General Formalism For Absorption Rate Calculations

DETECTABILITY OF AXION DARK MATTER WITH PHONON POLARITONS AND MAGNONS

6.2 General Formalism For Absorption Rate Calculations

We take f(v) to be a boosted Maxwell-Boltzmann distribution, f_χ^MB(v) = 1

N₀e^{−(v+ve)2/v02}Θ v_esc− |v+v_e|

, (6.8)

N₀ = π^3/2v₀²

v₀erf v_esc/v₀

− 2v_esc

√π exp −v_esc² /v²₀

. (6.9)

with parameters v0 = 230 km/s, v_e = 240 km/s, v_esc = 600 km/s. The local axion DM density ρa, which enters the effective couplings f_j (see Table 6.1), 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.

Phonon excitations

We begin by calculating the absorption rate from couplings to phonons. The target Hamilto- nian results from expanding the potential energy of the crystal around equilibrium positions of atoms:

Hˆ =X

lj

p²_lj 2m_j +1

2 X

ll⁰jj⁰

u_lj·Vll⁰jj⁰ ·u_l⁰_j⁰ +O u³

, (6.10)

wherep_lj =m_ju˙_lj, and the force constant matrices,Vll⁰jj⁰, can be calculated from ab initio density functional theory (DFT) methods [294–297].

To diagonalize the Hamiltonian, we expand the atomic displacements and their conjugate momenta in terms of canonical phonon modes:

u_lj =

3n

X

ν=1

X

k

1

p2N m_jω_ν,k ˆa_ν,k+ ˆa^†_ν,−k

e^ik·x0lj_ν,k,j, (6.11)

p_lj = i

3n

X

ν=1

X

k

rm_jω_ν,k

2N aˆ^†_ν,−k−ˆa_ν,k

e^ik·x0lj_ν,k,j, (6.12) where ν labels the phonon branch (of which there are 3n for a three-dimensional crystal with n atoms per primitive cell), k labels the phonon momentum within the 1BZ, N is the total number of primitive cells, m_j is the mass of the jth atom in the primitive cell, and ˆ

a^†_ν,k,ˆa_ν,k are the phonon creation and annihilation operators. The phonon energies ω_ν,k and eigenvectors _ν,k,j = ^∗_ν,−k,j are obtained by solving the eigensystem of Vll⁰jj⁰, for which we use the open-source code phonopy [50]. The target Hamiltonian then reads

Hˆ =

3n

X

ν=1

X

k

ων,kˆa^†_ν,kˆaν,k+O ˆa³

. (6.13)

Figure 6.2: Dispersion of phonon polaritons in GaAs near the center of the 1BZ, k ∼ ω. The mixing between the photon and TO phonons is maximal at ω ∼k. At k ω, the TO phonon-like modes are degenerate with the LO phonon mode (blue line), while at ω k they approach their unperturbed value (dotted blue line), and an LO-TO splitting is present.

For a polar crystal, since the ions are electrically charged, some of the phonon modes mix with the photon. This mixing has a negligible impact in most of the 1BZ wherek ω, and in particular does not affect the DM scattering calculations in Refs. [6, 7, 28,64]. However, near the center of the 1BZ where k .ω – relevant for DM absorption – the photon-phonon mixing modifies the dispersions to avoid a level crossing. The true energy eigenstates are linear combinations of photon and phonon modes, known as phonon polaritons. This is shown in Fig.6.2 for gallium arsenide (GaAs) as a simple example. For an isotropic diatomic crystal like GaAs, the two degenerate, (mostly) transverse optical (TO) phonon modes at k ω continue to photon-like modes at k ω, and vice versa. The phonon-like modes at k ωdo not have the same energies as away from the polariton regime: the TO phonon-like modes become degenerate with the longitudinal optical (LO) phonon mode atω_LOask→0, whereas there is an LO-TO splitting, ω_TO 6=ω_LO, at k ω. For more complex crystals like sapphire (Al2O3), quartz (SiO2) and calcium tungstate (CaWO4), the mixing involves more phonon modes, and in general shift all their energies with respect to the eigenvalues ω_ν,k computed from diagonalizing just the lattice Hamiltonian.

