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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bohm, D.: Note on a theorem of Bloch concerning possible causes of superconductivity. Phys. Rev. 75, 502–504 (1949)
Sewell, G.L.: Stability, equilibrium and metastability in statistical mechanics. Phys. Rep. 57, 307–342 (1980)
Vignale, G.: Rigorous upper bound for the persistent current in systems with toroidal geometry. Phys. Rev. B 51, 2612–2615 (1995)
Ohashi, Y., Momoi, T.: On the Bloch theorem concerning spontaneous electric current. J. Phys. Soc. Jpn. 65, 3254–3259 (1996)
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)
Tada, Y.: Equilibrium surface current and role of \(U(1)\) symmetry: sum rule and surface perturbations. Phys. Rev. B 92, 104502 (2015)
Yamamoto, N.: Generalized Bloch theorem and chiral transport phenomena. Phys. Rev. D 92, 085011 (2015)
Kim, Y.B., Hempstead, C.F., Strnad, A.R.: Critical persistent currents in hard superconductors. Phys. Rev. Lett. 9, 306–309 (1962)
Elitzur, S.: Impossibility of spontaneously breaking local symmetries. Phys. Rev. D 12, 3978–3982 (1975)
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)
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)
Fröhlich, J., Morchio, G., Strocchi, F.: Higgs phenomenon without symmetry breaking order parameter. Nucl. Phys. B 190(3), 553–582 (1981)
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)
Seiler, E.: On the Higgs-confinement complementarity. Preprint, arXiv:1506.00862
Mahan, G.D.: Many-Particle Physics, 2nd edn. Plenum Press, New York (1990)
Koma, T.: Topological current in fractional chern insulators. Preprint, arXiv:1504.01243
Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics II, 2nd edn. Springer, New York (1981)
Roepstorff, G.: Correlation inequalities in quantum statistical mechanics and their application in the Kondo problem. Commun. Math. Phys. 46, 253–262 (1976)
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)
Kennedy, T., King, C.: Spontaneous symmetry breakdown in the abelian Higgs model. Commun. Math. Phys. 104, 327–347 (1986)
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)
Koma, T., Tasaki, H.: Symmetry breaking in Heisenberg antiferromagnets. Commun. Math. Phys. 158, 191–214 (1993)
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.”
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tada, Y., Koma, T. Two No-Go Theorems on Superconductivity. J Stat Phys 165, 455–470 (2016). https://doi.org/10.1007/s10955-016-1629-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-016-1629-2