Broken Hermiticity phase transition in Bose-Hubbard model

DOI10.1103/PHYSREVA.98.052102arXiv1810.05771WikidataQ128984025 ScholiaQ128984025MaRDI QIDQ6308156

Miloslav Znojil

Publication date: 12 October 2018

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 $varepsilo{n}^{(before)}o0$. In the hiddenly Hermitian continuation $H^{(after)}=H^{(BH)}(ildevarepsilon)$ the imaginary part of the purely imaginary $ilde{varepsilon}^{(after)}$ grows. The smoothness of the transition occurring at the interface $varepsilon=ildevarepsilon=0$ is then guaranteed by an {it ad hoc,} amendment of the inner product in Hilbert space "after". The trivial Hilbert-space metric $Thet{a}^{(before)}=I$ must match $Thet{a}^{(after)}eqI$ smoothly. This is confirmed and illustrated by the explicit constructions of a few $Thet{a}^{(after)}$s in closed form.

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