Ground State Wave Function Overlap in Superconductors and Superfluids
GGround State Wave Function Overlap in Superconductors and Superfluids
Mark P. Hertzberg ∗ and Mudit Jain † Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, MA 02155, USA
In order to elucidate the role of spontaneous symmetry breaking in condensed matter systems, we explicitlyconstruct the ground state wave function for a nonrelativistic theory of a two-fluid system of bosons. This canmodel either superconductivity or superfluidity, depending on whether we assign a charge to the particles or not.Since each nonrelativistic field Ψ j ( j = ,
2) carries a phase θ j and the Lagrangian is formally invariant undershifts θ j → θ j + α j for independent α j , one can investigate whether these symmetries are spontaneously brokenby the ground state condensate of particles. We explicitly compute the wave function overlap between a pair ofground states (cid:104) G | G (cid:48) (cid:105) that differ by these phase shifts. We show that the ground state spontaneously breaks mostpairs of shifts, including θ j → θ j + m j ε , where m j is the mass of each species. This is associated with a singleGoldstone boson (phonon) and is associated with the conservation of mass. However, we show that the groundstate is unchanged under the transformation θ j → θ j + q j ε , where q j is the charge of each species, and hencethere is no spontaneous symmetry breaking of the U ( ) em associated with electromagnetism in the bulk of thematerial. Hence the bulk ground state wave function overlap correctly predicts the Goldstone mode structure. I. INTRODUCTION
The role of spontaneous symmetry breaking (SSB) in na-ture is a highly important subject, with applications to theStandard Model, cosmology, and condensed matter systems.In the context of the Standard Model, there are various formsof (approximate) SSB, including the breakdown of chiral sym-metry in QCD, etc. However, its role within the Higgs mecha-nism often involves some confusion, since it is often describedas being tied to the notion of gauge symmetry, which is in facta type of redundancy in the description. This has led to vari-ous conclusions in the literature; see Refs. [1–10].To elucidate the actual behavior of SSB in the StandardModel, we recently constructed the (approximate) vacuumwave function of the Standard Model and explicitly foundthe vacuum to be unique [11]. This shows that in addition toElitzur’s theorem [12] – that one cannot spontaneously breaka local, or “small”, gauge symmetry (which is obvious sincethey are always only redundancies) – there is in fact no globalsub-group that is spontaneously broken in the Standard Modelvacuum either.In this work, we turn our attention to nonrelativistic con-densed matter systems. In particular we will focus on multi-fluid systems of bosons. Some of the most familiar applica-tions are to collections of helium atoms, which can organizeinto a superfluid at low temperatures, and to collections ofCooper pairs of electrons, which can organize into a super-conductor at low temperatures. The general topic of superflu-ids and superconductors will be the subject of this paper (forsome foundations and reviews, see Refs. [13–26]).In the context of superfluidity, it is well known that thereis a global U ( ) phase rotation symmetry of the Schrodingerfield ψ that is spontaneously broken by the ground state; itscorresponding Goldstone is a phonon associated with gaplesssound waves. In the context of superconductivity, it is wellknown that the system can exhibit plasma oscillations; and ∗ [email protected] † [email protected] there is a gapped spectrum provided by the plasma frequency.For a discussion of Goldstones in condensed matter systems,see Refs. [27–30].This absence of a Goldstone mode in the latter case, hasled to various contradictory statements in the literature sur-rounding the fate of the U ( ) em of electromagnetism. For ex-ample, in the textbook “The quantum theory of fields” [31] itis claimed “ A superconductor is simply a material in whichelectromagnetic gauge invariance is spontaneously broken .”However, one cannot spontaneously break gauge invariance,as it is a mere redundancy, as mentioned above, and all statesare gauge invariant. However, one can wonder about the fateof the global sub-group of U ( ) em , which after all is the ac-tual symmetry of electromagnetism, associated with conser-vation of electric charge. Is it possible that this symmetry isspontaneously broken in superconductors, despite the absenceof the gapless Goldstone mode? In the review paper [32], itwas claimed that this is precisely what happens “ global U ( ) phase rotation symmetry, and not gauge symmetry, is sponta-neously violated ”. But how could there possibly be SSB whenthere is no associated Goldstone mode due to plasma oscilla-tions? Ref. [33] claimed “ in a superconductor the superfluiddensity fluctuations carry charge density fluctuations, whichhave long-range Coulomb interactions, whereas Goldstone’stheorem only applies to local interactions ”. Thus claimingthe Goldstone theorem