Phonon spectrum of QCD vacuum in magnetic-field-induced superconducting phase

# Phonon spectrum of QCD vacuum in magnetic-field-induced superconducting phase

M. N. Chernodub CNRS, Laboratoire de Mathématiques et Physique Théorique, Université François-Rabelais Tours,
Fédération Denis Poisson, Parc de Grandmont, 37200 Tours, France
Department of Physics and Astronomy, University of Gent, Krijgslaan 281, S9, B-9000 Gent, Belgium
Jos Van Doorsselaere CNRS, Laboratoire de Mathématiques et Physique Théorique, Université François-Rabelais Tours,
Fédération Denis Poisson, Parc de Grandmont, 37200 Tours, France
Department of Physics and Astronomy, University of Gent, Krijgslaan 281, S9, B-9000 Gent, Belgium
Henri Verschelde Department of Physics and Astronomy, University of Gent, Krijgslaan 281, S9, B-9000 Gent, Belgium
July 12, 2019
###### Abstract

In the background of a sufficiently strong magnetic field the vacuum was suggested to become an ideal electric conductor (highly anisotropic superconductor) due to an interplay between the strong and electromagnetic forces. The superconducting ground state resembles an Abrikosov lattice state in an ordinary type–II superconductor: it is an inhomogeneous structure made of a (charged vector) quark-antiquark condensate pierced by vortices. In this paper the acoustic (phonon) vibrational modes of the vortex lattice are studied at zero temperature. Using an effective model based on a vector meson dominance, we show that in the infrared limit the longitudinal (transverse) acoustic vibrations of the vortex lattice possess a linear (quadratic) dispersion relation corresponding to type I (type II) Nambu–Goldstone modes.

###### pacs:
12.38.-t, 13.40.-f, 74.90.+n
thanks: On leave from ITEP, Moscow, Russia.

## I Introduction

The quantum vacuum may exhibit quite unusual properties in a strong magnetic field background if the magnetic field exceeds the hadronic-scale, . A strong magnetic field enhances the chiral symmetry breaking in QCD due to the magnetic catalysis phenomena ref:magnetic:catalysis (). As a result, magnetic field affects the finite-temperature phase diagram of QCD ref:Tc:rising () by shifting the transition temperature in a quite unexpected way ref:lattice:results (). The phase structure of the QCD quark matter is also very sensitive to the presence of a strong magnetic field ref:magnetic:matter (). In addition, the magnetized quark matter should exhibit new transport phenomena, such as the chiral magnetic effect ref:CME (); ref:CME:cond:matt (), while the magnetic field background may affect standard transport phenomena ref:transport:related ().

The interest in physics of extreme magnetic fields is justified by the fact that such strong fields may emerge in noncentral heavy-ion collisions. For example, in lead-lead collisions at the LHC, the strength of the generated magnetic field may reach  ref:field:estimation (). The magnetic field may affect the quark-gluon plasma created by overlapping heavy ions as well as the vacuum between the ions if these ions near-miss each other in ultraperipheral collisions.

It was recently suggested that the strong magnetic field may cause the QCD vacuum to behave as an anisotropic perfect conductor (superconductor) at low temperatures Chernodub:2010qx (); Chernodub:2011mc () if the strength of the magnetic field exceeds the critical value

 Bc≃1016TeslaoreBc≃0.6GeV2. (1)

The transition from the insulating to superconducting regimes is caused by the condensation of the quark-antiquark pairs, which carry the quantum numbers of the electrically charged mesons (we call it later “-meson condensate”). The condensate is an anisotropic and inhomogeneous structure which may lack, in thermodynamics sense, a local order parameter beyond the mean field approximation ref:comment ().

In contrast to the standard phenomenon of superconductivity the magnetic field penetrates the condensate: the Meissner effect is absent in the new phase because of the vector rather than scalar nature of the condensate Chernodub:2010qx ().

