Quantum phase transitions in nonhermitian harmonic oscillator

TL;DR

This paper explores quantum phase transitions in non-Hermitian, PT-symmetric harmonic oscillators, demonstrating how to reconstruct the physical Hilbert space during level crossings to understand phase transitions.

Contribution

It provides a non-numerical method to reconstruct the physical Hilbert space in PT-symmetric systems at level crossings, facilitating the study of quantum phase transitions.

Findings

Reconstruction of the physical Hilbert space is feasible during unavoided level crossings.

The method enables treating exceptional points as genuine quantum phase transitions.

Application to PT-symmetric spiked harmonic oscillator demonstrates practical utility.

Abstract

The Stone theorem requires that in a physical Hilbert space ${\cal H}$ the time-evolution of a stable quantum system is unitary if and only if the corresponding Hamiltonian $H$ is self-adjoint. Sometimes, a simpler picture of the evolution may be constructed in a manifestly unphysical Hilbert space ${\cal K}$ in which $H$ is nonhermitian but ${\cal PT}-$symmetric. In applications, unfortunately, one only rarely succeeds in circumventing the key technical obstacle which lies in the necessary reconstruction of the physical Hilbert space ${\cal H}$. For a ${\cal PT}-$symmetric version of the spiked harmonic oscillator we show that in the dynamical regime of the unavoided level crossings such a reconstruction of ${\cal H}$ becomes feasible and, moreover, obtainable by non-numerical means. The general form of such a reconstruction of ${\cal H}$ enables one to render every exceptional unavoided-crossing point tractable as a genuine, phenomenologically most appealing quantum-phase-transition instant.