Axion/Hidden-Photon Dark Matter Conversion into Condensed Matter Axion

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 condensed matter axions. We discuss the relationship between the condensed matter axion and a collective 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 or hidden photon, into the condensed matter axion is estimated for the discovery of the dark matter signals.


Introduction
The QCD axion is a hypothetical elementary particle that solves the strong CP problem [1][2][3] and is a candidate of dark matter (DM) of the universe [4][5][6] (see Refs. [7][8][9] for reviews). Recently people often consider axion-like particles (ALPs) in a broad sense, partly motivated by the developments in string theory [10][11][12]. ALPs do not necessarily address the strong CP problem, but they are also good DM candidates and may be experimentally probed through, e.g., the axion-photon coupling of the form L ∝ a E · B where a denotes the ALP field and E ( B) denotes the electric (magnetic) field respectively. There are many experimental ideas to search for ALPs including the QCD axion, #1 although still it is not discovered yet .
On the other hand, the axion-like excitation also appears in the condensed matter physics [34,35] (see Refs. [36,37] for reviews). To distinguish it from the elementary particle 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 electromagnetic field as L ∝ δθ E · B, similar to the ALP. We call such an insulator an axionic insulator.
Let us briefly mention a relation between the topological insulator and axionic insulator. In general, topological electromagnetic responses of a three-dimensional insulator are described by the topological term in the Lagrangian: (1.1) For example, it implies that there appears a magnetization (electric polarization) proportional to the applied electric (magnetic) field: M ∝ θ E ( P ∝ θ B). If the Hamiltonian of the system is invariant under the time-reversal symmetry, the coefficient θ can only take a value either 0 or 1/2: i.e., such an insulator is classified by a discrete Z 2 index [38][39][40][41][42][43]. #2 The case of θ = 1/2 corresponds to the topological insulator, in which the existence of gapless surface states is ensured and it causes topological electromagnetic effects. On the other hand, if there is no time-reversal symmetry, θ does not have to be quantized but can take arbitrary values possibly with a space-time dependence: θ = θ( x, t). If θ is a dynamical field, it is called the CM axion. Although it is often helpful to start from the topological insulator for understanding the origin of CM axion, the existence of CM axion does not necessarily require that the insulator is topological. One can generally write θ( x, t) = θ 0 + δθ( x, t) so that δθ( x, t) expresses the CM axion while θ 0 is the background value. The value of θ 0 depends on the properties of the material and can be zero. It has been known that in a class of magnetically doped topological insulators, the fluctuation of the anti-ferromagnetic order parameter (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 and estimate the conversion rate. Such a process has been considered in Ref. [33] for the detection #1 In the following we use the terminology "ALP" for general elementary axion-like particles including the QCD axion.
#2 Time-reversal invariant topological insulators have been first considered in two-dimensional systems [44,45]. of axion-like DM. One of the main purposes of this paper is to discuss the origin of CM axion in a comprehensive and self-consistent manner for particle physicists. We will explicitly show the relationship between the CM axion and the spin-wave fluctuation (magnon) based on a model presented in Ref. [46]. Another purpose is to provide a useful method to calculate the DM conversion rate into the CM axion in a quantum mechanical way. As an illustration, we will consider the case of ALP DM and hidden-photon DM.
This paper is organized as follows. In Sec. 2 we review the (anti-ferromagnetic) Heisenberg model of the localized electron spin system on the lattice. It gives a basis of the collective spin-wave excitation (magnon) and its dispersion relation, which will turn out to be identified with the CM axion in a certain setup. In Sec. 3 the so-called (half-filling) Hubbard model is briefly introduced. Electrons in solids are often modeled by a tight-binding Hamiltonian 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 lattice may describe both the electron energy band structure as well as the anti-ferromagnetic order and magnon excitation around it. In Sec. 4 we introduce the Fu-Kane-Mele-Hubbard model as a concrete setup and show that it contains an excitation that is regarded as the CM axion along the line of Ref. [46]. It will become clear that the CM axion is described by the use of anti-ferromagnetic magnon and its dispersion can be estimated as explained in Sec. 2. In Sec. 5 we estimate the conversion rate of light bosonic DM into the CM axion. We consider two DM models: ALP and hidden photon. We conclude in Sec. 6.

