A universal black-box quantum Monte Carlo approach to quantum phase transitions

TL;DR

This work introduces a universal black-box finite-temperature quantum Monte Carlo framework that derives exact estimators for energy susceptibility $\chi_E(\lambda)$ and fidelity susceptibility $\chi_F(\lambda)$ within the PMR-QMC formalism, enabling order-parameter-free detection of quantum phase transitions for essentially arbitrary Hamiltonians. By automatically generating ergodic PMR-QMC updates and providing convergence diagnostics, the method is demonstrated on the 2D TFIM, the 2D XXZ model, and a random-unitary ensemble using a single code base. The estimators achieve efficient performance, including a constant-time $\chi_E$ estimator in diagonal-driving cases and a cubic-time $\chi_F$ estimator in general settings, with extensions to off-diagonal driving terms and mixed or high-spin systems discussed. The open-source implementation and validation against exact results and SSE-QMC benchmarks highlight the approach's broad applicability, scalability, and practical impact for locating QCPs in complex quantum many-body systems.

Abstract

We derive exact, universal, closed-form quantum Monte Carlo estimators for finite-temperature energy susceptibility and fidelity susceptibility, applicable to essentially arbitrary Hamiltonians. Combined with recent advancements in Monte Carlo, our approach enables a black-box framework for studying quantum phase transitions--without requiring prior knowledge of an order parameter or the manual design of model-specific ergodic quantum Monte Carlo update rules. We demonstrate the utility of our method by applying a single implementation to the transverse-field Ising model, the XXZ model, and an ensemble of models related by random unitaries.

Figures (11)

• Figure 1: FS for the two spin model, Eq. (\ref{['eq:prl-model']}), considered in Ref. zhang2008detection for different inverse temperatures, $\beta$. Data from QMC are averaged over 5 independent random seeds, whereas direct numerical calculations are shown as lines.
• Figure 2: We compute ES and FS restricted to the positive parity subspace (see \ref{['app:tfim-parity-details']}) of a $4 \times 4$ square TFIM with PBC and $h = 1$ as a function of $\beta$. QMC data are averaged over 100 independent runs. Our FS results are consistent with a similar computation performed with SSE-QMC albuquerque2010QuantumCriticalScaling.
• Figure 3: We compute ES and FS for the XXZ model in \ref{['eq:fidsus-xxz']} on a square lattice with PBC as a function of temperature, $T$, for different model sizes. QMC data are averaged over at least 200 independent runs. Our FS results are consistent with a similar computation performed with SSE-QMC wang2015FidelitySusceptibilityMade.
• Figure 4: Comparison of QMC estimates with exact values for an ensemble of 100--spin Hamiltonians generated by random rotations of Eq. \ref{['eq:prl-model']}. Detailed information on the ensemble can be found in \ref{['app:random-U']}. QMC estimates are averaged across 10 random models. In all cases, the QMC estimates show excellent agreement with exact finite $\beta$ curves. (a) The imaginary time correlator, $G(\tau)$, has the expected symmetry around $\beta/2 = 10.$ (b) The ES correctly predicts critical points despite the $\beta = 20$ curve not quite converging to the $\beta = \infty$ curve. (c) The FS correctly predicts critical points more sharply than ES, and the $\beta = 20$ curve matches the $\beta = \infty$ curve.
• Figure 5: Simulation time used to generate \ref{['fig:prl-comparison-1']}.

Theorems & Definitions (25)

• Theorem 1: An $O(1)$ ES estimator
• Theorem 2: An $O(q^3)$ estimator for FS
• Lemma 1: Convolution lemma zeng2025inequalities
• proof
• Proposition 1: Re-scaling relation zeng2025inequalities
• proof
• Lemma 2: A weighted, repeated argument sum simplification zeng2025inequalities
• proof
• Proposition 2: Parametric derivative
• proof