PHYSICAL REVIEW D 102, 095005 (2020)

Detectability of axion dark matter with phonon polaritons and magnons

Andrea Mitridate,1 Tanner Trickle ,2,3,1 Zhengkang Zhang ,1 and Kathryn M. Zurek1 1Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA 2Department of Physics, University of California, Berkeley, California 94720, USA 3Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

(Received 23 June 2020; accepted 5 October 2020; published 6 November 2020)

Collective excitations in condensed matter systems, such as phonons and magnons, have recently been proposed as novel detection channels for light dark matter. We show that excitation of (i) optical phonon polaritons in polar materials in an Oð1 TÞ magnetic field (via the axion-photon coupling), and (ii) gapped magnons in magnetically ordered materials (via the axion wind coupling to the electron spin), can cover the difficult-to-reach Oð1–100Þ meV mass window of QCD axion dark matter with less than a kilogram-year exposure. Finding materials with a large number of optical phonon or magnon modes that can couple to the axion field is crucial, suggesting a program to search for a range of materials with different resonant energies and excitation selection rules; we outline the rules and discuss a few candidate targets, leaving a more exhaustive search for future work. Ongoing development of single photon, phonon, and magnon detectors will provide the key for experimentally realizing the ideas presented here.

DOI: 10.1103/PhysRevD.102.095005

I. INTRODUCTION The QCD axion mass window ma ∼ Oð1–100Þ meV 1 The QCD axion [1–4] remains one of the best-motivated remains, however, unconstrained. The current best limits and predictive models of dark matter (DM) [5–7]. The are provided by CAST, but this could be outperformed in search for the axion has a decades long history, and it is the future by fourth generation helioscopes like IAXO [45], still ongoing. At the moment, only the axion dark matter and dish antennas [46] or multilayer films [47] (both of experiment [8,9] has sensitivity to the QCD axion in a which are related to MADMAX in concept but can reach narrow mass range around 2–3 μeV. The HAYSTAC [10] higher axion masses). These are limited by current single and ORGAN [11] experiments are seeking to extend these photon detection technology, which is rapidly improving. results to higher frequencies. The ABRACADABRA [12] Recently the use of axionic topological antiferromagnets and CASPEr [13] experiments have also recently achieved has been proposed to detect axions in this region [48], their first limits for very light masses (though with although such materials have not been fabricated yet in sensitivity still far above that needed to reach the QCD the lab, and even then, this proposal is limited to ≲ 10 axion). The CERN axion solar telescope (CAST) [14] is ma meV. searching for axions emitted by the Sun, and can constrain Collective excitations, such as phonons and magnons, O 1–100 the QCD axion for masses above ∼1 eV. Many more have resonance energies in the ð Þ meV range, as experiments plan to join this search. These include the shown in Fig. 1. They have been proposed as an excellent magnetized disk and mirror axion experiment [15,16], way to detect light dark matter through scattering (if the which uses a layered dielectric in an external magnetic dark matter is heavier than a keV) or absorption [if the dark O 1–100 – field, and the quaere axion (QUAX) experiment [17–19], matter is in the ð Þ meV mass window] [49 60]. which searches for axion-induced classical spin waves While previous work has shown the reach to dark photon inside a magnetic target. See also Refs. [20–29] for recent absorption, an open question is whether phonon and axion DM search proposals. magnon excitations possess a sufficiently strong coupling to reach the QCD axion. In this paper, we investigate axion absorption onto phonons and magnons, and demonstrate the potential of Published by the American Physical Society under the terms of these processes to cover the ma ∼ Oð1–100Þ meV QCD the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, 1There are many well-motivated models that accommodate a and DOI. Funded by SCOAP3. QCD axion in this mass window—see, e.g., Refs. [30–44].

2470-0010=2020=102(9)=095005(19) 095005-1 Published by the American Physical Society MITRIDATE, TRICKLE, ZHANG, and ZUREK PHYS. REV. D 102, 095005 (2020)

FIG. 1. Spectra of gapped phonon polaritons and magnons at zero momentum for several representative targets considered in this work. These collective excitations have typical energies of Oð1–100Þ meV, and can be utilized to search for axion DM in the mass window ma ∼ Oð1–100Þ meV. Longer lines with darker colors correspond to the resonances in Figs. 3–5, while the shorter ones with lighter colors represent modes with suppressed couplings to axion DM due to selection rules.

ˆ ˆ −iωt axion mass window. The particle-level axion interactions of δH ¼ δH0e þ c:c:; 8 P interest are as follows: ip·x0 > lj ⇒ <> e f j · ulj phonons; 1 X g ˆ lj ˜ μν aff ¯ μ 5 δH0 ¼ P ð4Þ L ¼ − gaγγaFμνF þ ð∂μaÞðfγ γ fÞ > ip·x0 4 2m :> e lj f · S ⇒ magnons; f¼e;p;n f j lj X lj gafγ ¯ μν 5 − aFμνðfiσ γ fÞ; ð1Þ 4 0 f¼p;n with the effective couplings, f j, proportional to a and the relevant axion coupling. While our focus here is axion where the three terms are the axion’s electromagnetic, DM, the same equations hold for general fieldlike DM wind, and electric dipole moment couplings, respectively. candidates. In the nonrelativistic limit, the effective interaction In Sec. II, we derive rate formulae for single phonon Hamiltonian is2 and magnon excitations starting from the general form of 3 Z couplings in Eq. (4). In the case of phonon excitation, the X g true energy eigenmodes in a polar crystal, at the low δ ˆ − 3 − aff ∇ H ¼ gaγγ d xaE · B a · sf momentum transfers relevant for dark matter absorption, mf X f¼e;p;n are phonon polaritons due to the mixing between the − gafγaE · sf: ð2Þ photon and phonons. We take this mixing into account f¼p;n while still often referring to the gapped polaritons as phonons since their phonon components are much larger. These couplings can be further matched onto axion The final results for phonon and magnon excitation rates couplings to low energy degrees of freedom in a crystal. are Eqs. (18) and (27). Depending on the couplings f j and In particular, phonon excitation results from couplings to symmetries of the target system, it often happens that − 0 atomic displacements ulj ¼ xlj xlj, where l labels the excitation of some of the phonon or magnon modes is 0 suppressed, reducing the sensitivity to DM. We discuss this primitive cell, j labels the atoms within each cell, and xlj are the equilibrium positions, while magnons can be problem and possible ways to alleviate it in Sec. III. excited via couplings to the (effective) spins of magnetic Then it remains to determine the effective couplings f j in terms of particle physics parameters—g γγ, g , etc.—in ions Slj. An axion field oscillating with frequency ω ¼ ma a aff the case of axion DM. The effective couplings can receive and wave number p ¼ mava is represented by multiple contributions, some of which rely on the presence

aðx;tÞ¼a0 cos ðp · x − ωtÞ; ð3Þ of an external field. We discuss the various possibilities for axion-induced single phonon or magnon production in where the field amplitude is related to the energy density Sec. IV. Among them, two are particularly promising: the 2 2 coupling of the gradient of the axion field to the electron via ρa ¼ maa0=2. The resulting effective Hamiltonian relevant for phonon and magnon production takes the general form 3Previous calculations of DM absorption rates often rely on rescaling of optical data. While this is in principle possible for some of the processes discussed in this paper, the full set of data 2The coupling to the axial current also generates a term needed to capture general anisotropic target responses are hard to proportional to masf · vf, we neglect this term since its coupling come by. Therefore, we take a first-principle approach and to collective spin excitations is suppressed compared to the one explicitly compute the DM absorption rates starting from the generated by the ∇a · sf term. Hamiltonian.

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TABLE I. Summary of the potentially detectable channels identified in Sec. IV. The axion field a is given by Eq. (3), ρa is its energy density, and va is its velocity. The axion couplings gaγγ and gaee are defined in Eqs. (1) and (2), and given by Eqs. (31) and (32) for the ε Z QCD axion. ∞ is the high-frequency dielectric constant due to electronic screening, j is the Born effective charge tensor of the ion, gj is the Land´e g factor, ρT is the target’s mass density, and γ is the resonance width.

