Higgs and Goldstone modes in crystalline solids
Marco Vallone






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Dipartimento di Elettronica e Telecomunicazioni, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy. 




In crystalline solids, the acoustic phonon can be described either as a Goldstone or as a non-Abelian gauge boson. However, the non-Abelianity of the related gauge group apparently makes the acoustic phonon a frequency-gapped mode, in contradiction with the other description. In a different perspective overcoming this contradiction, both acoustic and optical phonon – the latter never appearing following the other two approaches – emerge respectively as the gapless Goldstone (phase) and the gapped Higgs (amplitude) fluctuation mode of an order parameter arising from the spontaneous breaking of a global symmetry, without invoking the gauge principle.  In addition, the Higgs mechanism describes all the phonon-phonon interactions, including a possible perturbation of the acoustic phonon's frequency dispersion relation induced by the eventual optical phonon, a peculiar behavior able to produce mini-gaps inside the phonon Brillouin zone.


Higgs and Goldstone Modes in Crystalline Solids
    Marco Vallone
    December 30, 2023
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§ INTRODUCTION




The spontaneous breakdown of a continuous symmetry implies the emergence of a massless bosonic particle for each broken generator of the involved symmetry group. This is the Nambu-Goldstone theorem <cit.> in a nutshell, forming the basis of the Standard Model of the fundamental interactions. In addition, the Higgs mechanism <cit.>, described as a case of spontaneous symmetry breaking (SSB), plays a prominent role in the Standard Model, providing the mass for the gauge bosons of the electroweak interactions and for the fermions <cit.>. 

These concepts are seemingly far removed from physics of conventional semiconductors and metals commonly employed in electronics and optoelectronics industry. However, the Higgs mechanism, as described by P. W. Higgs in 1964 for particle physics <cit.>, is the relativistic analog of the plasmon phenomenon described by P. W. Anderson one year before in superconductivity <cit.>, and the existence of Anderson-Higgs modes in condensed matter physics  – in superconductors, cold-atoms in periodic lattices, in Bose-Einstein condensates, in antiferromagnets, in charge density waves <cit.> etc.  – seems pervasive. However, beside plasmons phenomenology <cit.> and related plasmonics <cit.>, presently the Higgs mechanism has been acknowledged in semiconductors physics only in somehow exotic materials, like e.g. in topological insulators, Weyl semimetals, cuprates <cit.>, etc., although its role could be much more diffuse even in standard semiconductors. 

On the other hand, the Goldstone theorem is very general and it holds also in the non-relativistic condensed state of matter, where the massless bosons correspond to collective excitations with wavevector k and gapless frequency dispersion relation ω_k, that is ω_k → 0→ 0, where k = |k|. As a few examples, spin waves in the Heisenberg model are bosons arising when the ground state of the Heisenberg Hamiltonian is magnetically ordered <cit.>; collective density excitations in superconductors arise from the spontaneous breaking of the electronic phase rotational U(1) symmetry <cit.>. Finally, as a more common example, in crystalline solids, collective excitations associated to lattice vibration modes are the acoustic phonons <cit.>, and correspond to Goldstone modes emerging from the breaking of a continuous spatial symmetry, the translational invariance, broken by the presence of the crystal lattice <cit.>. It is said that all these collective excitations originate from one of the so-called emergence principles, in this case the Goldstone theorem, since they emerge from the very beginning.

Nevertheless, in condensed matter systems as well as in high energy particle physics, interactions are mediated by gauge bosons, appearing when local gauge invariance with respect to a given symmetry group is requested for the system's Lagrangian density <cit.>. 


In a seemingly similar way, it has been shown <cit.> that by gauging the spatial translational group T(3) in crystal lattice, three gauge bosons appear to provide the local gauge invariance for the Lagrangian under the action of T(3). They can be identified with the three acoustic phonons and, as a major feature of this approach, the elastic properties of solids and the acoustic phonon's dynamical equations can be described in close analogy with General Relativity field equations. In fact, it turns out that the acoustic phonon travels in the crystal acting as a wavelike perturbation of the lattice, similarly to the graviton in vacuum, that travels as a wavelike perturbation of a locally flat differentiable manifold, both obeying very similar field equations <cit.>. Acoustic phonons arise in this case not as Goldstone, but as gauge bosons. 

A major concern regards whether the two descriptions could possibly be in contradiction and to what extent. Ref. [2014Dartora_JPA] describes in detail the linear limit of the acoustic phonon's gauge theory, where the two descriptions appear in agreement,  in particular providing in the long wavelength limit (k → 0) the same gapless  frequency dispersion law ω_k = c_s k, where c_s is the sound velocity in the given medium. However, the cited work does not explore in depth the consequences of the non-Abelianity of the involved gauge group on the ensuing dispersion relation ω_k. 

When more than one ion is present in the lattice elementary cell, another kind of phonon – the optical phonon <cit.> – constitutes a further and independent vibration mode. In polar semiconductors, the electron–longitudinal optical (LO) phonon emission is the dominant intersubband scattering mechanism responsible for electrons thermalization. However, although LO-phonons are of crucial importance in semiconductors transport theory and electron dynamics <cit.>, in the gauge theory of crystal lattice's interactions they are left apart, not arising as gauge bosons.  

In order to clarify these important points, in section <ref> we give a short recap about the emergence of acoustic phonon as Goldstone boson. In section <ref>, after an introduction about  the issues behind gauging a spatial symmetry, we go beyond the linear limit of the theory, finding that the obtained dispersion relation may result gapped (that is, ω_k → 0≠ 0), in sharp contrast with the gapless dispersion relation characteristic of Goldstone bosons.