Magnon in anti-ferromagnet
Let us start with the Heisenberg anti-ferromagnet model [47][48][49]. #3 Suppose a bipartite lattice consisting of sublattices A and B, and on each lattice point ∈ A or ∈ B there is an electron spin S. Applying an external magnetic field B 0 along the z direction, the model Hamiltonian is given by where J < 0 is the exchange interaction, g = 2 and µ B = e/(2m e ) is the Bohr magneton, and B A is the anisotropy field. The collective excitation of the spin-wave around the ground state, called magnon, is analyzed through the Holstein-Primakoff transformation, As explained in Sec. 3, the Heisenberg anti-ferromagnet model may be understood from the Hubbard model in the limit of strong electron self-interaction at each site.
where we have defined S ± = S x ± iS y and S ± = S x ± iS y , and the creation-annihilation operators satisfy the commutation relation In addition, s is the spin quantum number; the eigenvalue of S · S is given by s(s + 1). The Hamiltonian is rewritten in terms of the creation-annihilation operators as 5) where N is the total number of sites in a sublattice, z denotes the number of adjacent lattice points (e.g. z = 6 for simple bipartite cubic lattice), and Now let us move to the Fourier space. We define the Fourier component as Substituting this into the Hamiltonian, we find where ω J ≡ −2zsJ and with δ being the vector connecting the adjacent lattice points. Finally, it is diagonalized through the Bogoliubov transformation: One can check that the canonical commutation relation is maintained if |u k | 2 − |v k | 2 = 1. The concrete expression is given by with arg(γ k ) = 2 arg(u k ) = −2 arg(−v k ). (Thus, u k v k is real and negative.) Note that, when ω A ω J , we have large Bogoliubov coefficients |u k | 2 ∼ |v k | 2 1 for | k · δ| 1. Then, one finds the diagonal Hamiltonian: Here, ω k represents the magnon dispersion relation (besides the overall offset coming from the Larmor frequency ω L ), In the low frequency limit | k · δ| 1, we obtain γ k 1 + i δ ( k · δ)/z − δ ( k · δ) 2 /z. It implies the linear dispersion relation, ω k ∝ | k| for large | k| (but still it satisfies | k · δ| 1), in contrast to the ferromagnetic magnon dispersion relation, which would show ω k ∝ k 2 . They are related to the so-called type-I and type-II Nambu-Goldstone boson dispersion relation as generally classified in Refs. [50,51].
3 Hubbard model as origin of anti-ferromagnet

Tight-binding model
A tight-binding model is one of the approaches to estimate the electron energy band structure in solids. In this approach, one starts with the picture that each electron is rather tightly bounded by each atom and then takes into account the overlap between the nearest electron wave function.
Let us consider only one electron orbital at each site and neglect the interaction among different orbits, spin-orbit coupling, electron self-interaction, etc. #4 In the second quantization picture, the tight-binding Hamiltonian is given by where c † iσ and c iσ denote the electron creation and annihilation operators at the site i with spin σ (↑ or ↓) and the summation is taken over the combination of adjacent sites i, j . The creation and annihilation operators satisfy the anti-commutation relation #4 Effects of the interaction among different orbitals and spin-orbit coupling are important for the topological insulator. The electron self-interaction will be taken into account in the next subsection.

The Fourier transformation is defined by
The Hamiltonian is rewritten in a diagonal form as This k denotes the electron energy band. In a simple cubic lattice, for example, we obtain 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, the energy band is filled and this becomes an insulator as far as there is an energy gap to the next energy band. If there is only one electron at each orbital, the energy band is not filled and it becomes a metal.