This phenomenon has a known counterpart in the solid state physics which is sometimes the “reentrant” superconductivity ref:Zlatko (). In an increasing magnetic field a type-II superconductor will eventually experience a transition to a normal phase and may then – according to the proposal of Ref. ref:Zlatko () – “reenter” the superconducting regime again. The reentrant superconductivity is associated with an inhomogeneous condensation of electron pairs and is suggested to be realized in certain materials. In the reentrant regime the emergent superconductivity does not screen the background magnetic field, while the breaking of the electromagnetic symmetry results in appearance of short-range modulations (inhomogeneities) of the condensate in the transverse plane. In scope of the –meson condensation the absence of the Meissner effect was discussed in Refs. Chernodub:2010qx (), and a related proposal for ferromagnetic superconductors is put forward in Ref. ref:ferromagnetic ().

Generally, a possible appearance of the -meson condensate in strong magnetic field is not very surprising: it is quite similar to the gluon condensation in QCD ref:YM () and to the -boson condensation in the electroweak model ref:EW (). The question on the possibility of –meson condensation in high magnetic fields in QCD was also briefly raised in Ref. ref:earlier (). Both mesons, gluons and bosons are vector particles, which are sensitive to high magnetic (chromomagnetic, in the case of gluons) fields. The closely related question of condensation of charged vector particles in the background of a magnetic field is discussed in Ref. ref:related:vector ().

The condensation of quark-antiquark pairs with –meson quantum numbers were also found in various holographic approaches ref:holographic (); ref:holographic:hexagonal (), in local Chernodub:2011mc () and nonlocal Frasca:2013kka () Nambu–Jona-Lasinio (NJL) models as well as in numerical simulations of quenched lattice QCD Braguta:2011hq (). Due to the anisotropic superconductivity the vacuum may become an exotic hyperbolic metamaterial which shares a similarity with diffractionless “perfect lenses” Smolyaninov:2011wc (). The –meson condensation is a subject of an ongoing discussion ref:comment (); ref:discussion ().

In the mean field approach the vortices form a hexagonal lattice which is similar to the mixed Abrikosov state in an ordinary type–II superconductor. Since the vortex lattice breaks the continuous spatial symmetries, it should give rise to Nambu-Goldstone modes. These massless modes are, in fact, phonons which correspond to elementary vibrational excitations of the vortex lattice. The presence of phonons is crucial for understanding the stability of the vortex lattice against quantum and thermal fluctuations: depending on the spectrum of the phonon modes the vortex lattice may experience a chain of deformations and, eventually, melt into a vortex liquid ref:type-II:Review ().

According to the numerical simulations of lattice QCD in strong magnetic field, the vortices appear in a liquid phase rather then in a form of the ordered vortex crystal Braguta:2013uc (). Since the phonon modes are responsible for the melting of the vortex lattice (crystal), the study of the vortex excitations is physically interesting. In this paper we describe the vibrations of the -vortex lattice following a well-developed calculation method for the phonon spectrum in Abrikosov vortex lattices in type II superconductors. A good review on this subject can be found in Ref. ref:type-II:Review ().

In order to address the problem of the phonon spectrum of the vortex lattice we work in the framework of an effective electrodynamics for the mesons Djukanovic:2005ag () based on the vector meson dominance Sakurai:1960ju (). This model describes the superconducting ground state in consistency with other effective approaches ref:holographic (); ref:holographic:hexagonal (); Chernodub:2011mc (); Frasca:2013kka (). The choice of this model is also justified by the analogy with usual superconductivity, where the standard Ginzburg-Landau approach describes well both the vortex-lattice ground state and its phonon excitations. Finally, we would like to mention that the use of the effective field theories in highly-magnetized zero-temperature QCD is generally supported by numerical lattice calculations. The relevant examples include a linear behavior of the chiral condensate (Refs. Shushpanov:1997sf () and Bali:2012zg (), respectively) and a peculiar nonlinear behavior of the chiral magnetization (Frasca:2011zn () and Buividovich:2009ih () respectively) as functions of magnetic field.

