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

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.

Figures (5)

• Figure 1: (a,c) Second-order RTE and its derivative DRTE, and (b,d) corresponding data collapse analysis near the QCP of the bilayer Ising–Heisenberg model [(2+1)D Ising universality]. We set $J=3.045$ and take $J_{z}$ as the tuning parameter. (e--h) Results for the dimerized anisotropic Heisenberg model with $\Delta=0.9$ [(2+1)D O(2) universality], using $J_{2}=2.1035$ and tuning $J_1$. (i--m) Results for the dimerized isotropic Heisenberg model with $\Delta=1$ [(2+1)D O(3) universality], using $J_{2}=1$ and tuning $J_1$. To achieve good data collapse, we include a subleading correction: $Q(g,L)=L^{k}[\tilde{S}((g-g_c)L^{1/\nu})+bL^{-\omega}]$. $\omega = 0.830$, $0.789$, and $0.782$ correspond to the (2+1)D Ising, O(2), and O(3) critical points, respectively Andrea2002CriticalCampostrini2001CriticalCampostrini2002CriticalDeng2005Surface. For $Q=S^{(2)}$, $k=0$. $k=1/\nu$ when $Q$ is DRTE.
• Figure 2: (a) Derivative of the RTE, and (b) corresponding data collapse analysis near the phase transition point of the 2D anisotropic Heisenberg model with anisotropy parameter $\Delta$ ( see Eq. (\ref{['anisotropicHeisenberg ']})). (c) and (d) show the data for the derivative of the partition function at temperatures $\beta$ and $2\beta$ near the transition. Here, $J_1 = J_2 = 1$, and the anisotropy $\Delta$ is taken as the tuning parameter. (e–h) Results for the checkerboard $J$-$Q$ model, using $Q=1$ and tuning $J$.
• Figure 3: We set $Q=1$ and use $J$ as the tuning parameter. (a) The derivative of the RTE: DRTE $\partial S^{(2)}/\partial J$. (b) Corresponding data collapse analysis of (a). (c) and (d) show the data for the derivative of the partition function at temperatures $\beta$ and $2\beta$ near the transition of the $J-Q_3$ model. (e--h) Data for the $J-Q_2$ model. The raw values of the partition function derivative vary significantly across system sizes (see Appendix \ref{['sec:appmodel']}), making direct comparison difficult. To resolve this, we normalize the data by $L^3$, corresponding to $\beta L^2 = L^3$.
• Figure S1: $J$-$Q$ models, in which the antiferromagnetic Heisenberg interaction ($J$-term) acts on all nearest-neighbor bonds. (a) The checkerboard $J$-$Q$ model on a square lattice with periodic boundary condition in both directions. The four-spin interaction ($Q$-term) acts on nearest-neighbor bond pairs that form a plaquette. (b) $J-Q_2$ model: the four-spin $Q$ interaction covers the entire lattice. (c) $J-Q_3$ model: the six-spin $Q$ interaction covers the entire lattice.
• Figure S2: We set $Q=1$ and use $J$ as the tuning parameter. (a) The derivative of the RTE: DRTE $\partial S^{(2)}/\partial J$. An inset shows the data on the right side of the phase transition point. (b) and (c) show the data for the derivative of the partition function at temperatures $\beta$ and $2\beta$ near the transition of the checkerboard $J$-$Q$ model. (d--f) Data for the $J-Q_3$ model. (g--i) Data for the $J-Q_2$ model.