Complete finite-size scaling theory of Renyi thermal entropy for second, first and weak first order quantum phase transitions

TL;DR

This paper develops a comprehensive finite-size scaling theory based on Renyi thermal entropy to accurately identify and distinguish between different types of quantum phase transitions, including weak first-order ones, in finite-size simulations.

Contribution

It introduces a unified finite-size framework using RTE and DRTE, deriving complete scaling theories for various quantum phase transitions and demonstrating their effectiveness through quantum Monte Carlo simulations.

Findings

DRTE isolates singular free energy parts and enhances transition features.

Accurate critical exponents are extracted at (2+1)D O(N) critical points.

DRTE signatures clearly distinguish weak first-order transitions in debated models.

Abstract

Establishing the nature of a quantum phase transition in finite-size simulations -- whether continuous, first-order, or weak first-order -- is a fundamental challenge in quantum many-body computation. Especially, the weak first-order phase transition is affected by a super large correlation length and always displays as a continuous critical point in simulated finite-sizes. The core difficulty lies in the fact that there is no effective finite-size theory to distinguish these phase transitions in the realistic simulations limited by the computational resource. In this work, we have fixed this problem by introducing a unified finite-size framework based on the Renyi thermal entropy (RTE) and its derivative (DRTE) to detect and characterize quantum phase transitions. We derive complete scaling theories for the RTE and DRTE at second-order, first-order, and weak first-order transitions, showing that the DRTE naturally isolates the singular part of the free energy and strengthens the characteristics of various phase transitions in finite sizes. Using quantum Monte Carlo simulations, we demonstrate accurate data collapse and extraction of critical exponents at (2+1)-dimensional O($N$) critical points. More importantly, the DRTE provides a smoking-gun signature of weak first-order transitions through a clear double-peak structure and a crossing at zero, which we unambiguously observe in debated deconfined quantum criticality candidates such as the $J$--$Q$ models. Our approach offers a general, unbiased, and numerically efficient tool for probing the universal properties of quantum phase transitions, resolving long-standing ambiguities between continuous and weak first-order scenarios.