To account for the photon-phonon mixing, we write the total Hamiltonian of electromagnetic fields coupling to the ions in the target crystal, and diagonalize its quadratic part via a Bogoliubov transformation. We explain this procedure in detail in AppendixG. The resulting

diagonal Hamiltonian is

Hˆ =

3n+2

X

ν=1

X

k

ω_ν,k⁰ ˆa^0†_ν,kaˆ⁰_ν,k+O ˆa⁰³

. (6.14)

At each k, there are (3n+ 2) modes, created (annihilated) by aˆ^0†_ν,k (ˆa⁰_ν,k), which are linear combinations of 3n phonon modes and 2 photon polarizations. Among them, 5 are gapless atk = 0, including 3 acoustic phonons and 2 photon-like polaritons. The number of gapped modes,3n−3, is the same as in the phonon-only theory, but their energy spectrum is shifted, {ων=(6,...,3n+2),k⁰ } 6={ων=(4,...,3n),k}. The original phonon modes are linear combinations of the phonon polariton eigenmodes:

ˆ a_ν,k =

3n+2

X

ν⁰=1

U_νν⁰_,kˆa⁰_ν0,k+V_νν⁰_,kˆa^0†_ν0,−k

. (6.15)

For DM coupling to the atomic displacements u_lj, the perturbing potential is given by Eq. (6.4) and therefore

δHˆ₀|0i=X

lj 3n

X

ν=1

X

k

e^{i(p−k)·x0lj} 1

p2N m_jω_ν,k f_j ·^∗_ν,k,j ˆaν,−k+ ˆa^†_ν,k

|0i

= rN

2

3n

X

ν=1

X

j

√ 1 mjων,p

f_j·^∗_ν,p,j ˆaν,−p+ ˆa^†_ν,p

|0i

= rN

2

3n

X

ν=1 3n+2

X

ν⁰=1

X

j

√ 1

m_jω_ν,pf_j ·^∗_ν,p,j U^∗_νν0,p+V_νν⁰,−p

|ν⁰,pi, (6.16) where |ν⁰,pi = ˆa^0†_ν0,p|0i. To arrive at the second equation, we have used the identity P

le^{i(p−k)·x0lj} =N δ_k,p (for k,p∈1BZ). It follows that hν,k|δHˆ₀|0i=δ_k,p

rN 2

3n

X

ν⁰=1

X

j

√ 1

m_jω_ν⁰_,p f_j·^∗_ν0,p,j U^∗_ν0ν,p+V_ν⁰ν,−p

, (6.17)

where we have swapped the dummy indices ν and ν⁰. The DM absorption rate per unit target mass, R=hΓi/(N m_cell), is therefore

R= 2ω m_cell

Z

d³v_af(v_a)

×

3n+2

X

ν=6

ω_ν,p⁰ γ_ν,p

(ω²−ω_ν,p⁰² )²+ (ωγ_ν,p)²

X

j 3n

X

ν⁰=1

√ 1

m_jω_ν⁰_,pf_j ·^∗_ν0,p,j U^∗_ν0ν,p+V_ν⁰_ν,−p

2

, (6.18) where ω =m_a, p = m_av_a, and m_cell is the total mass of the atoms in a primitive cell. For our numerical calculations, we use the phonopy code [50] to process DFT output [6, 7, 28]

to obtain the unmixed phonon energies and eigenvectors ω_ν⁰_,p, _ν⁰_,p,j as mentioned above, and then compute the polariton-corrected energy eigenvalues ω⁰_ν,p and mixing matricesU,V via the algorithm of Refs. [268, 298]. We relegate the technical details to Appendix G, and review the diagonalization algorithm [268,298] in AppendixI.

Magnon excitations

We now move to the case of magnons. The target Hamiltonian is a spin lattice model, with the following general form:

Hˆ = X

ll⁰jj⁰

S_lj·Jll⁰jj⁰ ·S_l⁰_j⁰ +µ_BB·X

lj

g_jS_lj, (6.19)

where l, l⁰ label the magnetic unit cells and j, j⁰ the magnetic ions inside the unit cell, µB = _2m^e

e is the Bohr magneton, B is an external uniform magnetic field, and gj are the magnetic ions’ Landé g-factors. In the simplest case of the Heisenberg model,Jll⁰jj⁰ ∝1 for pairs of lj and l⁰j⁰ on (nearest, next-to-nearest, etc.) neighboring sites. One material which is well described by this simple model is yttrium ion garnet (YIG) [264, 265], which has already been considered for DM detection [8,32,91–93,164]. However, as we will see below, materials with spin-spin interactions beyond the simplest Heisenberg type can be useful for enhancing DM-magnon couplings.

