An Effective Field Theory for Acoustic and Pseudo-Acoustic Phonons in Solids

We present a relativistic effective field theory for the interaction between acoustic and gapped phonons, in the limit of small gap. While the former are the Goldstone modes associated to the spontaneous breaking of spacetime symmetries, the latter are pseudo-Goldstones, associated to some (small) explicit breaking. We hence dub them"pseudo-acoustic"phonons. In this first investigation, we build our effective theory for the cases of one and two spatial dimensions, two atomic species, and assuming large distance isotropy. As an illustrative example, we show how the theory can be applied to compute the total lifetime of both acoustic and pseudo-acoustic phonons. This construction can find applications that range from the physics of bi-layer graphene to sub-GeV dark matter detectors.


I. INTRODUCTION
Many properties of solids are dictated by the dynamics of their simplest collective excitations: the phonons. These are localized vibrational modes that, when characterized by wavelengths much larger than the atomic spacing, can be described in terms of quasi-particles. In a solid with a single atom per unit cell the phonons' dispersion relation is gapless, i.e. when its wavevector vanishes so does its frequency. In this case one talks about "acoustic" phonons. However, for more complicated (and common) solids, some phonons can be gapped, with a frequency that tends to a finite positive value at zero wavevector. When the gap is large enough (at least comparable to the maximum frequency of the acoustic phonons), these modes are typically called "optical" phonons.
It is well known that the acoustic phonons are nothing but the Goldstone bosons associated to the spontaneous breaking of spatial translations induced by the solid background [1]. 1 Taking this idea as a starting point, recent years have witnessed the development of effective field theory (EFT) techniques, based on symmetry breaking and its consequences, applied to the study of collective excitations in different states of matter (see [3][4][5] and references therein). An EFT description of the phonon degrees of freedom, organized in a low energy/long wavelength expansion, has the advantage of being universal, i.e. not to rely on the often complicated microscopic physics, up to a finite number of effective coefficients. The latter must be obtained from experiment or determined in other ways, as for example Density Functional Theory calculations-see e.g. [6][7][8][9][10]. Such an EFT approach has already proven to be useful to a number of phenomenologically relevant problems, covering a wide range of fields, from the physics of 4 He to cosmology (see e.g. [11][12][13][14][15][16][17][18][19][20][21]).
In this paper we develop a new EFT for the description of acoustic and gapped phonons and their interactions in a solid, in the regime where the gap is small compared to the typical frequency characterizing the microscopic system. For reasons that will be clear soon, we dub these collective excitations as "pseudo-acoustic" phonons. To the best of our knowledge, no bottom-up effective description of pseudo-acoustic phonons has been presented so far. 2 The construction we present here can be applied to any number of spatial dimensions and any number of atomic species within the solid. However, as we will show, we expect a small gap for the pseudo-acoustic phonons to be achieved when the different species are weakly coupled to each other. Although this might be hard to envision in three-dimensional materials, it does find examples in two dimensional ones, most notably bi-layer graphene [7,[24][25][26][27][28][29][30], where the two sheets are coupled via van der Waals forces.
In this first study, we focus on solids that are both homogeneous and isotropic at large distances. While for large enough wavelengths the first property is always true, the second one is a simplifying assumption. The extension of our EFT to the case of solids which preserve only discrete rotations at large distance is straightforward, but tedious.
In constructing the EFT for acoustic and pseudoacoustic phonons we impose relativistic Lorentz invariance. Although this might not be common in solid state physics, there are reasons why this approach is worth exploring. First of all, it is technically easier to impose the Lorentz symmetry rather than the nonrelativistic Galilei one, by simply contracting covariant indices. Moreover, given that the Lorentz group is more fundamental, one is always free to require invariance under it, and hence allow our EFT to include relativistic effects on the phonon dynamics. The nonrelativistic limit can always be taken afterwards, by reintroducing the speed of light with simple dimensional analysis, and formally sending it to infinity (see [16] for an example). That being said, if the system of interest is nonrelativistic, there is no conceptual obstruction in imposing invariance under Galilei boosts right from the beginning.