is avoided and there is still SSB. Onthe other hand, in another work Ref. [34] claimed that theSSB pattern is just the opposite of this “ in the superconduct-ing phase the symmetry is unbroken ”, while in Ref. [35] thesymmetry breaking language is said to be just “linguistics”.In this paper our goal is to address these issues in a di-rect and clear fashion, building on the ideas we developed inRef. [11]. In particular, we will explicitly compute the groundstate wave function | G (cid:105) , and then perform the (global) phaserotation shifts θ j → θ j + α j to obtain other possible groundstates | G (cid:48) (cid:105) . We then explicitly compute the overlap (cid:104) G | G (cid:48) (cid:105) todetermine which, if any, phase shifts lead to new states. Thisin fact is the precise definition of SSB. To our knowledge theexplicit construction of the wave function overlap in this con-text does not appear directly in the literature. Our focus herewill be on the properties within the bulk of the material. We a r X i v : . [ c ond - m a t . s up r- c on ] M a y will comment on effects from the boundary in the conclusions.The class of models we will study is a two-fluid systemof bosons in the nonrelativistic approximation. Each bosonicfield is described by a Schrodinger field Ψ j ( j = , θ j formally displaying a pair of phase rotation sym-metries θ j → θ j + α j , for arbitrary choices of α j . We allowfor a coupling to electromagnetism by endowing each specieswith charge q j (but in such a way that the homogeneous back-ground charge density is zero). We find that the overlap ofthe wave functions is zero for most choices of α j , including α j = m j ε where m j is the mass of each of the species. Thisimplies SSB and is associated with the conservation of mass.However, we find that for the special choice of α j = q j ε ,the overlap of the wave functions is 1. This implies there isno SSB of U ( ) em (neither gauge nor global) associated withelectric charge, at least in the bulk of the material.Our paper is organized as follows: In Section II we presentthe two-fluid nonrelativistic model. In Section III we discussthe condensate background. In Section IV we present the La-grangian and Hamiltonian governing fluctuations. In SectionV we explicitly compute the ground state wave function. InSection VI we explicitly compute the overlap of the groundstate wave functions. In Section VII we discuss which quan-tities are conserved and the associated symmetries. Finally, inSection VIII we conclude. II. NONRELATIVISTIC FIELD THEORY
We are interested in systems of nonrelativistic bosons. Wewill allow for several species that are distinguishable, labelled j , but will often specialize to the case of two species ( j = , a † k , j and destruction operators ˆ a k , j that produce N parti-cle states. Since we are interested in exploring condensates, itis convenient to pass to the field representation. This is de-fined by Fourier transforming the destruction operator to afield in position space, the so-called Schrodinger field ˆ Ψ j ( x ) (with conjugate field ˆ Ψ † j ( x ) ) as followsˆ Ψ j ( x ) = (cid:90) d k ( π ) ˆ a k , j e i k · x . (1)The corresponding particle number density operator forspecies j is ˆ n j ( x ) = ˆ Ψ † j ( x ) ˆ Ψ j ( x ) , (2)with particle number operator ˆ N j = (cid:82) d x ˆ n j ( x ) .In order to explore superconductivity, we minimally couplethe fields to electromagnetism A µ = ( − φ , A ) . For conveniencewe will use the Lagrangian formalism (though later we willmove to the Hamiltonian formalism). Since ordinary super-conductors and superfluids involve electrons and nuclei mov-ing much slower than the speed of light, one can often use aneffective nonrelativistic description. In order to build this, wenote that the leading order scalar field sector is essentially spe-cific uniquely by the Galilean symmetry. Furthermore, there is a unique way to couple to photons from the minimal cou-pling procedure. This uniquely specifies the nonrelativisticfield theory. For the leading relativistic corrections, the readermay see Appendix A for the sake of completeness. But for themost part, the nonrelativistic effective field theory will suffice,and is given by L = ∑ j (cid:20) i Ψ ∗ j ( ˙ Ψ j + iq j φΨ j ) + c . c − m j | ∇Ψ j − iq j A ψ j | (cid:21) − V ( Ψ ) − F µ ν F µ ν , (3)where m j is the mass of each of the species and q j is thecharge of each of the species. For the potential V , we allow4-point self-interactions. However, for simplicity we assumethat each species does not directly scatter off other species.For instance, one can imagine that the underlying fermionicdescription, which introduces Pauli exclusion, gives rise torepulsion among the indistinguishable particles. The gener-alization to other couplings is straightforward, but will not bestudied here. We also include a chemical potential µ j for eachof the species, to make it simpler to describe a background(this can also be obtained from a redefinition of the fields as Ψ j → Ψ j e iµ j t ). Together we write the potential as V ( Ψ ) = ∑ j (cid:20) − µ j | Ψ j | + λ j | Ψ j | (cid:21) , (4)where λ j are (positive) self-couplings.We note that the above theory carries the following set of(global) symmetries Ψ j → Ψ j e i α j , (5)for independent α j . When expanding around the vacuum,these are associated with the conservation of each of thespecies particle number, and includes the special case of α j = q j ε corresponding to electric charge. In the following we willexamine which, if any, of these symmetries is spontaneouslybroken by a condensate (ground state) solution. III. HOMOGENEOUS BACKGROUND
