Eigenstate entanglement scaling for critical interacting spin chains

Qiang Miao and Thomas Barthel

Department of Physics, Duke University, Durham, North Carolina 27708, USA

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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.

In contrast to classical systems, the state-space for a quantum many-body system grows exponentially in the number of its components. The goal of quantum computation is to exploit this enormous complexity to solve problems that are intractable for our usual classical computers. A decisive resource for quantum information processing is entanglement. Beyond that, measures of entanglement are used to understand and quantify the complexity of quantum matter.

Entanglement in ground states — the lowest energy states that the system approaches at very low temperatures — had been studied previously. For quantum many-body ground states, one generally finds that the entanglement between a subsystem and the rest is proportional to the surface area of the subsystem; this is called the area law. Much less was known about excited states, just that excited states are generally much more entangled and that their entanglement entropies should be proportional to the subsystem volume.

In recent contributions, the authors found that, rather generically, entanglement entropies of energy eigenstates are captured by a single crossover function. These functions capture the full crossover from the groundstate entanglement regime at low energies and small subsystem size (area or log-area law) to the extensive volume-law regime at high energies or large subsystem size. Furthermore, in quantum-critical regimes, i.e., a temperature and parameter regime that is dominated by a zero-temperature phase transition point, the crossover functions assume universal scaling forms. These universal scaling functions are shared by large classes of systems, irrespective of their microscopic differences.

These somewhat surprising results are based on the applicability of the eigenstate thermalization hypothesis. This hypothesis asserts that energy eigenstates are basically indistinguishable from thermal states when our observations are restricted to small subsystems of a quantum many-body system. Hence, the entanglement entropies in eigenstates can be deduced from subsystem entropies of corresponding thermal equilibrium states.

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