Finally, the formulation of an EFT in relativistic language also provides a useful connection to high energy physics, for example enabling observables to be computed using techniques more familiar to this community. One promising application of this is in the use of EFTs for describing the interactions of sub-GeV dark matter with a detector: when the de Broglie wavelength of the dark matter is large, an effective description of the coupling between dark matter and collective modes is necessary. For example, the relativistic approach could provide useful insights into cancellation mechanisms that have been observed in low-mass dark matter scattering rates in various detector materials [21,[31][32][33]. 3

II. EFT FOR ACOUSTIC AND PSEUDO-ACOUSTIC PHONONS
We now describe the EFT for pseudo-acoustic and acoustic phonons. We will focus on the simple cases of one and two spatial dimensions, as well as two atomic species. The generalization of this setup is conceptually simple. In particular, we will use the one-dimensional case as a simple instance to illustrate our construction and test its consistency. We will then transport many of 3 The rate of dark matter scattering by pseudo-acoustic phonons is also subject to the coupling-to-mass suppression that was observed in the scattering by optical phonons [32].
these considerations to the two-dimensional case, which instead constitutes the most relevant part of our work. In Figure 1 we report a schematic representation of the different phonon modes.

A. 1D Case
Consider for the moment a monoatomic solid [4,5,16]. Its volume elements can be labeled by a single scalar field, φ(x)-the comoving coordinate-which at equilibrium can always be taken to be proportional to the physical spatial coordinate, φ(x) = αx, where α is a constant determining the degree of compression/dilation of the solid [34]. From an EFT viewpoint this vacuum expectation value (vev) breaks boosts and spatial translations. Since all solids are homogeneous at large enough distances, one also postulates an internal U (1) shift symmetry, φ → φ + c, which is broken together with part of the Poincaré group 4 down to time translations and a diagonal U (1), i.e. ISO(1, 1) × U (1) → R t × U (1). It is this last unbroken U (1) that one uses to define large distance homogeneity.
The fluctuation of the comoving coordinate around its equilibrium configuration, φ(x) = αx + π(x), is the Goldstone boson associated to the broken symmetries, and corresponds to the phonon of the solid. Since the symmetries are spontaneously broken, the dynamics of the phonon can be described via a Lagrangian that is invariant under the full unbroken group. In the long wavelength limit (i.e. at lowest order in a derivative expansion), the only quantity that is invariant under both the Poincaré group and the internal shift symmetry is X = ∂ µ φ∂ µ φ, and the most general Lagrangian is F (X), with F an a priori generic function. Upon inspecting the stress-energy tensor of the theory, one finds that F is nothing but minus the energy density [4]. For a strongly coupled system its analytical expression is hard (or even impossible) to compute, and one must obtain it from experimental or numerical data.
Expanding the action in small fluctuations one obtains all possible interactions for the acoustic phonon which, being a Goldstone boson, is gapless-see e.g. [4] for details.
Let us now consider a second atomic species in our solid. One can introduce two comoving coordinates, φ A,B (x), one for each species, featuring two independent shift symmetries. At equilibrium both of them are proportional to space, φ A (x) = αx and φ B (x) = βx, and the symmetry breaking pattern is then (1). Despite the number of broken generators, the above symmetry breaking pattern leads to only two Gold-stones, as dictated by the inverse Higgs constraints [35][36][37]. In the following we will set α = β = 1 for simplicity.
Analogously to the monoatomic case, at lowest derivative order one can build three quantities that are invariant under the Poincaré group and the two internal U (1)'s, i.e.
In the ideal case of two non-interacting solids the effective Lagrangian is simply F 1 (X 1 ) + F 2 (X 2 ). In the most general case, however, the two solids couple to each other and the action will depend on all the invariants in Eq. (1), describing two interacting acoustic phonons, both gapless. This is, however, not the end of the story. In fact, it is now possible to build one more quantity, ∆ = (φ A − φ B ) 2 , which explicitly breaks the initial U A (1) × U B (1) but preserves their final diagonal combination. 5 This operator generates a gap for one of the two degrees of freedom, which then becomes a pseudo-Goldstone boson, hence the name pseudo-acoustic phonon. 6 If such a gap is smaller than the UV cutoff, the pseudo-acoustic phonon can still be treated perturbatively within the EFT. 7 This means that, while the A and B solids can be separately arbitrarily strongly coupled at the microscopic level, we expect this regime to be achieved when they are weakly coupled to each other. A priori, the most general Lagrangian that incorporates a small explicit breaking (and hence a small gap) can be written as with δf f . However, for the most common systems we expect that, if the two solids are weakly coupled to each other, all interactions between the phonons of the two sectors will be weak, i.e.
where f A,B are (minus) the energy densities of the two solids in the limit where they are exactly decoupled.
