Ground state wave function overlap in superconductors and superfluids

1 Introduction

The structure of the ground state of a field theory is a highly important subject, with applications to the Standard Model, cosmology and condensed matter systems. As an example, in the context of the Standard Model, there are various forms of (approximate) spontaneously broken symmetries (SSB) in the vacuum, including the breakdown of chiral symmetry in QCD, etc. However, the structure of the vacuum within the Higgs mechanism often involves some confusion, since it is often described as being tied to the notion of a breakdown of gauge symmetry, which is in fact a type of redundancy in the description. This has led to various conclusions in the literature; see Refs. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10].

To elucidate the structure of the vacuum in the Standard Model, we recently constructed the (approximate) vacuum wave function of the Standard Model and explicitly found the wave function overlap between vacua [11]. There is other important older work, including the famous Elitzur's theorem [12] that one cannot spontaneously break a local, or "small", gauge symmetry (which is obvious since they are always only redundancies).

In this work, we turn our attention to nonrelativistic condensed matter systems. In particular we will focus on multifluid systems of bosons. Some of the most familiar applications are to collections of helium atoms, which can organize into a superfluid at low temperatures, and to collections of Cooper pairs of electrons, which can organize into a superconductor at low temperatures. The general topic of superfluids and superconductors will be the subject of this paper (for some foundations and reviews, see Refs. [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26]).

In the context of superfluidity, it is well known that there is a global U(1) phase rotation symmetry of the Schrodinger field ψ that is spontaneously broken by the ground state; it’s corresponding Goldstone is a phonon associated with gapless sound waves. In the context of superconductivity, it is well known that the system can exhibit plasma oscillations, and there is a gapped spectrum provided by the plasma frequency. For a discussion of Goldstones in condensed matter systems, see Refs. [27], [28], [29], [30].

This absence of a Goldstone mode, in the latter case, has led to various contradictory statements in the literature surrounding the ground state of a superconductor.^[1] In this paper our goal is to address a very specific aspect of the structure of the ground state in a direct and clear fashion, building on the ideas we developed in Ref. [11]. In particular, we will explicitly compute the ground state wave function |G〉 in a particular well defined prescription, and then perform the (global) phase rotation transformations θ_j → θ_j + α_j to obtain other possible ground states |G′〉. We then explicitly compute the overlap 〈G|G′〉 to determine which, if any, phase rotations lead to new states. This will be the particular aspect of the ground state structure which we will focus on. We will make use of a very specific prescription to define the ground state, as it is subtle in the infinite volume limit. Importantly, we will use a prescription that treats the superfluid and superconducting cases altogether in the same computation. Of course, there are other important aspects of the ground state, including other issues surrounding SSB. This includes the existence of nonzero expectation values of certain kinds of field operators which can provide novel order parameters; however, these other issues have been very well studied elsewhere and are not the focus of the present study.

To our knowledge the explicit construction of the wave function overlap in this context does not appear directly in the literature. Our focus here will be on the properties within the bulk of the material; we will then comment on effects from the boundary in the conclusions. By defining the ground state through a kernel representation in position space and carefully regulating the UV modes, we will find a beautiful connection between the structure of the overlap 〈G|G′〉 and the Goldstone boson pattern, which is the primary finding of this work.

The class of models we will study is the effective theory of bosons (since Cooper pairs organize into bosons in the condensed phase) in the nonrelativistic approximation (see Appendix A for leading relativistic corrections). For generality we will study a multicomponent system, specializing to a two-fluid system, as this will allow us to identify more details of the ground state structure, as we will explain. Each bosonic field is described by a Schrodinger field Ψ_j (j = 1, 2), with phase θ_j formally displaying a pair of phase rotation symmetries θ_j → θ_j + α_j, for arbitrary choices of α_j. We allow for a coupling to electromagnetism by endowing each species with charge q_j (but in such a way that the homogeneous background charge density is zero). We find that the overlap of the wave functions is zero for most choices of α_j, including α_j = m_jϵ where m_j is the mass of each of the species. This implies SSB and is associated with the conservation of mass. On the other hand, we find that for the special choice of α_j = q_j ϵ, the overlap of the wave functions is 1. This makes the overlap wave function we construct to be directly associated with the Goldstone boson structure and a useful diagnostic of the physical phase that the system is in.

