Learning quantum phase transitions through Topological Data Analysis

TL;DR

This study addresses locating quantum phase transitions in fermionic lattice models where traditional finite-size scaling is challenging. It introduces a Topological Data Analysis pipeline that converts determinant QMC Hubbard-Stratonovich field snapshots into persistence diagrams, computes diagram distances, and uses fuzzy spectral clustering to identify phase boundaries, yielding estimates for critical points $V_c/t$ and $U_c/t$. Applied to the 2D periodic Anderson model and the half-filled honeycomb Hubbard model, the approach reproduces literature critical points with good accuracy across system sizes, though finite-size effects remain. The work showcases TDA as a promising tool for challenging quantum many-body problems, with potential extensions to sign-problem–afflicted regimes and higher dimensions.

Abstract

We implement a computational pipeline based on a recent machine learning technique, namely the Topological Data Analysis (TDA), that has the capability of extracting powerful information-carrying topological features. We apply such a method to the study quantum phase transitions and, to showcase its validity and potential, we exploit such a method for the investigation of two paramount important quantum systems: the 2D periodic Anderson model and the Hubbard model on the honeycomb lattice, both cases on the half-filling. To this end, we have performed unbiased auxiliary field quantum Monte Carlo simulations, feeding the TDA with snapshots of the Hubbard-Stratonovich fields through the course of the simulations The quantum critical points obtained from TDA agree quantitatively well with the existing literature, therefore suggesting that this technique could be used to investigate quantum systems where the analysis of the phase transitions is still a challenge.

Figures (6)

• Figure 1: Using the PAM data as point-cloud, for $n=12$$U=6.0$ and $V/t=1.15$, we show the persistence diagram for the 0-$th$ and first homological dimensions.
• Figure 2: Heatmap representing the distance matrix $M$ for the PAM, for $n=12$ and $U=6.0$, using the Betti distance, with $p=2$; column labels are the different values of the hybridization $V/t$.
• Figure 3: Line-plot representing the membership degree vectors $\bar{l}$to the fuzzy cluster 1 for the PAM, for $n=12$ and $U=6.0$, using the Betti distance (blue solid line) and the Wasserstein distance (orange dashed line), with $p=2$; labels on the $x$-axis are the different values of the hybridization $V/t$.
• Figure 4: Quantum critical hybridizations for the Periodic Anderson model from Topological Data Analysis. Panel (a) exhibits the TDA $V_{c}(U)/t$ (black square symbols) for fixed lattice size $L=12$, and $\beta t = 20$, comparing them with the results for the FSS analysis in Ref. hu17b (red triangle symbols). Panel (b) displays the TDA $V_{c}/t$ (red circle symbols) for fixed $U/t=4$, and $\beta t = 20$, and different lattice sizes. The blue hatched area corresponds to the result (within error bars) for the FSS analysis in Ref. hu17b.
• Figure 5: Quantum critical points of the Hubbard model on the honeycomb lattice from Topological Data Analysis as function of the inversion of temperature $\beta$, and for different lattice sizes. The black hatched area corresponds to the result (within error bars) for the FSS analysis in Ref. otsuka16.