Working with the most general action S = d 2 x F X i , ∆ , and expanding in small fluctuations, up to cubic order one gets where α, β, γ = A, B, and (...) represents symmetric indices. The effective couplings, g and y, are given in terms of the derivatives of F in Appendix A. In general, the coefficients of the quadratic terms depends on derivatives up to the second, those of the cubic ones up to the third, and so on. The spectrum of the theory is obtained by looking at the eigenmodes of the quadratic part of the action. For small momenta one gets the following dispersion relations for acoustic and pseudo-acoustic phonons: where the gap is given by The expression for c s and γ are also reported in Appendix A. The gap of the pseudo-acoustic phonon indeed goes to zero with vanishing g ∆ , this being the only parameter encoding the dependence of the theory on explicit breaking at quadratic order.
To make contact with physical systems, let us now show how the most general theory (4) can describe different instances of a linear diatomic chain. To do that, we will look at the following two example, and focus for now on the spectrum of the theory.
• Two non-interacting monoatomic chains with different atomic masses: A system of this kind is described, as already mentioned, by a Lagrangian as in Eq. (3) in the δf → 0 limit. At the level of the quadratic action this implies g AB = g AB = g ∆ = 0, and one obtains two gapless modes with two generically different sound speeds, c 2 A = g AA /g AA and c 2 B = g BB /g BB . Note that since the q → 0 and δ → 0 limits do not commute, one cannot obtain the above sound speed as a limit of the dispersion relations (5).
• Two identical chains with weak coupling between them: Being the two separate chain identical, the system is obtained from the action (3) imposing symmetry under X 1 ↔ X 2 , which implies g In this case the small gap survives, but γ c 2 s .
Before proceeding, let us briefly discuss how the effective couplings of the theory can be determined from the static properties of the solid, e.g. by experiment or numerical simulations. It is clear the the structure of the action (4) could have been found also by simply writing down all possible interactions compatible with the unbroken symmetries. Nevertheless, to express the couplings in terms of derivatives of the Lagrangian with respect to the invariants allows to determine them in terms of the nonlinear stress-strain curve of the solid [18]. Imagine for example, statically stretching or compressing only one of the solids, say solid A, while keeping the other at its equilibrium configuration. This corresponds to exciting a time-independent mode π A (x) while keeping π B = 0, i.e. a deformation of solid A. Clearly, the mode π A (x) must have a nontrivial spatial gradient, otherwise it would simply correspond to a global U A (1) transformation, which does not affect the system. At linear order in the deformation π A , this induces a variation in X 1 , X 3 , while X 2 remains unchanged. Exciting a mode π B (x) has the same effect but with X 1 ↔ X 2 . If instead we statically deform the two solids in opposite directions, π A (x) = −π B (x), this will generate a variation in X 1 , X 2 , but not in X 3 .
Recalling that the Lagrangian, F (X i , ∆), is minus the energy density of the solid, one deduces that the effective couplings can be obtained by studying the nonlinear response of the energy density under the mechanical deformations described above. For example, by measuring the linear change in the free energy following the deformations described above one can determine the first derivatives of the Lagrangian with respect to the X i invariants. To obtain higher derivatives of the Lagrangian with respect to X i , as well as the dependence on ∆, one can study the the nonlinear response.

B. 2D Case
Building on the results of the previous section, we now describe the case of a two-dimensional diatomic solid. This presents no conceptual novelty with respect to the previous section, but it does involve some technical aspects worth addressing.
In two spatial dimensions the comoving coordinates are described by two scalar fields for each species of solid, φ I α (x) with I = 1, 2 and α = A, B. At equilibrium they can be aligned to the physical coordinates, i.e. they acquire the vev φ I α (x) = x I . This again breaks boosts and spatial translations, but also spatial rotations. If, beside homogeneity, one also restricts oneself to solids that are isotropic at large distances, then it is necessary to impose an internal ISO(2) symmetry [4]. Under this, the comoving coordinates transform as (2) matrix. This internal Euclidean group is again spontaneously broken, but a diagonal combination of it with the spacetime Euclidean group is preserved, Let us now build all the independent operators that are invariant under the symmetry group. Imposing first Poincaré and shift invariance one obtains the following matrices The B matrices transform linearly under the initial where O α is an orthogonal matrix belonging to SO α (2). The matrix Σ, instead, only transforms linearly under the final unbroken SO (2), and is only invariant under the unbroken U (1).