Let us note that in this broader class of models, including multiple species of bosons, in addition to the electric symmetry, it also possesses other global symmetries. There is no reason a priori to expect that those symmetries cannot be spontaneously broken in the usual way. The electric symmetry on the other hand, since it does not have a local order parameter, is different and therefore we expect its realization to be distinct. It is useful to see exactly how these different properties are realized on the ground state wave function; this is non-obvious because these various transformations θ_j → θ_j + α_j are mixed up with one another, and are coupled because the electric interaction couples the multiples species to one another. The idea that the electric symmetry is expected to be realized differently is for a very physical reason that we shall return to: the conserved charge Q in the superconductor case behaves differently to the conserved number N of a regular global symmetry in the superfluid case. The reason being that in a superconductor the electric field is screened, so that when one integrates the electric flux over a surface at infinity, one formally obtains a vanishing charge; while the conserved number of a superfluid does not have any analogous screening. Since conserved quantities generate symmetries, this again leads to the expectation that the structure of the ground state is distinct. However, we will simply compute the overlap at infinite volume, using the same prescription in both superfluid and superconducting cases, and see the outcome directly.

Our paper is organized as follows: In Section 2 we present the two-fluid nonrelativistic model. In Section 3 we discuss the condensate background. In Section 4 we present the Lagrangian and Hamiltonian governing fluctuations. In Section 5 we explicitly compute the ground state wave function. In Section 6 we explicitly compute the overlap of the ground state wave functions. In Section 7 we discuss which quantities are conserved and the associated symmetries. Finally, in Section 8 we conclude.

2 Nonrelativistic field theory

We are interested in systems of nonrelativistic bosons. We will allow for several species that are distinguishable, labeled j, but will often specialize to the case of two species (j = 1, 2) where needed for simplicity. We are interested in systems of many particles. In this case, it is convenient to utilize creation aˆk,j† and destruction operators aˆk,j that produce N particle states. Since we are interested in exploring condensates, it is convenient to pass to the field representation. This is defined by Fourier transforming the destruction operator to a field in position space, the so-called Schrodinger field Ψˆj(x) (with conjugate field Ψˆj†(x)) as follows

The corresponding particle number density operator for species j is

with particle number operator Nˆj=∫d3x nˆj(x).

In order to explore superconductivity, we minimally couple the fields to electromagnetism A_μ = (−ϕ, A). For convenience we will use the Lagrangian formalism (though later we will move to the Hamiltonian formalism). Since ordinary superconductors and superfluids involve electrons and nuclei moving much slower than the speed of light, one can often use an effective nonrelativistic description. In order to build this, we note that the leading order scalar field sector is essentially specific uniquely by the Galilean symmetry. Furthermore, there is a unique way to couple to photons from the minimal coupling procedure. This uniquely specifies the nonrelativistic field theory. For the leading relativistic corrections, the reader may see Appendix A for the sake of completeness. But for the most part, the nonrelativistic effective field theory will suffice, and is given by

where m_j is the mass of each of the species and q_j is the charge of each of the species. For the potential V, we allow four-point self-interactions. However, for simplicity we assume that each species does not directly scatter off other species. For instance, one can imagine that the underlying fermionic description, which introduces Pauli exclusion, gives rise to repulsion among the indistinguishable particles. The generalization to other couplings is straightforward, but will not be studied here. We also include a chemical potential μ_j for each of the species, to make it simpler to describe a background (this can also be obtained from a redefinition of the fields as Ψj→Ψj eiμjt). Together we write the potential as

where λ_j are (positive) self-couplings.

We note that the above theory carries the following set of (global) symmetries

for independent α_j. When expanding around the vacuum, these are associated with the conservation of each of the species particle number, and include the special case of α_j = q_j ϵ corresponding to electric charge. In the following we will examine their behavior for a condensate (ground state) solution.