Half-filling Hubbard model
Let us add the effect of interaction between electrons at the same site i to the tight-binding Hamiltonian. The resulting Hamiltonian is called the Hubbard model: where U > 0 represents the interaction energy and n i↑ = c † i↑ c i↑ and n i↓ = c † i↓ c i↓ . The Hubbard model is characterized by several parameters: the relative interaction strength U/t and the number of electrons per site, N e /N s . The case of N e /N s = 1 is called the half-filling (it is "half" because of the spin degree of freedom) and its properties are well understood. Below, we consider the half-filling case. Naively, one may consider that the 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 an insulator. Such an insulator is called the Mott insulator. Now we consider the large interaction limit: U/t 1. In this limit, the tight-binding part is regarded as a perturbation. In the ground state, one electron is localized at each site to minimize the Hubbard interaction energy (hence it is expected that it behaves as an insulator rather than metal). Thus, the ground state is expressed as where σ schematically represents the array of spin, e.g., σ = (. . . , ↑, ↑, ↓, . . . ) and so on. There are 2 Ne degenerate ground states corresponding to the spin degree of freedom at each site. We want to consider an effective Hamiltonian regarding H t as a perturbation. Noting σ| H t | σ = 0, the nontrivial effect appears at the second-order in H t . The effective Hamiltonian is given by where P denotes the projection operator to the Hilbert space spanned by the ground state (3.6). The physical meaning is that, for σ = σ , 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 where we have defined Since the coefficient t 2 /U is positive, it represents the Heisenberg anti-ferromagnet model with J = −t 2 /U . Thus, the half-filling Hubbard model may describe both the metal phase in the limit U → 0 and the anti-ferromagnetic insulator phase in the large U limit.

Energy band in Fu-Kane-Mele-Hubbard model
A three-dimensional topological insulator has been proposed in Refs. [39,40]. An example is the diamond lattice with a strong spin-orbit coupling. On the other hand, taking account of the Hubbard on-site interaction between electrons may lead to the anti-ferromagnetic phase, 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 is a topological anti-ferromagnetic phase depending on the interaction strength, in which the spin-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. We assume the half-filling case, i.e., there is only one electron at the electron orbitals of our interest at each site. The model Hamiltonian is given by H = H 0 + H U : where c i ≡ (c i↑ , c i↓ ) T . Here, d 1 ij and d 2 ij are the two vectors that connect two adjacent sites: , with a being the lattice constant and λ represents the strength of the spin-orbit coupling. Note that the diamond lattice consists of two sublattices (which we call A and B) both of which are face-centered cubic. i, j denotes a set of the next-nearest neighbor sites, and hence sites i and j belong to the same sublattice. (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 with a 1 = a 2 (0, 1, 1), a 2 = a 2 (1, 0, 1), a 3 = a 2 (1, 1, 0) and These α matrices are Hermite and satisfy the anti-commutation relation {α µ , α ν } = 2δ µν . Then, it is easy to show that the energy eigenvalues are given by (4.10) This gives the dispersion relation of the bulk electron. It is found that, at the so-called a (0, 0, 1), which are located at the boundary of the Brillouin zone, we obtain E ± = 0 in the limit of δt = 0. Thus, this material is regarded as a semimetal in this limit. For example, the dispersion relation around k = k X 1 is given by where we have taken k = k X 1 + q. Thus, nonzero δt gives the energy gap between two energy bands, which makes the material the bulk insulator (topological insulator, actually).

Axionic excitation in anti-ferromagnetic phase
It is expected that the inclusion of the Hubbard interaction H U may lead to the antiferromagnetic ordering. Actually, it is found that the anti-ferromagnetic phase appears for sizable U/t in the mean field approximation [46]. Under this approximation, the Hubbard interaction term can be rewritten as with O being the ensemble average of the operator O. We use the operator equations with S i being spin operators in the coordinate system used in the previous subsection, with which three Dirac points are defined. Note that, in the U → ∞ limit of a half-filling model, we can safely restrict ourselves to states with n i↑ + n i↓ = 1. Then, neglecting constant terms, the Hubbard interaction becomes with m r are defined through which characterizes the anti-ferromagnetic ordering. Under this background and assuming U | m| λ, the X r points (r = 1, 2, 3) are slightly shifted as a .

