Rényi entropy helps distinguish weak first-order quantum phase transitions

Researchers at Westlake University in Hangzhou have developed a method using Rényi thermal entropy and its derivative to reliably identify weak first-order quantum phase transitions, a type of transition that appears continuous in standard computer simulations due to its extremely large correlation length. The team established a complete finite-size scaling theory for detecting and characterizing these transitions, addressing a fundamental challenge in quantum many-body computation where weak first-order transitions are often missed.

Their work demonstrates accurate data collapse and extraction of critical exponents, and provides a “smoking-gun signature” for weak first-order transitions through a distinct double-peak structure observed in models like the J-Q models. This approach offers a general, unbiased and numerically efficient tool for probing the universal properties of quantum phase transitions, resolving long-standing ambiguities.

Rényi Thermal Entropy Detects Quantum Phase Transitions

Researchers at Westlake University in Hangzhou report that their framework overcomes a core difficulty in quantum many-body computation: the inability to effectively classify transitions as second-order, first-order, or weak first-order within the constraints of realistic simulations. The approach centers on isolating the singular part of the free energy, a key step in characterizing critical behavior, and provides a means of detecting transitions even in systems with a super large correlation length.

The team’s method addresses a long-standing challenge stemming from the dominance of analytic terms that obscure the singular contributions indicative of a phase transition. These terms make numerical detection difficult. By constructing a quantity where these analytic contributions cancel, the researchers aimed to reveal the underlying critical behavior.

Inspired by techniques used to extract topological entanglement entropy, they proposed that the difference between the logarithm of the partition function at two inverse temperatures and twice the logarithm at a single inverse temperature serves this purpose. The derivative of the Rényi thermal entropy, or DRTE, exhibits strong characteristics in finite-size systems. Using quantum Monte Carlo simulations, they demonstrate accurate data collapse and extraction of critical exponents at (2+1)-dimensional O(N) critical points.

More importantly, the DRTE provides a signature of weak first-order transitions through a clear double-peak structure and a crossing at zero, which they unambiguously observe in debated deconfined quantum criticality candidates such as the J-Q models. This 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.

This scaling behavior, detailed in their work, provides a robust method for identifying the type of transition occurring, even when the correlation length is exceptionally large, a common issue with weak first-order transitions. Their approach proved particularly effective in analyzing debated deconfined quantum criticality candidates, specifically the J-Q models, which represent 2D quantum spin systems. These models, free from the sign problem that plagues many quantum simulations, are ideal for testing the new method.

Previous attempts to definitively classify the phase transition between antiferromagnetic Néel order and valence bond solid (VBS) phase in these models have been inconclusive, often appearing continuous in simulations. However, the DRTE, even in systems as small as L = 56, clearly captured both continuous critical fluctuations and first-order discontinuities.

The success of the DRTE stems from its ability to naturally cancel leading analytic contributions to the free energy, isolating the singular part that dictates critical behavior. This cancellation is achieved through a carefully constructed quantity, the difference between ln Z(2β) and 2 ln Z(β), where β is proportional to the system size. The team’s complete scaling theory extends to all three fundamental types of phase transitions: continuous, strong first-order and weak first-order, offering a unified framework for their detection and characterization.

The ability to accurately classify these transitions, even in finite-size simulations, is important for advancing the field of quantum many-body computation and understanding the behavior of complex quantum materials. The team’s findings represent a step forward in the development of robust and reliable methods for studying quantum phase transitions.

Finite-Size Scaling Theory for Second-Order Transitions

The ability to definitively classify quantum phase transitions, as continuous, first-order, or weak first-order, remains a significant hurdle in quantum many-body computation, particularly when simulations are limited by computational resources. The approach isolates the singular component of free energy, amplifying the distinguishing characteristics of each transition type within the constraints of finite system sizes. This new framework tackles a specific problem: weak first-order transitions, often appearing continuous in simulations due to their extremely large correlation length, are difficult to identify.

Challenges in Identifying Weak First-Order Transitions

A key obstacle in characterizing quantum phase transitions lies in the difficulty of definitively identifying weak first-order transitions, particularly given their tendency to mimic continuous transitions within the constraints of computational simulations. The behavior of these transitions is complicated by an exceptionally large correlation length, a factor that obscures their true nature when examined using finite-size simulations. This approach moves beyond simply detecting a transition. It provides a means to extract critical exponents through data collapse, with the exponents for weak first-order transitions reflecting those of the nearest critical fixed point.

The result is a method for unambiguously identifying the nature of quantum phase transitions through finite-size scaling of RTE and DRTE, offering a tool to resolve debates in quantum criticality. Unlike the derivative of the logarithm of the partition function, which exhibits smoother behavior, the DRTE provides a more robust signal for identifying phase transitions.

The team’s scaling theory accounts for the competition between the underlying first-order mechanism and the influence of a nearby critical fixed point, a key consideration for accurately characterizing weak transitions. For standard first-order transitions, the DRTE displays a discontinuity with a double-peak structure, scaling with the volume law. This hybrid form allows for a more subtle and accurate description of the transition process. This cancellation is achieved through the use of RTE and its derivative, physical observables that naturally isolate the critical behavior.

The researchers emphasize that this approach is not limited to continuous phase transitions, but also is a powerful probe for first-order transitions, providing a more comprehensive understanding of quantum criticality. The framework allows for the unambiguous identification of the transition’s nature through finite-size scaling of the RTE and DRTE, a capability previously hindered by computational limitations and the difficulty of distinguishing between continuous and weak first-order transitions.

Double-Peak DRTE Signature of Weak First-Order Transitions

This isolation allows for accurate data collapse and critical exponent extraction at (2+1)-dimensional O(N) critical points, as demonstrated through quantum Monte Carlo simulations. However, weak first-order transitions present a unique challenge due to the influence of quantum fluctuations originating from a nearby critical point. These fluctuations introduce an effective singular contribution that mimics continuous transitions, complicating identification.

To account for this, the researchers propose a modified DRTE expression incorporating this effect: (∂ g) = [2L^(d + z)Δ $f^{\text{′}}(g)$ + S^′(Δ gL^(1/ν))L^(1/ν)][w(2β)-w(β)]. This equation predicts that the DRTE at a weak first-order phase transition will still exhibit a double-peak structure, as the w(2β)-w(β) term remains, while also demonstrating a zero crossing at the transition point.

The DRTE curves for different system sizes are predicted to intersect at zero at the transition point, indicating the absence of singular behavior typically seen in continuous phase transitions. These findings provide compelling evidence against a genuine continuous transition in these models, which have been the subject of ongoing debate. The zero crossing and sign change of the peaks in this scenario provide a clear indication of phase coexistence.

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