3 Homogeneous background

We now expand around a homogeneous background. At the classical field level, the ground state is determined by minimizing the above potential V. We can use the chemical potential to obtain whatever background number density of particles we desire. Let’s denote the background number density of each species nj0. By minimizing the potential, this value of number density can be immediately obtained by choosing the chemical potential to be

We will ensure that the background charge density ρ₀ = 0, so that we have a neutral superconductor to expand around. Hence we assume that the background number densities are related by the condition

where we have included a possible qnnn0 term, and this should be included in the case of a superconductor: it refers to nuclei, which carry positive charge q_n > 0. The nuclei provide the compensating charge, so there is a well-defined neutral superconductor that we may then add charge density fluctuations to. This is the standard physical starting point. In the case of the superconductor, we will not need to track the dynamics of the nuclei (although they will inevitably play a role when talking about mass density perturbations), as they are not accurately described by bosons, and are very heavy; so they will only be relevant at this background level. Instead, as is well known, in a BCS superconductor, the relevant dynamics is provided by the much lighter Cooper pairs, with charge q = −2e. One can have more general systems with multiple types of effective bosons. We will leave our analysis in terms of multiple species for the sake of pedagogy, and in fact the case of two species of bosons will often be our focus, since we can then discuss the fate of multiple types of symmetries. On the other hand, in the case of a superfluid, then the relevant species 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. Hence our framework is quite general.

The corresponding background field value for each of the relevant dynamical species (which is quite different depending on whether it is a superconductor or a superfluid) is given by

As is well known, the phase of the ground state condensate Ψj0 is not determined by this condition. This suggests there is a family of distinct ground state solutions labeled by a set of constant phase parameters θj0 as

which all formally spontaneously break the symmetry given above in Eq. (5). Since these symmetry transformations include the global sub-group of U(1)em, one should be extra careful in aspects of this conclusion. In this work, we will examine the specific issue of the structure of the ground state systematically; first we quantize the fluctuations and actually computing the ground state of the quantum theory precisely.

4 Perturbations

Let us expand around the homogeneous background by decomposing the fields into a perturbation in modulus η_j(x, t) and phase parameter θ_j(x, t) as

We then treat η_j and (derivatives of) θ_j as small to study small perturbations. Expanding the Lagrangian density to quadratic order in the fluctuations we obtain

Now the electromagnetic field includes the non-dynamical Coulomb potential ϕ. We can solve for this from Gauss law as follows

where the charge density (to linear order in perturbations) is

We now decompose the vector potential A into its longitudinal AL and transverse AT components

However, we can now exploit gauge invariance to simplify our results by operating in Coulomb gauge ∇⋅A = 0. So AL=0 and A=AT is purely transverse. One should bear in mind that all of our results can be trivially rewritten in a gauge invariant way be replacing

if desired. The Lagrangian density decomposes into a sum of longitudinal ℒL and transverse ℒT pieces that decouple at the quadratic order

The final term in (17) shows the familiar fact that within a superconductor the magnetic field acquires an effective mass. It is given by the sum of squares of the plasma frequencies as

This means the magnetic field is short-ranged, which is the famous Meissner effect (for example, see [37], [38], [39]). This is analogous to the Higgs mechanism in the Standard Model. However there is an important difference: In the Standard Model the Lorentz symmetry ensures that the Coulomb potential A0=−ϕ also acquires the same effective mass. However, in this nonrelativistic setup that is not the case. As we will later discuss, despite appearances, the Coulomb potential (in Coulomb gauge) and the associated electric field gets screened more strongly than the magnetic field. This has important ramifications for the behavior of the charge and the fate of symmetries, as we will discuss in Section 7.

Our interest is in the behavior of the longitudinal modes, as these involve the phase parameters θ_j, and enjoy the symmetries θ_j → θ_j + α_j. To study these modes in more detail, it is convenient to now pass to the Hamiltonian formalism. The appropriate phase space variables are the phase θ_j and momentum conjugates π_j given by

Furthermore, we diagonalize the problem by passing to k-space. We write the Hamiltonian for the longitudinal modes as

and find the k-space Hamiltonian density to be

Note that as expected it is the charges q_j that couple the different species to one another, as seen in the final term.

5 Ground state wave function

For the sake of simplicity, let us now specialize to the case of two species j = 1, 2. We can readily write the above Hamiltonian density in matrix notation, by defining the following vector fields

and the following matrices

This gives the following Hamiltonian density

We are now in a position to construct the ground state wave function. Recall that for a single harmonic oscillator with Hamiltonian H = Kp²/2 + Fx²/2 the ground state wave function in position space is well known to be ψ(x)∝exp(−F/Kx2/2). For the above Hamiltonian HL we need to generalize this to take into account the nontrivial matrix structure. Some matrix algebra reveals that the result in the field basis is

where M_k is the following matrix

Let us now examine this result in some important limits.