Process Fundamental interaction f j in Eq. (4) Rate formula Rate estimate(on resonance) ffiffiffiffi p 2 e ρ ρ 2 2 2 Axion þ B field → phonon aE · B 1ffiffi a ε−1 Z Eq. (18) gaγγ a Z e B p gaγγ B · ∞ · 3 2 2 [Eq. (40)] 2 ma j m ε∞m γ ffiffiffiffi a ion pρ 2 ρ 2 Axion → magnon ∇a · s iffiffi a Eq. (27) gaee ava ns e − p gaeeðgj − 1Þ va 2 ρ γ [Eq. (36)] 2 me me T

spin, gaee, allows for magnon excitation, while the axion- subscript j. We will keep f j general in this section, and induced electric field in the presence of an external derive their expressions for the case of axion DM in Sec. IV. magnetic field, due to the axion-photon coupling gaγγ, We assume the target system is prepared in its ground can excite phonon polaritons. These processes are sum- state j0i at zero temperature. The transition rate from marized in Table I. It is in fact not difficult to roughly standard time-dependent perturbation theory reads estimate the rates based on simple considerations about the X physical quantities that should enter each process. We show ˆ 2 Γ ¼ jhfjδH0j0ij 2πδðω − ω Þ: ð5Þ these estimates in the last column of the table, which will f f allow us to quickly assess the projected sensitivity in Eqs. (40) and (36). We present our full numerical results for the projected Strictly speaking, since phonons and magnons are unstable reach via these processes in Sec. V. We find that, consistent particles, the sum over final states f should include with the parametric estimates, when the axion mass is well multiparticle states resulting from their decays. In practice, ω matched to phonon polariton or magnon resonances in the however, when is close to a phonon/magnon resonance, we can simply smear the delta function to the Breit-Wigner target material, the QCD axion can be easily within reach. 4 The sensitivity is inherently narrowband for any specific function and sum over single phonon/magnon states : target material, with the axion masses covered limited by X 4ωω γ the resonance widths. However, combining the reach of a ˆ 2 ν;k ν;k Γ ¼ jhν; kjδH0j0ij ; ð6Þ set of judiciously chosen materials with different phonon ω2 − ω2 2 ωγ 2 ν;k ð ν;kÞ þð ν;kÞ and magnon frequencies can offer a broader coverage. Finally, we conclude in Sec. VI and discuss future ν ν interdisciplinary work needed to better understand and where j ; ki is the single phonon/magnon state on branch γ realize the potential of the ideas presented in this work. with momentum k, and ν;k is its decay width. Away from a resonance, the line shape deviates from Breit-Wigner, and depends on the details of phonon/magnon interactions. II. GENERAL FORMALISM FOR ABSORPTION Since we are interested in sub-eV DM candidates, the RATE CALCULATIONS momentum deposited is limited to mava ≲ meV—the DM In this section, we adapt the DM scattering calculations field drives phonon/magnon modes close to the center in Refs. [56,58] to the present case of bosonic DM of the first Brillouin zone (1BZ). Finally, averaging over absorption. Unlike the scattering case, light bosonic DM the DM velocity distribution fðvÞ, we obtain the expected (denoted by a in what follows) should be treated as a total rate: τ 2 −1∼ classical field. Within the coherence time a ¼ðmavaÞ Z −7 10 sð10 meV=m Þ, its effect can be modeled as a 3 a hΓi¼ d vfðvÞΓðvÞ: ð7Þ harmonic perturbation on the target system as in Eq. (4). In this work, we focus on configurations with no external ω ac electromagnetic fields, so that ¼ ma. An ac external We take fðvÞ to be a boosted Maxwell-Boltzmann dis- ω field with frequency e would generate perturbations with tribution, ω ¼jma ωej, for which the calculations in this section also apply. 4In deriving Eq. (6) we have assumed the observation time Phonons and magnons arise from quantizing crystal ≳ γ−1 Γ t ν;k, for which the transition rate is time independent. We lattice degrees of freedom, displacements ulj and effective 2 also assume that the line width of the axion, Δω ∼ mava ∼ spins S , respectively, which DM can couple to, as −8 lj 10 eVðma=10 meVÞ is smaller than excitation linewidth, γν;k, — −6 mentioned in the Introduction see Eq. (4). The effective which is true as long as γν;k is greater than ∼10 times the couplings f j depend on the atom/ion types, hence the resonance frequency.

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1 − 2 2 MB ðvþveÞ =v0 Θ − fχ ðvÞ¼ e ðvesc jv þ vejÞ; ð8Þ N0   2v π3=2 2 − escffiffiffi − 2 2 N0 ¼ v0 v0erfðvesc=v0Þ pπ expð vesc=v0Þ ; ð9Þ

230 240 with parameters v0 ¼ km=s, ve ¼ km=s, and 600 ρ vesc ¼ km=s. The local axion DM density a, which enters the effective couplings f j (see Table I), is assumed to be 0.3 GeV= cm3. In the following subsections, we derive the rate formulae for single phonon and magnon excitations, respectively. For easy comparison, we present both derivations in as similar ways as possible. FIG. 2. Dispersion of phonon polaritons in GaAs near the center of the 1BZ, k ∼ ω. The mixing between the photon and TO A. Phonon excitations phonons is maximal at ω ∼ k.Atk ≪ ω, the TO phononlike modes are degenerate with the LO phonon mode (blue line), We begin by calculating the absorption rate from ω ≫ couplings to phonons. The target Hamiltonian results from while at k they approach their unperturbed value (dotted blue line), and an LO-TO splitting is present. expanding the potential energy of the crystal around equilibrium positions of atoms: k ≫ ω, and in particular does not affect the DM scattering X 2 1 X ˆ plj 3 calculations in Refs. [53,54,58,59]. However, near the cen- H ¼ þ u · V 0 0 · u 0 0 þ Oðu Þ; ð10Þ 2m 2 lj ll jj l j ter of the 1BZ where k ≲ ω—relevant for DM absorption— lj j ll0jj0 the photon-phonon mixing modifies the dispersions to _ where plj ¼ mjulj, and the force constant matrices, Vll0jj0 , avoid a level crossing. The true energy eigenstates are can be calculated from ab initio density functional theory linear combinations of photon and phonon modes, known (DFT) methods [61–64]. as phonon polaritons. This is shown in Fig. 2 for gallium To diagonalize the Hamiltonian, we expand the atomic arsenide (GaAs) as a simple example. For an isotropic displacements and their conjugate momenta in terms of diatomic crystal like GaAs, the two degenerate (mostly) canonical phonon modes: transverse optical (TO) phonon modes at k ≫ ω continue to photonlike modes at k ≪ ω, and vice versa. The phonon- X3n X 1 0 ≪ ω † ik·x like modes at k do not have the same energies as away pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆ ˆ lj ϵ ulj ¼ ðaν;k þ aν;−kÞe ν;k;j; ð11Þ from the polariton regime: the TO phonon-like modes 2Nm ων ν¼1 k j ;k become degenerate with the longitudinal optical (LO) rffiffiffiffiffiffiffiffiffiffiffiffiffi phonon mode at ω as k → 0, whereas there is an LO- X3n X ω LO mj ν;k † ik·x0 ω ≠ ω ≫ ω p ¼ i ðaˆ − aˆν Þe lj ϵν ; ð12Þ TO splitting, TO LO,atk . For more complex lj 2N ν;−k ;k ;k;j ν¼1 k crystals like sapphire (Al2O3), quartz (SiO2), and calcium tungstate (CaWO4), the mixing involves more phonon where ν labels the phonon branch (of which there are 3n for modes, and in general shift all their energies with respect a three-dimensional crystal with n atoms per primitive cell), to the eigenvalues ων;k computed from diagonalizing just k labels the phonon momentum within the 1BZ, N is the the lattice Hamiltonian. total number of primitive cells, mj is the mass of the jth To account for the photon-phonon mixing, we write the ˆ† ˆ atom in the primitive cell, and aν;k; aν;k are the phonon crea- total Hamiltonian of electromagnetic fields coupling to the tion and annihilation operators. The phonon energies ων;k ions in the target crystal, and diagonalize its quadratic part ϵ ϵ via a Bogoliubov transformation. We explain this pro- and eigenvectors ν;k;j ¼ ν;−k;j are obtained by solving the cedure in detail in the Appendix A. The resulting diagonal eigensystem of V 0 0 , for which we use the open-source ll jj Hamiltonian is code PHONOPY [65]. The target Hamiltonian then reads 3 2 3 Xnþ X Xn X † 3 † 3 ˆ ω0 ˆ0 ˆ0 O ˆ0 ˆ ω ˆ ˆ O ˆ H ¼ ν;ka ν;kaν;k þ ða Þ: ð14Þ H ¼ ν;kaν;kaν;k þ ða Þ: ð13Þ ν¼1 k ν¼1 k 3 2 For a polar crystal, since the ions are electrically charged, At each k, there are ( n þ ) modes, created (annihilated) ˆ0† ˆ0 3 some of the phonon modes mix with the photon. This by a ν;k (aν;k), which are linear combinations of n phonon mixing has a negligible impact in most of the 1BZ where modes and two photon polarizations. Among them, five are

095005-4 DETECTABILITY OF AXION DARK MATTER WITH PHONON … PHYS. REV. D 102, 095005 (2020) gapless at k ¼ 0, including three acoustic phonons and two 3Xnþ2 ˆ ˆ0 ˆ0† photonlike polaritons. The number of gapped modes, aν;k ¼ ðUνν0;kaν0;k þ V νν0;kaν0;−kÞ: ð15Þ 3n − 3, is the same as in the phonon-only theory, but ν0¼1 ω0 ≠ their energy spectrum is shifted, f ν¼ð6;…;3nþ2Þ;kg ω f ν¼ð4;…;3nÞ;kg. The original phonon modes are linear For DM coupling to the atomic displacements ulj, the combinations of the phonon polariton eigenmodes: perturbing potential is given by Eq. (4) and therefore