In section <ref> we present a different and more general approach, showing that both acoustic and optical phonons may arise from a SSB which sets out an order parameter ϕ. It follows that the Higgs mechanism plays an important role: the amplitude fluctuations of ϕ are Higgs modes, for which a mass-like term appears in the Lagrangian, that makes the mode gapped, and we identify them with the optical phonons. Conversely, the phase fluctuations of ϕ (Goldstone modes) are the acoustic phonons, here revisited in a much more general way that also sheds light on the seeming contradiction arisen when interpreting the acoustic phonon either as Goldstone or as gauge boson. Finally, in section <ref> main ideas and findings are summarized.

In this work A^α and B_β are contravariant and covariant four-vectors, ∂_μ is the partial derivative ∂/∂ x^μ, where standard Greek indices α, β, μ,... = 0,... 3 are indices of space-time coordinates on a four-dimensional differentiable manifold with metric 
g_μν and connection Γ^α_μν, latin indices (i, j, k, ... = 1, 2, 3) mark spatial components, whereas Greek indices with an “hat” 
μ̂, ν̂,... =0, ...3 are used for indices of local four-dimensional frames (vierbeins or tetrad indices) on a flat Lorentzian space-time with Minkowski metric 
η_μ̂ν̂ = (+1, -1, -1, -1). 
The Einstein's summation over repeated index is always understood, and not-italicized “i” is the imaginary unit.




§ GOLDSTONE THEOREM AND ACOUSTIC PHONONS




Apart from superconductivity and exotic materials, things are quite complicated even in the well-known world of solid-state crystals. Let us consider the Lagrangian density

    ℒ = ψ^†_k∂_t ψ_k - 1/2 m^*∇ψ^†_k·∇ψ_k


describing the dynamics of a non-relativistic Pauli electron with wavefunction ψ_k(t,r) and effective mass m^*, free to move in the crystal and obeying the Bloch theorem <cit.>. ℒ is symmetric (i.e. invariant) under global, continuous transformations described by the Galilei group G <cit.> (the corresponding relativistic formulation of ℒ is Lorentz-invariant). If H is the group of time-translations, the generators of the quotient group G/H are the momentum p (translations), the angular momentum J (rotations), and the boosts K. Concerning the group of spatial translations T(3) in the ordinary space (one of the subgroups of G/H), the elements of the infinitesimal form of T(3) are the operators U_T = 1 - k·δR, where δR is an infinitesimal displacement of the crystal ions in space around a Bravais lattice <cit.> translation vector R. U_T may be written as U_T = 1 - ϵ^j p_j, where ϵ_j are real parameters, and the momentum components p_j = -ħ∂_j are the three generators of T(3), whose Lie algebra is described by [p_i , p_j] = 0. Similar considerations could be made for the other two subgroups.

The underlying crystal lattice can be described <cit.> as a potential -V(ϕ) to be inserted into ℒ and eventually depending from several fields ϕ = {ϕ^j}. If ϕ_0 is a the value of ϕ that minimizes V(ϕ), the system's ground state (the system's true vacuum) is the state for which ϕ = ϕ_0. 
We can expand the potential V around its minimum, obtaining at the second order 


    V(ϕ) = V(ϕ_0) + 1/2(ϕ - ϕ_0 )^k (ϕ - ϕ_0 )^j M_k j ,


where 


    M_k j = ( ∂^2 V/∂ϕ^k  ∂ϕ^j )_ϕ_0


is a symmetric matrix that in the Lagrangian plays the role of a mass term, whose eigenvalues give the eventual masses of the fields {ϕ^j}.

The crystal lattice breaks both translational and rotational invariance making the system's ground state not symmetric, and this can happen even if the Lagrangian is symmetric. To be more clear, if we again consider the translations, the fields ϕ^j transform under the action of U_T as ϕ^j →ϕ^j + α(ϕ) p^j, where α(ϕ) is an infinitesimal parameter. Nevertheless, the symmetry of the Lagrangian can remain exact, provided 


    V(ϕ^j) = V(ϕ^j + α(ϕ) p^j) .


Taylor expanding the Eq. (<ref>), the same condition can be written as 


    α(ϕ) ∂ V(ϕ)/∂ϕ^j = 0 ,


and differentiating the Eq. (<ref>) with respect to ϕ^k around the potential minimum ϕ_0, we obtain


    ( ∂α(ϕ)/∂ϕ^k∂ V/∂ϕ^j)_ϕ_0  + α(ϕ_0) ( ∂^2 V/∂ϕ^k  ∂ϕ^j )_ϕ_0 = 0 .


The first term vanishes, since ϕ_0 is a minimum of V. Regarding the second term, it can be zero if α(ϕ_0) = 0, but this would be a trivial case: the symmetry would be exact, V(ϕ) would be merely an arbitrary constant, it would not be necessary to introduce a field ϕ, and the crystal lattice simply would not exist. Instead, if α(ϕ_0) ≠ 0, it must be


    ( ∂^2 V/∂ϕ^k  ∂ϕ^j )_ϕ_0 = M_k j = 0 ,


stating that the field is massless.
In summary, the crystal lattice makes a massless field ϕ^j to arise, the Goldstone excitation associated to p^j. In the end, we expect three Goldstone bosons, one for each broken translations' generators, and the ground state's symmetry is said to be spontaneously broken. It is possible to see that apart from p, also J and K are broken generators (i.e. the system's ground state is not symmetric under the corresponding transformations they generate), but it has been shown that they do not give rise to Goldstone bosons <cit.>. Furthermore, it should be remarked that at this level nothing can be said about phonon-phonon interactions.