The structure of the paper is as follows. The effective electrodynamics of mesons and the ground state of this model in strong magnetic field are described in Section II. We introduce the basis of crystal wavefunctions and study their basic properties in Section III. In Section IV we derive the dispersion relation for the phonons. We show that in the infrared limit the transverse acoustic vibrations of the -vortex lattice possess a quadratic “super-soft’ dispersion relation corresponding to type II Goldstone modes, while the longitudinal modes are always linear in momentum similarly to type I Goldstone bosons. Our main result for the dispersion relation for the low-energy phonons is given in Eq. (77):

 ω2k = k2z+f(B)(k2)2+…, (2)

where and are the longitudinal and transverse momenta, respectively, and the field-dependent prefactor is given explicitly in Eq. (78). Apart from the prefactor, this dispersion relation for the low-energy phonons in the -vortex ground state has the same qualitative form as the dispersion relation for the acoustic phonons in Abrikosov vortex lattices in conventional superconductors. The presence of the supersoft (quadratic) transversal mode in the phonon spectrum (2) may be responsible for the melting of the mean-field vortex lattice into the vortex liquid. The latter state was indeed observed in lattice simulations Braguta:2013uc (). Our conclusions are given in the last section.

## Ii Model, phases and approximations

The mesons are charged and neutral vector particles made of light ( or ) quarks and antiquarks. A self-consistent quantum electrodynamics for the mesons can be described by the Djukanovic–Schindler–Gegelia–Scherer (DSGS) Lagrangian Djukanovic:2005ag ():

 L = −14 FμνFμν−12 ρ†μνρμν+m2ρ ρ†μρμ −14 ρ(0)μνρ(0)μν+m2ρ2 ρ(0)μρ(0)μ+e2gs Fμνρ(0)μν,

where

 ρμ≡ρ−=ρ(1)μ−iρ(2)μ√2, (4)

and are charged meson fields, is the neutral meson field and is the electromagnetic field. The field strengths in Eq. (II) are as follows:

 Fμν = ∂μAν−∂νAμ, (5) f(0)μν = ∂μρ(0)ν−∂νρ(0)μ, (6) ρ(0)μν = f(0)μν−igs(ρ†μρν−ρμρ†ν), (7) ρμν = Dμρν−Dνρμ, (8)

where is the covariant derivative and the phenomenological coupling is

 gs≡gρππ≈5.88. (9)

The -meson mass at the vanishing magnetic field is and the mass of the neutral meson is as follows:

 m0≡mρ(0)=mρ(1−e2g2s)−12. (10)

The DSGS model (II) is basically the vector meson dominance model Sakurai:1960ju () coupled to electromagnetism with the following Abelian gauge transformations:

 U(1)e.m.:⎧⎪ ⎪⎨⎪ ⎪⎩ρμ(x)→eieω(x)ρμ(x),ρ(0)μ(x)→ρ(0)μ(x),Aμ(x)→Aμ(x)+∂μω(x). (11)

We consider the model (II) in a background of a uniform static magnetic field parallel to the axis. The corresponding gauge potential is as follows:

 Ax=−B2y,Ay=B2x,Az=At=0. (12)

We ignore quantum fluctuations of the gauge and meson fields, thus treating the effective model (II) at the classical level. We always assume for simplicity.

The model (II) predicts that if the magnetic field exceeds the critical value (1) then the vacuum enters a new phase where positively and negatively charged –meson fields condense Chernodub:2010qx (). In this phase the vacuum becomes a perfect electric conductor: the electric current can be carried by the condensates along the lines of the magnetic field without dissipation. The transport of electric charge along the magnetic field is described by a one-dimensional London equation, therefore we refer to the perfectly ideally conducting phase as “the superconductor phase” Chernodub:2010qx (); Chernodub:2011mc ().

The electric superconductivity of the vacuum in strong magnetic field is caused by a –wave condensation of the charged –meson field with . Other vector components of the vector condensate are zero, . In the mean–field approach and at the classical level of the effective model (II), the superconducting ground state of the vacuum resembles the Abrikosov lattice state in an ordinary type–II superconductor Chernodub:2011mc (); Chernodub:2010qx ().