X X3n X 0 1 ˆ iðp−kÞ·x † δH0 0 e lj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f · ϵ aˆν − aˆ 0 j i¼ 2 ω j ν;k;jð ; k þ ν;kÞj i lj ν¼1 k Nmj ν;k rffiffiffiffi 3 N Xn X 1 ϵ ˆ ˆ† 0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi f j · ν;p;jðaν;−p þ aν;pÞj i 2 m ων ν¼1 j j ;p rffiffiffiffi 3 3 2 N Xn Xnþ X 1 ϵ ν0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi f j · ν;p;jðUνν0;p þ V νν0;−pÞj ; pi; ð16Þ 2 m ων ν¼1 ν0¼1 j j ;p P † i p−k ·x0 ν0 ˆ0 0 ð Þ lj δ ∈ 1 where j ; pi¼a ν0;pj i. To arrive at the second equation, we have used the identity l e ¼ N k;p (for k; p BZ). It follows that rffiffiffiffi 3 N Xn X 1 ν δ ˆ 0 δ ϵ h ; kj H0j i¼ k;p pffiffiffiffiffiffiffiffiffiffiffiffiffiffi f j · ν0;p;jðUν0ν;p þ V ν0ν;−pÞ; ð17Þ 2 m ων0 ν0¼1 j j ;p

ν ν0 Γ where we have swapped the dummy indices and . The DM absorption rate per unit target mass, R ¼h i=ðNmcellÞ,is therefore Z 3Xnþ2 0 X X3n 2 2ω ων γν 1 3 ;p ;p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϵ 0 R ¼ d vafðvaÞ 2 02 2 2 p f j · ν0;p;jðUν0ν;p þ V ν ν;−pÞ ; ð18Þ m ðω − ων Þ þðωγν Þ m ων0 cell ν¼6 ;p ;p j ν0¼1 j ;p

ω J ∝ 1 0 0 where ¼ ma, p ¼ mava, and mcell is the total mass of the of the Heisenberg model, ll0jj0 for pairs of lj and l j atoms in a primitive cell. For our numerical calculations, on (nearest, next-to-nearest, etc.) neighboring sites. One we use the PHONOPY code [65] to process DFT output material which is well described by this simple model is [54,58,59] to obtain the unmixed phonon energies and yttrium ion garnet (YIG) [68,69], which has already been eigenvectors ων0;p, ϵν0;p;j as mentioned above, and then considered for DM detection [17–19,24,56,70]. However, 0 compute the polariton-corrected energy eigenvalues ων;p as we will see below, materials with spin-spin interactions and mixing matrices U, V via the algorithm of Refs. [66,67]. beyond the simplest Heisenberg type can be useful for We relegate the technical details to Appendix A, and review enhancing DM-magnon couplings. the diagonalization algorithm [66,67] in Appendix C. The spin-spin interactions in Eq. (19) can result in a ground state with magnetic order. Here we focus on the B. Magnon excitations simplest case of commensurate magnetic dipole orders, for which a rotation on each sublattice can take Slj to a We now move to the case of magnons. The target local coordinate system where each spin points in the þzˆ Hamiltonian is a spin lattice model with the following direction: general form: X X S R S0 ; S0 0; 0;S : ˆ lj ¼ j · lj h lji¼ð jÞ ð20Þ H ¼ Slj · Jll0jj0 · Sl0j0 þ μBB · gjSlj; ð19Þ ll0jj0 lj YIG and many other magnetic insulators have commensu- rate magnetic order. The calculation can be easily gener- 0 0 where l, l label the magnetic unit cells and j, j the alized to single-Q incommensurate orders, as we discuss in magnetic ions inside the unit cell, μ ¼ e is the Bohr B 2me Appendix B. magneton, B is an external uniform magnetic field, and gj For a magnetically ordered system, the lowest energy are the magnetic ions’ Land´e g factors. In the simplest case excitations are magnons. To obtain the canonical magnon

095005-5 MITRIDATE, TRICKLE, ZHANG, and ZUREK PHYS. REV. D 102, 095005 (2020) rffiffiffiffi modes, we first apply the Holstein-Primakoff transforma- Xn X pffiffiffiffiffi ν δ ˆ 0 δ N tion to write the Hamiltonian in terms of bosonic creation h ; kj H0j i¼ k;p 2 Sj f j and annihilation operators: ν¼1 j · ðUjν;prj þ V jν;−prj Þ: ð26Þ 0þ 2 − ˆ† ˆ 1=2 ˆ 0− ˆ† 2 − ˆ† ˆ 1=2 Slj ¼ð Sj aljaljÞ alj;Slj ¼ aljð Sj aljaljÞ ; We can now obtain the DM absorption rate per unit target 0z − ˆ† ˆ Slj ¼ Sj aljalj; ð21Þ mass: Z 0 0x 0y Xn ω γ where Slj ¼ Slj iSlj. The Holstein-Primakoff transfor- 2ω 3 ν;p ν;p R ¼ d vafðvaÞ 2 2 2 2 mation ensures that the spin commutation relations m ðω − ων Þ þðωγν Þ cell ν¼n0þ1 ;p ;p 0α 0β δ δ ϵαβγ 0γ ½Slj ;Sl0j0 ¼ ll0 jj0 i Slj are preserved when the usual X pffiffiffiffiffi 2 † ˆ ˆ δ 0 δ 0 × S f · U r V ν − r ; 27 canonical commutation relations ½alj; al0j0 ¼ ll jj are j j ð jν;p j þ j ; p j Þ ð Þ imposed. As in the phonon case, translation symmetry j instructs us to go to momentum space: where n0 is the number of gapless modes, which depends X 1 0 on the material (in particular, on the symmetry breaking ik x ffiffiffiffi · lj aˆlj ¼ p aˆj;ke ; ð22Þ pattern). Similarity to the phonon formula Eq. (18) is N k apparent. We again use the algorithm of Refs. [66,67], reviewed in Appendix C, to solve the diagonalization where k ∈ 1BZ. The quadratic Hamiltonian, whose problem to obtain the magnon energies ων;p and mixing detailed form can be found in Appendix B, only couples matrices U, V. modes with the same momentum, i.e., aˆj;k and aˆj0;k, † aˆ 0 . A Bogoliubov transformation takes the quadratic j ;−k III. SELECTION RULES AND WAYS Hamiltonian to the desired diagonal form: AROUND THEM

n X X Depending on the DM couplings f j, excitation rates for ˆ ω ˆ0† ˆ0 O ˆ03 H ¼ ν;kaν;kaν;k þ ða Þ: ð23Þ some of the phonon or magnon modes can be suppressed. ν¼1 k In the context of DM scattering, it has been known that acoustic and optical phonons are sensitive to different types At each k, there are n magnon modes with n the number of of DM couplings [58,59]: if DM couples to the inequivalent spins per magnetic unit cell. These energy eigenmodes are atoms/ions with the same sign (different signs), the single ˆ0† ˆ0 created (annihilated) by aν;k (aν;k), which are related to the phonon excitation rate is dominated by acoustic (optical) unprimed creation and annihilation operators by phonons, corresponding to in-phase (out-of-phase) oscil- lations of the atoms/ions. Xn 0 0† The same considerations apply to absorption of DM, aˆ ¼ ðU ν aˆ þ V ν aˆ Þ: ð24Þ j;k j ;k ν;k j ;k ν;−k though here the gapless acoustic phonons are kinematically ν¼1 inaccessible, and therefore only gapped optical phonons can be excited. Thus, the rate is suppressed if all f point For DM coupling to the effective spins Slj, the inter- j action is given by Eq. (4), and we find, in complete analogy in the same direction. As an extreme example, consider with Eq. (16), f j ¼ mjf, with f a constant vector. Up to the photon- rffiffiffiffiffiffiffi phonon mixing (which mostly shifts the energy eigenvalues δ X X while leaving the factor ðUν0ν þ V ν0ν;−pÞ close to ν0ν), iðp−kÞ·x0 Sj † ;p P δ ˆ 0 lj ˆ ˆ 0 pffiffiffiffiffiffi 2 H0j i¼ e f j · ðrj aj;−k þ rja Þj i; ϵ 2N j;k the rate in Eq. (18) is proportional to j j mjf · ν;p;jj . lj k rffiffiffiffi However, one can show from translation symmetry that X pffiffiffiffiffi pffiffiffiffiffiffi N † mj f can be written as a linear combination of the δ S f r aˆ − r aˆ 0 ; ¼ k;p 2 j j · ð j j; k þ j j;kÞj i ϵ ν ∈ — j polarization vectors ν;p→0;j with acoustic see, for rffiffiffiffi example, the explicit discussion of [71], consistent with the Xn X pffiffiffiffiffi δ N earlier results in Refs. [53,54]. This is not surprising since ¼ k;p 2 Sj f j ν¼1 j gapless acoustic phonons are Goldstone modes of the broken translation symmetries. Thus, by the orthogonality ν · ðUjν;prj þ V jν;−prj Þj ; pi; ð25Þ of the phonon polarization vectors, optical phonons do not contribute to the rate in the p → 0 limit, and only higher ≡ xx yx zx xy yy zy ν where rj ðRj ; Rj ; Rj ÞþiðRj ; Rj ; Rj Þ, and j ; pi¼ order terms in the DM velocity can give a nonzero 0† aˆν;pj0i. Therefore, contribution. In the next section, in the context of axion