We now expand around a homogeneous background. At theclassical field level, the ground state is determined by mini-mizing the above potential V . We can use the chemical po-tential to obtain whatever background number density of par-ticles we desire. Lets denote the background number densityof each species n j . By minimizing the potential, this value ofnumber density can be immediately obtained by choosing thechemical potential to be µ j = λ j n j . (6)We will ensure that the background charge density ρ = ρ = ∑ j q j n j + ( q n n n ) = . (7)where we have included a possible q n n n term; this shouldbe included in the case of a superconductor: it refers to nu-clei, which carry positive charge q n >
0. The nuclei providethe compensating charge, so there is a well defined neutralsuperconductor, that we may then add charge density fluctu-ations to. This is the standard physical starting point. In thecase of the superconductor, we will not need to track the dy-namics of the nuclei (although they will inevitably play a rolewhen talking about mass density perturbations), as they arenot accurately described by bosons, and are very heavy; sothey will only be relevant at this background level. Instead, asis well known, in a BCS superconductor, the relevant dynam-ics is provided by the much lighter Cooper pairs, with charge q = − e . One can have more general systems with multipletypes of effective bosons. We will leave our analysis in termsof multiple species for the sake of pedagogy, and in fact thecase of 2 species of bosons will often be our focus, since wecan then discuss the fate of multiple types of symmetries. Onthe other hand, in the case of a superfluid, then the relevantspecies all carry no charge. In this case, we will just use the j index to refer to the appropriate bosonic degrees of freedom,which are all heavy neutral bosons, such as helium, etc. Henceour framework is quite general.The corresponding background field value for each of therelevant dynamical species (which is quite different dependingon whether it is a superconductor or a superfluid) is given by v j ≡ | Ψ j | = √ n j . (8)As is well known, the phase of the ground state condensate Ψ j is not determined by this condition. This suggests thereare a family of distinct ground state solutions labelled by a setof constant phases θ j as ψ j = v j e i θ j , (9)which all appear to spontaneously break the symmetry givenabove in Eq. (5). Since these symmetry transformations in-clude the global sub-group of U ( ) em , one should be extracareful in reaching such conclusions. In this work, we willexamine this issue systematically by quantizing the fluctua-tions and actually computing the ground state of the quantumtheory precisely. IV. PERTURBATIONS
Let us expand around the homogeneous background by de-composing the fields into a perturbation in modulus η j ( x , t ) and phase θ j ( x , t ) as Ψ j ( x , t ) = ( v j + η j ( x , t )) e i θ j ( x , t ) . (10)We then treat η j and (derivatives of) θ j as small to study smallperturbations. Expanding the Lagrangian density to quadraticorder in the fluctuations we obtain L = ∑ j (cid:104) − v j η j ˙ θ j − v j q j φ η j − µ j η j − m j (( ∇η j ) + v j ( ∇θ j − q j A ) ) (cid:105) − F µ ν F µ ν . (11) Now the electromagnetic field includes the non-dynamicalCoulomb potential φ . We can solve for this from Gauss law asfollows − ∇ φ = ∇ · ˙ A + ρ , (12)where the charge density (to linear order in perturbations) is ρ = ∑ j v j q j η j . (13)We now decompose the vector potential A into its longitu-dinal A L and transverse A T components A = A L + A T . (14)However, we can now exploit gauge invariance to simplify ourresults by operating in Coulomb gauge ∇ · A =
0. So A L = A = A T is purely transverse. One should bear in mind thatall of our results can be trivially re-written in a gauge invariantway be replacing θ j ( x ) → θ j ( x ) − q j ∇ · A L ∇ (15)if desired. The Lagrangian density decomposes into a sum oflongitudinal L L and transverse L T pieces that decouple at thequadratic order L L = ∑ j (cid:104) − v j η j ˙ θ j − µ j η j − m j (( ∇η j ) + v j ( ∇θ j ) ) (cid:105) − (cid:32) ∑ j q j v j ∇η j ∇ (cid:33) , (16) L T = ( ˙ A T ) − ( ∇ × A T ) − ∑ j v j q j m j ( A T ) . (17)The final term in (17) shows the familiar fact that within asuperconductor the magnetic field acquires an effective mass.It is given by the sum of squares of the plasma frequencies as m eff = ∑ j v j q j m j . (18)This means the magnetic field is short ranged, which is thefamous Meissner effect (for