Quantum phase transitions and entanglement entropy in a non-Hermitian spin-boson system

TL;DR

The paper analyzes a spin-boson system with non-Hermitian coupling, revealing an infinite set of two-dimensional invariant subspaces plus a singlet ground state. It formalizes pseudo-Hermiticity and its breaking via metric operators and intertwiners, showing real spectra in the pseudo-Hermitian phase and complex spectra in the broken phase, separated by exceptional points with a 1/2 critical exponent. It introduces biorthogonal and Dirac-normalized entanglement measures, demonstrating that the spin-boson entanglement entropy on each subspace distinguishes the phases: it ranges from 0 to ln 2 in the pseudo-Hermitian phase and saturates at ln 2 in the broken phase, with maximal entanglement at the EP. The results connect phase transitions to coherence-to-decoherence transitions in each invariant subspace, highlighting information-theoretic signatures of non-Hermitian quantum criticality and potential applications in open quantum systems.

Abstract

In this paper, we describe some interesting properties of a spin-boson system with non-Hermitian coupling. For this particular model, it is known that the Hilbert space can be described by infinitely-many two-dimensional invariant (closed) subspaces, together with the global ground state. We expose the appearance of exceptional points on such two-dimensional subspaces, together with quantum phase transitions marking the transition from real to complex eigenvalues. We also compute the spin-boson entanglement entropy on each invariant subspace to show that the two phases can be distinguished by their distinct entanglement-entropy profiles.

Figures (3)

• Figure 1: Eigenvalues of $H_1$ as a function of $\delta_1 = \gamma$. The eigenvalue $R^{\rm I}_1$ is indicated by the solid-orange lines while the eigenvalue $R^{\rm II}_1$ is indicated by the black-dashed lines. The vertical grey line at $\delta_1 = 2$ indicates the exceptional point.
• Figure 2: Figures showing the regions of the $\gamma\epsilon$-parameter space for $\omega = 1$ in which pseudo-Hermiticity is broken (grey) and unbroken (blue) for $n=0,1,2,3$. Exceptional points lie on the dashed lines separating the two regions.
• Figure 3: Variation of the spin-boson entanglement entropy for $\omega = 1$ and $\epsilon =5$, as a function of $\delta_{n+1}^2 = (n+1) \gamma^2$. The orange-solid curve represents $S^{\rm I}$ while the black-dotted curve represents $S^{\rm II}$. The blue-dashed vertical line represents the exceptional point at which pseudo-Hermiticity breaks down.