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DM, we will encounter both cases where the suppression gapless modes cannot be excited due to kinematics. As a due to the f js being aligned is present and absent, and will result, the detection rate is severely suppressed by powers identify a process free of the suppression as a viable of DM velocity. We have checked this explicitly for several detection channel. target materials. In what follows, let us expand on the two Additional selection rules may be present among the examples. optical phonons. For example, sapphire has 27 optical phonon branches, but we find that near the 1BZ center, only A. YIG (Y3Fe5O12) 10 of them couple to the axion-induced electric field (in the presence of an external magnetic field). Furthermore, 8 of The crystal primitive cell of YIG consists of four copies 3þ the 10 modes are degenerate in pairs, reducing the total of Y3Fe5O12. The magnetic ions are Fe , each of which 5 2 number of distinct resonances to six, as seen in Fig. 1. This has spin = . The magnetic unit cell coincides with the is consistent with the well-known fact that, due to crys- crystal primitive cell, and contains 20 magnetic ions. The talline symmetries, sapphire has six infrared-active phonon spin Hamiltonian has antiferromagnetic Heisenberg inter- modes [72–74]. Thus, despite the existence of many optical actions, which we include up to third nearest neighbors phonon modes, sapphire does not really offer broadband [69]. The ground state has ferrimagnetic order, where the coverage of the axion mass. The same observation, that 12 magnetic ions on the tetrahedral sites and the 8 magnetic only a subset of gapped phonon modes couple to axion ions on the octahedral sites have spins pointing in opposite zˆ DM, also holds for the other targets considered: SiO2 and directions [68], taken to be . The symmetry breaking 3 → 2 CaWO4—see Fig. 1 (GaAs has only three optical phonon pattern is SOð Þ SOð Þ, and hence there are two broken modes, which are all degenerate and can couple to axion generators Sx, Sy. There is however just one Goldstone DM). To broaden the mass coverage, it is therefore mode (with quadratic dispersion) due to the nonvanishing necessary to run experiments with several target materials expectation value of the commutator between the broken 2 ≠ 0 – with distinct phonon frequencies. generators, h½Sx;Syi ¼ ði= ÞhSzi [76 79]. Thus, There are also selection rules in the case of magnon among the 20 magnon branches, only one is gapless. excitations. It has been pointed out that, assuming the We find that at zeroP momentum,pffiffiffiffiffi the j sum in the rate absence of an external magnetic field, for a target system formula Eq. (27), j Sj · ðUjν;0rj þ V jν;0rj Þ, indeed described by the Heisenberg model with quenched orbital vanishes for all but the gapless mode (ν ¼ 1), confirming angular momentum, such as YIG, only gapless magnons the argument above that the DM field only couples to can be excited in the zero momentum transfer limit [56,75]. gapless magnons. To understand why, let us review and quantify the semi- classical argument given in Ref. [56]. Within a coherence B. Ba NbFe Si O length, the DM field couples to the spins as a uniform 3 3 2 14 magnetic field, causing all the spins to precess. As a result, This is an example of materials with incommensurate the rate of change in the Heisenberg interaction energy magnetic order. We discuss the generalization needed in the between any pair of spins is proportional to rate calculation in Appendix B, with the final result given in Eq. (B11). The magnetic unit cell contains three magnetic 3þ 5 2 − d dSlj dSl0j0 ions Fe with spin = , which form a triangle in the x y ðS · S 0 0 Þ¼ · S 0 0 þ S · dt lj l j dt l j lj dt plane. The crystal consists of layers stacked in the z direction. The antiferromagnetic Heisenberg interactions ¼ðf × S Þ · S 0 0 þ S · ðf 0 × S 0 0 Þ; j lj l j lj j l j result in a frustrated order with 120° between the three spins ¼ðf j − f j0 Þ · ðSlj × Sl0j0 Þ; ð28Þ that are nearest neighbors. Further, the chiral structure of interlayer Heisenberg exchange couplings results in a which vanishes for f j ¼ f j0. Therefore, if the target system rotation of the order in the z direction, with a wave vector is described by the Heisenberg Hamiltonian, and all f j are that is irrational, Q ≃ 0.1429ð2π=cÞzˆ where c ≃ 5.32 Åis equal (which is quite generic since they all originate from the interlayer lattice spacing [66]. This is known as a single- DM-electron spin coupling), the total energy cannot change Q incommensurate order. All three generators of SO(3) are in response to the DM field, and no gapped magnons can be broken while the ground state has zero total magnetization, excited. In other words, the DM field only couples to so there are three Goldstone modes. These appear at gapless magnons. In the case of scattering, the rate is not k ¼ 0; Q, which are also the momenta near which the severely suppressed by this fact since the scattering axion coupling is nonzero due to (generalized) momentum kinematics allows us access to finite momentum magnons conservation. We find that at all three momenta, the j sum and, hence, sufficient energy deposition to be detected. in the generalized rate formula Eq. (B11) is nonzero only The situation is much worse in the case of absorption, for ν ¼ 1; i.e., the gapless modes, again confirming the because the momentum transfer is small in comparison to argument above that the DM field only couples to gapless the DM mass due to its small velocity v ∼ 10−3. Therefore, magnons.

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Nevertheless, there are several possibilities to alleviate requires the magnetic ions to have different spin and orbital the problem. First, one can consider targets involving angular momentum compositions. We demonstrate how additional, non-Heisenberg interactions. These additional this allows for axion couplings to gapped magnons in terms can explicitly break the rotational symmetries, Sec. VB. causing the otherwise gapless Goldstone modes to become As in the case of phonons, there are usually additional gapped, and match the DM absorption kinematics. selection rules due to crystalline symmetries. As a result, Concretely, we can identify two ways of implementing the strategies discussed above usually generate axion this idea: couplings to only a subset of gapped magnon modes— (1) An external magnetic field B ≠ 0 can generate a gap see Fig. 1. Therefore, multiple target materials which for the lowest magnon branch equal to the Larmor cover complementary ranges of magnon frequencies are frequency, desirable.   B IV. AXION COUPLINGS AND DETECTION ω ¼ 2μ B ¼ 0.12 meV ; ð29Þ L B T CHANNELS The derivation and discussion in the previous two sec- assuming g ¼ 2 for all j. The QUAX experiment j tions apply to general fieldlike DM candidates. We now [17–19] makes use of this to search for axion DM specialize to the case of axion DM. Our goal in the present with m ∼ Oð0.1 meVÞ, currently in the regime a section is to identify the most promising detection channels where the magnon number is large and a classical involving phonon or magnon excitation via order-of- description can be used. Recently, a calculation to magnitude estimates. We then examine these processes exploit this effect in the quantum regime, similar to quantitatively in the next section. the derivation in Sec. II B, was carried out in The axion couplings of interest are already given in Ref. [70]. However, sensitivity of such a setup is Eqs. (1) and (2). For the QCD axion, we have [81] limited to sub-meV DM by the achievable magnetic   field strengths, and only the lowest magnon mode(s) α 2 03 10−12 ma −1 can be excited. gaγγ ¼ Cγ 2π ¼ . × Cγ 10 GeV ; (2) There are materials with anisotropic interactions fa meV where the number of gapless Goldstone modes is ð31Þ reduced. In Sec. VB, we consider a concrete    example, NiPS3, where the Jll0jj0 matrices in the mf −13 mf ma gaff ¼ Cf ¼ 1.18 × 10 Cf ; ð32Þ spin Hamiltonian Eq. (19) have unequal diagonal fa MeV meV entries [80]. In this case, two gapped magnon modes at 12 and 44 meV can couple to axions. There are −3 1 1 g γ ¼ −g γ ¼ð3.7 1.5Þ × 10 also materials with nonzero off-diagonal entries in an ap f GeV  a Jll0jj0 , arising from, e.g., Dzyaloshinskii-Moriya −12 ma −2 interactions, which could be used to achieve the ¼ð6.5 2.6Þ × 10 GeV ; ð33Þ 10 meV same result. An orthogonal route to solve the problem is to use targets where we have denoted Cγ ≡ E=N − 1.92ð4Þ with E=N ¼ where the DM-spin couplings f are nondegenerate. j 0 (8=3) in the Kim-Shifman-Vainstein-Zakharov (KSVZ) For materials with nondegenerate Land´e g factors, even a [Dine-Fischler-Srednicki-Zhitnitsky (DFSZ)] model. The uniform magnetic field can drive the magnetic ions differ- axion-fermion couplings Cf are also model dependent. In ently and excite gapped magnon modes; the same is true 2 particular, the axion-electron coupling is C ¼ sin β=3 in for a uniform DM field. The basic reason for this is the e the DFSZ model, where tan β is the ratio of the vacuum presence of spin-orbit couplings that break the degeneracy expectation values of the two Higgs doublets giving masses between the DM couplings to the magnetic ions’ total to the up and down-type quarks. In the KSVZ model, on the effective spins. Concretely, the Land´e g factors are given by 2 other hand, Ce is Oðα Þ suppressed. In the following subsections we consider axion cou- 3 1 sjðsj þ 1Þ − ljðlj þ 1Þ gj ¼ 2 þ 2 1 ; ð30Þ plings independent of external fields and in the presence of SjðSj þ Þ a magnetic field.5 In each case, we discuss the phonon and magnon excitation processes that are allowed, and identify where sj and lj are respectively the spin and orbital angular momentum components of the total effective spin Slj. In the 5 simplest and most common case of magnetic ions with An external electric field shifts the equilibrium positions of l ≃ 0 the ions such that there is no net electric field at the new quenched orbital angular momenta (i.e., j ), we re- equilibrium positions, so it does not generate new axion cou- cover the usual result gj ¼ 2. Breaking the degeneracy plings at leading order.

