Fidelity and Quantum phase transition for the Heisenberg chain with the next-nearest-neighbor interaction

TL;DR

This study explores how fidelity measures, especially between first excited states, can effectively identify quantum phase transitions in the Heisenberg chain with next-nearest-neighbor interactions, overcoming limitations of ground state fidelity.

Contribution

The paper introduces the use of first excited state fidelity to better characterize quantum phase transitions in the $J_1-J_2$ model, especially for finite systems.

Findings

Ground state fidelity fails to detect the phase transition in finite systems.

First excited state fidelity successfully characterizes the quantum transition.

The method is effective even for small system sizes.

Abstract

In this paper, we investigate the fidelity for the Heisenberg chain with the next-nearest-neighbor interaction (or the $J_1-J_2$ model) and analyze its connections with quantum phase transition. We compute the fidelity between the ground states and find that the phase transition point of the $J_1-J_2$ model can not be well characterized by the ground state fidelity for finite-size systems. Instead, we introduce and calculate the fidelity between the first excited states. Our results show that the quantum transition can be well characterized by the fidelity of the first excited state even for a small-size system.