A Proof of the Bloch Theorem for Lattice Models
The Bloch theorem is a powerful theorem stating that the expectation value of the U(1) current operator averaged over the entire space vanishes in large quantum systems. The theorem applies to the ground state and to the thermal equilibrium at a finite temperature, irrespective of the details of the Hamiltonian as far as all terms in the Hamiltonian are finite ranged. In this work we present a simple yet rigorous proof for general lattice models. For large but finite systems, we find that both the discussion and the conclusion are sensitive to the boundary condition one assumes: under the periodic boundary condition, one can only prove that the current expectation value is inversely proportional to the linear dimension of the system, while the current expectation value completely vanishes before taking the thermodynamic limit when the open boundary condition is imposed. We also provide simple tight-binding models that clarify the limitation of the theorem in dimensions higher than one.
KeywordsBloch theorem Persistent current Many-body systems
I would like to thank S. Bachmann, A. Kapustin, and L. Trifunovic for useful discussions on this topic. In particular, I learned the generalization to the Gibbs state from the authors of Ref.  and I am indebted to their email correspondence. I also thank T. Momoi for informing us of Ref. . This work is supported by JST PRESTO Grant No. JPMJPR18LA.
- 6.Bachmann, S., Bols, A., Roeck, W.D., Fraas, M.: A many-body index for quantum charge transport. Commun. Math. Phys. (2019). https://doi.org/10.1007/s00220-019-03537-x
- 9.Takahiro, S.: Second law-like inequalities with quantum relative entropy: an introduction. In: Nakahara, M. (ed.) Lectures on Quantum Computing, Thermodynamics and Statistical Physics, pp. 125–190. World scientific, Singapore (2013)Google Scholar
- 16.Watanabe, H., Oshikawa, M.: Inequivalent Berry phases for the bulk polarization. Phys. Rev. X 8, 021065 (2018)Google Scholar