095005-8 DETECTABILITY OF AXION DARK MATTER WITH PHONON … PHYS. REV. D 102, 095005 (2020) those with potentially detectable rates. The results of this C2ρ m v4 2 ∼ f a a a ns exercise are summarized in Table I. R 2 2 f ρ γν a T     100 2 μ ∼ −1 2 GeV ma eV A. Axion couplings independent of external fields ðkg · yrÞ Cf ; ð37Þ fa 10 meV γν The axion wind coupling to electron spin leads to a l coupling to the spin component of Slj. From slj þ lj ¼ Slj can be sizable only for uninterestingly low f . Therefore, 2 l a and slj þ lj ¼ gjSlj, we see that the axion wind couples we conclude that axion wind couplings to nucleon spins do − 1 to slj ¼ðgj ÞSlj. Thus, not offer a viable detection channel. X Beyond the axion wind couplings, the axion field also ˆ gaee δH ¼ − ∇a · ðg − 1ÞS turns magnetic dipole moments from sf into oscillating m j lj e lj EDMs, and one may consider phonon and magnon exci- X gaee a0 ip·x0 −iωt tation by the resulting electromagnetic radiation fields. − v − 1 S lj ¼ ðima aÞ 2 · ðgj Þ lje þ H:c: In the case of the electron, since the EDM coupling is me lj perturbatively generated by the aFF˜ coupling, the process ð34Þ mentioned above is essentially converting the crystal magnetic field into electromagnetic radiation, and should In the notation of Eq (4), we thus have be less efficient than applying an external magnetic field ffiffiffiffiffi pρ (discussed below in Sec. IV B). In the case of nucleons, iffiffiffi a the EDM coupling in Eq. (1) is not much larger than the f j ¼ − p gaeeðgj − 1Þ va: ð35Þ 2 me perturbative contribution from aFF˜ , so the same conclu- sion applies. For an order of magnitude estimate of the rate, let us note In sum, for axion couplings independent of external that theffiffiffi mixing matrices U, V in Eq. (27) generically scale fields, we have identified magnon excitation via the axion 1 p as = n with n the number of magnetic ions in a primitive wind coupling to electrons as the only viable detection cell. The maximum rate is obtained on resonance, which is channel. parametrically given by     2 ρ 2 2 μ B. Axion couplings in a magnetic field ∼ gaee ava ns ∼ −1 gaee eV R 2 ρ γ ðkg · yrÞ −15 γ ; ð36Þ me T 10 In the presence of a dc magnetic field B, the axion field induces oscillating electromagnetic fields via the aFF˜ ρ where ns and T are the spin and mass densities of the coupling. Solving the modified Maxwell equations (see, target, taken to be ð5 ÅÞ−3 and 5 g=cm3, respectively (close e.g., Refs. [82,83]), we find the induced electric field is −1 to the values for YIG), in the estimate. We see that, with Ea ¼ −gaγγaε∞ · B. Note that the high frequency dielectric single magnon sensitivity, interesting values of gaee may be constant, ε∞, which takes into account screening effects reached with less than a kilogram-year exposure. from the fast-responding electrons while excluding ionic The axion wind also couples to nucleon spins. However, contributions, should be used when solving the macro- these couplings do not excite magnons, since magnetic scopic Maxwell equations—essentially, since we are con- order originates from electron-electron interactions, mean- cerned with how the axion-induced electric field acts on ing that the effective spins of magnetic ions that appear in the ions, the ion charges should appear in the source term the spin Hamiltonian Eq. (19) come from electrons. On the rather than being coarse grained into a macroscopic electric other hand, if the nuclear spins SN are ordered (e.g., by polarization. In the long-wavelength limit, the axion- applying an external magnetic field that does not affect the induced electric field Ea couples to charged ions via an axion-nucleon couplings) and form a periodic structure, effective dipole coupling: the axion wind couplings could excite phonons. However, X X the rate suffers from multiple suppressions. First, coupling ˆ −1 δH ¼ −e E · Z · u ¼ eg γγa B · ε∞ · Z · u ; to atomic displacements relies on the spatial variation of a j lj a a j lj lj lj ∇a · SN, which brings in an additional factor of va on top of 2 ð38Þ the gradient: f j ∼ ðCf=faÞmaaðva · SN;jÞva. Second, there is a further suppression for exciting optical phonons since where Z is the Born effective charge tensor of the jth ion f j are approximately aligned with acoustic phonon polar- j — izations if all SN;j point in the same direction (see in the primitive cell it captures the change in macroscopic δ Z discussion in Sec. III). Even without taking into account polarization due to a lattice displacement, P ¼ e j · δ Ω Z ≃ the second suppression, the estimated on-resonance rate ulj= and is numerically close to the ionic charge, j using Eq. (18), Qj1 (where Ω is the primitive cell volume). It follows that

095005-9 MITRIDATE, TRICKLE, ZHANG, and ZUREK PHYS. REV. D 102, 095005 (2020) pffiffiffiffiffi 1 e ρ anharmonic phonon couplings, theoretically it is most ffiffiffi a ε−1 Z f j ¼ p gaγγ B · ∞ · j : ð39Þ 2 ma efficient to read out phonons (heat) in the final state. However, phonon readout (e.g., through a transistor edge pffiffiffi Noting that the phonon polarization vectors scale as 1= n sensor) is complicated by the strong external magnetic field with n the number of ions in the primitive cell, and needed for the axion absorption process. One possibility is assuming photon-phonon mixing gives just a small cor- to detect the phonon via evaporation of helium atoms rection, we can estimate the on-resonance rate from deposited on the surface of the crystal. The helium atoms Eq. (18) as follows: are then detected well away from the crystal, such that the   magnetic field is isolated from the sensor. Alternatively, if a 2 2 2 2 2 g γγρ Z e B g γγ target material can be found in which the photon yield from R ∼ a a ∼ −1 a 3 ε2 2 γ ðkg · yrÞ 10−13 −1 phonon polariton decays is substantial, photon readout ma ∞mion GeV       becomes a viable option. In this case, the photons may be 100 meV 3 B 2 meV focused by a mirror and lens and directed by a waveguide × 10 γ ; ð40Þ ma T onto a single photon detector (e.g., a superconducting nanowire) placed in a region of zero magnetic field. Both ε ∼ 1 ∼ 20 where we have taken Z = ∞ and mion GeV. We the phonon and photon readout possibilities sketched above see that, with single phonon sensitivity, there is excellent will be studied in future work. In what follows, we simply potential for reaching the QCD axion coupling if the axion assume three single phonon polariton excitation events per mass is close to a phonon resonance. kilogram-year exposure (corresponding to 90% C.L. for a The axion-induced magnetic field is much smaller, background-free counting experiment) when presenting B ∼ E v . An order of magnitude estimate tells us that a a a the reach. the magnon excitation rate by Ba is much smaller than the μ 2 To compute the projected sensitivity to the axion-photon Rmagnon ∼ BBaSlj 2 ∼ mjmava ∼ phonon excitation rate by Ea: ð Þ 2 Rphonon eZ Eaulj me coupling, we use the rate formula Eq. (18), with the effec- −8 10 for a 100 meV axion. tive couplings f j given in Eq. (39). We consider several In sum, we have identified phonon excitation via the example target materials and different orientations of the axion-photon coupling in a magnetic field as the only novel external magnetic field. The results are shown in Figs. 3 viable detection channel when considering an external dc and 4. As we discussed in Sec. III, materials with different magnetic field. phonon frequencies play important complementary roles in covering a broader axion mass range. In these plots, we −2 V. PROJECTED SENSITIVITY have taken the resonance widths to be γ=ω ¼ 10 , con- sistent with the order of magnitude of the measured We now compute the projected sensitivity for the two numbers in sapphire. As predicted by Eq. (40), the on- detection channels identified in the previous section (see resonance phonon production rate scales as R ∝ γ−1. From Table I). In both phonon and magnon calculations, an this follows that near-resonance constraints on gaγγ scales important but elusive parameter is the resonance width, γν;p, of each mode. While all other material parameters entering the rate calculation (equilibrium positions, phonon energies and eigenvectors, magnon energies, and mixing matrices) can be computed within the quadratic Hamiltonian, the widths involve anharmonic interactions and are not always readily available. In what follows, we present the reach with reasonable assumptions for γν;p. Our goal here is to demonstrate the viability of phonons and magnons for detecting axion DM and motivate further study in the condensed matter and materials science community on phonon and magnon interactions. This will be crucial both for having more accurate inputs to the DM detection rate calculation and for designing detectors to read out these excitations.

