Entanglement transitions from restricted Boltzmann machines

TL;DR

This paper investigates whether entanglement transitions can occur in neural network-inspired quantum states, specifically restricted Boltzmann machines, beyond classical statistical mechanics explanations, revealing the importance of long-range correlations.

Contribution

It demonstrates that long-range correlated phases in RBM wave functions are essential for entanglement transitions, extending understanding beyond traditional statistical mechanics models.

Findings

No entanglement transition with uncorrelated random phases.

Long-range correlations with decay as 1/r^α lead to different entanglement scaling regimes.

Numerical evidence suggests critical behavior at phase boundaries.

Abstract

The search for novel entangled phases of matter has lead to the recent discovery of a new class of ``entanglement transitions'', exemplified by random tensor networks and monitored quantum circuits. Most known examples can be understood as some classical ordering transitions in an underlying statistical mechanics model, where entanglement maps onto the free energy cost of inserting a domain wall. In this paper, we study the possibility of entanglement transitions driven by physics beyond such statistical mechanics mappings. Motivated by recent applications of neural network-inspired variational Ans\"atze, we investigate under what conditions on the variational parameters these Ans\"atze can capture an entanglement transition. We study the entanglement scaling of short-range restricted Boltzmann machine (RBM) quantum states with random phases. For uncorrelated random phases, we analytically demonstrate the absence of an entanglement transition and reveal subtle finite size effects in finite size numerical simulations. Introducing phases with correlations decaying as $1/r^\alpha$ in real space, we observe three regions with a different scaling of entanglement entropy depending on the exponent $\alpha$. We study the nature of the transition between these regions, finding numerical evidence for critical behavior. Our work establishes the presence of long-range correlated phases in RBM-based wave functions as a required ingredient for entanglement transitions.