(4.18)
For example, the energy dispersion around the X 1 point is given by where we have taken k = k X 1 + q. It is seen that there is an additional gap due to the anti-ferromagnetic order.
The Hamiltonian around the X 1 point is expressed as where we have rescaled the momentum as tq x → q x /a, 2λq y → q y /a, 2λq z → q z /a. In deriving Eq. (4.20), we have performed an appropriate change of the basis of the α matrices through a unitary transformation, with which α 1 ↔ α 5 (see App. B). The Hamiltonian around the X 2 and X 3 points can also be reduced to the same form except for the last term, which becomes U m 2 α 5 and U m 3 α 5 , respectively. From this Hamiltonian, we can infer the effective action for the electron which mimics the action of the relativistic Dirac fermion as One can make a chiral rotation of the fermion to eliminate the γ 5 dependent term, ψ r → e iγ 5 θr/2 ψ r . Then, there appears a topological term: #5 where θ 0 is either 0 or 1/2 depending on the sign of δt. (See App. C for another derivation of θ.) Note that the background magnetization m can fluctuate: it is a spin-wave or magnon excitation, m( x). Then, θ( x) is not a constant but a dynamical field and it has an axionic coupling to the electromagnetic field. Therefore, in this model, the magnon effectively behaves as an axion-like field (CM axion).

Axionic excitation as magnons
To relate the axionic excitation (or the CM axion) θ to the conventional magnons defined in Sec. 2, we repeat the analysis in the previous subsection, taking into account the fluctuation of the background magnetization in terms of magnon operators. We focus only on the spatially homogeneous spin fluctuations and consider their interaction with electrons at around a Dirac point k ∼ k Xr . Then, the relevant part of the Hubbard interaction term is schematically expressed as where the Fourier transform of operators O i is defined as #5 Eq. (4.22) may not be applicable when U m r /δt 1 [46]. F L in Eq. (4.23) is determined by the magnetization, which may fluctuate around the average value. We again use the operator equations Eqs. (4.13)-(4.15) to rewriteF L in terms of spin operators S i . The relationship between S i and S i , which are defined in Sec. 2 and directly related to magnon operators, is given by Taking everything into consideration, the magnon-Dirac electron interaction term is, up to some constant and quadratic terms of magnons, expressed as with α rr ≡ −iα r α r . Coefficients are given by 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 R r is proportional to the r-th component of the order parameter ( S i,A − S i,B )/2, while that of R rr to the average magnetization ( S i,A + S i,B )/2. The R rr terms induce interactions between magnon and electron/hole. It may cause, for example, the decay of a magnon into an electron-hole pair when the gap is small. Because we are interested in the magnon interaction with electromagnetic fields, which is not induced by the R rr terms, we neglect them from now on. Repeating the same procedure as Sec. 4.2, we obtain the relationship between the axionic excitation and magnons. Finally, the electromagnetic interaction of magnons is described by being an O(1) factor, assuming only a moderate hierarchy between U and δt. Note that (u 0 −v 0 ) is real because γ 0 = 1. The interaction Hamiltonian shows that a linear combination of magnon states is excited by a non-zero value of E · B. #7

Dark matter conversion into condensed matter axion
Now we discuss the detection of the elementary-particle DM axion (or ALPs) and hidden photon through the interaction with CM axion. (To avoid confusion between the DM and CM axions, hereafter, the DM axion and ALPs are both called ALPs.)