A. Phonon excitation via the axion-photon coupling FIG. 3. Projected reach on gaγγ from axion absorption onto phonon polaritons in Al2O3, CaWO4, GaAs, and SiO2 in an When axion absorption excites a phonon polariton, it external 10 T magnetic field, averaged over the magnetic field cascade decays to a collection of lower energy phonons directions, assuming three events per kilogram-year. Also shown −1 and photons on a timescale of γ ∼ ps ðmeV=γÞ. Since are predictions of the KSVZ and DFSZ QCD axion models, and the polariton is a phononlike state which decays via horizontal branch (HB) star cooling constraints [84].

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FIG. 4. Similar to Fig. 3, but with the external magnetic field oriented in the xˆðzˆÞ direction in the left (right) panel. The strength of axion-phonon couplings depends on the orientation of the magnetic field, and different resonances can be selected by changing the magnetic field direction. like γ1=2 making them stronger (weaker) for smaller (larger) 1. External magnetic field value of the width, while at the same time narrowing The idea of using an external magnetic field to lift the (broadening) the peak structure. We should also note that gapless mode is the one adopted in the QUAX experiment our projected constraints are derived assuming a Breit- [17–19]. In Ref. [18], a classical calculation was used to Wigner line shape, which is expected to be a good estimate the axion absorption rate. Our formalism allows to approximation near resonance; details of phonon self- compute the same rate in the quantum regime, and agrees interactions are needed to obtain more accurate results with the recent computation carried out in Ref. [70]. The away from resonance. projected reach we obtain for a YIG sample in a 1 T field is It is also worth noting that, since the effective couplings shown in Fig. 5, where the resonance is at 0.12 meV [see bˆ f j depend on the direction of the magnetic field , the Eq. (29)]. For comparison, we also overlay the projection in strengths of the resonances vary as bˆ is changed, as we can see in Fig. 4. For example, for a sapphire target, when bˆ is parallel (perpendicular) to the crystal c axis, which is chosen to coincide with the z axis here, only two (four) out of the six resonances appear. This observation provides a useful handle to confirm a discovery by running the same experiment with the magnetic field applied in different directions.

B. Magnon excitation via the axion wind coupling To compute the magnon excitation rate, we substitute the coupling f j in Eq. (35), into the rate formula Eq. (27). In Sec. III, we discussed three strategies to alleviate the suppression of axion-magnon couplings due to selection rules: external magnetic fields, anisotropic interactions, and nondegenerate g factors. In this subsection, we show the FIG. 5. Projected reach on gaee from axion-to-magnon con- projected reach for each of these strategies. The results are version, compared with DFSZ (assuming 0.28 ≤ tan β ≤ 140) summarized in Fig. 5, assuming three single magnon events and KSVZ model predictions, as well as white dwarf (WD) per kilogram-year exposure. Absent a detailed study of constraints from Ref. [85]. The suppression of axion-magnon anharmonic magnon interactions, we take the resonance couplings is alleviated by using the three strategies discussed in widths to be a free parameter, and show results for γ=ω ¼ the main text: lifting gapless magnon modes by an external 10−2 10−5 magnetic field (YIG target in a 1 T magnetic field, compared to and , consistent with measured phonon width the scanning scheme of Ref. [70]), anisotropic interactions values on the high end and YIG’s Kittel magnon width on (NiPS3 target), and using targets with nondegenerate g factors the low end. We see that, on resonance, all methods could (hypothetical toy models based on YIG, referred to as YIGo and reach axion-electron couplings predicted by QCD axion YIGt). For all the cases considered we assume three events per models. In the following, we expand on the calculation for kilogram-year exposure, and take the magnon width to frequency each strategy. ratio γ=ω to be 10−2 (solid) or 10−5 (dashed).

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Ref. [70] based on scanning the resonance frequency by In the phonon polariton case, we considered several changing the magnetic field, with an observation time of example targets—Al2O3, CaWO4, GaAs, and SiO2—and 104 s per frequency interval and a total integration time showed that on resonance, per kilogram-year exposure, −12 −1 of 10 years. In deriving their constraints, the authors of they can reach gaγγ ∼ 10 GeV , as shown in Figs. 3 Ref. [70] also include an estimate for the expected back- and 4. This outperforms the leading constraint in this mass ground noise. This, together with the smaller observation window from stellar cooling, and reaches below the QCD time per resonant frequency, is the origin of the difference axion band. Carefully choosing a set of target materials between their constraints and our predicted peak sensitivity. with different phonon frequencies is key to covering a broad range of axion masses. 2. Anisotropic interactions Previous proposals for probing gaee via absorption onto magnons, which underlies the QUAX experiment, As another way to lift the gapless magnon modes, so considered targets in an external magnetic field, which that the absorption kinematics are satisfied, we may use would lift the lowest magnon mode and kinematically materials with anisotropic exchange couplings. As an allow sub-meV axion absorption [17–19,24,70].Incon- example, we consider NiPS3, which has a layered crystal trast, we focused on the Oð1–100Þ meV axion mass 2þ structure [86]. The magnetic ions are spin-1 Ni . window, and showed that without an external magnetic Following Ref. [80], we model the system as having field, materials with anisotropic exchange interactions, intralayer anisotropic exchange couplings up to third e.g., NiPS3, and materials with nondegenerate g factors nearest neighbors, as well as single-ion anisotropies. All can host gapped magnons coupling to the axion. On re- four magnon branches are gapped, two of which are found sonance and with kilogram-year exposure, they can have to have nonzero couplings to the axion wind. These −15 sensitivity to the DFSZ model and down to gaee ∼ 10 , correspond to the resonances at 12 and 44 meV in Fig. 5. shown in Fig. 5. Realizing the exciting potential of discovering 3. Nondegenerate g factors Oð1–100Þ meV axion DM via phonons and magnons Finally, we consider coupling the axion to gapped hinges upon the ongoing effort to achieve low threshold magnon modes in the presence of nondegenerate g factors. single quanta detection. One possibility is to evaporate We are not aware of a well-characterized material with helium atoms from phonon interactions at the surface of nondegenerate g factors so, as a proof of principle, we the crystal and then detect the evaporated helium atoms entertain a few toy models, where a nondegenerate l in a region separated from the magnetic field region; component is added to the effective spins S in YIG. In research and development is underway for this direction. 3 Another route is to read out photons produced from the reality, all the magnetic ions Fe þ in YIG have ðl;s;SÞ¼ decay of a phonon polariton. Single photon detectors ð0; 5=2; 5=2Þ; the orbital angular momenta of 3d electrons (e.g., superconducting nanowires) may operate in a field- are quenched. In Fig. 5, we show the reach for two toy free region, away from the crystal target and connected to models, with either the octahedral sites or the tetrahedral it via a waveguide. Finally, magnons are read out in a sites modified to have ðl;s;SÞ¼ð1; 5=2; 7=2Þ. In each resonant cavity in the QUAX setup in the classical regime case, only one of the 19 gapped magnon modes, at 7 and [17–19,24,87,88]. Work is underway to detect single 76 meV, respectively, is found to contribute to axion magnons in YIG by coupling cavity modes to a super- absorption. This is because, to preserve the lattice sym- conducting qubit [89], though as in other resonant cavity metries, we have modified all the effective spin composi- searches, axion masses are best produced near the inverse tions on tetrahedral or octahedral sites in the same way. cavity size; for larger axion masses the virtual cavity The sole gapped mode that couples to axion DM corre- modes will be off shell and readout efficiency is sup- sponds to out-of-phase precessions of the tetrahedral and pressed. On the materials side, we would like to make octahedral spins. more accurate predictions for the detection rates via an improved understanding of phonon and magnon reso- VI. CONCLUSIONS nance line shapes and explore the possibility of scanning In this paper we showed multiple ways collective the resonance frequencies by engineering material proper- excitations in crystal targets are, in principle, sensitive to ties in order to fully exploit the discovery potential of an QCD axion DM. Specifically, we identified two novel axion DM search experiment based on phonon and detection possibilities: axion-induced phonon polariton magnon excitations. excitation in an external magnetic field, and axion-induced magnon excitation in the absence of external fields. These ACKNOWLEDGMENTS paths are complementary as each probes a different axion We thank Rana Adhikari, Maurice Garcia-Sciveres, coupling, gaγγ and gaee in the phonon case and magnon Sin´ead Griffin, Thomas Harrelson, David Hsieh, Stephen case, respectively. Lyon, Matt Pyle, and Thomas Schenkel for discussions.

