Entanglement transitions in translation-invariant tensor networks

TL;DR

This paper investigates the entanglement transition in translation-invariant tensor networks, linking spectral properties of the transfer matrix to contraction complexity and entanglement scaling.

Contribution

It uncovers a transition between volume- and area-law entanglement phases by analyzing the spectral behavior of transfer matrices in tensor networks.

Findings

Deep in the volume-law phase, the transfer matrix spectrum forms a dense ring with a sharp outer edge.

In the area-law phase, a distinct leading eigenvalue dominates the spectrum.

The study connects spectral properties of the transfer matrix to entanglement scaling and contraction complexity.

Abstract

We study the complexity of approximately contracting translation-invariant tensor networks. The computational cost of row-by-row tensor network contraction, which defines a discrete time evolution governed by a fixed transfer matrix, is associated with the entanglement of the state of a row. By analyzing a family of tensor networks whose transfer matrices interpolate between chaotic Floquet and strongly non-unitary limits, we uncover a transition between volume- and area-law entanglement in states evolved under the transfer matrix. We show that deep in the volume-law phase the spectrum of the transfer matrix in the complex plane consists of a dense ring with a sharp outer edge, reminiscent of behavior identified for non-unitary random matrices. At late times an evolving row state therefore has significant contributions from many eigenvectors with nearly degenerate eigenvalue magnitudes. In the area-law phase, there is instead a distinct leading eigenvalue. Our results establish connections between contraction complexity, spectral properties of the transfer matrix, and purification under non-unitary dynamics.