The mean-field solution for the ground state does not depend on the longitudinal coordinate. Therefore it is convenient to combine the transverse vectors into complex scalars: , , or in general , . As an exception to this general convention we put

 ρ=ρx−iρy, (13)

as the other combination simply vanishes in the LLL approximation:

 ρx+iρy≡0(LLL). (14)

The static (potential) energy density corresponding to the DSGS Lagrangian (II) is – assuming (14) – as follows:

 E⊥[ρ,ρ0,A]=12|Dρ|2+12F2xy+12(ρ(0)xy)2 +12m2ρ(|ρ(0)|2+|ρ|2)−egsFxyρ(0)xy, (15)

where

 Dρ = ∂ρ+igsρ(0)ρ−ieAρ, (16) Fxy = −i2(¯∂A−∂¯A)=B , (17) ρ(0)xy = gs2|ρ|2−i2(¯∂ρ(0)−∂¯ρ(0)). (18)

We consider the spontaneous condensation of the meson fields in the background of the magnetic field (12) in the vicinity of the transition temperature: with . One can show that in this case the -meson condensate is very small, , and the equations of motion for -meson fields can be linearized Chernodub:2010qx (). The smallness parameter is:

 ϵ=gs|ρ|maxmρ≃√B−Bc2Bc,B>Bc, (19)

where is the maximal value of the inhomogeneous condensate of the field.

The classical equations of motion in the magnetic field background are discussed in details in Ref. Chernodub:2010qx (). The equation for the charged –meson condensate can be written as follows:

 Dρ=0, (20)

where the correction to this equation is of the order of with given in Eq. (19). Basically, the solutions of Eq. (20) reduce possible condensate solutions to the space of the lowest Landau levels.

The neutral –meson field can be expressed via the charged –meson condensate in the following form:

 ρ(0)=igs2∂−Δ+m20|ρ|2, (21)

where is the two–dimensional Laplacian,

 1−Δ+m20(x⊥)=12πK0(m|x⊥|), (22)

is the two-dimensional Euclidean propagator of a scalar particle with the mass of the neutral vector meson (10) and is a modified Bessel function.

Using the equations of motion for our model (II), the mean energy density (15) can be expressed via a nonlocal function of the –meson condensate ,

 ⟨E⊥[ρ,ρ0,A]⟩ = 12B2+⟨E⊥[ρ]⟩, (23) ⟨E⊥[ρ]⟩ = e28⟨|ρ|2⟩2+12(m2ρ−eB)⟨|ρ|2⟩ (24) +g2s8m2ρ⟨|ρ|21−Δ+m20|ρ|2⟩,

where field is a minimum-energy solution of Eq. (20) and the brackets denote the average over the transverse plane:

 ⟨O⟩=1Area⊥∫dxdyO(x,y), (25)

and is the area of the transverse plane.

The ground state solution of the condensate in the phase is given by the following expression Chernodub:2011gs (),

 ρ(x,y) = ∑n∈ZCnhn(x,y), (26)

where

 hn(x,y)=exp{−π(x−nνLB)2−iy(x−2nνLB)L2B}, (27)

are the eigenstates corresponding to the Lowest Landau Level and

 LB=√2πeB (28)

is the magnetic length.

The parameters and should be chosen to minimize the energy (24) gained due to the condensation of the –meson fields (the condensation energy), at a fixed value of magnetic field. The minimization of the energy functional is usually done by assuming that the (generally, complex) parameters obey the symmetry for an integer .

The global minimum of the energy functional (24) corresponds to a global minimum of the dimensionless quantity

 βρ=⟨|ρ2|⟨|ρ2|⟩m20−Δ+m20|ρ2|⟨|ρ2|⟩⟩, (29)

which is an analogue of the Abrikosov ratio

 βA=⟨|ϕ|4⟩/⟨|ϕ|2⟩2, (30)

used in the Ginzburg-Landau theory of ordinary superconductors Chernodub:2011gs (). Notice that the limit .

In Ref. Chernodub:2011gs () we have found that the condensation energy density (24) – with given in Eq. (24) – is minimized for with and

 ν=4√3√2=0.9306…. (31)

This choice of the parameters corresponds to an equilateral triangular (called also hexagonal) lattice with , similarly to the case of the Abrikosov lattice in the Ginzburg–Landau model. All lattices with odd values of possess higher energies than the case while all even– lattices are reduced to the case.

The parameters of Eq. (26) corresponding to the ground state are as follows:

 Cn=C0αn,α2Z=1,α2Z+1=i. (32)

The overall coefficient of the solution (26) and (32) can then be calculated by a requirement of the minimization of the condensation energy (24) or the ratio (29). The mean-field behavior of the coefficient is calculated in Ref. Chernodub:2011gs () and is also presented in Eq. (III.3).