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This material is based upon work supported by the U.S. APPENDIX A: PHOTON-PHONON MIXING Department of Energy, Office of Science, Office of High Energy Physics, under Award No. DE-AC02-05CH11231. As outlined in Sec. II A, mixing between long- A. M., T. T., Z. Z., and K. Z. are supported by the Quantum wavelength photon and phonon states needs to be taken Information Science Enabled Discovery (QuantISED) for into account when computing DM absorption rates in a High Energy Physics (KA2401032). Z. Z.’s work was also polar crystal. Our starting point is the Lagrangian for an supported in part by the NSF Grant No. PHY-1638509. ionic lattice coupling to electromagnetism:

  Z   X 1 1 X 1 1 2 ð2Þ 0 3 μν μν L m u_ − u · V 0 0 · u 0 0 eE x · Z · u d x − F Fμν AμΠ Aν : A1 ¼ 2 j lj 2 lj lj;l j l j þ ð ljÞ j lj þ 4 þ 2 ð Þ lj l0j0

In the long wavelength limit, the leading electromagnetic coupling is via the electric dipole, as shown in theR third term. _ 3 μ PluggingR in E ¼ −∇A0 − A and integrating by parts, we can write it in the familiar form of − d xJ Aμ ¼ 3 d xð−ρA0 þ J · AÞ, with X X ρ − ∇δð3Þ − 0 Z Z _ δð3Þ − 0 ðxÞ¼ e ð ðx xljÞÞ · j · ulj; JðxÞ¼e j · ulj ðx xljÞ: ðA2Þ lj lj when expanded to linear order in u. The last term in Eq. (A1) results from integrating out electron response. As explained in detail in Ref. [58], the photon self-energy Πμν can be related to the dielectric tensor of the medium, ε (taken to be the electronic contribution, usually denoted by ε∞, in the present case), and the photon Lagrangian can be written as 1 1 1 1 1 1 − μν Πμν _ ε _ − ∇ ε ∇ _ ∇ ε − ∂ j 2 ∇ 2 4 F Fμν þ 2 Aμ Aν ¼ 2 A · · A 2 A0ð · · ÞA0 þ A0ð · · AÞ 2 ð iA Þ þ 2 ð · AÞ : ðA3Þ

It is convenient to choose a generalized Coulomb gauge, ∇ · ε · A ¼ 0, and since the A0 field is nondynamical, it can be immediately integrated out. We thus obtain X 1 1 X 2 0 L m u_ − u V 0 0 u 0 0 eA x Z u_ ¼ 2 j lj 2 lj · ll jj · l j þ ð ljÞ · j · lj lj ll0jj0 Z   1 1 1 1 1 þ d3x A_ · ε · A_ − ð∂ AjÞ2 þ ð∇ · AÞ2 þ ρ ρ : ðA4Þ 2 2 i 2 2 ∇ · ε · ∇

To derive the Hamiltonian, we note the canonical momenta are these:

∂L ∂L p m u_ eA x0 Z; P x ε A_ x : lj ¼ ∂ _ ¼ j lj þ ð ljÞ · j ð Þ¼ _ ¼ · ð Þ ðA5Þ ulj ∂AðxÞ

Therefore, X Z _ 3 _ − H ¼ plj · ulj þ d xP · A L ¼ Hph þ HCoulomb þ HEM þ Hmix; ðA6Þ lj where

X 2 X plj 1 H ¼ þ u · V 0 0 · u 0 0 ; ðA7Þ ph 2m 2 lj ll jj l j lj j ll0jj0 Z 1 1 H ¼ − d3xρ ρ; ðA8Þ Coulomb 2 ∇ · ε · ∇ Z   1 X 1 H d3x P ε−1 P ∂ Aj 2 − ∇ A 2 e2 δð3Þ x − x0 A Z 2 ; EM ¼ 2 · · þð i Þ ð · Þ þ ð ljÞð · j Þ ðA9Þ lj mj

095005-13 MITRIDATE, TRICKLE, ZHANG, and ZUREK PHYS. REV. D 102, 095005 (2020) Z   d3k 1 1 X 1 ˜ ε−1 ˜ ˜ K2 ˜ − 0 Z HEM ¼ 3 PðkÞ · · PðkÞþ AðkÞ · · AðkÞ ; Hmix ¼ e AðxljÞ · j · plj: ðA10Þ ð2πÞ 2 2 lj mj ðA16Þ Note that while we have written A and P as three-vectors, where they are implicitly assumed to satisfy the gauge condition. This means that, in Eq. (A9), ε should be projected onto the 2 e X ZZT subspace satisfying the gauge condition before the inverse K2 ¼ k21 þ j j − kk: ðA17Þ Ω m is taken. j j We now consider the four terms in turn. First, as ˜ mentioned in Sec. II A, we use the PHONOPY code with We decompose the photon field A into two orthogonal 2 Vð Þ linear polarizations e1;k ⊥ e2;k, which satisfy the gauge DFT calculations of the force constants lj;l0j0 to diago- condition, k · ε · eλ ¼ 0 (λ ¼ 1, 2). We choose the basis nalize the lattice (phonon) Hamiltonian H , giving ;k ph in which the projection of ε onto the two-dimensional subspace, eλ0 · ε · eλ, is diagonal, with eigenvalues ε1, ε2. X3n X K2 ω ˆ† ˆ Denote the projection of in this basis by Hph ¼ ν;kaν;kaν;k: ðA11Þ ν 1 ¼ k K2 ≡ 2 eλ0;k · · eλ;k Kλ0λ: ðA18Þ Second, the Coulomb term, H , becomes more Coulomb We introduce phonon creation and annihilation operators transparent when written as a momentum space integral: ˆ† ˆ bλ;k, bλ;k satisfying the usual (discretized) commutation Z † 3 2 ˆ ˆ δ δ 1 d k jρ˜ðkÞj relations, ½bλ;k; bλ0;k0 ¼ λ;λ0 k;k0 , etc., based on the diagonal H ¼ ; ðA12Þ Coulomb 2 ð2πÞ3 k · ε · k piece of the Hamiltonian: pffiffiffiffiffiffiffiffiX 1 ˜ Ω qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆ ˆ† with the charge density AðkÞ¼ N ðbλ;k þ bλ;−kÞeλ;k; ðA19Þ 1=2 λ 2ε Kλλ Z X λ 3 − −ik·x0 ρ˜ ik·xρ − Z lj sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðkÞ¼ d xe ðxÞ¼ ie k · j · ulje : 1 2 lj ffiffiffiffiffiffiffiffiX ε = 1 ˜ p λ Kλλ ˆ ˆ† P k NΩ bλ − b eλ : ð Þ¼ 2 ð ;k λ;−kÞ ;k ðA20Þ ðA13Þ λ i

ˆ ˆ† The photon Hamiltonian then becomes Expanding ulj in ðaν;k0 þ aν;−k0 Þ as in Eq. (11) and sum- 0  ming over l picks out the kR ¼ k modes.P With the X X2 2 d3k 1 Kλλ ˆ† ˆ K12 momentum integral discretized, 3 → , we find ffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2πÞ NΩ k HEM ¼ p bλ;kbλ;k þ pffiffiffiffiffiffiffiffiffi ελ 2 ε1ε2K11K22 k λ¼1  2 X ξ ξ e 1 ðk · ν0 Þðk · ν;kÞ ˆ ˆ† ˆ ˆ† ;k × ðb1 − þ b1 Þðb2 þ b2 − Þ ; ðA21Þ HCoulomb ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; k ;k ;k ; k 4Ω ων0 ων k · ε · k ν;ν0;k ;k ;k † † ˆ ˆ ˆ ˆ where we have used K12 ¼ K21. × ðaν0;−k þ aν0;kÞðaν;k þ aν;−kÞ; ðA14Þ Finally, the photon-phonon mixing term Hmix can be written in terms of the creation and annihilation operators where according to Eqs. (12) and (A19): X 1 0 ξ ≡ Z ϵ X X ik·xlj ν;k pffiffiffiffiffiffi · ν;k;j: ðA15Þ e e m j H ¼ − A˜ ðkÞ · Z · p j j mix NΩ m j lj k lj j sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X X3n X2 As a technical note, while there is an option in PHONOPY ie ων;k pffiffiffiffi eλ − · ξν (nonanalytic correction) to also include H in the ¼ 1=2 ð ; k ;kÞ Coulomb 2 Ω ν 1 λ 1 ε Kλλ diagonalization calculation, it seems to work only at k ≳ ω. k ¼ ¼ λ † † PHONOPY ˆ − ˆ ˆ ˆ Therefore, in our calculation, we use to diago- × ðaν;k aν;−kÞðbλ;−k þ bλ;kÞ: ðA22Þ nalize only Hph, and include HCoulomb separately. Next, we also write the photon Hamiltonian HEM in The total quadratic Hamiltonian, given by the sum of momentum space: Eqs. (A11), (A14), (A21), and (A22), involves the 3n