We now need to build operators that are also invariant under internal rotations. There is a total of eleven independent invariants. 8 The following eight are invariant under the full SO A (2) × SO B (2) group, and hence are compatible with its spontaneous breaking: Here with [ . . . ] we represent the trace. The remaining three operators are only invariant under the unbroken group, and hence explicitly break the initial symmetry: The expression for ∆ 3 has been chosen so that it does not contribute to the quadratic action. Again, the phonons will be the fluctuations of the comoving coordinates around their equilibrium configuration, φ I α (x) = x I + π I α (x), and the action for their dynamics is S = d 3 x F (X i , ∆ i ), which can be determined by the nonlinear response of the system to shear and stress, as discussed in the previous section. We can now expand the action up to cubic order in small fluctuations. Moreover, we perform a field redefinition, π α = O αβ S βγ χ γ , where S is a matrix that brings the temporal kinetic term to its canonical form, and O an orthogonal matrix that diagonalizes the mass term. The result is with M 2 = diag(0, δ 2 ). As in the 1D case, the gap δ vanishes when the effective Lagrangian is independent of ∆ 2 , while the couplings λ (8) and λ (9) vanish with vanishing ∆ i dependence. The expressions for the effective coefficients in terms of the derivatives of the energy density are very cumbersome and we will spare them to the reader. Nonetheless, they can be obtained straightforwardly with simple linear algebra starting from the original Lagrangian.
To read off the dispersion relation, it is convenient to split the fields into longitudinal and transverse components, χ α = α + t α , such that ∇ i j α = ∇ j i α and ∇ · t α = 0. By looking at the quadratic action, it is simple to show that the dispersion relations one obtains are BB and γ T ≡ K (2) BB . Interestingly, as far as the pseudo-acoustic modes are concerned, the longitudinal and transverse ones share the same gap. This is a consequence of the isotropic approximation, which forces the mass term to be proportional to the identity, (M 2 ) ij αβ = M 2 αβ δ ij . Nonetheless, this is also what happens, for example, in bi-layer graphene [29], which features an hexagonal symmetry. Indeed, in two spatial dimensions, when the discrete rotation symmetry is larger than a simple reflection, the quadratic action is identical to that of a fully isotropic material. 9

III. PHONON DECAY IN 2D
We can now employ our EFT to analytically compute the decay rates of both acoustic and pseudo-acoustic phonons in the large wavelength limit, which are directly related to the thermal conductivity of the material [39]. In particular, we can quantize the χ α fields and use them to evaluate the relativistic matrix elements starting from the action (10). The phase space is the standard relativistic one [40]. In Appendix B we report the details on the canonical quantization, which is slightly unusual given the non-diagonal form of the quadratic action.
Let us start with the decay of an acoustic phonon which, for a 2D material, requires some care. In fact, if the dispersion relations (11a) were exactly linear, the decay of a longitudinal (transverse) acoustic phonon into two longitudinal (transverse) acoustic phonons would produce decay products which are exactly collinear. In two spatial dimensions, the phase space for this configuration is singular, since it diverges as ∼ 1/θ, with θ the angle between the decay products. Now, while the matrix element for the T, ac → T, ac+T, ac decay vanishes when the outgoing phonons are collinear, the one for the L, ac → L, ac + L, ac does not, making the total rate formally divergent. This is cured recalling that the dispersion relation is not exactly linear, but it features a nontrivial curvature: ω L,ac (q) = c L q 1 + q 2 + . . . , (12) with q 2 1 at small momenta. From the EFT viewpoint, such a (small) curvature is due to operators with a higher number of derivatives, which we did not include in the lowest order action (10). Once this is taken into account, energy and momentum conservation forces the angle between the two outgoing phonons to be θ √ 6 (q − q 1 ), where q is the momentum of the decaying phonon and q 1 that of one of the final products. For this reason, at small momenta the L, ac + L, ac channel dominates the decay rate.
One can now use the action (10) and the canonical field (B1) to compute the three-phonon matrix elements, and obtain the following spin-averaged total decay rates 10 for an acoustic phonon of initial momentum q: Two comments are in order about the previous expression. First of all, one immediately sees that, when the gap grows, the third term in parenthesis can be neglected, and the rates becomes what one would obtain from an EFT for a single acoustic phonon [4], in agreement with the idea that the pseudo-acoustic phonon can be integrated out at large gap. Secondly, because of the considerations made above, the decay width of acoustic phonons in two spatial dimension is less suppressed at small momenta than what one would expect from naïve scaling which, instead, would suggest a ∼ q 4 behavior. This is a consequence of the well-known extra infrared divergences arising in low-dimensional systems.
Note also that our analysis applies to an ideal twodimensional system, since it only involves in-plane phonons. Out-of-plane modes in two spatial dimensions have been shown to have peculiar properties in absence of external strain, and to contribute sensibly to the decay rate of an acoustic phonon, modifying the behavior at small momenta from q 3 to q 0 [41,42]. Their dispersion relation in absence of strain is, in fact, gapless but quadratic, and they consequently contribute to a large part of the available phase space. Nonetheless, it has also been shown that in presence of an arbitrarily small strain, the dispersion relation quickly approaches a linear one, and a scaling of the decay rate like the one in Eq. (13) is observed [42].