For uniformity, it is also possible to introduce a pseudo-relativistic notation, describing the acoustic phonons emerged by the Goldstone theorem as sound-like four-momenta p_μ states in a locally flat manifold ℳ with metric g_μν. The tangent space to any point of ℳ is a four-dimensional manifold T_ℳ with the same local pseudo-Minkowski metric expression η_μ̂,ν̂ as in special relativity, and known as acoustic or sound metric <cit.>. Space-time coordinates on ℳ are expressed as  x^μ =  (c_s t, r), and the world “pseudo” means that the relevant velocity is the sound speed c_s  in the considered medium (that of course is not a limit velocity).  

Following this convenient formalism, instead of T(3) we can consider the translations group T(4) with elements


    U_T = 1 - ϵ^μ p_μ


on the space-time ℳ, where the parameters ϵ^μ are point dependent. The four  T(4) generators are p_μ = -ħ∂_μ, and the crystal lattice breaks the three spatial components p_1...3, but not p_0. Since p_μ p^μ =0 (they are sound-like four-momenta in ℳ), a wave-like solution for them yields the gapless dispersion relation ω_k = c_s k, valid for small k, that makes the Goldstone bosons massless quasi-particles, as required <cit.>. The periodic structure of the lattice itself  is known to produce Brillouin-zone folding and the appearance of gaps in the phonon spectrum, i.e., phonon stop bands, for wave vectors satisfying the Bragg condition <cit.>. It must be stressed that the present approach does not address this aspect of the problem, since at this level the lattice is treated as a continuum, an approximation valid for wavevectors k ≪π/a, where a is the lattice constant.






§ CHALLENGES IN GAUGING SPATIAL SYMMETRY



The existence of Goldstone bosons is closely related to the gauge theories of interactions. The possibility to choose freely a local parameter without changing the physics of a system was declared in 1929 by H. Weyl as a general principle <cit.>, known as local gauge invariance. It turns out that each time a Lagrangian density ℒ(ψ_k) is requested to be invariant under the action of an element U_G of a Lie group G of local transformations ψ_k→ U_G ψ_k, one or more compensating fields arise, mediating an interaction (see the description for the classic case of electromagnetism e.g. in Ref. [2010Mandl], for which the compensating – or gauge – field is the vector potential A_μ that mediates the electromagnetic interaction, and the involved gauge group is the Lie group U(1), whose elements induce phase rotations on ψ_k).

In somehow similar way, it is tempting to describe acoustic phonons as gauge fields <cit.>, like in electromagnetism or, more generally, in Yang-Mills theories <cit.>, considering the local (coordinate-dependent) form of U_T, that is U_T = 1 - ϵ^j(r) p_j. Although U_T is unitary, the Lagrangian ℒ in the Eq. (<ref>) is manifestly not invariant under the action of U_T on ψ_k because of the spatial derivatives in the expression of the generators.
However, the naive pathway to request the Lagrangian ℒ to be invariant under local spatial translations promoting T(4) to the role of gauge group is not trivial. Considering its importance, before going further it is worth clarifying why gauging a spatial translation is so different from gauging e.g. a wavefunction's phase transformation.

In the gauge theory of electromagnetism, since the involved symmetry transformation is a phase rotation, it is said that an internal symmetry is gauged, because the transformation does not involve space-time coordinates. In addition, all this has a simple geometrical meaning: with reference to Figure <ref>, given three differentiable manifolds  ℳ (the base space), L(P) (the fiber) and ℰ (the total space), and a Lie group G (the structure group), a fiber bundle is defined as a topological structure (π :  ℰ→ℳ, L, G) where, for each point P ∈ℳ a fiber L(P) is associated to P, such that a neighborhood V_P of P is mapped in ℰ according to π^-1(V_P), and that π^-1(V_P) is homeomorphic to the Cartesian product V_P × L. 



The fiber bundle is defined in conjunction with a group G which acts as a transformation group on the fiber, representing the different ways the fiber can be viewed as equivalent. The covariant derivative D_μ is a connection on ℰ allowing to parallel transport <cit.> a vector in ℰ. We may also visualize the n-dimensional manifold ℳ as a n-surface, with an internal space L associated to each point P ∈ℳ with given topological structure. Specifically, in electromagnetism ℳ is the ordinary four-dimensional space-time of the special relativity, a phase transformation on ψ_k is a “vertical” automorphism of the bundle and a diffeomorphism of L, such that the fiber above each point in ℳ is unchanged (Figure <ref>(a)). In fact, since only the phase is changing, the point in the total space ℰ does move, but just “vertically”  along L(P). This is the gauge freedom carried by  the fiber bundle within its fibers.



By contrast, trying to gauge a space-time transformation group like T(4), it is quite clear that translations in ℳ induced by T(4) are “horizontal” diffeomorphisms of ℳ, since a translation by definition changes the point  P in the base space, involving spatial coordinates. For this reason, an invariance under the action of T(4) is said to be an external symmetry. In this context, it is not easy to identify a fiber as an internal space L for which  ℰ = ℳ× L locally holds, preventing a straighforward definition of a fiber bundle (π :  ℰ→ℳ, L, T(4)) (Figure <ref>(b)). 





 §.§ Gauge theory of spatial translations



Gauging T(4) in the pseudo-Lorentzian, locally flat manifold, equipped with the acoustic metric and possibly suitable to describe acoustic phonon dynamics, propagation and interactions, is mathematically not too different from the gauging of T(4) on the standard Lorentzian manifold of the General Relativity <cit.>. Nevertheless, the latter revealed a hard goal to obtain <cit.>. 

Everything in this field started from the pionieering work by R. Utiyama <cit.> in 1956, a good starting point, but not entirely convincing, especially because Utiyama did not address the central problem: translations are a diffeomorphism in ℳ, and not in a fiber, as a gauge theory would require, a problem not solved neither by subsequent works by T. Kibble <cit.> and D. Sciama <cit.>.
Following the usual Yang-Mills formalism, they defined the covariant derivative as D_ν̂ = ∂_ν̂ + W_ν̂, where W_ν̂ is a gauge (compensating) field that, written in terms of its components R_ν̂^μ, reads


    W_ν̂ = R_ν̂^μ p_μ .



