Continuum field theory of matchgate tensor network ensembles

TL;DR

This paper develops a continuum field theory description for random fermionic matchgate tensor network ensembles, revealing universal long-distance physics governed by a nonlinear sigma-model with topological features.

Contribution

It introduces a continuum description of random matchgate tensor networks, connecting them to the thermal quantum Hall problem and continuum quantum field theories.

Findings

Disorder induces universal long-distance behavior in tensor networks.

The phase diagram includes localized, critical, and metallic phases.

Weak non-Gaussian deformations suppress long-range correlations.

Abstract

Tensor networks provide discrete representations of quantum many-body systems, yet their precise connection to continuum field theories remains relatively poorly understood. Invoking a notion of typicality, we develop a continuum description for random ensembles of two-dimensional fermionic matchgate tensor networks with spatially fluctuating parameters. As a diagnostic of the resulting universal physics, we analyze disorder-averaged moments of fermionic two-point functions, both in flat geometry and on a hyperbolic disk, where curvature reshapes their long-distance structure. We show that disorder drives universal long-distance behavior governed by a nonlinear sigma-model of symmetry class D with a topological term, placing random matchgate networks in direct correspondence with the thermal quantum Hall problem. The resulting phase structure includes localized phases, quantum Hall criticality, and a robust thermal metal with diffusive correlations and spontaneous replica-symmetry breaking. Weak non-Gaussian deformations reduce the symmetry to discrete permutations, generate a mass for the Goldstone modes, and suppress long-range correlations. In this way, typicality offers a route from ensembles of discrete tensor networks to continuum quantum field theories.