Quantum phase transitions in nonhermitian harmonic oscillator

Abstract

The Stone theorem requires that in a physical Hilbert space $\mathcal{H}$ 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 $\mathcal{K}$ in which $H$ is nonhermitian but $\mathcal{P}\mathcal{T}-$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 $\mathcal{H}$. For a $\mathcal{P}\mathcal{T}-$symmetric version of the spiked harmonic oscillator we show that in the dynamical regime of the unavoided level crossings such a reconstruction of $\mathcal{H}$ becomes feasible and, moreover, obtainable by non-numerical means. The general form of such a reconstruction of $\mathcal{H}$ enables one to render every exceptional unavoided-crossing point tractable as a genuine, phenomenologically most appealing quantum-phase-transition instant.

18 pp, 1 figure

References in corpus (3)

• Making Sense of Non-Hermitian Hamiltonians
• Three-Hilbert-Space Formulation of Quantum Mechanics
• Non-Hermitian coupled cluster method for non-stationary systems and its interaction-picture reinterpretation

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