Employing Eq. (<ref>), D_ν̂ was written as


    D_ν̂ = h_ν̂^μ∂_μ


where


    h_ν̂^μ = δ_ν̂^μ-  ħ R_ν̂^μ .


The field h_ν̂^μ defines a set of four orthonormal vectors h_ν̂^μ∂_μ, the tetrad or vierbein, provided the covariant index ν̂ is a local Lorentz index. 
The correct understanding of h_ν̂^μ was given by Y. M. Cho <cit.>, who developed a gauge theory of translations with a Yang-Mills–type Lagrangian, where the gauge potentials were correctly interpreted as translational connections (in particular, they are the nontrivial part of the vierbein fields h_ν̂^μ), and not as general coordinate transformations on the base manifold (as e.g. in Utiyama <cit.>), that would have been not correct. D. and G. Grensing <cit.> came to similar conclusions, obtaining the gauging of the Poincaré group in a form that allowed to express General Relativity as a gauge theory of this symmetry group. More recently, the Yang-Mills theory of the affine group (the semidirect product
of translations T(4) and general linear transformations GL(4,R)) was formulated <cit.>, where tetrads have been identified with nonlinear translational connections, for which the given h_ν̂^μ∂_μ expression is a simplified yet correct version of the general formulation.  

These concepts were not developed for solid-state physics, but rather to describe gravity, now correctly formulated as a Yang–Mills theory with T(4) as gauge group. Nevertheless, very interestingly the same formulation holds and can be exploited to develop a Yang-Mills theory of elasticity in solid crystals. In fact, Eqs.  (<ref>), (<ref>) and (<ref>) also hold when considering, instead of the standard space-time, the crystal lattice's manifold ℳ with metric g_μν, and with its pseudo-Minkowski tangent space T_ℳ equipped with the sonic metric η_μ̂ν̂. In this case, the field components R_ν̂^μ 
can be identified with the tensor components of the crystal elasticity field, whereas in theory of gravity they are related to the Ricci's tensor <cit.>. This allows to formulate the physics of lattice vibrations in crystals (acoustic phonons) in a way that results somehow similar to the physics of gravitational waves in the ordinary space-time, as described in part in Ref. [2014Dartora_JPA] and developed more in detail here in section <ref>.




 §.§ Yang-Mills theory of acoustic phonons in crystal



The Lie group T(4) can be promoted to the role of gauge group <cit.>, defining a gauge-covariant derivative D_μ̂ as a connection that makes ℒ gauge invariant, and consistent with the meaning expressed by Eq. (<ref>).  
Following the proposed pseudo-relativistic formalism and the standard gauge prescriptions, the infinitesimal displacements of the crystal lattice induce local translations ψ_k(x^μ) →ψ'_k(x^μ) = U_T(x^μ) ψ_k(x^μ) of the electronic wavefunction, where all the ordinary derivatives must be replaced by gauge-covariant derivatives D_ν̂= h_ν̂^μ∂_μ. The field W_ν̂ must transform in turn according to


    W_ν̂→ W'_ν̂ = U_T W_ν̂ U_T^† - (∂_ν̂ U_T) U_T^† ,


and Eq. (<ref>) implies 


    (D_ν̂ψ_k)' = U_T (D_ν̂ψ_k) ,


that makes the Lagrangian ℒ gauge invariant under the local action of T(4). 

The translational symmetry manifests itself through the occurrence of conserved currents j_μ. In the present case, the relevant gauge charges are the three generators p_k = ∫ d^3x    j_k(x) = ∫ d^3x   T_0 k(x) of the spatial translations, whose corresponding currents are the components of the symmetric energy-momentum tensor T_0 k. Furthermore, the total energy (Hamiltonian) operator H = ∫ d^3x   c_s^2   T_0 0(x) is written in terms of p_0, the unbroken generator of translations in the time coordinate. 

A major difference with respect to electromagnetism is the fact that the elements of T(4) do not commute, even if the group generators on a flat space-time commute, [p_μ̂, p_ν̂] = 0. In fact, if U_T(x) and U_T(y) are two elements of T(4), it is 


    [U_T(x), U_T(y)] = -ħ^2 [ϵ^μ(x)  ∂_μϵ^ν(y)  - ϵ^μ(y) ∂_μϵ^ν(x)   ]  ∂_ν ,


that in general is nonzero for x ≠ y, making non-Abelian the local form of the gauge group. The intrinsic difference between the structure of the global and local versions of T(4) can be stated more formally writing the currents algebra


    [ j_μ(x), j_ν(y)] = - ħ( ∂_μ j_ν(y) -  ∂_ν j_μ(x) ) δ^3(x-y) ,


where the commutator in general is nonzero for x ≠ y. Hence the generators' algebra can be written as <cit.>


    [p_μ , p_ν] =  g f^γ _μν p_γ


where f^γ _μν are the structure constants for the currents' Lie algebra and g is the coupling constant of the theory. It is f^γ _μμ = 0, and f^γ _μν = - f^γ _νμ = 1 for μ≠ν. In a flat space-time T_ℳ of course it is [p_μ̂ , p_ν̂]  = 0, but this is not true on ℳ.
This is a central point of the theory, and the algebra expressed by Eq. (<ref>) makes non-Abelian the local symmetry, and it is responsible for the phonon-phonon interactions. On this basis, it also follows


    [W_μ̂, W_ν̂] =  g f^α _βγ  R_μ̂^β R_ν̂^γ p_α


Making use of Eq. (<ref>), the commutator [D_μ̂, D_ν̂] can be written in the alternative forms


    [D_μ̂, D_ν̂] = G_μ̂ν̂^α p_α  = G_μ̂ν̂


where the field strength tensor G_μ̂ν̂ and its components G_μ̂ν̂^α are given by


    G_μ̂ν̂ =      ∂_μ̂ W_ν̂ - ∂_ν̂ W_μ̂ + [W_μ̂, W_ν̂]  
    
    G_μ̂ν̂^α =      ∂_μ̂ R_ν̂^α - ∂_ν̂ R_μ̂^α + g f^α_βγ  R_μ̂^β R_ν̂^γ .


