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...

Full description

Saved in:
Bibliographic Details
Published inJournal of statistical physics Vol. 177; no. 4; pp. 717 - 726
Main Author Watanabe, Haruki
Format Journal Article
LanguageEnglish
Published New York Springer US 01.11.2019
Springer
Springer Nature B.V
Subjects
Analysis
Boundary conditions
Mathematical and Computational Physics
Physical Chemistry
Physics
Physics and Astronomy
Quantum Physics
Statistical Physics and Dynamical Systems
Theorems
Theoretical
Thermodynamics
Persistent current
Bloch theorem
Many-body systems
Online AccessGet full text

Cover

More Information
Summary: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.
ISSN:0022-4715
1572-9613
DOI:10.1007/s10955-019-02386-1