Symmetry-resolved entanglement in symmetry-protected topological phases

Abstract

Symmetry protected topological phases (SPTs) have universal degeneracies in the entanglement spectrum in one dimension (1D). Here, we formulate this phenomenon in the framework of "symmetry resolved entanglement" using cohomology theory. We develop a general approach to compute entanglement measures of SPTs in any dimension and specifically symmetry resolved entanglement (SRE) via a discrete path integral on multi-sheet Riemann surfaces with generalized defects. The resulting path integral on nontrivial manifolds is expressed in terms of group cocycles describing the topological actions of SPTs. Their cohomology classification allows to identify universal entanglement properties. Specifically, we demonstrate entanglement equipartition for all 1D topological phases protected by finite abelian unitary symmetries. The method opens the way to explore higher dimensions and additional symmetries.

Boundary effects on symmetry resolved entanglement

“…The characterisation of the interplay between entanglement and internal symmetries has recently become the focus of an intense research activity [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] aimed to have a deeper resolution of the structure of the reduced density matrix of many-body systems and quantum field theories. The theoretical work in this area has almost entirely focussed on systems with periodic boundary conditions (PBC).…”

Symmetry resolution in non-Lorentzian field theories

“…These computations have been extended for the Wess-Zumino-Witten model, as well as for theories with non-invertible symmetries [82,83]. Furthermore, SREE has been studied for disordered systems, many-body localized systems, and for certain topological phases [84][85][86][87][88][89][90]. Not just in computing SREE, but the idea of symmetry resolution has also been extended to exploration of other measures of entanglement, e.g., Computable Cross-Norm (CCNR) negativity, relative entropies, and fidelities [91][92][93][94][95][96][97].…”

Boundary effects on symmetry resolved entanglement

“…The characterisation of the interplay between entanglement and internal symmetries has recently become the focus of an intense research activity [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] aimed to have a deeper resolution of the structure of the reduced density matrix of many-body systems and quantum field theories. The theoretical work in this area has almost entirely focussed on systems with periodic boundary conditions (PBC).…”