It is worth noting that Eqs. (<ref>) resembles expressions typical of classical Yang-Mills theories, although it should not be forgotten that the symmetry group rules an external symmetry, and the gauge field W_μ̂ contains spatial partial derivatives, since the infinitesimal group generators act as differential operators. 

The Lagrangian density for the free gauge field is 


    ℒ_R = 1/4 G_μ̂ν̂^α G^μ̂ν̂_α ,


that explicitly reads


    ℒ_R =      1/2(∂_μ̂ R_ν̂^α∂^μ̂ R^ν̂_α  -  ∂_μ̂ R_ν̂^α∂^ν̂ R^μ̂_α)   
          +  g f_α^γδ   R^μ̂_γ R^ν̂_δ (∂_μ̂ R_ν̂^α)  
          - g^2/4 f^σ_βγ f_σ^δϵ R_μ̂^β R_ν̂^γ R^μ̂_δ R^ν̂_ϵ


and describes the dynamics of a field with cubic and quartic (self-interacting) terms. Writing the Eulero-Lagrange equation 


    ∂^μ̂∂ℒ_R/∂(∂^μ̂ R^ν̂_α) = ∂ℒ_R/∂ R^ν̂_α


and imposing the Lorenz gauge (that allows for great simplification of all the expressions), the equation of motion for the free field R_ν̂^α results (defining λ = g^2 and = ∂_μ̂∂^μ̂)


    R_ν̂^α + λ   f^σ_βγ f_σ^δα R_μ̂^β R_ν̂^γ R^μ̂_δ  = 0 .


This equation does not include any mass-term (it would be a term linear in R_ν̂^α), but a cubic self-interacting term is present, coming from the quartic term in the Lagrangian. 

All the components are highly coupled and difficult to manage. In order to understand the underlying physics, it is particularly interesting to consider in better detail the case of a simple one-dimensional atomic chain along x, assuming – without loosing generality – that the perturbation is along the longitudinal direction (longitudinal acoustic phonon), hence R_0^0 = 0. From Eq. (<ref>), that expresses the group commutation rules, the motion equations for the nonzero field components R̂ = R_0^1 (the longitudinal acoustic phonon field) and Ŝ = R_1^1 (the spatial strain field) decouple, yielding


    ( ∂^2_t  - c_s^2 ∂^2_x ) R̂ + λ c_s^2  |R̂|^2 R̂ = 0




    ( ∂^2_t  - c_s^2 ∂^2_x ) Ŝ = 0   .


Ŝ(x,t) describes the fluctuations of the crystal strain field, that in higher dimensional domains are coupled to phonons, which may get scattered by them. Regarding the equation for the acoustic phonon R̂(x,t), the parameter λ leads to a nonlinear oscillatory solution for R̂ with a dispersion relation given by <cit.> 


    ω_k^2 = √(λ c_s^2 /2)ρ_R^2 +  c_s^2 k^2


where ρ_R is an integration constant depending on the cell characteristics. Eq. (<ref>) can be derived by the Eulero-Lagrange equation from a Lagrangian that is a simplified form of Eq. (<ref>),


    ℒ_R = 1/2(η^μ̂ν̂∂_μ̂R  ∂_ν̂R - 1/2λ R^4 ) ,


the well-known Lagrangian with ϕ^4-potential, describing a massless self-interacting scalar field <cit.>.  

Before concluding this section, we recap the path followed so far: the local gauge invariance imposed for the electron's Lagrangian in Eq. (<ref>) makes three gauge fields to arise, identified with acoustic phonon modes, one longitudinal and two transverse. Their Lagrangian is the Eq. (<ref>), for which a simplified, one-dimensional form is the Eq. (<ref>), describing phonon-phonon interactions through the nonlinear (quartic) term proportional to λ.   


It is also worth stressing that Eq. (<ref>) describes a possible frequency gap √(λ c_s^2/2)ρ_R^2 at the long-wavelength side of the spectrum. This gap, not described by the classical theory of acoustic phonons, is due to the fields' self-interaction term coming from the nonzero commutators in Eq. (<ref>) and Eq. (<ref>), in turn originating from the current algebra in Eq. (<ref>) and Eq. (<ref>). However, since the coupling constant can be very small, the frequency (energy) gap may be a very tiny mini-gap, probably beyond practical observability limit. Moreover, it is also worth remarking that the present gauge theory is a massless, non-Abelian gauge theory, not a massive one. Apart from issues in its renormalizability, a mass-term in Eq. (<ref>) would be a term proportional to R^2, not existent in the present formulation, beside the quartic term coming from the nonlinearities. Nevertheless, the final result in the phonon dispersion relation would be similar (a mini-gap in the spectrum), but its very origin would be very different: a massive boson, and not a self-interacting massless boson as in the present case. 

In summary, we may say that in solid crystals a Goldstone mode is a massless excitation, which very existence is due to the broken, continuous translational symmetry caused by the lattice, and its dynamics can be described in the framework of a non-Abelian gauge theory. It is worth considering that Eq. (<ref>) reduces to the well known Debye form<cit.> ω_k =  c_s |k|, provided the nonlinear term in the Lagrangian can be neglected. Nevertheless, a relevant and quite unexpected result is that, in its general form, ω_k results gapped (i.e. ω_k ≠ 0 in the limit k → 0) owing to self-interactions, if the latter are relevant, even though the gauge field is a massless boson. 