ALP dark matter
The dynamics of the ALP DM a and the photon in a material is described by where and µ are the permittivity and permeability of the material. Hereafter, we treat the ALP field as a classical background . When the ALP explains the total amount of the dark matter ρ DM ∼ 0.3 GeV/cm 3 , we obtain m 2 a a 2 0 /2 = ρ DM . We consider applying a constant magnetic field B 0 = B 0ẑ to the system, whereẑ is a unit vector along the z-axis. This magnetic field, combined with the ALP background, generates an oscillating electric field with The target mass range of this set up is m a ∼ O(10 −3 ) eV, which has a de-Broglie length ∼ 1/m a | v a | ∼ O(10) cm. We assume that is larger than the material size and neglect the x dependence of the ALP background inside the material. Since E · B is uniform in this case, only the magnon zero-modes may be excited, which are considered in Sec. 4.3. Substituting the value of E · B generated by the ALP background, the interaction Hamiltonian is rewritten as with ω 0 being the magnon frequency at k = 0 and V unit = V /N the volume of the magnetic unit cell.
with V being the material volume. H int describes the generation of both α-and β-modes of the magnon. However, as we will see below, one of them is highly enhanced when the corresponding excitation energy matches with the ALP mass; in such a case, we may expect an observable signal rate at the laboratory. Accordingly, we will estimate a signal rate of the magnon excitation assuming that a single mode is selectively excited. #8 We start from the α-mode, while the discussion for the β-mode is parallel, as we will comment later. We define the ground and the one-magnon states of the material through α q |0 = β q |0 = 0 for any q and |1 ≡ α † 0 |0 , respectively. Also, we express the state of the material at the time t as #9 and consider its time evolution described by where H and H int are given in Eqs. (2.12) and (5.5), respectively. We treat H int as a perturbation and evaluate the time evolution perturbatively. Expressing the time derivative with a dot, the evolution of coefficients a 0 (t) and a 1 (t) is described as where the magnon mass is defined as m m ≡ ω 0 + ω L . By solving these equations, we obtain The probability that we find a one-magnon state |1 at the time t is given by P (t) ≡ |a 1 (t)| 2 . P (t) is highly enhanced when m m m a , with which we obtain For the β-mode, we can repeat the discussion by defining |1 ≡ β † 0 |0 , and all the calculations are 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 several reasons; one of them is the ALP coherence time τ a ∼ 1/m a v 2 a and another is the magnon dissipation time τ m . Neglecting other possible sources of limitation for simplicity, we define #8 Precisely speaking, the α-and β−modes are not mass eigenstates since they mix with a photon, forming the so-called axionic polariton [35]. However, since the mixing is expected to be small for a small momentum, we neglect it in our analysis.
#9 The occupation number can be larger than 1. In the present case, however, the expectation value of the occupation number is much smaller than 1, and the states with higher occupation numbers are irrelevant. the effective coherence time τ ≡ min(τ a , τ m ). Then, the average magnon excitation rate is evaluated as Numerically, the signal rate is evaluated as where V /N = V unit with V unit being the volume of the magnetic unit cell. Note that, from Eq. (2.11), a straightforward calculation shows and hence the signal rate is enhanced if ω J ω A . In Fig. 1, we show the sensitivity on the ALP parameter space taking (u 0 − v 0 ) = 1 and 10, V unit = (0.3 keV) −3 , and |D| 2 = = 1 as the material properties and postulating V = (10 cm) 3 . We also assume τ a < τ m and use where the plus (minus) sign is selected for the α-(β-)mode. The magnetic field is assumed to be scanned within the range 1 T < B 0 < 10 T. The β-mode is used for our analysis only when B 0 < 7 T to avoid the instability or the enhanced noise rate according to the low frequency. For each step of the scan, we can search for a mass range of ∆m a ∼ 2/τ ∼ 10 −8 eV and we use ∆t scan ∼ 10 2 s for an observation. Accordingly, in order to cover all the accessible ALP mass, it takes ∼ 1 year to scan the magnetic field. We do not discuss in detail the detection method of generated magnons in this paper; they might be observed through the conversion into photons at the boundary of the material as in [33], or might be detected using some specific features for axionic insulators, such as the dynamical chiral magnetic effect [36]. For the estimation of the sensitivity, we just assume the noise rate for the detection dN noise /dt ∼ 10 −3 s −1 as is adopted in [33], which is an already demonstrated value for a single photon detector in the THz regime at the temperature T = 0.05 K [52]. We estimate the sensitivity by requiring the signal-to-noise ratio (SNR) 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 0 − v 0 = 10, while the dot-dashed line in each region shows the sensitivity of the corresponding mode with u 0 − v 0 = 1. The other colored regions show existing constraints from the Light-Shining-through-Walls (LSW) experiments such as the OSQAR [53] (yellow), the measurement of the vacuum magnetic birefringence at the PVLAS [54] (pink), and the observation of the ALP flux from the sun using the helioscope CAST [55] (blue). We also show the predictions of the KSVZ and DFSZ axion models with black solid and dashed lines, respectively. We can see that the use of both α-and β-modes gives a detectability over a broad mass range of 10 −3 -10 −2 eV and the sensitivity may reach both the KSVZ and DFSZ model predictions for some mass range. It is also notable that the sensitivity becomes much better for the lighter (heavier) mass region with the β-mode (α-mode), both of which correspond to larger B 0 , due to the B 4 0 dependence of the signal rate.

