# Two No-Go Theorems on Superconductivity

## Abstract

We study lattice superconductors such as attractive Hubbard models. As is well known, Bloch’s theorem asserts absence of persistent current in ground states and equilibrium states for general fermion systems. While the statement of the theorem is true, we can show that the theorem cannot exclude possibility of a surface persistent current. Such a current can be stabilized by boundary magnetic fields which do not penetrate into the bulk region of a superconductor, provided emergence of massive photons, i.e., Meissner effect. Therefore, we can expect that a surface persistent current is realized for a ground/equilibrium state in the sense of stability against local perturbations. We also apply Elitzur’s theorem to superconductors at finite temperatures. As a result, we prove absence of symmetry breaking of the global *U*(1) phase of electrons for almost all gauge fields. These observations suggest that the nature of superconductivity is the emergence of massive photons rather than the symmetry breaking of the *U*(1) phase of electrons.

### Keywords

Superconductivity Persistent current*U*(1) symmetry breaking Bloch’s theorem Elitzur’s theorem Meissner effect

## 1 Introduction

Superconductivity still has some mysteries to solve although it has been one of the central issues in condensed matter physics. Bloch’s theorem [1, 2, 3, 4, 5, 6, 7] states absence of bulk persistent current in ground states and equilibrium states for general fermion systems, whereas experiments of superconductors [8] show persistent current with toroidal geometry. Elitzur’s theorem [9, 10, 11] states that local gauge symmetries including quantum mechanical *U*(1) symmetry cannot be broken spontaneously, whereas superconductivity has been often explained by using the idea that the *U*(1) symmetry is spontaneously broken with a nonvanishing expectation value of an order parameter such as the Cooper pair amplitude. Although these two theorems have a long history, it is still unclear how the two theorems are reconciled with the conventional theory of superconductivity.

If the superconducting state which carries the persistent current in the experiments is neither a ground state nor a thermal equilibrium state, then the state must be an excited state or a nonequilibrium state which has a very long lifetime. Such a state is often said to be metastable [2]. What mechanism makes the lifetime of the metastable state carrying the persistent current very long? One of the important stuffs is multiply-connected geometry such as toroid. In fact, we can expect that, when magnetic fields are wound up the surface of the large toroid in a stable state which is realized by Meissner effect, changing the topology of the global structure needs a large energy which is determined by the system size.

In the present paper, we consider possibility of a ground/equilibrium state which carries a persistent current, and show that Bloch’s theorem cannot exclude such a state if the persistent current is localized near the surface of the sample with boundary magnetic fields which are stabilized by Meissner effect. Therefore, we can expect that a surface persistent current is realized for *a ground/equilibrium state* in the sense of stability against local perturbations. Such a current carrying state has a long lifetime even in a finite size system, because a transition from the corresponding state to other states costs larger and larger energy as the system size increases.

In order to explain Meissner effect in the above argument, one has to deal with the interactions between electrons and electro-magnetic fields. As mentioned above, however, Elitzur’s theorem [9] implies that the local *U*(1) gauge symmetry cannot be broken spontaneously. Actually, we can prove, by relying on the argument of [9], that the fluctuations of the gauge fields yield absence of the *U*(1) symmetry breaking for superconductors at finite temperatures. Surprisingly, even for the systems with a fixed configuration of gauge fields, we can prove absence of symmetry breaking of the global *U*(1) phase of electrons for almost all configurations of gauge fields in the sense of macroscopic spontaneous magnetization. This is slightly different from the former consequence of Elitzur’s theorem.

These observations in the present paper strongly suggest that the nature of superconductivity is the emergence of massive photons rather than the symmetry breaking of the *U*(1) phase of electrons. In fact, it is known that, in Higgs models coupled with gauge fields [11, 12, 13, 14], a transition from massless photons to massive photons is possible for varying the coupling constant, irrespective of whether or not the *U*(1) symmetry breaking occurs.

The present paper is organized as follows: In Sect. 2, we give the precise definition of the models which we consider, and examine Bloch’s theorem in a mathematically rigorous manner. As a result, we show that Bloch’s theorem cannot exclude the possibility of surface persistent current. In Sect. 3, we apply Elitzur’s theorem to two situations, the present models with annealed and quenched gauge fields. In both cases, we prove absence of the *U*(1) symmetry breaking.

## 2 Absence of Bulk Persistent Current

In this section, we show that Bloch’s theorem does not exclude possibility of a surface current in a ground/equilibrium state, by examining the proof in a mathematically rigorous manner.

