Broken Hermiticity phase transition in Bose-Hubbard model

TL;DR

This paper introduces a novel framework for quantum phase transitions involving a switch from Hermitian to non-Hermitian Hamiltonians, ensuring smooth physical and mathematical continuity, exemplified through a Bose-Hubbard model.

Contribution

It proposes a consistent method to model phase transitions with Hermitian to non-Hermitian Hamiltonians, including explicit constructions of the necessary Hilbert space metrics.

Findings

Demonstrates smooth transition at the operator and metric level

Provides explicit forms of the Hilbert space metric after transition

Validates the approach using the Bose-Hubbard model example

Abstract

A new version of the change of the "phase" (i.e., of the set of observable characteristics) of a quantum system is proposed. In a general scenario the evolution is assumed generated, before the phase transition, by some standard Hermitian Hamiltonian $H^{(before)}$, and, after the phase transition, by one of the recently very popular non-standard, non-Hermitian (but hiddenly Hermitian, i.e., still unitarity-guaranteeing) Hamiltonians $H^{(after)}$. For consistency, a smoothness of matching between the two operators as well as between the related physical Hilbert spaces must be guaranteed. The feasibility of the idea is illustrated via the two-mode $(N-1)-$bosonic Bose-Hubbard Hamiltonian. In $H^{(before)}=H^{(BH)}(\varepsilon)$ we use the decreasing real $\varepsilon^{(before)} \to 0$. In the hiddenly Hermitian continuation $H^{(after)}=H^{(BH)}(\tilde{\varepsilon})$ the imaginary part of the purely imaginary $\tilde{\varepsilon}^{(after)}$ grows. The smoothness of the transition occurring at the interface $\varepsilon=\tilde{\varepsilon}=0$ is then guaranteed by an {\it ad hoc\,} amendment of the inner product in Hilbert space "after". The trivial Hilbert-space metric $\Theta^{(before)}=I$ must match $\Theta^{(after)} \neq I$ smoothly. This is confirmed and illustrated by the explicit constructions of a few $\Theta^{(after)}$s in closed form.