Symmetry resolved entanglement: Exact results in 1D and beyond

TL;DR

This paper derives exact results for symmetry-resolved entanglement in 1D quantum chains using advanced mathematical conjectures, confirming theoretical predictions and exploring topological effects.

Contribution

It applies the generalized Fisher-Hartwig and Widom conjectures to obtain exact symmetry-resolved entanglement results in 1D and higher dimensions, including topological phases.

Findings

Exact characteristic functions for 1D spin chains

Periodic structure in gapless tight binding chain

Degeneracy in entanglement spectrum due to Majoranas

Abstract

In a quantum many-body system that possesses an additive conserved quantity, the entanglement entropy of a subsystem can be resolved into a sum of contributions from different sectors of the subsystem's reduced density matrix, each sector corresponding to a possible value of the conserved quantity. Recent studies have discussed the basic properties of these symmetry-resolved contributions, and calculated them using conformal field theory and numerical methods. In this work we employ the generalized Fisher-Hartwig conjecture to obtain exact results for the characteristic function of the symmetry-resolved entanglement ("flux-resolved entanglement") for certain 1D spin chains, or, equivalently, the 1D fermionic tight binding and the Kitaev chain models. These results are true up to corrections of order $o(L^{-1})$ where $L$ is the subsystem size. We confirm that this calculation is in good agreement with numerical results. For the gapless tight binding chain we report an intriguing periodic structure of the characteristic functions, which nicely extends the structure predicted by conformal field theory. For the Kitaev chain in the topological phase we demonstrate the degeneracy between the even and odd fermion parity sectors of the entanglement spectrum due to virtual Majoranas at the entanglement cut. We also employ the Widom conjecture to obtain the leading behavior of the symmetry-resolved entanglement entropy in higher dimensions for an ungapped free Fermi gas in its ground state.