### 2.1 Models

Consider a *d*-dimensional connected graph \(G:=(\Lambda ,\mathcal{B})\), where \(\Lambda \) is a set of lattice sites and \(\mathcal{B}\) is a set of bonds, i.e., pairs of lattice sites. The graph-theoretic distance, \(\mathrm{dist}(x,y)\), between two lattice sites *x*, *y* is defined as the minimum number of bonds in \(\mathcal{B}\) that one needs to connect *x* and *y*.

*x*. We assume that both of the hopping amplitudes and the interactions are of finite range in the sense of the graph-theoretic distance, and assume that all of the strengths are uniformly bounded as

### 2.2 Current Operators

*R*as

*L*. Namely, the region \(\Omega \) consists of

*L*-layers of the segments \(D_j\) of the \((d-1)\)-dimensional hyperplanes. We assume that the distance between two discs \(D_i\) and \(D_j\) satisfies

*M*. We assume that the support of the wavepacket \(\varphi \) is contained in the region \(\tilde{\Omega }\). We define the position operator

*X*as

*X*is written as

*X*, we have

### 2.3 Bloch’s Theorem

In order to realize persistent current in the ground state of the present model, we consider a toroidal geometry for the graph as an example. We impose a boundary condition which mimics the plus boundary condition for ferromagnetic Ising models leading to the symmetry breaking phases with plus spontaneous magnetization. To be specific, we apply a magnetic field tangential to the surface \(\partial \Lambda \) of the toroidal lattice \(\Lambda \) so that surface current perpendicular to the magnetic field can appear along the boundary of the toroid.

*N*-fermion ground state \(\Phi _{\Lambda ,0}^{(N)}\) of the Hamiltonian \(H_\Lambda \) with the above boundary condition. The ground-state expectation is given by

*a*which we consider.

The statement of Bloch’s theorem is as follows:

### Theorem 2.1

*L*and radius

*R*.

### Remark

- (i)
For the two or higher dimensional systems, the order of the double limit in (2.4) is not interchangeable. In fact, for a fixed

*R*, the upper bound of the expectation value in the limit \(L\nearrow \infty \) becomes infinity and meaningless in the proof below. If we want to measure the strength of surface current which is localized near the surface of the sample, we must take*R*to be finite so that the support \(\Omega \) of the current operator \(J_\Omega \) is localized near the surface. Then, Bloch’s theorem tells us absolutely nothing about the surface current.

- (ii)
Clearly, in one dimensional systems, the surface current itself is meaningless, and the net current along the direction of the chain always vanishes, in contrast to higher dimensional systems. This is a consequence of lack of a mechanism which stabilizes the current against local perturbations. In other words, any current is destroyed by local perturbations in one dimension.

- (iii)
The boundary magnetic fields are expected to be realized by Meissner effect, and once it is generated, the surface current remains stable in a ground states, i.e., a persistent current.

- (iv)The extension of the statement of Bloch’s theorem to the systems at finite temperatures is straightforward by relying on “passivity” which is a stability property of thermal equilibrium states. More precisely, an infinite-volume state \(\omega \) is said to be passive iffor any local unitary operator$$\begin{aligned} \omega (U^*[H,U])\ge 0 \end{aligned}$$
*U*, where*H*is the Hamiltonian. As is well known, all of thermal equilibrium states are passive [17]. This fact was pointed out to us by Hal Tasaki.

### Proof

*R*is the radius of the discs. Then, the volume \(|\Omega |\) of the region \(\Omega \) satisfies \(|\Omega |=\mathcal{O}(R^{d-1}\times L)\). We also assume that the area of the side of the cylindrical region \(\Omega \) satisfies \(|\partial \Omega \backslash (D_0\cup D_L)|=\mathcal{O}(R^{d-2}\times L)\). Since the hopping range is of finite, we have

*x*,

*y*satisfying \(k=\ell (y)-\ell (x)\) appear

*k*times in the sums. From these observations, we have

## 3 Absence of *U*(1) Symmetry Breaking

In this section, we extend Elitzur’s theorem to fermionic models. Consider the Hamiltonian \(H_\Lambda \) of (2.1) on a *d*-dimensional finite lattice \(\Lambda \subset \mathbb {Z}^d\). For simplicity, we assume that the Hamiltonian \(H_\Lambda \) contains only the nearest neighbor hopping, and that the fermions have only spin-1/2 as the internal degree of freedom, i.e., we consider usual electrons.