example, see [36–38]). This isanalogous to the Higgs mechanism in the Standard Model.However there is an important difference: In the StandardModel the Lorentz symmetry ensures that the Coulomb po-tential A = − φ also acquires the same effective mass. How-ever, in this nonrelativistic setup that is not the case. As wewill later discuss, despite appearances, the Coulomb poten-tial (in Coulomb gauge) and the associated electric field getsscreened more strongly than the magnetic field. This has im-portant ramifications for the behavior of the charge and thefate of symmetries, as we will discuss in Section VII.Our interest is in the behavior of the longitudinal modes,as these involves the phases θ j , and enjoy the symmetries θ j → θ j + α j . To study these modes in more detail, it is con-venient to now pass to the Hamiltonian formalism. The appro-priate phase space variables are the phase θ j and momentumconjugates π j given by π j = ∂ L ∂ ˙ θ j = − v j η j . (19)Furthermore, we diagonalize the problem by passing to k -space. We write the Hamiltonian for the longitudinal modesas H L = (cid:90) d k ( π ) H Lk (20)and find the k -space Hamiltonian density to be H Lk = ∑ j (cid:34)(cid:32) k m j v j + µ j v j (cid:33) | π j | + v j k m j | θ j | (cid:35) + k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∑ j q j π j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (21)Note that as expected it is the charges q j that couple the dif-ferent species to one another, as seen in the final term. V. GROUND STATE WAVE FUNCTION
For the sake of simplicity, let us now specialize to the caseof two species j = ,
2. We can readily write the above Hamil-tonian density in matrix notation, by defining the followingvector fields (cid:126) θ k = (cid:18) θ θ (cid:19) , (cid:126) π k = (cid:18) π π (cid:19) , (22)and the following matrices K k = q k + µ v + k m v q q k q q k q k + µ v + k m v , (23) F k = v k m v k m . (24)This gives the following Hamiltonian density H Lk = (cid:126) π ∗ k K k (cid:126) π k + (cid:126) θ ∗ k F k (cid:126) θ k . (25)We are now in a position to construct the ground statewave function. Recall that for a single harmonic oscilla-tor with Hamiltonian H = K p / + Fx / ψ ( x ) ∝ exp ( − (cid:112) F / K x / ) . For the above Hamiltonian H L we needto generalize this to take into account the non-trivial matrixstructure. Some matrix algebra reveals that the result in thefield basis is ψ ( θ j ) ∝ exp (cid:20) − (cid:90) d k ( π ) (cid:126) θ ∗ k M k (cid:126) θ k (cid:21) , (26) where M k is the following matrix M k = K − k F k K − k . (27)Let us now examine this result in some important limits. A. Superfluid
Firstly, consider the simple case in which the species areneutral q j =
0. In this case the matrices becomes diagonal,the modes decouple, and we have a set of superfluids. Theargument of the exponential in the wave function simplifiesinto the following form (cid:126) θ ∗ k M k (cid:126) θ k = ∑ j v j m j k ω jk | θ j | , (28)where ω jk = (cid:115) µ j k m j + k m j (29)is the usual dispersion relation in a superfluid for each species.For long wavelength modes the effective sound speed c j is c j = (cid:114) µ j m j . (30)Note that for small k , the pre-factor of | θ j | in Eq. (28) is linear in k (since ω jk is itself linear in k ) (cid:126) θ ∗ k M k (cid:126) θ k = ∑ j v j √ µ j m j k | θ j | + O ( k ) . (31)This will be very important when we come to compute thewave function overlap in the next Section. B. Superconductor
Our main interest is the case in which the species arecharged q j (cid:54) = q j (cid:54) = k expansion in the exponentof the wave function (this can also be viewed as a large q j ex-pansion). We expand the above matrix M k to the first severalleading terms and obtain M k = (cid:32) a k + b k + c k − a k + ˜ bk + c k − a k + ˜ bk + c k a k − b k + c k (cid:33) (32)plus corrections that are O ( k ) . The coefficients a i j , b i j , ˜ b , c i j are defined as a i j ≡ q i q j a , b i j ≡ q i q j b , ˜ b ≡ b ( q − q ) , c i j ≡ q i q j c , where a ≡ v v ( √ m q v + √ m q v )( q + q ) (cid:113) m m ( q v µ + q v µ ) , (33) b ≡ q q ( v √ m v − v √ m v )( q + q ) √ m m ( q v µ + q v µ ) , (34) c ≡ √ m q v + √ m q v √ m m ( q + q ) . (35)If we then expand out the matrix structure that appears in theargument of the exponent of ψ we find (cid:126) θ ∗ k M k (cid:126) θ k = ak ( q θ − q θ ) + bk ( q θ − q θ )( q θ + q θ )+ ck ( q θ + q θ ) + O ( k ) . (36)Note the important phase dependencies here: The first term ∼ k has dependence on q θ − q θ , while the last term ∼ k has dependence on q θ + q θ , while the second term ∼ k depends on both. Recall that the U ( ) em phase transforma-tions are θ j ( x ) → θ j ( x ) + q j ε ; this evidently does not affectthe first term, but only the final terms. VI. WAVE FUNCTION OVERLAP
