Anomaly in Ehrenfest’s theorem in lattice quantum mechanics

Robert Nyden Hill

Emmanuel Fromager

J. G. Esteve

We present an analysis of anomalies and its relation with the Ehrenfest theorem from the Hamiltonian point of view. It is shown that when an operator becomes anomalous it is due to the fact that it does not keep invariant the domain of definition of the Hamiltonian, proving a theorem which states that an operator which keeps invariant such a domain cannot have anomalies.

Jerzy Czerwonko

Konstantinos Moulopoulos

For real potentials and for arbitrary boundary conditions we point to previously noted paradoxes in applications of the Ehrenfest and Hellmann-Feynman theorems, their resolution and their role on the consistency of quantum mechanical uncertainty relations. The paradoxes originate from a hidden non-Hermitian character of the kinetic energy operator and are resolved if the proper boundary terms (almost always discarded in the literature) are taken seriously. These non-Hermitian contributions (reflections at a deeper level of topological anomalies) follow their own patterns (for any dimensionality, for both Schroedinger and Dirac/Weyl Hamiltonians and for either continuous or lattice (tight-binding) models): they can always be written as global fluxes of certain generalized current densities. In continuous nonrelativistic cases, these have the forms that had earlier been used by Chemists to describe atomic fragments of polyatomic molecules - while for Dirac/Weyl or lattice models they appear to have the corresponding relativistic forms. In spite of the fact that these boundary terms originate from a deep mathematical anomaly, we point out examples where such non-Hermiticities have physical significance in Quantum Condensed Matter Physics (for both conventional and topologically nontrivial materials). In all stationary state examples considered, these non-Hermitian boundary terms have turned out to be quantized, this quantization being either of conventional or of a topological (Quantum Hall Effect (QHE)-type) origin. The latter claim is substantiated through direct application to a simple QHE arrangement (2-D Landau system in an external electric field). Finally, the non-Hermitian terms are also shown to be crucial for the consistency of the standard uncertainty relations (of Kennard/Robertson-type) in multiply-connected space or, generally, in any system that satisfies a Bloch theorem.

Konstantinos Moulopoulos

For real potentials and for arbitrary boundary conditions we point to previously noted paradoxes in applications of the Ehrenfest and Hellmann-Feynman theorems, their resolution and the role played on the consistency of quantum mechanical uncertainty relations. The paradoxes originate from a hidden non-Hermitian character of the kinetic energy operator and are resolved if the proper boundary terms (almost always discarded in the literature) are taken seriously. These non-Hermitian contributions (reflections at a deeper level of topological anomalies) follow their own patterns (for any dimensionality, for both Schroedinger and Dirac/Weyl Hamiltonians and for either continuous or lattice (tight-binding) models): they can always be written as global fluxes of certain generalized current densities. In continuous nonrelativistic cases, these have the forms that had earlier been used by Chemists to describe atomic parts of polyatomic molecules while for Dirac/Weyl or lattice models they appear to have the corresponding relativistic forms. In spite of the fact that these boundary terms originate from a deep mathematical anomaly, we point out examples where such non-Hermiticities have physical significance in Quantum Condensed Matter Physics (for both conventional and topologically nontrivial materials). In all stationary state examples considered, these non-Hermitian boundary terms have turned out to be quantized, this quantization being either of conventional or of a topological (Quantum Hall Effect (QHE)-type) origin. The latter claim is substantiated through direct application to a simple QHE arrangement (2-D Landau system in an external electric field). Finally, the non-Hermitian terms are also shown to be crucial for the consistency of the standard uncertainty relations (of Kennard/Robertson type) in multiply-connected space or, generally, in any system that satisfies the Bloch theorem.