### 3.1 Annealed Gauge Fields

*U*(1) gauge field

*A*as follows: For each nearest neighbor pair \(\langle x,y\rangle \) of sites \(x,y\in \Lambda \), the gauge field \(A_{x,y}\) takes the value \(A_{x,y}\in \mathbb {R}\ \text{ mod }\ 2\pi \), and satisfies the conditions,

*A*is given by

*U*(1) symmetry breaking, we consider the Cooper pair \(c_{u,\uparrow }c_{v,\downarrow }\) of electrons for fixed two sites

*u*,

*v*, where we assume that the site

*v*is in a neighborhood \(\mathcal {N}_u\) of the site

*u*. The total energy containing the energy of the gauge fields and the terms of the symmetry breaking fields is

*p*(unit square cell); the Hamiltonian \(H_{\mathrm{ext},\Lambda }\) of the symmetry breaking field is given by the sum of the local order parameters as

*A*. Here, \(\mathcal {B}\) is the set of the bonds (the nearest neighbor pairs of sites), and we have written \(A_b=A_{x,x+e_i}\) with the unit vector \(e_i\) in the

*i*-th direction. The three parameters, \(\kappa , h\) and \(\tilde{h}\), are taken to be positive.

*p*are vanishing. We stress that our result still holds for a finite \(\kappa \). But we impose this strong condition because a nonvanishing magnetic flux generally is believed to suppress superconducting states. In the infinite limit \(\kappa \uparrow \infty \), the gauge fixing degree of freedom still remains. Therefore, in order to lift the gauge fixing degree of the freedom, we have introduced the Hamiltonian \(\tilde{H}_{\mathrm{ext},\Lambda }(A)\) of (3.3) which prefers the gauge fixing \(A_b=0\) for all the bonds

*b*.

We have:

### Theorem 3.1

### Remark

- (i)
Instead of the Cooper pair amplitude, the statement of Theorem 3.1 holds for any local observable which transforms nontrivially under the local gauge transformations.

- (ii)
Consider the situation without the symmetry breaking fields. But, instead of the symmetry breaking fields, we introduce gauge fixing terms into the Hamiltonian as in Kennedy and King [20] and Borgs and Nill [11, 13]. For the Higgs models, they proved that the two-point correlations for the Higgs fields do not exhibit long-range order in dimensions \(d\le 4\) except for the Landau gauge. In the Landau gauge, Kennedy and King [20, 21] proved that the noncompact

*U*(1) Higgs model has a phase transition in dimensions \(d\ge 3\). However, when there appears the remaining gauge degree of freedom which is called the Gribov ambiguity, the two-point correlations vanish at noncoinciding points even in the Landau gauge as Borgs and Nill [11, 13] pointed out. In the same situation for the present lattice fermion systems, we can prove that the two-point Cooper pair correlations do not show long-range order in spatial dimensions \(d\le 4\) except for the Landau gauge. Further, if the corresponding Gribov ambiguity appears, then the two-point correlations vanish at noncoinciding points even in the Landau gauge. We will publish the details elsewhere.

### Proof

*u*for the Cooper pair \(c_{u,\uparrow }c_{v,\downarrow }\) as

*O*be an observable. Note that

*a*and \(\tilde{a}\) are some positive constant.

### 3.2 Quenched Gauge Fields

*A*, let us consider the Hamiltonian,

*A*is given by

We obtain:

### Theorem 3.2

Both of the spontaneous magnetization \(M(\beta ,A)\) and the long-range order \(\sigma (\beta ,A)\) for the Cooper pair are vanishing for almost all gauge fields *A*.

### Remark

- (i)
Theorem 3.2 does not exclude the possibility that the

*U*(1) symmetry breaking occurs for some particular gauge fixing. As mentioned in Remark (ii) of Theorem 3.1, Kennedy and King [20, 21] showed that there appears*U*(1) symmetry breaking in a noncompact*U*(1) Higgs model in Landau gauge. However, there are some differences between noncompact and compact gauge theories. In fact, except for this special case which was treated by Kennedy and King, several cases in a certain gauge fixing show absence of the*U*(1) symmetry breaking in Higgs models [11, 12, 13].

- (ii)
Remark (i) of Theorem 3.1 holds for the present case, too.

### Proof

*a*. This implies

*u*,

*v*. Similarly, for \(\mathcal {M}_\Lambda \) of (3.17), we have

*U*(1) transformation contains the reversal of the orientation of the external symmetry breaking field. Under this transformation, the free energy is invariant. Combining this fact with the concavity of the free energy, one can show the positivity of the magnetization, i.e.,

## Notes

### Acknowledgments

We would like to thank Peter Fulde, Hosho Katsura, Masaaki Shimozawa, Hal Tasaki and Masafumi Udagawa for helpful discussions. YT was partly supported by JSPS/MEXT Grant-in-Aid for Scientific Research (Grant No. 26800177) and by Grant-in-Aid for Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers (Grant No. R2604) “TopoNet.”