Hidden photon dark matter
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 where m H is the hidden photon mass. Below, we use the convention that the expressions such 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 µ ≡ B µ − Y H µ and H µ ≡ 1 − 2 Y H µ , we can rewrite the kinetic terms in the canonical form and obtain After the electroweak symmetry breaking, there appear additional mass terms and further mixing occurs. The mass terms are given by where m Z is the Z-boson mass, W 3 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 are approximately diagonalized by performing the unitary transformation for Z µ , A µ , and H µ fields, respectively. According to the mixing among gauge bosons described above, the interaction between H µ and electrons is induced as where H ≡ 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 can replace a photon field in the interaction at the cost of a factor H , leading to the magnonhidden photon-photon interaction with E H ≡ − ∇H 0 −˙ H being the hidden electric field. From now on, let us resort to the abbreviation of H µ and m H for the mass eigenstate and eigenvalue of the hidden photon for notational simplicity. We consider the light hidden photon to explain the whole amount of the DM. #10 Taking into account the equation of motion ( + m 2 H )H µ = 0 and ∂ µ H µ = 0, we can express each component of the hidden photon field as In this parametrization, the hidden electric field is expressed as By repeating the same analysis as in the previous subsection, we can estimate the magnon excitation rate from the existence of the hidden photon coherent oscillation. The rate is given by θ is defined as an angle between H and B 0 . Numerically, we obtain the estimation Note that the signal rate is proportional to a different power of the magnetic field and the DM mass compared with that for the ALP (5.14). In Fig. 2, we show the sensitivity in the hidden photon parameter space. The assumptions for the material properties are the same as those used in the previous subsection, while we assume τ = τ H ∼ 1/m H v 2 H and cos 2 θ = 1/2 in this case. Again, the orange and green regions correspond to the sensitivity of β-and α-modes, respectively. The gray region shows existing constraints taken from [63], while the magenta region shows a sensitivity of the proposal with a polar material [61]. The purple and green lines correspond to the sensitivity of the Dirac material [62] with gap sizes ∆ = 2.5 meV and 0, respectively. We can see that the use of magnons gives a good sensitivity over a mass range 10 −3 -10 −2 eV of the hidden photon. The sensitivity has a smaller mass dependence compared with the result for the ALP because of the smaller power of B 0 in the expression of the signal rate. #10 The correct relic abundance of hidden photon DM of meV mass range is reasonably explained by the gravitational production mechanism [56][57][58][59] or the production from cosmic strings [60].   Fig. 1. The gray region is a combination of existing constraints, while the magenta region shows a sensitivity of the polar material [61]. The purple and green lines correspond to the sensitivity of the Dirac material [62] with gap sizes ∆ = 2.5 meV and 0, respectively.