6 Wave function overlap

We now come to the main issue of comparing the set of ground state wave functions that differ by the symmetry transformations

for different choices of phase rotations α_j. Note that if we choose αj=qjα(x), with α(x) → 0 at infinity, then this represents a (small) gauge transformation and the above wave function, like all wave functions, is in fact gauge invariant; this can be made manifest by simply reinstating θj(x)→θj(x)−qj∇⋅AL/∇2, which is a well-defined operation for (small) gauge transformations. As emphasized earlier in the paper, the interesting issue is that of global transformations, with α_j constant.

However, performing constant phase rotations is awkward in k-space, since it would formally involve shifting θ_k by a delta-function θjk→θjk+αj(2π)3δ3(k). This means we then need to deal with factors of k multiplying delta-functions. So as k → 0, this means we formally have zero times infinity. Since delta functions are a kind of distribution and all distributions are defined in terms of their Fourier transform, then, as we did in Ref. [11], it is much more transparent to pass to position space. Since we are integrating over all of space ∫​d3x, it will be useful to be precise; mathematically an integral over an infinite domain is of course defined as ∫​d3x=limV→∞∫Vd3x, i.e., as a limit over a volume V as the volume is taken to infinity. In position space, we will then define our ground state wave function by its kernel representation

(where V‾ is the domain that x′ is integrated over, with volume V centered at x; again all understood in the limiting sense of lim_V→∞). Here M_ϵ is a matrix of kernels, defined by

Here ϵ is a UV regulator. It will be convenient to first regulate the UV modes, then send ϵ → 0 limit at the end of the calculation. This makes good physical sense, since the physics associated with the structure of ground state is the infrared behavior of the theory, and should not be sensitive to the UV. While there may be other prescriptions to define the ground state, which is subtle due to the fact that one is in infinite volume, what is important is that the above definition is quite natural and we will use it self-consistently throughout this paper. In particular, we will use it for both the superfluid and superconductor cases, and we shall see that it allows the distinct realizations of its symmetry structure to be made manifest, while other prescriptions can hide these important distinctions and so are less useful for our purposes.

In position space we also note that the complete wave function should be periodic under θ_j → θ_j + 2πn_j, where n_j is an integer. This is easily ensured by defining the improved wave function as

and furthermore, the final result is to be normalized appropriately.

Let us now consider a pair of ground state wave functions: One of them, |G〉, centered around θ_j = 0 and the other one, |G′〉, centered around θ_j = α_j. The (normalized) overlap between these two wave functions is given by the integral

We can readily compute this integral as it is Gaussian. Since M_ϵ(x−x′) is only a function of the difference between the position vectors, we can easily perform one of the spatial integrals to obtain a volume factor V, leaving a single spatial integral left over. We obtain

where we defined a vector of constant phase rotations

The above integral ∫​d3r Mϵ(r) can be more readily understood by rewriting Mϵ(r) in terms of the Laplacian of another matrix of kernels Jϵ(r) defined implicitly by

In terms of a Fourier transform we can define this by

where we inserted an extra factor of 1/k² in the integrand (and we used the fact that ϵ is very small).

Using the divergence theorem, the wave function overlap can then be given by the following boundary term

where dS is an infinitesimal surface area vector that points radially outward, which bounds the domain of integration V. In this representation it is now clear that the UV has decoupled, as the boundary term is purely an IR effect. In other words, we can now send ϵ → 0 to evaluate the above kernels, since we know that we do not need to evaluate J(r) as r → 0, we only need to evaluate J(r) at large r.

7 Conserved quantities

To understand this result further, let us examine the possible conserved quantities in the system. Naively there is a conserved quantity for each rotation α_j. Indeed this is true when expanding around the vacuum. However, when expanding around the superconducting condensate, it is more subtle.

