Relating non-Hermitian and Hermitian quantum systems at criticality

TL;DR

This paper explores how transformations connect Hermitian and non-Hermitian quantum systems at criticality, revealing conditions under which their conformal field theory descriptions, including central charges, are preserved or altered.

Contribution

It introduces three types of transformations linking Hermitian and non-Hermitian systems at criticality, analyzing their effects on spectra and central charges, with explicit examples and models.

Findings

Transformations can preserve both energy and entanglement spectra, maintaining identical central charges.

A non-Hermitian system with c = -4 maps to a Hermitian system with c = 2 via dilation.

Galois conjugation relates models with negative and positive central charges through entanglement entropy scaling.

Abstract

We demonstrate three types of transformations that establish connections between Hermitian and non-Hermitian quantum systems at criticality, which can be described by conformal field theories (CFTs). For the transformation preserving both the energy and the entanglement spectra, the corresponding central charges obtained from the logarithmic scaling of the entanglement entropy are identical for both Hermitian and non-Hermitian systems. The second transformation, while preserving the energy spectrum, does not perserve the entanglement spectrum. This leads to different entanglement entropy scalings and results in different central charges for the two types of systems. We demonstrate this transformation using the dilation method applied to the free fermion case. Through this method, we show that a non-Hermitian system with central charge $c = -4$ can be mapped to a Hermitian system with central charge $c = 2$. Lastly, we investigate the Galois conjugation in the Fibonacci model with the parameter $\phi \to - 1/\phi$, in which the transformation does not preserve both energy and entanglement spectra. We demonstrate the Fibonacci model and its Galois conjugation relate the tricritical Ising model/3-state Potts model and the Lee-Yang model with negative central charges from the scaling property of the entanglement entropy.