Universal entanglement of typical states in constrained systems

TL;DR

This paper develops a diagrammatic formalism to exactly evaluate bipartite entanglement spectra of random states in large constrained quantum systems, revealing a rich phase diagram with implications for thermalization.

Contribution

It introduces a novel large N diagrammatic approach to analyze entanglement in constrained systems, uncovering a phase diagram with a Marchenko-Pastur phase and critical points.

Findings

Identifies a non-trivial phase diagram for entanglement spectra.

Finds the Rydberg-blockaded chain lies in the MP phase with a specific Page correction.

Provides a baseline for numerical studies of entanglement in constrained systems.

Abstract

Local constraints play an important role in the effective description of many quantum systems. Their impact on dynamics and entanglement thermalization are just beginning to be unravelled. We develop a large $N$ diagrammatic formalism to exactly evaluate the bipartite entanglement of random pure states in large constrained Hilbert spaces. The resulting entanglement spectra may be classified into `phases' depending on their singularities. Our closed solution for the spectra in the simplest class of constraints reveals a non-trivial phase diagram with a Marchenko-Pastur (MP) phase which terminates in a critical point with new singularities. The much studied Rydberg-blockaded/Fibonacci chain lies in the MP phase with a modified Page correction to the entanglement entropy, $\Delta S_1 = 0.513595\cdots$. Our results predict the entanglement of infinite temperature eigenstates in thermalizing constrained systems and provide a baseline for numerical studies.