It was numerically found that in the ground state close to the critical magnetic field (1) at the ratio (29) takes the following value Chernodub:2011gs ():

 βρ(B=1.01Bc)=1.0192…. (33)

The –meson condensate is an inhomogeneous function of the transverse coordinates and and is independent of the longitudinal coordinate (26). The inhomogeneities are caused by the presence of an infinite periodic lattice of the so-called vortices which are parallel to the magnetic field. The –vortex is a stringy topological defect in the condensate as the phase of the field winds by around the vortex center. There is one vortex per unit area of the transverse plane. According to calculations in DSGS model Chernodub:2011gs (), supported also by the holographic models Bu:2012mq (), the vortices arrange themselves in the transverse plane in the form of a hexagonal (equilateral triangular) lattice, Fig. 1. The hexagonal periodic lattice is encoded in the particular form of the parameters  and in Eq. (32).

For the hexagonal lattice solution (26) the basic lattice vectors are as follows:

 d1=LBν(0,1),d2=LBν(√32,12). (34)

The solution (26) is periodic, up to a phase factor, with respect to the shifts and :

 ρ(x+da) = eiπx×da/L2Bρ(x),a=1,2, (35)

where is the vector product, and the latin superscript labels the basic vectors (34).

## Iii Crystal wavefunctions

### iii.1 Definition

According to Eq. (35), the ground state solution (26), (27), (32) is not invariant under the translational shifts along basic vectors (34) in the transverse plane. In order to study the vibrations of the vortex lattice it is convenient to describe the full basis of the eigenstates of the Lowest Landau Level (27) by the so-called “magnetically translated” states. We follow the standard procedure which is used to study the vibrations of the vortex lattice in usual superconductors ref:type-II:Review ().

We will perturb the vortex lattice both in transverse and longitudinal dimensions by slightly deformed configurations carrying so called “crystal quasimomentum” where and are the transverse and longitudinal quasimomenta, respectively. These quasimomentum states are space dependent functions, close to the original lattice configuration, but shifted to hold a relative momentum in a special way ref:type-II:Review ():

 ρk(x,z)=eikzzρk(x),ρk(x)=eikxψk(x), (36)

where

 ψk(x)=ρ(x+2~keB)≡ρ(x+L2Bπ~k), (37)

and is the conjugate momentum. The shifted ground state function  satisfies the classical equations of motion, in particular, Eq. (20). By construction, the momentum states (36) and (37) satisfy the periodicity rule (35) for arbitrary momentum :

 ρk(x+da) = eiπx×da/L2Bρk(x),a=1,2. (38)

The unperturbed ground–state function (26) corresponds to the function (36) with zero quasimomentum, .

It is convenient to define new dimensionless transversal coordinates:

 x′=1LBx,k′=LBπk, (39)

and then drop the primes, so that the the explicit form of the momentum eigenfunctions (36) is as follows:

 ρk(x,y) = eiπ(xkx+yky)ρ(x−kx,y+ky) ≡ C0eiπ(πxy−kxky)+2πiyky ⋅∑n∈Zαne−π(x−nν−ky)2+2πinνy+2πinνkx.

In subsequent sections we will define the excited vortex states in terms of the quasimomentum wavefunctions (36). These wavefunctions will eventually enter the perturbed energy density (24), which consists of two– and four–point functions with respect to the average over the transverse space (25). For the sake of further convenience, in the rest of this section we define the two– and four–point functions of the quasimomentum wavefunctions (36) in the transverse plane. The dependence of the wavefunctions on the longitudinal coordinate is omitted below since it gives rise to trivial phase factors only.

It is well known that the periodicity of the functions (37) is inherited by the quasi-momenta, which can therefore be restricted to an elementary region in momentum space, the Brillouin-zone. This zone is then defined by the reciprocal vectors from (34):

 (~da)i=eB2ϵij(da)j→1LBϵij(da)j, (41)

taking into account (39). While this periodicity allows a freedom in choosing the Brillouin zone, it is customary to choose it symmetrically around the origin and such that it respects the symmetries of the original lattice. In this way one can clearly identify the small momenta corresponding to the infrared regime we’re interested in. Since we are mainly interested in the infrared behavior of the phonon spectrum, in the remainder we will always assume all quasi-momenta sufficiently small, more explicitly:

 kx,lx∈[−14ν,14ν],ky,ly∈[−ν2,ν2]. (42)

Given the symmetries of the system, one can extend the region above to a small hexagon inside the Brillouin zone as shown in Fig. 2. We have not tested the validity of our calculations outside this range, nor have we extended the calculations to include the full Brillouin zone. Since our results coincide in the limiting case of Abrikosov lattices with the known result, we expect good agreement in the whole Brillouin zone.

