − iσ i − iσ i (cid:19) . (B.3)24 α (cid:101) α (cid:101) α (cid:101) α (cid:101) α U α α α α − α U α α α α − α U α α α α − α Table 3: Transformation law of α -matrices under the unitary transformation by U , U and U .for i = 1 , ,
3. One can easily show that they yield U † i α j U i = (cid:40) α j for i (cid:54) = j − α for i = j . (B.4)The Dirac representation for the α matrices is given by α i = (cid:18) σ i σ i (cid:19) , α = (cid:18) − (cid:19) , α = (cid:18) − ii (cid:19) , (B.5)The chiral and Dirac representations are related by the unitary transformation as α (Dirac) µ = U † α (chiral) µ U, U = 1 √ (cid:18) − (cid:19) . (B.6) C Berry connection and topological term
C.1 Dimensional reduction of (4 + 1) -dimensional quantum Hallinsulator
In Sec. 4, we derived θ using the Lagrangian formulation following Ref. [46]. On the otherhand, θ can also be expressed in terms of the Berry connection [67, 68].It is well known that the general (2 + 1)-dimensional quantum Hall insulator is charac-terized by the first Chern number N (1)ch in terms of the integration of the Berry connectionover the Brillouin zone [69]. Its electromagnetic response is described by the action S = N (1)ch π (cid:90) dtd x (cid:15) µνρ A µ ∂ ν A ρ . (C.1)Similarly, the (4 + 1)-dimensional quantum Hall insulator is characterized by the secondChern number N (2)ch and described by the action S = N (2)ch π (cid:90) dtd x (cid:15) µνρστ A µ ∂ ν A ρ ∂ σ A τ , (C.2)25here N (2)ch = 132 π (cid:90) BZ d k (cid:15) ijkl Tr [ F ij F kl ] , (C.3)with F ij ≡ ∂ i A j − ∂ j A i + i [ A i , A j ] . (C.4)Here we used a shorthand notation like ∂ i ≡ ∂/∂k i and so on ( k may be rather understoodas ϕ ≡ k + A ) and A i denotes the Berry connection matrix in the momentum space givenby A αβi = − i (cid:104) u αk | ∂∂k i | u βk (cid:105) . (C.5)with | u αk (cid:105) being the Bloch state with α representing the band index, and the trace in Eq. (C.4)is taken over the occupied bands. Note that N (2)ch is expressed as N (2)ch = 12 π (cid:90) ∂θ∂ϕ dϕ, (C.6)where θ ≡ π (cid:90) BZ d k (cid:15) ijk Tr (cid:20) A i ∂ j A k + i A i A j A k (cid:21) . (C.7)Now let us perform a dimensional reduction. The action (C.2) is written as S = 18 π (cid:90) dtd x (cid:15) µνρσ ∂θ∂ϕ ∂ µ ϕA ν ∂ ρ A σ = − π (cid:90) dtd x θ (cid:15) µνρσ ∂ µ A ν ∂ ρ A σ , (C.8)where we used ∂ µ θ = ( ∂θ/∂ϕ ) ∂ µ ϕ . This is an action that describes the electromagneticresponse of (3 + 1)-dimensional topological insulator. C.2 Hamiltonian expression of θ Let us assume the four-band model whose (momentum space) Hamiltonian is given by H = c † k,α H αβ c k,β , H = (cid:88) µ =1 R µ ( (cid:126)k ) α µ , (C.9)where c † k,α and c k,α with α = 1–4 denote the electron creation and annihilation operator withthe wavenumber k and R µ are real coefficients. Here we take the Dirac representation forthe α matrices (B.5). The Hamiltonian (C.9) is diagonalized by the unitary matrix U : U = N + ( − R + iR ) N + ( − R + iR ) N − ( R − iR ) N − ( R − iR ) N + ( R + iR ) N + ( − R − iR ) N − ( − R − iR ) N − ( R + iR )0 N + ( R + R ) 0 N − ( R − R ) N + ( R + R ) 0 N − ( R − R ) 0 , (C.10)26here N ± ≡ / (cid:112) R ( R ± R ) and R ≡ (cid:113)(cid:80) µ =1 − ( R µ ) . One finds U † H U = diag( − R, − R, R, R ) . (C.11)The lower two energy bands and upper two bands are degenerate and we assume that thelower bands are occupied and upper bands are empty. One can define the creation/annihilationoperator in the diagonal basis through d k,α ≡ U † αβ c k,β , d † k,α ≡ c † k,β U βα . (C.12)The Bloch state may be given by | u αk (cid:105) = d † k,α | (cid:105) = c † k,β U βα | (cid:105) . Thus the Berry connection iscalculated as A αβi = − i (cid:104) | U † αγ c k,γ ∂∂k i ( c † k,δ U δβ ) | (cid:105) = − iU † αγ ∂U γβ ∂k i . (C.13)Note that A αβi is a 2 × A i = (cid:88) a =1 A ia σ a , (C.14)where A i = − N [( R ∂ i R − R ∂ i R ) + ( R ∂ i R − R ∂ i R )] , (C.15) A i = − N [( R ∂ i R − R ∂ i R ) + ( R ∂ i R − R ∂ i R )] , (C.16) A i = − N [( R ∂ i R − R ∂ i R ) + ( R ∂ i R − R ∂ i R )] . (C.17)Note that the term proportional to the unit matrix is canceled.Using the trace formula Tr [ σ a σ b ] = 2 δ ab and Tr [ σ a σ b σ c ] = 2 i(cid:15) abc , the first and secondterms of θ in (C.7) are calculated as (cid:15) ijk Tr [ A i ∂ j A k ] = − R ( R + R ) (cid:15) µνρσ R µ ( ∂ x R ν )( ∂ y R ρ )( ∂ z R σ ) , (C.18) (cid:15) ijk Tr (cid:20) i A i A j A k (cid:21) = R − R R ( R + R ) (cid:15) µνρσ R µ ( ∂ x R ν )( ∂ y R ρ )( ∂ z R σ ) , (C.19)where µ, ν, ρ, σ = 1 , , ,
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