Conclusions and discussion
Motivated by recent developments in the axion electrodynamics in the context of condensed matter physics, we considered a possibility of DM detection through DM conversion into the condensed-matter (CM) axion. We formulated a way how the CM axion degree of freedom appears starting from the tight-binding model of the electrons on the lattice. In a particular example, we have taken the model in [46], in which the CM axion may be interpreted as the spin wave or the (linear combination of) magnons in an anti-ferromagnetic insulator. #11 For the convenience of readers of particle physics side, we have reviewed the Heisenberg model and half-filling Hubbard model in a self-consistent and comprehensive manner. Based on these basic ingredients, we can derive the CM axion dispersion relation and its interaction with electromagnetic fields.
As DM models, we considered two cases: the elementary particle axion (or ALP) and the hidden photon. We calculated the DM conversion rate into the CM axion in a quantum mechanical way and estimated the signal rate. It is possible to cover the parameter regions which have not been explored so far in the DM mass range of about meV. It may be possible to reach the QCD axion. One should note, however, that our calculation is just based on an idealized theoretical model of the electron system in the anti-ferromagnetic insulator. It is nontrivial how well such a description is when it is applied to a real material. We have not provided a concrete way to detect the CM axion excitation. One possible way is to use the photon emission through the CM axion-photon mixing (axionic polariton) and detect it by the dish antenna as discussed in Ref. [33]. In any case, it is important to understand the origin of CM axion and its properties, and we believe our formulation gives a basis of the estimation of the CM axion production rate from background DM and is useful for future developments of this field.
A final comment is that the physics of CM axion is very rich and the CM axion in a different material may have a different microphysical origin [36,37]. It would be interesting to 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].  [65]. n ≡ r j − r i denotes the lattice displacement vector. We omitted the principal quantum numbers associated with µ and ν since a different choice only results in different numerical values of V -factors such as V ssσ . The left bottom elements with * markers can be obtained by the relationship T ij µν = T ji νµ = T ij νµ n→− n .
lattice point. In this case, we can neglect the overlap between two sites unless they are the nearest neighbors of each other. Accordingly, we obtain where µ denotes the energy level of the electron orbital µ of a single atom. #12 The offdiagonal elements T ij µν are calculated by Slater and Koster [65] as summarized in Table 1 for several important choices of electron orbitals. One of the important features of these results is the directional dependence (i.e., the existence of n ≡ r j − r i in the expressions), which is sourced from the directional dependence of orbitals. Information of the shape of the lattice comes into the Hamiltonian due to this dependence.
Next, we take into account the effects of the spin-orbit interaction. Due to the relativistic motion of an electron inside an atom, it feels a magnetic field whose size and direction are proportional to its angular momentum L. As a result, we obtain the on-site spin-orbit interaction Hamiltonian where V (r) is the centrifugal potential in which the electron moves, while L and S are the electron angular momentum and spin operators, respectively. Given that the operator L · S does not change the principal and azimuthal quantum numbers, this interaction induces the term where n and are the common principal and azimuthal quantum numbers of µ and ν, respectively, while ξ n denotes the radial average of the coefficient in Eq. (A.2). Some of the matrix elements L · S µν are shown in Table 2 as examples.

A.2 Graphene
Graphene is made of carbon atoms that are located on the two-dimensional honeycomb lattice on the xy plane. Three out of four electrons of the outermost shell of each carbon in 2s, 2p x , and 2p 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 2p z orbital is also shared and called the 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 integrating out all the other states. Among the full hamiltonian H ≡ H TB + H SO , we treat the off-diagonal elements, i.e., the second 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 , both of which are constructed only from c pz,i and c † pz,i . Then, the matching condition of the full theory to the effective theory is given by where |µ, i ≡ c † µi |0 with |0 being the vacuum state, while U and U eff are the time evolution operators in the full and effective theories, respectively. Working in the interaction picture, they are given by with T being the time-ordering operator, and while 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 order of V I since L · S does not have a non-zero matrix element. Also, there are contributions only with even numbers of L · S at the second order of perturbation. Such contributions just slightly modify µ and T ij µν and do not qualitatively change the physics, so we just neglect it. The third order contribution can be rewritten as According to [66], it is known that the contributions from the spin-orbit interaction among 3d orbitals are numerically large in this model, so we may focus only on them. As a result, we deform (A.7) to obtain forces the matrix element to be zero when i = j and the only non-zero matrix elements are those with (i, j) being a pair of next-nearest neighbors. Therefore, the subscript k in the definition of d 1,2 should be understood as the lattice site in between i and j. The corresponding matrix element in the right-handed side of Eq. (A.4) is given by This agrees with (4.1) when we set λ = a 2 ξ 3d V 2 pdπ /4( 3d − 2p ) 2 .

