Learning phases with Quantum Monte Carlo simulation cell

TL;DR

The paper tackles the challenge of extracting phase information from large, high-dimensional QMC data by introducing spin-opstring, a compact representation that combines the initial spin state with the SSE operator string. It demonstrates its effectiveness for phase classification, nonlocal observable regression, and transfer learning across related models and system sizes, while enabling interpretable ML via SHAP and LRP. The results show accurate phase boundaries, compatible predictions of nonlocal observables, substantial memory and compute savings, and robust generalization beyond training domains. The work suggests spin-opstring as a versatile input for ML in quantum many-body physics, with promising avenues for unsupervised learning and generative modeling.

Abstract

We propose the use of the ``spin-opstring", derived from Stochastic Series Expansion Quantum Monte Carlo (QMC) simulations as machine learning (ML) input data. It offers a compact, memory-efficient representation of QMC simulation cells, combining the initial state with an operator string that encodes the state's evolution through imaginary time. Using supervised ML, we demonstrate the input's effectiveness in capturing both conventional and topological phase transitions, and in a regression task to predict non-local observables. We also demonstrate the capability of spin-opstring data in transfer learning by training models on one quantum system and successfully predicting on another, as well as showing that models trained on smaller system sizes generalize well to larger ones. Importantly, we illustrate a clear advantage of spin-opstring over conventional spin configurations in the accurate prediction of a quantum phase transition. Finally, we show how the inherent structure of spin-opstring provides an elegant framework for the interpretability of ML predictions. Using two state-of-the-art interpretability techniques, Layer-wise Relevance Propagation and SHapley Additive exPlanations, we show that the ML models learn and rely on physically meaningful features from the input data. Together, these findings establish the spin-opstring as a broadly-applicable and interpretable input format for ML in quantum many-body physics.

Figures (22)

• Figure 1: Schematic diagram of the simulation cell of SSE QMC algorithm for a spin-$1/2$ system with six sites. Here $i$ signifies the lattice site number and $\sigma(i)$ represents the initial/final spin state. At each level $p$ of the simulation cell, the spin state is depicted by filled (+1)/open (-1) circles. The diagonal or off-diagonal operator at each level acting on a bond is represented by open or filled bars respectively. The information in the operator string is compactly codified in $S(p)$ ($a(p)$ and $b(p)$ are defined in the text). The "spin-opstring" for the above simulation cell is constructed by combining $\sigma(i)$ and $S(p)$ together.
• Figure 2: Phase diagram of the $t_2-V_1$ model in Eq. \ref{['Eq:Hamiltonian']} in terms of particle-density $\rho$ and NN repulsion $V_1$, where the NNN hopping $t_2$ is set to unity. The dashed and solid blue lines represent mixed and continuous phase transitions, respectively, between the SF and cSS phases. The green filled circle denotes the tricritical point separating the mixed transition line from the continuous one. The shaded vertical and horizontal bands indicate the lines where we apply our ML model to predict the phase transitions.
• Figure 3: Top: ML prediction for the two phases, cS and SF, as a function of the NN repulsion $V_1$ for a fixed particle density $\rho= 1/2$. Bottom: QMC results for the observables, superfluid density $\rho_s$ and structure factor $S(\pi,\pi)$ as a function of $V_1$ for a $12\times12$ square lattice at half-filling.
• Figure 4: Top: ML prediction for the three phases, cS, cSS and MI, as a function of the chemical potential $\mu$ for a fixed NN repulsion $V_1=4$. The vertical dashed lines indicate the ML predicted phase boundaries. Bottom: QMC results for the observables, particle-density $\rho$, superfluid density $\rho_s$ and structure factor $S(\pi,\pi)$ with varying chemical potential $\mu$, on a $12\times12$ square lattice. The left $y$-axis shows the variation of $\rho_s$ and $S(\pi,\pi)$, while the right $y$-axis represents the variation of $\rho$.
• Figure 5: Comparison of superfluid density obtained from the ML regression model with QMC results on a $12\times 12$ square lattice. The error band of the ML data has been obtained from 100 independently trained ML models.