### References

- 1.Bohm, D.: Note on a theorem of Bloch concerning possible causes of superconductivity. Phys. Rev.
**75**, 502–504 (1949)ADSCrossRefMATHGoogle Scholar - 2.Sewell, G.L.: Stability, equilibrium and metastability in statistical mechanics. Phys. Rep.
**57**, 307–342 (1980)ADSCrossRefMathSciNetGoogle Scholar - 3.Vignale, G.: Rigorous upper bound for the persistent current in systems with toroidal geometry. Phys. Rev. B
**51**, 2612–2615 (1995)ADSCrossRefGoogle Scholar - 4.Ohashi, Y., Momoi, T.: On the Bloch theorem concerning spontaneous electric current. J. Phys. Soc. Jpn.
**65**, 3254–3259 (1996)ADSCrossRefGoogle Scholar - 5.Kusama, Y., Ohashi, Y.: Effect of the BCS supercurrent on the spontaneous surface flow in unconventional superconductivity with broken time-reversal-symmetry. J. Phys. Soc. Jpn.
**68**, 987–993 (1999)ADSCrossRefGoogle Scholar - 6.Tada, Y.: Equilibrium surface current and role of \(U(1)\) symmetry: sum rule and surface perturbations. Phys. Rev. B
**92**, 104502 (2015)ADSCrossRefGoogle Scholar - 7.Yamamoto, N.: Generalized Bloch theorem and chiral transport phenomena. Phys. Rev. D
**92**, 085011 (2015)ADSCrossRefGoogle Scholar - 8.Kim, Y.B., Hempstead, C.F., Strnad, A.R.: Critical persistent currents in hard superconductors. Phys. Rev. Lett.
**9**, 306–309 (1962)ADSCrossRefGoogle Scholar - 9.Elitzur, S.: Impossibility of spontaneously breaking local symmetries. Phys. Rev. D
**12**, 3978–3982 (1975)ADSCrossRefGoogle Scholar - 10.De Angelis, G.F., de Falco, D., Guerra, F.: Note on the abelian Higgs-Kibble model on a lattice: absence of spontaneous magnetization. Phys. Rev. D
**17**, 1624–1628 (1978)ADSCrossRefGoogle Scholar - 11.Borgs, C., Nill, F.: The phase diagram of the abelian lattice Higgs model. A review of rigorous results. J. Stat. Phys.
**47**, 877–904 (1987)ADSCrossRefMathSciNetGoogle Scholar - 12.Fröhlich, J., Morchio, G., Strocchi, F.: Higgs phenomenon without symmetry breaking order parameter. Nucl. Phys. B
**190**(3), 553–582 (1981)ADSCrossRefMathSciNetGoogle Scholar - 13.Borgs, C., Nill, F.: Gribov copies and absence of spontaneous symmetry breaking in compact \(U(1)\) lattice higgs models. Nucl. Phys. B
**270**(16), 92–108 (1986)ADSCrossRefMathSciNetGoogle Scholar - 14.Seiler, E.: On the Higgs-confinement complementarity. Preprint, arXiv:1506.00862
- 15.Mahan, G.D.: Many-Particle Physics, 2nd edn. Plenum Press, New York (1990)CrossRefGoogle Scholar
- 16.Koma, T.: Topological current in fractional chern insulators. Preprint, arXiv:1504.01243
- 17.Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics II, 2nd edn. Springer, New York (1981)CrossRefMATHGoogle Scholar
- 18.Roepstorff, G.: Correlation inequalities in quantum statistical mechanics and their application in the Kondo problem. Commun. Math. Phys.
**46**, 253–262 (1976)ADSCrossRefMathSciNetGoogle Scholar - 19.Dyson, F.J., Lieb, E.H., Simon, B.: Phase transition in quantum spin systems with isotropic and nonisotropic interactions. J. Stat. Phys.
**18**, 335–383 (1978)ADSCrossRefMathSciNetGoogle Scholar - 20.Kennedy, T., King, C.: Spontaneous symmetry breakdown in the abelian Higgs model. Commun. Math. Phys.
**104**, 327–347 (1986)ADSCrossRefMATHMathSciNetGoogle Scholar - 21.Borgs, C., Nill, F.: Symmetry breaking in Landau gauge. A comment to a paper by T. Kennedy and C. King. Commun. Math. Phys.
**104**, 349–352 (1986)ADSCrossRefMATHMathSciNetGoogle Scholar - 22.Koma, T., Tasaki, H.: Symmetry breaking in Heisenberg antiferromagnets. Commun. Math. Phys.
**158**, 191–214 (1993)ADSCrossRefMATHMathSciNetGoogle Scholar