Excited State Quantum Phase Transitions Studied from a Non-Hermitian Perspective

TL;DR

This paper investigates the relationship between exceptional points in non-Hermitian quantum mechanics and excited state quantum phase transitions in the Lipkin-Meshkov-Glick model, revealing how EPs signal ESQPTs and how they approach the real axis as system size grows.

Contribution

It demonstrates the connection between EPs and ESQPTs using the non-Hermitian LMG model and introduces a method to extract critical parameters via Padé approximants.

Findings

EPs are linked to avoided crossings in ESQPTs.

EPs approach the real axis as system size increases.

Padé approximants can determine critical control parameters.

Abstract

A main distinguishing feature of non-Hermitian quantum mechanics is the presence of exceptional points (EPs). They correspond to the coalescence of two energy levels and their respective eigenvectors. Here, we use the Lipkin-Meshkov-Glick (LMG) model as a testbed to explore the strong connection between EPs and the onset of excited state quantum phase transitions (ESQPTs). We show that for finite systems, the exact degeneracies (EPs) obtained with the non-Hermitian LMG Hamiltonian continued into the complex plane are directly linked with the avoided crossings that characterize the ESQPTs for the real (physical) LMG Hamiltonian. The values of the complex control parameter $\alpha$ that lead to the EPs approach the real axis as the system size $N\rightarrow \infty$. This happens for both, the EPs that are close to the separatrix that marks the ESQPT and also for those that are far away, although in the latter case, the rate the imaginary part of $\alpha$ reduces to zero as $N$ increases is smaller. With the method of Pad\'e approximants, we can extract the critical value of $\alpha$.