Emergent statistical mechanics in holographic random tensor networks

TL;DR

This paper demonstrates that random tensor network states, especially those modeling holographic dualities, tend to equilibrate over time under generic Hamiltonians, linking tensor networks, statistical mechanics, and holography.

Contribution

It establishes the dynamical equilibration of RTN states under broad conditions, extending static ensemble averages to time-dependent behavior in holographic models.

Findings

RTN states equilibrate at large bond dimension.

Equilibration occurs in hyperbolic, MPS, and black hole geometries.

Hierarchy of equilibration linked to entanglement and geometry.

Abstract

Recent years have enjoyed substantial progress in capturing properties of complex quantum systems by means of random tensor networks (RTNs), which form ensembles of quantum states that depend only on the tensor network geometry and bond dimensions. Of particular interest are RTNs on hyperbolic geometries, with local tensors typically chosen from the unitary Haar measure, that model critical boundary states of holographic bulk-boundary dualities. In this work, we elevate static pictures of ensemble averages to a dynamical one, to show that RTN states exhibit equilibration of time-averaged operator expectation values under a highly generic class of Hamiltonians with non-degenerate spectra. We prove that RTN states generally equilibrate at large bond dimension and also in the scaling limit for three classes of geometries: Those of matrix product states, regular hyperbolic tilings, and single "black hole" tensors. Furthermore, we prove a hierarchy of equilibration between finite-dimensional instances of these classes for bulk and boundary states with small entanglement. This suggests an equivalent hierarchy between corresponding many-body phases, and reproduces a holographic degree-of-freedom counting for the effective dimension of each system. These results demonstrate that RTN techniques can probe aspects of late-time dynamics of quantum many-body phases and suggest a new approach to describing aspects of holographic dualities using techniques from statistical mechanics.