### iii.2 Two-point function

The two-point function of the transverse wavefunctions can be calculated explicitly using Eq. (III.1):

 ⟨ρ∗lρk⟩ = |C0|2∑m,n∈Zαnα∗me−π[(x−nν−ky)2+(x−mν−ly)2] (43) ⋅e2iπν(n−m)y+2πiν(kxn−lxm)+2πi(ky−ly)y.

By shifting the coordinate, , we arrive to the following equation:

 ⟨ρ∗lρk⟩ = L2B|C0|2Area⊥∫dxdye−2πx2∑n∈Z|αn|2e2πinν(kx−lx) (44) = |C0|2√2νδ(k−l).

### iii.3 Four-point function

The (normalized) nonlocal correlation functions,

 Ql2,k2,l1,k1=1|C0|4⟨ρ∗l2ρk2M2−Δ+M2ρ∗l1ρk1⟩, (45)

correspond to the last term in the transverse energy functional (24). Inserting Eq. (III.1) into Eq. (45) and after an extensive algebra we find the following expression:

 Ql2,k2,l1,k1 = (46) ≡Q(b,c,M),

depending only on two linear combinations of the momenta and given below and the dimensionless mass

 M=m0LB, (47)

with is defined in Eq. (28). The vectors , and in Eq. (46) are as follows:

 b = k1+l1−k2−l22, (48) c = l1−k1−l2+k22, (49) Xm,n = m~d2+n~d1 (50) = mν−1(12ex−ν2ey)+nν−1ex, (51)

where and are the unit vectors in the and directions, respectively. A detailed calculation of the correlation function (46) is presented in Appendix A.

Notice that in our gauge (12) the function (46) is always a real function, which is even in both arguments:

 Ql2,k2,l1,k1≡Qk2,l2,k1,l1,ImQl2,k2,l1,k1=0. (52)

One can check Eq. (46) by calculating the ratio (29):

 βρ≡2ν2Q0,0,0,0=∑m,n∈ZM2e−πX2m,n4π2X2m,n+M2. (53)

The ratio depends on magnetic field via the dependence of the mass parameter according to Eqs. (28) and (47). An explicit calculation gives us that Eq. (53) at the critical magnetic field (1) reproduces the known numerical value (33). In the vicinity of the transition, , the parameter depends on the value of magnetic field very weakly. Parametrically,

 βρ(B)=1.01937−0.01702(BBc−1)+…, (54)

for . In the limit we recover the standard result for the value of the Abrikosov ratio for an equilateral triangular (hexagonal) lattice ref:type-II:Review ().

Substituting the ground state wavefunction, Eq. (III.1) with , into the energy functional (24) we compute, term by term, the ground-state energy:

 E(0)⊥ = e2|C0|416ν2+m2ρ−eB2√2ν|C0|2 (55) +g2sm2ρ8m20Q0,0,0,0|C0|4,

where we used the definition (45).

The value of is then determined by the minimum of the energy (55). Given the phase degeneracy of the prefactor of the condensate , we choose this prefactor to be a real number, . Then the ground state for is defined by Eqs. (26), (27) and (32) with the prefactor

 C0(B) =  ⎷2√2(eB−m2ρ)νe2+(g2s−e2)βρ ≃ 0.2733√eB−eBc≃0.1504√B−Bc.

where we took into account the first equality in Eq. (53) and Eq. (10). In numerical estimation of the prefactor (III.3) we used , the phenomenological value (9) for the coupling , and the values (31) and (33) for, respectively, the parameter and for the ratio in the ground state.

Equation (55) allows us to compute the energy of the ground state at :

 E(0)⊥(B)=−(eB−eBc)22[e2+(g2s−e2)βρ] (57) ≃−0.0142(eB−eBc)2≃−1.3×10−3(B−Bc)2.