We now come to the main issue of comparing the set ofground state wave functions that differ by the symmetry trans-formations θ j ( x ) → θ j ( x ) + α j (37)for different choices of phases α j . Note that if we choose α j = q j α ( x ) , with α ( x ) → θ j ( x ) → θ j ( x ) − q j ∇ · A L / ∇ ,which is a well defined operation for (small) gauge transfor-mations. As emphasized earlier in the paper, the interestingissue is that of global transformations, with α j constant.However, performing constant phase shifts is awkward in k -space, since it would formally involve shifting θ k by a delta-function θ jk → θ jk + α j ( π ) δ ( k ) . Hence, as we did inRef. [11], it is much more convenient to pass to position spaceand define the field theory in a volume V , then take the large V limit. In position space the ground state wave function isgiven by ψ ( θ j ) ∝ exp (cid:20) − (cid:90) V d x (cid:90) V d x (cid:48) (cid:126) θ ( x ) M ε ( x − x (cid:48) ) (cid:126) θ ( x (cid:48) ) (cid:21) , (38)where M ε is a matrix of kernels, defined by M ε ( r ) = (cid:90) d k ( π ) M k e − i k · r e − k ε . (39)Here ε is a UV regulator. It will be convenient to first regulatethe UV modes, then send ε → θ j → θ j + π n j , where n j is an integer. This is easily ensured by defining the improvedwave function as˜ ψ ( θ j ) ∝ ∑ n , n ψ ( θ + π n , θ + π n ) , (40)and furthermore, the final result is to be normalized appropri-ately.Let us now consider a pair of ground state wave functions:One of them, | G (cid:105) , centered around θ j = | G (cid:48) (cid:105) , centered around θ j = α j . The (normalized) overlap be-tween these two wave functions is given by the integral (cid:104) G | G (cid:48) (cid:105) = (cid:82) D θ D θ ˜ ψ ( θ j ) ˜ ψ ( θ j + α j ) (cid:82) D θ D θ ˜ ψ ( θ j ) ˜ ψ ( θ j ) . (41)We can readily compute this integral as it is Gaussian. Weobtain (cid:104) G | G (cid:48) (cid:105) ∝ ∑ n , n exp (cid:20) − V (cid:126) α n n (cid:90) V d r M ε ( r ) (cid:126) α n n (cid:21) , (42)where we defined a vector of constant phase shifts (cid:126) α n n = (cid:18) α + π n α + π n (cid:19) . (43)The above integral (cid:82) d r M ε ( r ) can be more readily under-stood by re-writing M ε ( r ) in terms of the Laplacian of anothermatrix of kernels J ε ( r ) defined implicitly by M ε ( r ) = − ∇ J ε ( r ) . (44)In terms of a Fourier transform we can define this by J ε ( r ) = (cid:90) d k ( π ) k M k e − i k · r e − k ε , (45)where we inserted an extra factor of 1 / k in the integrand (andwe used the fact that ε is very small).Using the divergence theorem, the wave function overlapcan then be given by the following boundary term (cid:104) G | G (cid:48) (cid:105) ∝ ∑ n , n exp (cid:20) V (cid:126) α n n (cid:73) d S · ∇ J ε ( r ) (cid:126) α n n (cid:21) , (46)where d S is an infinitesimal surface area vector that pointsradially outward. In this representation it is now clear thatthe UV has decoupled, as the boundary term is purely an IReffect. In other words, we can now send ε → J ( r ) as r →
0; we only need to evaluate J ( r ) at large r . A. Superfluid
In the case of the superfluid, recall that the matrix of kernels M k is diagonal, and hence the J ( r ) will be diagonal too. Usingthe leading order result from Eq. (31), in which M k ∼ k , wehave J ( r ) i j = δ i j v j √ µ j m j f ( r ) . (47)Here we have defined the function f ( r ) , which is a specialcase of the Fourier transform of inverse powers of k , definedby f p ( r ) ≡ lim ε → (cid:90) d k ( π ) k p e − i k · r e − k ε . (48)For the special case of p =
1, it is readily found to be f ( r ) = π r . (49)Note that this has significant support at large r . By takingthe gradient of J ( r ) , inserting into Eq. (46), and defining ourtheory in a sphere of radius R , we obtain the following resultfor the overlap (cid:104) G | G (cid:48) (cid:105) = N ∑ n n exp (cid:34) − ∑ j v j √ µ j m j R ( α j + π n j ) (cid:35) , (50)where the normalization factor N is simply equal to the nu-merator with α j =
0. For any finite α j in the domain 0 < α j < π we can readily approximate the sum over n , n by just the n = n = (cid:104) G | G (cid:48) (cid:105) = exp (cid:34) − ∑ j v j √ µ j m j R α j (cid:35) . (51)Evidently for large R , the wave function overlap falls off ex-ponentially towards (cid:104) G | G (cid:48) (cid:105) → B. Superconductor
In the case of the superconductor with non-zero charges, wereturn to our expression in Eq. (32) for the leading IR contri-bution to the kernel M k . In this case the leading dependencefor small k include k , k / , and k . Which of these dominateswill depend on the particular choice of phase shifts, as seenin Eq. (36). To compute the various contributions to J ( r ) , wetherefore need to divide by a factor of k (recall Eq. (45) com-pute the Fourier transform of 1 / k , 1 / √ k , and 1. The Fouriertransform of 1 / k is denoted f ( r ) and was reported earlier inEq. (49); it has significant support at large r and scales as ∼ / r . Similarly the Fourier transform of 1 / √ k is f ( r ) = √ π / r / , (52) which also has somewhat significant support at large r . Onthe other hand, the Fourier transform of 1 is known to be justa delta-function f ( r ) = δ ( r ) , (53)and has no support at all at large r . Hence the terms in M k thatinvolve k do not contribute at all to the wave function overlapat large volume. In fact these are