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arXiv:2107.05735 · Physical review. B./Physical review. B

Entanglement transitions from restricted Boltzmann machines

Raimel A. Medina, Romain Vasseur, Maksym Serbyn

Published Sep 30, 2021• 35 citations
arXivHTMLPDF

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 $1/{r}^{\ensuremath{\alpha}}$ in real space, we observe three regions with a different scaling of entanglement entropy depending on the exponent $\ensuremath{\alpha}$. 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.

DOI
10.1103/physrevb.104.104205

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Entanglement is a key quantum phenomena and understanding transitions between phases of matter with different entanglement properties are an interesting probe of quantum mechanics. We numerically study a model of a 2D tensor network proposed to have an entanglement entropy transition first considered by Vasseur et al.[Phys. Rev. B 100, 134203 (2019)]. We find that by varying the bond dimension of the tensors in the network we can observe a transition between an area and volume phase with a logarithmic critical point around D≈2D\approx 2D≈2. We further characterize the critical behavior measuring a critical exponent using entanglement entropy and the tripartite quantum mutual information, observe a crossover from a `nearly pure' to entangled area law phase using the the distributions of the entanglement entropy and find a cubic decay of the pairwise mutual information at the transition. We further consider the dependence of these observables for different Rényi entropy. This work helps further validate and characterize random tensor networks as a paradigmatic examples of an entanglement transition.