Obviously, in the low- phase the condensation energy is zero, .

## Iv Dispersion relation for phonons

We perturb the mean-field solution for the ground state (III.1), , by adding the states which carry a nonzero quasimomentum in the transverse plane:

 ρph(x)=∑kckρk(x),c|k|≠0≪c0=1. (58)

Our strategy is to substitute the perturbed wavefunction (58) into the transverse energy functional (24), and expand the latter expression over the coefficients :

 E⊥[ρph]=E(0)⊥+E(2)⊥+O(c4k), (59)

with is the ground state energy (57) and is the quadratic term of the phonon contribution to the energy. Then, the phonon eigenfunctions may be found by diagonalisation of the quadratic part with respect to the coefficients .

### iv.1 Propagation in the transverse plane

Due to the orthogonality of the magnetically translated wavefunctions (44) the quadratic term of the energy functional (24) is diagonal in the coefficients . However, the quartic term of this functional can, in general, mix four independent phonon modes of the perturbed wavefunction (58). Using the properties of the four-point function (52), one can show that number of independent mixing modes is reduced to a half of that upon complex conjugation. The mixing of the remaining two modes can be described by the following two-component vector,

 vk=(ck,c∗−k)T, (60)

and the mixing term can be written as follows:

 ⟨ρ∗ρM2−Δ+M2ρ∗ρ⟩=12∑kv†k⋅^Q⋅vk, (61)

with the matrix

 ^Q=2(Qk,k,0,0+Qk,0,0,kQ∗k,0,−k,0Qk,0,−k,0Qk,k,0,0+Qk,0,0,k). (62)

The eigenvalues of the matrix (62) are as follows:

 λk,± = 2(Qk,k,0,0+Qk,0,0,k±|Qk,0,−k,0|) (63) ≡ 2(Q(k,0,M)+Q(0,k,M)±|Q(k,k,M)|),

Notice that

 Q0,0,0,0=λ0,+6=λ0,−2≡βρ2ν2. (64)

The eigenvectors of the matrix (62),

 ok=ck+c∗−k2,ak=ck−c∗−k2i, (65)

define the optical (massive) and acoustic (massless) phonon modes with the eigenvalues and , respectively.

To the lowest order the energy functional of the optical and acoustic transverse modes reads as follows:

 E(2)⊥=∑kE(2)⊥(k), (66)

where the contributions from the individual modes are diagonal in the coefficients :

 E(2)⊥(k) = c∗kck(e24⟨|ρ0|2⟩⟨ρ∗kρk⟩+12(m2ρ−eB)⟨ρ∗kρk⟩)+g2sm2ρ|C0|24m20(Qk,k,0,0+Qk,0,0,k)ckc∗k +g2sm2ρ|C0|28ckc−kQ0,k,0,−k+g2sm2ρ|C0|28c∗kc∗−kQk,0,−k,0 = o2k⋅(e28ν2|C0|2+m2ρ−eB√8ν+g2s8m2ρm20|C0|2λk,+)+a2k⋅(e28ν2|C0|2+m2ρ−eB√8ν+g2s8m2ρm20|C0|2λk,−).

The above expression can be simplified further using Eq. (64) and the fact that the parameter , given explicitly in Eq. (III.3), corresponds to the minimum of energy (55):

 E(2)⊥(k) = E(2)⊥,O(k)+E(2)⊥,A(k)=g2s−e28|C0|2 (68) ⋅[o2k(λk,+−λ0,−)+a2k(λk,−−λ0,−)].

The phonon contribution to the transverse energy (68) has a massless mode in the spectrum – the last term in Eq. (68) vanishes if – which are therefore called the “acoustic” modes. The modes represent the massive “optical” modes. Below we concentrate on massless acoustic modes.

According to Eqs. (68) and (64) the contribution of the acoustic modes to the energy can be rewritten as follows:

 E(2)⊥,A(k)=a2kg2s−e28|C0|2(λk,−−βρν2), (69)

with given in Eq. (III.3). The eigenvalue is explicitly given by Eqs. (63) and (46):

 λ−(k)≡λk,−=M22ν2(∑Xe−π(k−X)24π2(k−X)2+M2 (70) +∑Xe−πX2+2πik×X4π2X2+M2−∣∣ ∣∣∑Xe−π(k−X)2+2πik×X4π2(k−X)2+M2∣∣ ∣∣).

and as given in Eq. (51) with running over the whole lattice.

