Journal of Statistical Physics

, Volume 165, Issue 3, pp 455–470 | Cite as

Two No-Go Theorems on Superconductivity

  • Yasuhiro Tada
  • Tohru Koma


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.


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



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.”


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute for Solid State PhysicsThe University of TokyoKashiwaJapan
  2. 2.Max Planck Institute for the Physics of Complex SystemsDresdenGermany
  3. 3.Department of PhysicsGakushuin UniversityToshima-kuJapan

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