precisely the terms that arisefrom electric transformations α j = q j ε , which is only non-zero for the k terms (as well as higher order terms, that all in-volve even powers of k ; all are associated with delta-functionsand do not contribute at large volume to the overlap).Using these results, we find that the matrix J ( r ) for non-zero r is given by J ( r ) = (cid:32) a π r + b √ π / r / − a π r + ˜ b √ π / r / − a π r + ˜ b √ π / r / a π r − b √ π / r / (cid:33) (54)Inserting this into the general expression for the wave functionoverlap Eq. (46) and again evaluating the integral on a sphereof radius R , we obtain our primary result (cid:104) G | G (cid:48) (cid:105) = N ∑ n n exp (cid:104) − a R ( q ˜ α − q ˜ α ) − b √ π √ R / ( q ˜ α − q ˜ α )( q ˜ α + q ˜ α ) (cid:105) (55)where ˜ α j ≡ α j + π n j . As before, the normalization factor issimply equal to the numerator with α j = α j , the argument of theexponent is non-zero. This leads to the wave function overlapapproaching zero exponentially fast. For 0 < α j < π and for q α − q α (cid:54) = n = n =
0, giving the leadingfall off of the wave function as (cid:104) G | G (cid:48) (cid:105) = exp (cid:20) − a R ( q α − q α ) (cid:21) . (56)This is evidently associated with SSB and there is a corre-sponding Goldstone mode, which we will describe in moredetail in the next Section.However the important fact is that there is one, and onlyone, special choice of shifts that does not lead to a new wavefunction; namely if we perform a U ( ) em phase shift α j = q j ε , (57)(where ε is a common factor). This is the one special combi-nation that sets q α − q α = , (58)leading towards vanishment of both terms in the argument ofthe exponent in the wave function overlap. In fact we havechecked that it vanishes for all higher order contributions tothe wave function too. Hence for U ( ) em we have | G (cid:48) (cid:105) = | G (cid:105) and there is no SSB in the bulk of the material (we commenton boundary effects in the conclusions). VII. CONSERVED QUANTITIES
To understand this result further, let us examine the pos-sible conserved quantities in the system. Naively there is aconserved quantity for each shift α j . Indeed this is true whenexpanding around the vacuum. However, when expandingaround the superconducting condensate, it is more subtle. A. Total Charge
Recall that each species has a corresponding particle num-ber given by N j = (cid:90) d x | Ψ j ( x , t ) | . (59)In this section we will study the classical field evolution forsimplicity. The leading order fluctuations in particle number ∆ N j around the homogeneous background are given by ∆ N j = v j (cid:90) d x η j ( x , t ) = v j η j | k → , (60)where in the second step we have expressed the spatial integralas the zero mode of the Fourier space representation. Thesecond time derivative of this is ∆ ¨ N j = v j ¨ η j | k → = v j k m j ˙ θ j | k → , (61)where in the second step we used the classical equation of mo-tion for η j that follows from the Hamiltonian Eq. (21). Thiscan be determined to be ∆ ¨ N j = − v j q j m j Q , (62)where we used the classical equation of motion for θ j . Herethe total electric charge Q is Q = ∑ j q j ∆ N j . (63)In general we see that the particle numbers in a superconduc-tor are typically not conserved if the total integrated chargefluctuation is non-zero (this is to be contrasted to the case ofexpanding around the vacuum). Related to this, we can com-pute the time evolution of the charge itself. The last 2 equa-tions give ¨ Q = − ∑ j v j q j m j Q . (64)Hence the electric charge is not conserved, but oscillates inany enclosed region if its total initial value is non-zero; theseare the familiar plasma oscillations. B. Charge Density Fluctuations and Screened Electric Field
As detailed in the appendix (where we also include leadingorder relativistic corrections for completeness) we can studythe charge density fluctuations also. For a multi-fluid system,these fluctuations are complicated, so it will suffice here to re-port on the case of a single species. One can readily show thatin this case the charge density ρ is related to ˙ θ (in Coulombgauge) by ρ k = − q k k mv + µk v + q ˙ θ k (65)Then one can show that the equation of motion for θ k is givenby ¨ θ k = − (cid:20) m eff + k m (cid:16) k + m ξ (cid:17)(cid:21) θ k (66)where m eff = v q / m and m ξ = µm . One can then replace k → − ∇ to turn this into a wave equation back in positionspace. By taking a time derivative of this equation, and usingEq. (65), we see that the exact same equation is obeyed bythe charge density itself. By integrating this over space anddropping boundary terms, we obtain the earlier Eq. (64) (ifwe simply generalize again to multiple species).However, this form of the fluctuations equation, revealssomething very important: Suppose we consider initial con-ditions provided by θ and ˙ θ that are localized, i.e., they havesupport in the bulk, but die away rapidly towards the bound-ary. These are associated with perfectly reasonably initialconditions, with finite energy, etc. Then we see that the to-tal charge is of a special form. Consider the low k limit ofEq. (65) ρ k = − k q ˙ θ k ( small k ) (67)If we then write out the expression for the total integratedcharge, it is Q = (cid:90) d x ρ ( x , t ) = − q (cid:90) d x ∇ ˙ θ = − q (cid:90) d S · ∇ ˙ θ (68)where (cid:82) dS indicates the integral over a surface out towardsinfinity, by use of the divergence theorem. For a localizedinitial condition, i.e., ˙ θ → Q →