### iv.2 Acoustic spectrum

The phonon contribution (69) to the energy is, in fact, a potential (time-independent) energy of essentially transverse modes with zero longitudinal momentum, . The propagation in the longitudinal direction and the time evolution of the acoustic modes can naturally be taken into account by the following expression for the phonon fluctuations:

 ρph(x,z,t)=∑k,kzck,kze−iωkt+ikzzρk(x), (71)

where according to Eq. (36).

The easiest way to obtain the dispersion relation for the phonon modes is to notice that the quadratic contribution to the time-dependent part of the Lagrangian (II) coming from the phonon fluctuations (71) is as follows:

 T = (72) = ω2ka2k4⟨|ρk|2⟩=ω2ka2k4√2ν.

The expression (72) plays a role of the kinetic energy of the phonon fluctuations. The potential energy is given by , where

 E(2)∥,A(kz)=a2k4√2νk2z, (73)

is the contribution from the longitudinal phonons which can be obtained by substituting the wavefunction of phonon fluctuations (71) into the expression for the full energy (15). Since the energy is diagonal in the longitudinal wavefunctions, there is no term which mixes the longitudinal modes with different .

Solving the equations of motion is then equivalent to putting . Then we obtain the following dispersion relation for the acoustic phonon modes:

 ω2k=2e(g2s−e2)(B−Bc)e2+(g2s−e2)βρ[ν2λ−(LBkπ)−βρ]+k2z, (74)

where the explicit form of is given in Eq. (70). In deriving Eq. (74) we have used Eqs. (III.3) and (69).

The dispersion relation (74) for the transverse (with ) acoustic modes in the Brillouin zone is shown in Fig. 3. Qualitatively, the phonon spectrum in the –vortex lattice is very similar to the spectrum of phonons in Abrikosov vortex lattices of conventional superconductors ref:type-II:Review ().

### iv.3 Low-energy spectrum of acoustic phonons

To obtain the infrared spectrum of the low-energy acoustic photons we expand the eigenvalue (70) over small transverse momenta111We remind that according to Eq. (39) the dimensionless momentum is related to the physical one a follows: . :

 (75) α(B=1.01Bc)≃0.662. (76)

Similarly to the ratio (54), the prefactor (76) is practically insensitive to the magnetic field in the transition region.

In the case of the Abrikosov vortex lattice in a type–II superconductor – which is achieved by taking the limit in Eq. (70) – the coefficient in front of the quartic term in Eq. (75) is 4 times bigger: .

Then we substitute the expansion (75) to the full phonon spectrum (74), and get the following infrared spectrum of the acoustic phonons:

 ω2k = k2z+f(B)(k2)2+…, (77) f(B) = Cf|eB|(1−BcB)+…, (78)

where and the higher-order corrections in and in are shown by the ellipsis. This expansion is valid in the vicinity of the transition .

The longitudinal part of the phonon spectrum (77) contains massless (“soft”) phonon modes with the linear dispersion relation . Thus, the acoustic phonons travel along the magnetic field axis with the speed of light and belong to type-I Nambu-Goldstone (NG) bosons.

The transverse part of the spectrum (77) corresponds to the “supersoft” phonon modes which are described by the quadratic dispersion relation . This mode corresponds to a type-II NG boson. Similar quadratic dispersion relations were found for kaon condensation in the color-flavor locked phase of QCD at a nonzero strange chemical potential Schafer:2001bq (); Miransky:2001tw (). Generalizations of the NG Amado:2013xya () and pseudo-NG Filev:2009xp () modes were found recently in the holographic approaches.

The purely transverse acoustic modes propagate with the following velocity (in units of speed of light ):

 v⊥(k,kz=0)=2√f(B)|k|. (79)

For example, at a transverse acoustic phonon carrying energy  MeV should travel with the velocity equal to 2% of speed of light.

The dispersion relation for the low-energy phonons (77) in the