The spin-spin interactions in Eq. (6.19) can result in a ground state with magnetic order.

Here we focus on the simplest case of commensurate magnetic dipole orders, for which a rotation on each sublattice can takeS_lj to a local coordinate system where each spin points in the +ˆz direction:

S_lj =Rj·S⁰_lj, hS⁰_lji= (0, 0, S_j). (6.20) YIG and many other magnetic insulators have commensurate magnetic order. The calcu- lation can be easily generalized to single-Q incommensurate orders, as we discuss in Ap- pendix H.

For a magnetically ordered system, the lowest energy excitations are magnons. To obtain the canonical magnon modes, we first apply the Holstein-Primakoff transformation to write the Hamiltonian in terms of bosonic creation and annihilation operators:

S_lj⁰⁺= 2S_j−aˆ^†_ljˆa_lj1/2

ˆ

a_lj, S_lj⁰⁻ = ˆa^†_lj 2S_j −ˆa^†_ljaˆ_lj1/2

, S_lj^0z =S_j−aˆ^†_ljˆa_lj, (6.21) whereS_lj^0± =S_lj^0x±iS_lj^0y. The Holstein-Primakoff transformation ensures that the spin commu- tation relations [S_lj^0α, S_l^0β0j⁰] =δ_ll⁰δ_jj⁰i^αβγS_lj^0γ are preserved when the usual canonical commu- tation relations[ˆalj,ˆa^†_l0j⁰] =δll⁰δjj⁰ are imposed. As in the phonon case, translation symmetry

instructs us to go to momentum space:

ˆ

a_lj = 1

√N X

k

ˆ

a_j,ke^ik·x0lj, (6.22)

where k ∈ 1BZ. The quadratic Hamiltonian, whose detailed form can be found in Ap- pendixH, only couples modes with the same momentum, i.e.ˆa_j,k andaˆ_j⁰_,k, ˆa^†_j0,−k. A Bogoli- ubov transformation takes the quadratic Hamiltonian to the desired diagonal form:

Hˆ =

n

X

ν=1

X

k

ων,kˆa^0†_ν,kˆa⁰_ν,k+O(ˆa⁰³). (6.23) At each k, there are n magnon modes with n the number of spins per magnetic unit cell.

These energy eigenmodes are created (annihilated) by ˆa^0†_ν,k (ˆa⁰_ν,k), which are related to the unprimed creation and annihilation operators by

ˆ a_j,k =

n

X

ν=1

U_jν,kˆa⁰_ν,k+V_jν,kˆa^0†_ν,−k

. (6.24)

For DM coupling to the effective spinsS_lj, the interaction is given by Eq. (6.4), and we find, in complete analogy with Eq. (6.16),

δHˆ₀|0i = X

lj

X

k

e^{i(p−k)·x0lj} rSj

2N f_j ·(r^∗_jˆaj,−k+r_jˆa^†_j,k)|0i

= δ_k,p rN

2 X

j

pS_jf_j·(r^∗_jaˆj,−k+r_jˆa^†_j,k)|0i

= δ_k,p rN

2

n

X

ν=1

X

j

pS_jf_j· U^∗_jν,pr_j +Vjν,−pr^∗_j

|ν,pi, (6.25) where r_j ≡(R^xx_j , R^yx_j , R^zx_j ) +i(R^xy_j , R^yy_j , R^zy_j ), and |ν,pi= ˆa^0†_ν,p|0i. Therefore,

hν,k|δHˆ₀|0i=δ_k,p rN

2

n

X

ν=1

X

j

pS_jf_j· U^∗_jν,pr_j +Vjν,−pr^∗_j

. (6.26)

We can now obtain the DM absorption rate per unit target mass:

R = 2ω m_cell

Z

d³v_af(v_a)

n

X

ν=n0+1

ω_ν,pγ_ν,p

(ω²−ω_ν,p² )²+ (ωγ_ν,p)²

X

j

pS_jf_j· U^∗_jν,pr_j+Vjν,−pr^∗_j

2

, (6.27) where n₀ is the number of gapless modes, which depends on the material (in particular, on the symmetry breaking pattern). Similarity to the phonon formula Eq. (6.18) is appar- ent. We again use the algorithm of Refs. [268, 298], reviewed in Appendix I, to solve the diagonalization problem to obtain the magnon energiesων,p and mixing matrices U,V.