0. Hence this gives a conservedcharge, but only in a trivial sense, i.e., Q = Q ,but it will no longer be conserved, as in Eq. (64).This phenomenon is closed related to the screening of theelectric field in the superconductor. For any number of fields,and using Gauss’ law, we have (cid:90) d S · E = Q (69)As we showed above, for local θ and ˙ θ , the integrated Q van-ishes. This is connected to the electric field being exponen-tially suppressed at large distances, which ensures a surfaceintegral over it vanishes at large distance. Hence we recoverthe idea that the electric field is screened in a superconduc-tor. In fact in Coulomb gauge, the Coulomb potential φ can beshown to also obey the same equation as Eq. (66). So we seethat if m ξ (cid:29) √ m m eff , the electric field is screened over lengths ∼ m ξ / ( m m eff ) to leading order, whereas if m ξ (cid:29) √ m m eff , it isscreened over lengths ∼ / √ m m eff . In either of the cases, wesee that it is shielded even more strongly than the magneticfield (which is screened over lengths ∼ / m eff ).Since the charge integrates to 0 (for localized sources), thishas ramifications for the properties of the vacuum. In thequantum theory, it will annihilate the vacuum. Since, in thequantum theory, charge is the generator of a symmetry. itmeans that the corresponding symmetry is the identity opera-tor, which maps ground states into themselves. This is to becontrasted to other conserved quantities and symmetries thatwe discuss in the next subsection. C. Other Combinations
For a two species system, there is one linear combination ofthe ∆ N j that is conserved. The enclosed perturbation in totalmass is given by ∆ M = ∑ m j ∆ N j . (70)This evolves according to ∆ ¨ M = − ∑ j v j q j Q = , (71)where in the last step we used the condition that the back-ground total charge density ρ = ∑ j v j q j =
0, so that weare expanding around a neutral superconductor (to be clear,one needs to extend the sum over j to include the positivelycharged heavy nuclei, as they cannot be ignored when we mul-tiply throughout by the mass of the particles). Hence there is asingle conserved quantity associated with these internal sym-metries; which is the conservation of mass. Its correspondingGoldstone is a phonon. The associated phase transformationsthat are generated by this conserved quantity are θ j ( x ) → θ j ( x ) + m j ε . (72)This is a symmetry transformation that is spontaneously bro-ken by the ground state. In fact the related Galilean symmetryof boosts is also spontaneously broken. This is as opposed tothe U ( ) em ( θ j ( x ) → θ j ( x ) + q j ε ) which is not spontaneouslybroken in the bulk of the material. More precisely, there isno conserved charge, so this symmetry is better understood asbeing removed in this phase. VIII. CONCLUSIONS
In this work we have explicitly computed the wave func-tion overlap between ground states in nonrelativistic systems of condensed bosons, either modeling superconductivity orsuperfluidity. We showed that while a generic phase trans-formation of the nonrelativistic Schrodinger field does indeedlead to small overlap and hence a new state and SSB, the onecombination of phase shifts that does not lead to a new stateis that of the electromagnetic phase shifts. Hence there is noSSB of U ( ) em by superconductors, neither gauge or global,at least in the bulk of a superconductor. Instead there doesexist SSB of symmetries associated with mass conservation,associated with phonons, among other possibilities dependingon the number of fields (related ideas appear in Ref. [41]).Our focus here has only been on the bulk and the directcomputation of the overlap wave function, which we believeto be a new result. It is nicely in one-to-one correspondencewith the presence, or lack thereof, of Goldstone modes. Onthe other hand, in a finite size superconductor, there can beinteresting boundary effects, including the Josephson effect[39, 40]. This can give a different point of view on the ultimatefate of SSB [32]. However, the main purpose of this work hasbeen to show that in the bulk of the material, there is no partof the wave function that actually differs between the groundstates; this was seen in a new explicit computation.These results are in accord with our earlier work in theStandard Model [11]. Other directions to consider are morecomplicated condensed matter systems, including those thatexhibit strong coupling, and various other phases. Further ap-plications may be to other areas in which SSB may play arole, including ideas in particle physics [42–44] and cosmol-ogy [45–49]. ACKNOWLEDGMENTS
MPH is supported in part by National Science Foundationgrant PHY-1720332.