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 structure, i.e., the two-dimensional honeycomb lattice. Here, we argue that the structure of the interaction, given in Eq. (4.1), can be understood by symmetries.
Here, we consider the interaction between next-nearest neighbor sites induced by the spin-orbit interaction H SO . For this purpose, we consider a set of next-nearest neighbor sites from the A-sublattice (called Aˆi and Aĵ), which share only one nearest neighbor site (called Here, we adopt a coordinate in which Aˆi, Aĵ, and B O are on the x vs. y plane; the position of B O is set to be the origin and the y axis is chosen to be parallel to d 1 iĵ − d 2 iĵ (see Fig. 3). Hereafter, we assume that the whole lattice is invariant under the following transformations 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 the electron on i-th site transforms as (L ). In addition, where R[µ] denotes the R-transformed orbital of µ. Starting with the model that is invariant under the P and R transformations, the effective theory 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 where i, j is a set of the next-nearest neighbor sites. (Here, we consider the effective theory containing only the electrons in the unique orbital of our interest, and the index for the electron orbital is omitted for the notational simplicity.) Now, we discuss the properties of the coefficient a,ij and show that, with P and R symmetries, ij is proportional to d 1 ij × d 2 ij . To see this, we can use the following relations: Eq. (A.14) results in 1,ij = 2,ij = 0 while Eq. (A.15) implies 3,ji = − 3,ij , and hence we can find that ij ∝ d 1 ij × d 2 ij .

B Transformation of α matrix
The chiral representation of α matrices are defined as where α 5 = α 1 α 2 α 3 α 4 . They satisfy the anti-commutation relation {α µ , α ν } = 2δ µν . Under the unitary transformation α µ → α µ = U † α µ U , the anti-commutation relation remains intact. For some choice of U , the α matrices are exchanged. Examples are summarized in Table. 3, where Note that they have the form of Table 3: Transformation law of α-matrices under the unitary transformation by U 1 , U 2 and U 3 .
for i = 1, 2, 3. One can easily show that they yield The Dirac representation for the α matrices is given by The chiral and Dirac representations are related by the unitary transformation as C Berry connection and topological term C.1 Dimensional reduction of (4 + 1)-dimensional quantum Hall insulator In Sec. 4, we derived θ using the Lagrangian formulation following Ref. [46]. On the other hand, 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 characterized by the first Chern number N (1) ch in terms of the integration of the Berry connection over the Brillouin zone [69]. Its electromagnetic response is described by the action Similarly, the (4 + 1)-dimensional quantum Hall insulator is characterized by the second Chern number N (2) ch and described by the action S = N (2) ch Here we used a shorthand notation like ∂ i ≡ ∂/∂k i and so on (k 4 may be rather understood as ϕ ≡ k 4 + A 4 ) and A i denotes the Berry connection matrix in the momentum space given by with |u α k 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 where Now let us perform a dimensional reduction. The action (C.2) is written as where we used ∂ µ θ = (∂θ/∂ϕ)∂ µ ϕ. This is an action that describes the electromagnetic response 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 where c † k,α and c k,α with α = 1-4 denote the electron creation and annihilation operator with the wavenumber k and R µ are real coefficients. Here we take the Dirac representation for the α matrices (B.5). The Hamiltonian (C.9) is diagonalized by the unitary matrix U : where N ± ≡ 1/ 2R(R ± R 4 ) and R ≡ µ=1−5 (R µ ) 2 . One finds U † HU = diag(−R, −R, R, R). (C.11) The lower two energy bands and upper two bands are degenerate and we assume that the lower bands are occupied and upper bands are empty. One can define the creation/annihilation operator in the diagonal basis through The Bloch state may be given by |u α k = d † k,α |0 = c † k,β U βα |0 . Thus the Berry connection is calculated as Note that A αβ i is a 2×2 matrix since only the two low energy states are occupied. Substituting the concrete expression (C.10), we obtain (C.14) where Note that the term proportional to the unit matrix 1 is canceled. Using the trace formula Tr [σ a σ b ] = 2δ ab and Tr [σ a σ b σ c ] = 2i abc , the first and second terms of θ in (C.7) are calculated as where µ, ν, ρ, σ = 1, 2, 3, 5. Note that terms proportional ∂ j N + , ∂ j ∂ k R µ , ∂ j R 1 ∂ k R 1 and so on vanish when contracted by ijk . Thus we obtain the following expression for θ, This expression is consistent with [70]. #13 #13 Note that the definition of α 3 and α 5 are reversed between us and Ref. [70] and hence there appears an extra minus sign in the final expression (C.20).