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arXiv:1807.07082 · Physical review. B./Physical review. B

Entanglement transitions from holographic random tensor networks

Romain Vasseur, Andrew C. Potter, Yi‐Zhuang You, Andreas W. W. Ludwig

Published Oct 9, 2019• 216 citations
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Abstract

Universal features in patterns of entanglement of quantum states provide valuable information-theoretical insights into complex quantum many-body phases and dynamics. This work uncovers a novel class of phase transition between weakly- and strongly- entangled quantum states, described holographically at the boundary of a random tensor network. The univeral scaling structure of these transitions is explored via an exact mapping onto a classical spin model, which also opens the door to efficient numerical methods to study these complex quantum phenomena.

DOI
10.1103/physrevb.100.134203

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The authors explore a class of random tensor network models with stabilizer local tensors and study the entanglement phase transitions in these models. They find that an entanglement phase transition can be induced by randomly breaking bulk tensor legs and they extract a set of universal data characterizing the conformal invariance at the critical points. The results clearly demonstrate regimes where a geometric bond percolation picture does or does not apply at the transitions. This work may open new connections to holographic quantum error correcting codes, built from tensor networks.
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The relation between loop quantum gravity (LQG) and tensor networks is explored from the perspectives of the bulk-boundary duality and holographic entanglement entropy. We find that the LQG spin-network states in a space $\mathrm{\ensuremath{\Sigma}}$ with boundary $\ensuremath{\partial}\mathrm{\ensuremath{\Sigma}}$ is an exact holographic mapping similar to the proposal in [X.-L. Qi, Exact holographic mapping and emergent space-time geometry, arXiv:1309.6282]. The tensor network, understood as the boundary quantum state, is the output of the exact holographic mapping emerging from a coarse-graining procedure of spin networks. Furthermore, when a region AAA and its complement A‾\overline{A}A are specified on the boundary $\ensuremath{\partial}\mathrm{\ensuremath{\Sigma}}$, we show that the boundary entanglement entropy S(A)S(A)S(A) of the emergent tensor network satisfies the Ryu-Takayanagi formula in the semiclassical regime, i.e., S(A)S(A)S(A) is proportional to the minimal area of the bulk surface attached to the boundary of AAA in $\ensuremath{\partial}\mathrm{\ensuremath{\Sigma}}$.