As a side feature already noticed in Ref. [2014Dartora_JPA], the commutator [D_μ̂, D_ν̂] satisfies a cyclic identity that can be written as


    D_σ G_μ̂ν̂ + D_ν̂G_σ̂μ̂ + D_μ̂G_ν̂σ̂ = 0  ,


an equation known as the Bianchi identity for the field strenght tensor. The tensor R_μ̂^ν plays the same role of the Ricci tensor in General Relativity, obeying the same dynamic equations implied by Eq. (<ref>), and describing the local perturbation of the sonic metric η_μ̂ν̂ of the solid crystal caused by the propagation of an acoustic phonon, in strict analogy with the space-time Lorentz metric perturbation induced by a gravitational wave propagating as described in General Relativity. 





§ OPTICAL PHONON: A HIGGS MODE





In several binary compounds of interest in electronics, as e.g. GaAs, InP, GaN and many others, the crystal can be considered as composed of two interacting sublattices, one for each atomic species. Textbook toy-models describe the relevant physics considering a one-dimensional Bravais lattice with two ions per primitive cell. When treated as a set of elementary classical oscillators, they show that two distinct oscillation modes exist <cit.>: 

a) the two atoms in the cell may oscillate at frequency ω_k around their equilibrium position with the same direction and phase (Figure <ref>(a)), keeping unchanged their relative displacement σ_0. The perturbation is  the acoustic mode, an elastic wave propagating in the lattice at the sound velocity c_s, and the corresponding collective ions excitation is the acoustic phonon, whose dispersion relation has an expression that in the long wavelength limit can be approximated as ω_k = c_s |k|, indicating that the acoustic mode can be excited with arbitrarily small energy (there is no gap in ω_k);




b) conversely, the two atoms in the cell may oscillate opposite to each other (Figure <ref>(b)), and their relative displacement σ oscillates around an equilibrium position σ_0 along a well-defined direction θ_0 respect to a reference, defining an oscillating cell's dipole moment, if the solid is ionic. This is the well known optical mode, and the associated collective excitation is the optical phonon, whose role in semiconductors transport theory and electron dynamics is of great importance. It is found that ω_k → 0→ω_0 ≠ 0: in other words, the optical phonon dispersion ω_k is gapped, and the optical mode's excitation costs a fixed nonzero energy ħω_0 also for k → 0. 

We have found that both acoustic and optical phonons may be described as arising from a spontaneous breaking of a global symmetry, and that they are respectively a Goldstone (phase) and a Higgs (amplitude) mode <cit.> of an order parameter oscillation. Beside its importance as a further sign of a widespread presence of the amplitude mode in condensed-matter systems with broken continuous symmetry, the present approach provides a unified formulation of phonons, where the coupling between the acoustic and the optical phonon modes arises in a natural way.

In order to show this important point, let us define the displacement between the two atoms in the cell as a complex field ϕ = ( σ_0+σ) exp(θ).  

We can describe the dynamics of the collective oscillation modes by the Lagrangian of the field ϕ  


    ℒ_ϕ = 1/2(∂_μ̂ϕ ∂^μ̂ϕ^* - V(ϕ) ) ,


where V(ϕ) is the interacting potential between the two atoms in the cell. 
With the choice V(ϕ) = (1/4) λ |ϕ|^4, we would obtain the Eq. (<ref>), able to describe acoustic phonons. We could assume for V(ϕ) the form 


    V(ϕ) =  1/2μ^2 c_s^2/ħ^2 |ϕ|^2+ 1/4λ |ϕ|^4 ,


typical of the ϕ^4 theories <cit.> with a mass term proportional to ϕ^2 . However, if the two parameters μ^2 (where μ has the dimension of a mass) and λ are both positive, the expectation value of ϕ in the physical vacuum state is <ϕ> = 0 (the minimum of V(ϕ)). 
In this case the system is in a disordered phase, since there is not a preferred direction for the atoms' displacement oscillation. 
Furthermore, the system's vacuum energy is invariant under a U(1) transformation, since ℒ_ϕ is invariant under a global U(1) transformation of the field ϕ. 
Consequenty, this model cannot describe optical phonons, for which θ = θ_0 results a preferred direction for the system. 
In addition, the introduction of the mass term is arbitrary, beside making the theory non-renormalizable. Furthermore, the theory would describe massive acoustic phonons, against all evidences. 


Instead, observing that the optical phonon defines an oscillating cell's dipole moment, we may inspire us to ferromagnets, assuming that the global phonon field might have a preferred direction, violating a symmetry of the Lagrangian. 
In this case, the related field theory has a continuous, hidden symmetry that is spontaneously broken by a term in the Lagrangian that cannot be derived, but only inserted in V(ϕ) in an ad hoc manner, as in the electroweak theory <cit.>. 
To this end, we proceed in the standard way according to the linear-sigma model <cit.>. If the crystal formation makes μ^2 negative, the potential V(ϕ) assumes the well known “Mexican-hat”  shape shown in Figure <ref>(c), and all the couples (σ, θ) on the circle with radius |ϕ| = √(-μ^2 c_s^2/(ħ^2 λ)) = σ_0 are V(ϕ) minima. They identify degenerate vacuum states, all with the same energy, and among them the system chooses the ordered state for which <ϕ> = (σ_0, θ_0), that becomes the true vacuum state of the system. 
The angle θ_0 identifies the preferred direction imposed by the cell structure and, without loosing generality, we can set θ_0=0 as reference x-axis. Hence, the Lagrangian in Eq.  (<ref>) with the potential in Eq. (<ref>) and negative μ^2 has a spontaneously broken symmetry. 