Appendix A: Relativistic Corrections
In order to study relativistic effects and the screening ofelectric field in a superconductor in more detail, in this ap-pendix we consider only one bosonic Schrodinger field ψ (theCooper pair field), and also include the relativistic term in theLagrangian (3) for illustration purposes: L → L + δ m (cid:12)(cid:12) ˙ Ψ + iq φψ (cid:12)(cid:12) . (A1)However, as we will see, the final result will be insensitive tothis term since we are always interested in the nonrelativisticlimit k / m (cid:28)
1, and to make this point clear, we have intro-duced a “switch” δ in the front; δ = δ = φ and inserting it back as be-fore, is therefore L L = − v η ∗ k (cid:18) k ω eff (cid:19) ˙ θ k + c . c − v k m | θ k | − η ∗ k (cid:18) m Λ (cid:19) η k + δ m (cid:20) | ˙ η k | + v k ω eff (cid:12)(cid:12) ˙ θ k (cid:12)(cid:12) (cid:21) . (A2)where ω eff ≡ k + δ v q m ≡ k + m eff δ , Λ ≡ k + µ m + m eff m ω eff ≡ k + m ξ + m eff m ω eff . (A3)From this Lagrangian, it is suggestive that the radial degreeof freedom η is super-massive because of the factor of 4 m in Λ . So in the limit when k (cid:28) m , we can neglect the ˙ η k termand then it becomes a constraint variable which we can solvefor η k ≈ − m v k ω eff Λ ˙ θ k (A4) and insert back into the Lagrangian to obtain the followingLagrangian for the longitudinal degree of freedom L L ≈ m v ω eff (cid:18) δ m + k Λ ω eff (cid:19) (cid:12)(cid:12) ˙ θ k (cid:12)(cid:12) − v m | θ k | . (A5)The equation of motion for this degree of freedom to leadingorder (the coherence length ξ has to be much larger than theCompton wavelength l cp of the bosonic degree of freedom, i.e.we can expand in m ξ / m ) and we recover¨ θ = − (cid:20) m eff − ∇ m (cid:16) − ∇ + m ξ (cid:17)(cid:21) θ . (A6)as we reported earlier when beginning in the exact non-relativistic theory. Note that there is no dependence on δ here.Now since the Coulomb scalar potential φ = ˆ O θ where ˆ O is alinear operator, so it too satisfies this. [1] F. Englert and R. Brout, “Broken Symmetry and the Mass ofGauge Vector Mesons,” Phys. Rev. Lett. , 321 (1964).[2] P. W. Higgs, “Broken Symmetries and the Masses of GaugeBosons,” Phys. Rev. Lett. , 508 (1964).[3] G. S. Guralnik, C. R. Hagen and T. W. B. Kibble, “Global Con-servation Laws and Massless Particles,” Phys. Rev. Lett. , 585(1964).[4] J. Bernstein, “Spontaneous symmetry breaking, gauge theo-ries, the Higgs mechanism and all that,” Rev. Mod. Phys. ,7 (1974). Erratum: [Rev. Mod. Phys. , 259 (1975)] Erratum:[Rev. Mod. Phys. , 855 (1974)].[5] F. Strocchi, “Spontaneous Symmetry Breaking in Local GaugeQuantum Field Theory: The Higgs Mechanism,” Commun.Math. Phys. , 57 (1977).[6] D. Stoll and M. Thies, “Higgs mechanism and symmetry break-ing without redundant variables,” hep-th/9504068.[7] A. Maas, “(Non-)Aligned gauges and global gauge sym-metry breaking,” Mod. Phys. Lett. A , 1250222 (2012)[arXiv:1205.0890 [hep-th]].[8] T. W. B. Kibble, “Spontaneous symmetry breaking in gaugetheories,” Phil. Trans. Roy. Soc. Lond. A , no. 2032,20140033 (2014).[9] F. Strocchi, “Symmetries, Symmetry Breaking, Gauge Symme-tries,” arXiv:1502.06540 [physics.hist-ph].[10] A. Maas, “Brout-Englert-Higgs physics: From foundations tophenomenology,” arXiv:1712.04721 [hep-ph].[11] M. P. Hertzberg and M. Jain, “Counting of States inHiggs Theories,” Phys. Rev. D , no. 6, 065015 (2019)[arXiv:1807.05233 [hep-th]].[12] S. Elitzur, “Impossibility of Spontaneously Breaking LocalSymmetries,” Phys. Rev. D , 3978 (1975).[13] L. Landau, “Theory of the Superfluidity of Helium II,” Phys.Rev. , 356 (1941).[14] V. L. Ginzburg and L. D. Landau, ”On the theory of supercon-ductivity”. J. Exp. Theor. Phys. , 1175 (1957). [16] J. R. Schrieffer, “Theory of Superconductivity,” New York: W.A. Benjamin (1964)[17] R. P. Feynman, “Superfluidity and Superconductivity,” Rev.Mod. Phys. , 205 (1957).[18] F. W. London, ”Macroscopic Theory of Superconductivity”.Dover (2005).[19] A. J. Leggett, “A theoretical description of the new phases ofliquid He-3,” Rev. Mod. Phys. , 331 (1975) Erratum: [Rev.Mod. Phys. , 357 (1976)].[20] J. C. Wheatley, “Experimental properties of superfluid He-3,”Rev. Mod. Phys. , 415 (1975).[21] M. Rabinowitz, “Phenomenological theory of superfluidity andsuperconductivity,” Int. J. Theor. Phys.
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