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R0 R 0 phonon and two photon creation/annihilation operators, Slj ¼ ðxljÞ · j · Slj; ðB1Þ ˆð†Þ ˆð†Þ aν¼ð1;…;3nÞ;k, bλ¼ð1;2Þ;k. For simplicity, let us write † † where the additional rotation R0ðx Þ brings to a reference aˆð Þ ≡ bˆð Þ . The quadratic Hamiltonian lj ν¼ð3nþ1;3nþ2Þ;k λ¼ð1;2Þ;k frame where the spin orientation looks the same in all the then has the form: unit cells (for commensurate ordered materials, R0 ¼ 1). X ˆ † Following the Holstein-Primakoff transformation, Eq. (21), H ¼ ak · hk · ak with we can rewrite Eq. (B1) as, at leading order, ∈1 k BZ rffiffiffiffiffi  † † T a ¼½aˆ1 ; …; aˆ3 2 ; aˆ ; …; aˆ : ðA23Þ S † † k ;k nþ ;k 1;−k 3nþ2;−k S R0 x j raˆ r aˆ t S − aˆ aˆ ; lj ¼ ð ljÞ 2 ð j lj þ j ljÞþ jð j lj ljÞ The matrix h can be written as: k ðB2Þ 0 † 1 Ak Bk Ak Bk where tj is a unit vector pointing along the direction of the B T T C B B C −B C C B k k k k C jth spin in the rotating frame (i.e., the reference frame hk ¼ @ A; ðA24Þ R0 Ak B−k Ak B−k defined by the rotation ), while rj and rj span an T − T orthogonal coordinate system. These vectors are related Bk Ck Bk Ck to the components of the matrix Rj by where A is a 3n × 3n matrix given by k α Rα1 Rα2 α Rα3 rj ¼ j þ i j ; tj ¼ j : ðB3Þ 2 1 ðk · ξ Þðk · ξν0 kÞ ω δ e ν;k ; Ak;νν0 ¼ ν νν0 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 4Ω ων;kων0;k k · ε · k By substituting Eq. (B2) into Eq. (19), and going to momentum space, we obtain the following expression ðA25Þ for the quadratic part of the Hamiltonian X while Bk is a 3n × 2 with the following structure † Hˆ ¼ a · h · a with sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k k k X2 k∈1BZ ie ων k − ffiffiffiffi ; ξ † † T Bk;νν0 ¼ p 1 2 ð ν;k · eν0;−kÞ; ðA26Þ ˆ … ˆ ˆ … ˆ 4 Ω ε = ak ¼½a1;k; ; an;k; a1;−k; ; an;−k : ðB4Þ λ¼1 ν0 Kν0ν0

The matrix hk can be written in terms of n × n sub- and C νν0 is a 2 × 2 matrix given by k; matrices as 2   Kνν0 A − CB C νν0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ðA27Þ k k k; 4 pε ε h ¼ ; ðB5Þ ν ν0 KννKν0ν0 k † − Bk A−k C

We review a general algorithm for diagonalizing such 0 0 T 0 where, by defining J 0 0 ¼ R ðx Þ · J 0 0 · R ðx 0 0 Þ,we Hamiltonians in Appendix C. lj;l j lj lj;l j l j have pffiffiffiffiffiffiffiffiffiffi S S 0 1 APPENDIX B: GENERAL FORM OF THE † j j T 0 T A 0 A 0 r · J 0 · r 0 − μ δ 0 B · g · t ; k;jj ¼ k;jj ¼ 2 j −k;jj j 2 B jj j j MAGNON HAMILTONIAN pffiffiffiffiffiffiffiffiffiffi S S 0 In this Appendix, by following Ref. [66], we review the j j T J0 Bk;jj0 ¼ Bk;jj0 ¼ rj · −k;jj0 · rj0 ; procedure to derive the quadratic Hamiltonian describing X 2 δ T J0 0 small fluctuations around the ground state of the spin lattice Ck;jj0 ¼ jj0 Sltj · jlð Þ · tl: ðB6Þ 6 described by Eq. (19). The results we review will apply to l both commensurate and single-Q incommensurate materi- The procedure to diagonalize Hamiltonian of this kind is als (such as Ba3NbFe3Si2O14 discussed in Sec. III). We reviewed in Appendix C. conclude the Appendix giving the generalization of the rate We conclude by generalizing to the incommensurate in Eq. (27) for the case of incommensurate materials. case the formula for magnon production given in Eq. (27). To include in our discussion the case of single-Q incom- We start by writing the rotation matrix R0ðx Þ in terms of mensurate materials, we need to generalize Eq. (20) to lj the propagation vector that characterizes the incommensu- rate order Q ¼ðτ1; τ2; τ3Þ: 6We adopt a different phase convention compared to ik·xlj ik·xl Ref. [66], by using e as oppose to e , in the Fourier R0 R transform Eq. (22). ðxljÞ¼ ðnjQ · xljÞ; ðB7Þ

095005-15 MITRIDATE, TRICKLE, ZHANG, and ZUREK PHYS. REV. D 102, 095005 (2020) where RðnjφÞ is the rotation matrix around the unit vector n by an angle φ:

R φ iφ 1 − n − T T R0 R0 iφ 0 −iφ ðnj Þ¼Re½e ð i × nn Þ þ nn ¼ 0 þ þe þ R−e ; ðB8Þ 0 1 0 −nz ny 1 B C R0 nnT; R0 1 ∓ in − nnT ; n @ n 0 −n A; 0 ¼ ¼ 2 ð × Þ × ¼ z x ðB9Þ −ny nx 0 n noting that × · v ¼ n × v for any vector v. Then, by using Eq. (B1) and the interaction given by Eq. (4), we find rffiffiffiffiffiffiffi X ˆ Sj −ik·x 0 ν; k δH0 0 e lj f R x ν − r r ; h j j i¼ 2 j · ð ljÞ · ðV j ; k j þ Ujν;k jÞ lj N rffiffiffiffi X pffiffiffiffiffi N 0 0 0 S f δ 0R δ R δ − R− ν − r r ; ¼ 2 j j · ð k; 0 þ k;Q þ þ k; Q Þ · ðV j ; k j þ Ujν;k jÞ ðB10Þ j from which we can obtain the generalized expression for the rate:

Xn X1 X 2 2ω ων λ γν λ pffiffiffiffiffi ; Q ; Q TR0 RQ≠0 ¼ 2 2 2 2 Sj f j λðV jν;−λQrj þ Ujν;λQrjÞ : ðB11Þ m ω − ω ωγν λ cell ν¼1 λ¼−1 ð ν;λQÞ þð ; QÞ j

APPENDIX C: DIAGONALIZATION OF equivalent to Eq. (14) and Eq. (23), with the U and V QUADRATIC HAMILTONIANS matrices implicitly defined as In this Appendix, by closely following Refs. [66,67],we   review the procedure to diagonalize quadratic Hamiltonians Ujν;k V jν;k : which have the form of Eqs. (A23) and (B4). The goal T k ¼ ðC5Þ V jν;−k Ujν;−k of the procedure is to find a homogeneous linear trans- formation Such diagonalization procedure (usually called para- 0 ˆ0 ˆ0 ˆ0† ˆ0† T unitary diagonalization) can be achieved if hðkÞ is ak ¼ T k · a ≡ T k · ½a1 ; …; a ; a1 − ; …; a − ; k ;k n;k ; k n; k positive definite. If the spectrum contains zero energy ðC1Þ modes, the hðkÞ matrix will be positive semidefinite and such paraunitary diagonalization may not exist. This ˆ0 ˆ0† such that the operators aν;k, aν;k satisfy the canonical problem can be cured by adding a small ϵ value to commutation relations the diagonal components of hðkÞ [67]. This introduces a   small and negligible shift in the spectrum but makes hðkÞ 1 0 positive definite and allows for a paraunitary diagonal- ½a0 ; a0 †¼ ≡ g; ðC2Þ k k 0 −1 ization. Reference [67] provides a simple three-step algorithm to find the linear transformation T and the and that rewrites the quadratic Hamiltonian as associated eigenvalues: X X (1) A Cholesky decomposition is applied to find a ˆ † 0† 0 † H ¼ akhkak ¼ a k · E · ak; ðC3Þ complex matrix Kk such that hk ¼ KkKk. k k (2) The eigenvalue problem for the Hermitian matrix † where KkgKk is solved, and the resulting eigenvalues used to form the columns of the matrix Uk. The order of † 1 the columns is chosen such that the first N elements ≡ ω …ω ω …ω † Ek T k · hk · T k ¼ diagð 1;k n;k; 1;k n;kÞ; † 2 of the diagonalized matrix L ¼ U KkgKkU are positive and the last N negative. with ων;k > 0: ðC4Þ (3) Finally, the matrix Ek in Eq. (C4) is simply related By using the commutation relations in Eq. (C2), it can be to Lk by Ek ¼ gLk, and the T k matrix is given by −1 1=2 easily shown that, up to constant terms, Eq. (C3) is T k ¼ Kk UkEk .

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