Symmetry resolved entanglement of excited states in quantum field theory. Part II. Numerics, interacting theories and higher dimensions

Abstract

In a recent paper we studied the entanglement content of zero-density excited states in complex free quantum field theories, focusing on the symmetry resolved entanglement entropy (SREE). By zero-density states we mean states consisting of a fixed, finite number of excitations above the ground state in an infinite-volume system. The SREE is defined for theories that possess an internal symmetry and provides a measure of the contribution to the total entanglement of each symmetry sector. In our work, we showed that the ratio of Fourier-transforms of the SREEs (i.e. the ratio of charged moments) takes a very simple and universal form for these states, which depends only on the number, statistics and symmetry charge of the excitations as well as the relative size of the entanglement region with respect to the whole system’s size. In this paper we provide numerical evidence for our formulae by computing functions of the charged moments in two free lattice theories: a 1D Fermi gas and a complex harmonic chain. We also extend our results in two directions: by showing that they apply also to excited states of interacting theories (i.e. magnon states) and by developing a higher dimensional generalisation of the branch point twist field picture, leading to results in (interacting) higher-dimensional models.

Symmetry resolved entanglement of excited states in quantum field theory. Part I. Free theories, twist fields and qubits

“…Indeed, the generalisation to states of many distinct quasi-particles is straightforward, and each particle contributes independently (multiplicatively) to the ratio of charged moments, similar to the structure found in [33,34] for the charged Rényi entropies. We highlight that the results (20) are identical for fermions with the definitions (12) and (16) when n is even.…”

“…In this section we derive the main formulae for the charged replica negativities by starting with a multi-qubit system. This "toy model" was already employed in [29,30,31,33,34] following the realisation that even if multi-qubit states are much simpler than the excited states of a QFT, they both produce the same universal contribution to entanglement entropies and negativities. Hence it is advantageous to obtain such contribution from qubit states rather than a vastly more involved field-theoretical approach.…”

“…Branch point twist fields sit at branch points from which branch cuts extend. Twist operators were introduced in a previous work [34], and have very similar exchange relations with respect to local fields as branch point twist fields, with the difference that they act on extended regions of space rather than points.…”

“…In [34] we showed that the analysis of the entanglement entropies of quasi-particle states relies on few algebraic properties, mainly their exchange relations w.r.t. local fields, while most of the theory-dependent features are hidden in the zero-point fluctuations (ground-state entanglement).…”

“…These states are "zero-density" in the sense that they contain a fixed and finite number of quasi-particle excitations within an infinite volume. The present work applies to the same kind of excited states considered in [29,30,31,32] for the standard entanglement measures and more recently in [33,34] for the symmetry resolved entropies.…”