The next important step is to write the Lagrangian ℒ_ϕ in terms of the fields σ and θ, small fluctuations of ϕ=(σ_0 + σ,θ) around the true vacuum ϕ_0 = (σ_0, 0):


    ℒ_ϕ = ℒ_θ + ℒ_σ + ℒ_σ-θ ,


where

    ℒ_θ =      1/2∂_μ̂θ ∂^μ̂θ - λθ^4 /4
    ℒ_σ =      1/2( ∂_μ̂σ ∂^μ̂σ  -  ω_0^2/c_s^2σ^2  )  - √(λω_0^2 /2 c_s^2)σ^3  - λσ^4 /4
    ℒ_σ-θ =        - √(λω_0^2 /2 c_s^2)σθ^2 -  λ/2θ^2σ^2   .


Here, for later convenience, we have introduced the real and positively defined parameter ω_0^2 = -2μ^2 c_s^4/ħ^2, a factor playing the dynamical role of a mass-term. The Lagrangians ℒ_θ and ℒ_σ respectively describe the dynamics of fields θ and σ. As expected and as it is typical for the Higgs' mechanism, θ is massless (Eq. (<ref>) does not include a term proportional to ω_0^2), whereas in Eq. (<ref>) a mass-term with the correct sign is associated to the field σ. The Lagrangian ℒ_σ-θ describes the σ-θ coupling terms and will be treated in section <ref>.

It is remarkable that the vacuum state is no more U(1) invariant (it is non-symmetric in θ), although Eq. (<ref>) and Eq. (<ref>) describe the same system: we only changed notations and chose a particular vacuum as system's ground state to expand the Lagrangian around. This is why we may speak of a symmetry spontaneously broken: no external agent provoked it and the symmetry of the system is still preserved, although “hidden” by a particular choice for the ground state.

Taking the limit λ→ 0 and plugging ℒ_ϕ into the Eulero-Lagrange equations, we obtain the motion equation for fields σ and θ for a one-dimensional diatomic chain along spatial coordinate x as


    (∂^2/∂ t^2 - c_s^2 ∂^2/∂ x^2)  θ(t,x) =  0 
       (∂^2/∂ t^2 - c_s^2 ∂^2/∂ x^2 + ω_0^2)  σ(t,x) =0


whose oscillatory solutions ∝ e^i (ω_θ,σ(k) t - k x) yield the dispersion relations 


    ω_θ^2 = c_s^2  k^2



    ω_σ^2 =  ω_0^2 +  c_s^2  k^2  .


They describe respectively a gapless and a gapped mode of the displacement field ϕ (and consequently a gapless and a gapped mode of the ensuing cell's dipole moment, if the solid is ionic), resembling respectively the acoustic and the optical phonon's dispersion law.

From a more formal point of view, we can state that the SSB has generated the real and positively defined mass-like term ω_0^2 σ^2 for the amplitude or Higgs mode σ (the optical phonon). From the present approach it is also clear that the optical phonon is not a gauge field, since the oscillation costs at least the energy ħω_0, and it is not possible to change ground state changing σ without spending energy. 

In addition, still owing to the spontaneous breaking of a continuous Lagrangian symmetry, a scalar field (here θ) appears and results massless, yielding the gapless dispersion relation ω_θ^2 =   c_s^2 k^2. In this limit, it is possible to change the ground state of θ without spending energy, just operating a gauge transformation along the valley of the Mexican hat (see Figure <ref>(c)). It follows that the field θ is a pure gauge field that we identify with the acoustic phonon, arising as the Goldstone mode associated to the breaking of a continuous symmetry. 

However, it must be stressed that the present approach also describes in an unitary way several nonlinearities, in particular the coupling between optical and acoustic phonons, an argument addressed in section <ref>.   





§ NONLINEARITIES AND PHONONS-COUPLING




An  important outcome of the present SSB approach is the natural emergence of phonon-phonon interactions and self-interactions, ensuing from the terms in λ in the expressions of Eq. (<ref>). The dynamics of θ alone, described by ℒ_θ in Eq. (<ref>), is the same we obtained gauging the spatial symmetry T(4) in the framework of non-Abelian gauge theories (see Eq. (<ref>)), revealing a deep link between the two approaches. The term (λ/4) θ^4 describes four-phonon processes, like a scattering between two acoustic phonons or a decay of one into three acoustic phonons (all them are impossible in an Abelian theory, like e.g. the electromagnetism). The same argument also applies to optical phonons, whose scattering is ruled by the corresponding term (λ/4) σ^4 in Eq. (<ref>).  





A further important point is the functional form of the Lagrangian ℒ_σ-θ, Eq. (<ref>), neglected during the preliminary analysis. It consists of two terms, and it confirms a possible coupling between acoustic and optical phonons, as shown in Refs. [2015Willatzen_PRB,2018Lu_PRB] for bulk GaAs, GaN, ZnO, MoS_2, and for BN monolayers. In Eq. (<ref>), the term proportional to σθ^2 describes the decay of an optical phonon σ into two acoustic phonons θ (or the reverse process), as outlined e.g. in Ref. [1966Klemens_PR] fitting the theory to silicon. In the same equation, the term proportional to θ^2σ^2  describes a scattering between two acoustic and two optical phonons, a four-phonon scattering term whose role has been recently recognized <cit.> in boron arsenide to be responsible for a substantial reduction of its expected thermal conductivity.

At the lowest order in λ, Eq. (<ref>) provides additional terms in the coupled equations of motion for θ and σ, that become


    (∂^2/∂ t^2 - c_s^2 ∂^2/∂ x^2 + √(2 λ) c_s ω_0 σ(t,x)  )  θ(t,x) =  0  
    
       (∂^2/∂ t^2 - c_s^2 ∂^2/∂ x^2 + ω_0^2)  σ(t,x) =0 .



