Eigenstate entanglement scaling for critical interacting spin chains

Qiang Miao,Thomas Barthel

doi:10.22331/q-2022-02-02-642

Abstract

With increasing subsystem size and energy, bipartite entanglement entropies of energy eigenstates cross over from the groundstate scaling to a volume law. In previous work, we pointed out that, when strong or weak eigenstate thermalization (ETH) applies, the entanglement entropies of all or, respectively, almost all eigenstates follow a single crossover function. The crossover functions are determined by the subsystem entropy of thermal states and assume universal scaling forms in quantum-critical regimes. This was demonstrated by field-theoretical arguments and the analysis of large systems of non-interacting fermions and bosons. Here, we substantiate such scaling properties for integrable and non-integrable interacting spin-1/2 chains at criticality using exact diagonalization. In particular, we analyze XXZ and transverse-field Ising models with and without next-nearest-neighbor interactions. Indeed, the crossover of thermal subsystem entropies can be described by a universal scaling function following from conformal field theory. Furthermore, we analyze the validity of ETH for entanglement in these models. Even for the relatively small system sizes that can be simulated, the distributions of eigenstate entanglement entropies are sharply peaked around the subsystem entropies of the corresponding thermal ensembles.

Highlights

Entanglement is a fundamental feature of quantum matter with far-reaching consequences for its macroscopic properties, complexity, and technological potential
We argued that, based on the eigenstate thermalization hypothesis (ETH), the crossover of eigenstate entanglement entropy from the groundstate scaling at low energies and small to a volume law at larger energies or is captured by a single crossover function which assumes a universal scaling form in quantum-critical regimes [40]
The long-range physics of critical condensed matter systems in equilibrium can usually be described by field theories, and their subsystem entropies are captured by scaling functions that depend on dimensionless parameters like energy ratios

Summary

Introduction

Entanglement is a fundamental feature of quantum matter with far-reaching consequences for its macroscopic properties, complexity, and technological potential. We argued that, based on the eigenstate thermalization hypothesis (ETH), the crossover of eigenstate entanglement entropy from the groundstate scaling at low energies and small to a volume law at larger energies or is captured by a single crossover function which assumes a universal scaling form in quantum-critical regimes [40]. The long-range physics of critical condensed matter systems in equilibrium can usually be described by field theories, and their subsystem entropies are captured by scaling functions that depend on dimensionless parameters like energy ratios. We demonstrated the scaling behavior and weak ETH for the entanglement in systems of non-interacting fermions in d = 1, 2, 3 [40], and for the harmonic lattice model (free scalar field theories) in d = 1, 2 [52]. We assess the applicability of ETH for entanglement entropies by computing entanglement for all eigenstates and comparing to the corresponding scaling function

Numerical simulations

XXZ models

Scaling of thermal subsystem entropy

Entanglement entropy and ETH

Next-nearest neighbor transverse Ising model and its dual

Conclusion