8 Conclusions

In this work, we have explicitly computed the wave function overlap between ground states, defined in a specific prescription through the position space kernel representation, in nonrelativistic systems of condensed bosons, either modeling superconductivity or superfluidity. We showed that while a generic phase rotation transformation of the nonrelativistic Schrodinger field does indeed lead to vanishing overlap for any phase rotations, associated with those of a superfluid. For a super-conductor, there is a special combination of phase rotations that does not lead to a new state which are the standard form of electromagnetic phase rotations. In the literature, this is often summarized by the language that for “gauge symmetry breaking, the apparently different ground states are related by gauge transformations” in the bulk of a superconductor.^[2] However, here we wish to move beyond language and emphasize that there is a very physical phenomenon occurring: the electric field gets screened; this leads to the charge Q=limS→∞∮​dS⋅E formally vanishing, and so as an operator, it maps a ground state into itself. Moreover, in the superfluid case, one can add a large number of massless Goldstones of zero momenta to generate a new |G′〉, while in the superconductor case, the spectrum is gapped, so adding massive photons would raise the system out of the ground state.

Note that this in no way undermines the idea of a phase transition from the normal phase to the superconductor. One can define order parameters, such as 〈|V−1∫Vd3x Ψˆi|〉 (note the absolute value; it is unchanged under a phase rotation transformation), which are nonzero in the superconducting phase, while vanishing in the regular phase. This connects to the more familiar notion of SSB in field theories. Furthermore, one can express this in entirely particle physics terms: the former phase has a massless photon with long-ranged electric/magnetic fields, while the latter phase has a massive photon with short ranged electric/magnetic fields.

It is important to note that when the phase transition to the super-conducting state occurs, our ground state overlap exhibits a standard notion of SSB of related global symmetries. In this context of multiple bosonic fields, another is associated with mass conservation with phonons acting as the Goldstones, among other possibilities depending on the number of fields. These additional symmetries that can undergo the standard form of global SSB can act as their own diagnostic to determine the physical phase that the theory is in (Higgs or Coulomb, etc.) since the phase transition is accompanied by their breaking; so they act as a kind of custodial symmetry breaking (related ideas appear in Ref. [42]).

Our primary result is that our overlap furnishes a clear mapping to the Goldstone structure. In particular, with two-bosonic fields: if neither phase rotation exhibits 〈G|G′〉=0, we are in a normal phase; if one combination of the phase rotations exhibits 〈G|G′〉=0, we are in a superconducting phase; if both phase rotations exhibits 〈G|G′〉=0, we are in a superfluid phase. We noted that in the case of the super-conductor, the different structure in the overlap is related to the fact that the electric field is screened in this phase. So by using Gauss’ law the corresponding charge, expressed as a boundary term, vanishing. Since it acts as a symmetry operator in the quantum theory, it maps a ground state into itself. However, for the superfluid, its conserved “charges” (or particle numbers) cannot be expressed in terms of vanishing boundary terms; they are non-vanishing and drive the overlap to vanish for any phase rotation. So this is all in accord with good physical principles.

Our focus here has only been on the bulk and the direct computation of the overlap wave function, which we believe to be a new result. This construction is useful because it is nicely in one-to-one correspondence with the presence, or lack thereof, of Goldstone modes. One of course should expect that the existence of Goldstones should be reflected in the structure of the ground state wave function. Our prescription to define the ground state at infinite volume through its position space kernel representation, has led to the Goldstone bosons being directly connected to whether the overlap 〈G|G′〉=0 or not. This is an especially useful diagnostic for the physical phase of the system as it is gauge invariant (unlike most other order parameters), and may be applied in more general settings with richer dynamics.

On the other hand, if one has multiple finite size superconductors, there can be interesting boundary effects, including the Josephson effect [40], [, 41], when multiple superconductors are brought in contact with each other. The system can release some energy, in the form of the Josephson current, to relax to an even lower energy state. This provides other important features of the ultimate fate of SSB [32]. However, the focus of this work has been on the wave function in the bulk of a single material and relates directly to the physical phase that the system is in, including its energy spectrum.

These results are in accord with our earlier work in the Standard Model [11]. Other directions to consider are more complicated condensed matter systems, including those that exhibit strong coupling, and various other phases. Further applications may be to other areas in which SSB may play a role, including ideas in particle physics [43], [44], [45] and cosmology [46], [47], [48], [49], [50].