The equation for σ(t,x) is the same Eq. (<ref>): without loosing generality, we can write its oscillatory solution as σ(t,x) ∝cos(ω_σ t - k x) and plug it in Eq. (<ref>). Approximating ω_σ≈ω_0, separating the variables as θ(t,x) = θ_t(t) θ_x(x), and supposing θ_x(x) ∝exp(i k x), the equation of motion for θ_t(t) is 


    d^2 θ_t(t)/dt^2 + (c_s^2 k_n^2 + q  ω_0  cos(ω_0 t )  )  θ_t(t) =  0


where q = √(2 λ). For simplicity, in the calculations it is also possible to set c_s=1. The phonon wavevector takes the discrete values k_n = n π /(N a), where n = 1, ...N, and N is the number of atoms in the chain. Eq. (<ref>) can be treated as a Mathieu's equation <cit.>, which describes time-dependent harmonic oscillators perturbed by a periodic load (the term in the cosine) representing the interaction with the optical phonon. For λ=0 the eigenfrequencies are ω_θ = c_s k_n, recovering the standard acoustic phonon dispersion law in the long wavelength limit. 

In the general case q ≠ 0, we treated the perturbation induced on the cell by the optical oscillation as a boundary value problem according to Ref. [1987Ascher]. In short, Eq. (<ref>) was written as θ_t” + (b_n + q  ω_0 cos(ω_0 t )  )  θ_t, where b_n is an unknown parameter, to be regarded as a discrete eigenvalue. Defining the domain as t ∈[0, T] and imposing as boundary conditions θ_t(0) = 1, and θ_t'(0) = θ_t'(T) = 0 (cosine-like solutions), the eigenvalues b_n were found iteratively, starting from a guess solution θ_t=cos(n π t/T) and a guess value b_n. The convergence is fast, and the obtained parameters b_n can be regarded as the discrete eigenvalues ω_θ^2 associated to eigenfunctions θ_t. 

In order to see how the optical-acoustic interaction qualitatively works, since the energy of optical phonons is usually higher than acoustics', it makes sense to set ω_0 = c_s k_N. It is interesting that for q  0 the present formulation predicts the existence of mini-gaps opening inside the Brillouin zone, for ω_θ≈ 0.5 ω_0, as visible in Figure <ref>. This is a feature typical of Mathieu's equation: a periodic modulation induces a Bragg-like condition that prevents the phonon propagation. Similar behavior was predicted and experimentally observed by  Raman spectroscopy in superlattices <cit.>: in both cases, there is a periodic modulation (caused by the optical mode according to the present work, or caused by the superlattice in the cited cases) that induces a Bragg condition which prevents the phonon propagation when its frequency is around half of the modulation frequency. Furthermore, piezoelectrical coupling of optical and acoustic phonon modes was found also in zinc-blende GaAs quantum-well slabs <cit.> and theoretically calculated for wurzites in the Lagrangian formalism in Ref. [2017Datta_ICN], confirming by different approaches the existence of the phenomenology described in the present work.






§ CONCLUSIONS



We investigated in depth the emergence of acoustic phonons in crystalline solids, both as Goldstone and as gauge bosons. The acoustic phonon was previously acknowledged as the gauge boson appearing when the electron's Lagrangian in a crystalline solid is requested to be locally invariant with respect to the group of spatial translations, whose generators appear broken due to the lattice itself. 

However, the gauging of the translations group is not trivial: in order to better explain the context, first we reviewed the differences between the gauging of spatial (external) and internal symmetries, as in Yang-Mills' theories. Then, exploiting the mathematical similarities between the sonic (or acoustic) metric and the Lorentzian metric of the ordinary space-time, we gauged the translations group in solid crystals following the same tetrad (or vierbein) formalism employed to express the General Relativity as a gauge theory with space-time translations as structure group. This allowed to compare the dynamical role of the crystal elasticity tensor in solid state physics and the Ricci's tensor in General Relativity at a more formal and rigorous level than before. The subsequent analysis showed that, beyond the linear limit, the translations group's non-Abelianity may generate a gap in the frequency dispersion relation of the acoustic phonon, as if a mass-like term were present in its free field Lagrangian, although this is not the case. 

In order to better investigate the latter point, we considered a different scenario, driving attention to the problem of understanding the nature of the optical phonons in standard solids. We showed that both the acoustic
and optical phonon emerge respectively as the gapless Goldstone (phase) and the gapped Higgs (amplitude) fluctuation mode of an order parameter arising from the spontaneous breaking of a global symmetry, without invoking the gauge principle. In greater detail, following the linear-sigma model approach the Higgs' mechanism is shown to generate a massive amplitude (Higgs) mode that we identify with the optical phonon, with gapped frequency dispersion relation due to the mass-like term. At the same time, the Higgs' mechanism generates a massless phase (Goldstone) mode, the acoustic phonon. A frequency-gap only appears in the strong nonlinear regime, and it is due to an anharmonic term, the same that arises from the gauging of the spatial translations group, an approach which did not provide any description of the optical phonon, though.

Although the SSB is a well known phenomenon, it turns out that it is able to describe not only novel aspects of material science (like e.g. Cooper pair plasmon modes in superconductors layered cuprates and thin films <cit.>), but also an old acquaintance of the solid state physics, as the acoustic and optical phonons in crystals. In particular, the Higgs mechanism describes all the phonon-phonon interactions, including a possible perturbation on the acoustic phonon's frequency dispersion relation induced by the eventual optical phonon, a peculiar behavior not described so far, at the best of author's knowledge, able to produce unexpected mini-gaps inside the Brillouin zone.














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