We can now move to the spin-averaged decay rate for the pseudo-acoustic phonon. Given the large number of effective couplings, let us focus on the case of a 2D solid in which the two species, A and B, are identical, physically relevant to a description of bi-layer graphene. In analogy to what we explained in Section II A, the Lagrangian for such a system can be written as where we used the fact that the free energies of the two solids must be the same in absence of coupling. Moreover, since the system must respond in the same way to modes where either π A or π B is excited, the Lagrangian must be symmetric under the exchanges X 1 ↔ X 2 , X 4 ↔ X 5 and X 7 ↔ X 8 . It is simple to show that this implies that the effective couplings are symmetric under the exchange A ↔ B, as one might have guessed from the beginning. Remarkably, the quadratic action becomes diagonal, i.e. K (i) AB = 0, and one also finds c 2 λ = γ λ + O(δf )-in agreement, for example, with [29]. Moreover, the gap (squared) and all the couplings with mixed A, B indices must arise from the coupling between the two solids, and are therefore of order O(δf ). Using the Feynman rules in Appendix B one obtains the following decay rate for a pseudo-acoustic phonon at rest: Note the interesting fact that, to this order in small explicit breaking, the decay rate is independent on the gap itself.

IV. CONCLUSIONS
In this work we have presented a relativistic effective field theory for the description of the low-energy degrees of freedom of a solid made of two species, weakly coupled to each other. In this regime the system features two distinct types of excitations: acoustic and pseudo-acoustic phonons. The first are the Goldstone bosons associated to the spontaneous breaking of spacetime symmetries and are consequently gapless. The second are instead pseudo-Goldstone bosons, and are characterized by a small gap arising from a perturbative explicit breaking operator.
An EFT formulation of the problem has the important advantage of putting on a firm ground several properties of these collective excitations by connecting them to those universal features of the system that only depend on lowenergy/large distance physics. It also allows analytical control over the observables, which can be computed for a large class of solids, only via symmetry arguments.
From this viewpoint there are several open questions of both conceptual and phenomenological relevance. One of them is to understand the nature of out-of-plane modes from a low-energy perspective. These modes are gapless but feature a quadratic dispersion relation and, as already commented, contribute to an important fraction of the total decay rate of acoustic phonons [42] and, in turns, to the thermal conductivity of two-dimensional materials [39]. In a similar direction, it would be interesting to understand what is the contribution to the latter due to pseudo-acoustic phonons. The contribution of optical phonons is typically negligible because of their large gap. However, pseudo-acoustic phonons having a perturbative gap, they could play a relevant role. It would also be interesting to understand if our action (10) captures any of the features of true optical phonons, despite their large gap. Finally, the generalization of the present construction to systems which only preserve a discrete subgroup of rotations at large distances is clearly phenomenologically relevant. We leave these and other interesting questions for future work. Here we report the explicit expressions for the couplings and parameters of the EFT written in terms of the derivatives of the Lagrangian evaluated on the background configuration, φ = x. The subscripts indicate derivatives with respect to a given invariant. The effective couplings appearing in the action (4) are for the quadratic ones, and for the cubic ones. The parameters appearing in the dispersion relation for the acoustic and optical phonons in one spatial dimension, Eqs. (5), are instead

Appendix B: Canonical quantization and Feynman rules for the 2D solid
In this appendix we perform the canonical quantization for the fields χ α appearing in the action (10). Following the standard canonical quantization procedure, we expand them in creation and annihilation operators, and require that they satisfy the equations of motion. Since the quadratic action is in general non-diagonal, χ A and χ B obey a set of coupled linear differential equation and, therefore, both of them contain creation/annihilation operators for the acoustic and pseudo-acoustic phonons. We thus write where λ is the phonon's polarization (longitudinal/transverse) and f its 'flavor' (acoustic/pseudoacoustic), and a λf q is the annihilation operator, normalized so that: Moreover, λ q is a polarization vector, which for longitudinal and trasverse phonons is given respectively by L,i q =q i and T,i q = ijqj , such that they satify the completeness relation, λ λ,i q λ,j q = δ ij . To determine the overlap functions, C λf α (q), we first require for the fields to satisfy the equal-time commutation relations, [χ i α (t, x),χ j α (t, y)] = iδ (2)  (B3) Note that, in the case in which the two solids are the same-see Eq. (14)-one has K (i) A,B = 0, and the two fields interpolate a single mode each, i.e. the acoustic phonon for χ A and the pseudo-acoustic one for χ B .