Entanglement spectra of stabilizer codes: A window into gapped quantum phases of matter

Abstract

The entanglement spectrum (ES) provides a barometer of quantum entanglement and encodes physical information beyond that contained in the entanglement entropy. In this paper, we explore the ES of stabilizer codes, which furnish exactly solvable models for a plethora of gapped quantum phases of matter. Studying the ES for stabilizer Hamiltonians in the presence of arbitrary weak local perturbations thus allows us to develop a general framework within which the entanglement features of gapped topological phases can be computed and contrasted. In particular, we study models harboring fracton order, both type-I and type-II, and compare the resulting ES with that of both conventional topological order and of (strong) subsystem symmetry protected topological (SSPT) states. We find that non-local surface stabilizers (NLSS), a set of symmetries of the Hamiltonian which form on the boundary of the entanglement cut, act as purveyors of universal non-local features appearing in the entanglement spectrum. While in conventional topological orders and fracton orders, the NLSS retain a form of topological invariance with respect to the entanglement cut, subsystem symmetric systems-fracton and SSPT phases-additionally show a non-trivial geometric dependence on the entanglement cut, corresponding to the subsystem symmetry. This sheds further light on the interplay between geometric and topological effects in fracton phases of matter and demonstrates that strong SSPT phases harbour a measure of quasi-local entanglement beyond that encountered in conventional SPT phases. We further show that a version of the edge-entanglement correspondence, established earlier for gapped two-dimensional topological phases, also holds for gapped three-dimensional fracton models.

Crystal-to-fracton tensor gauge theory dualities

“…The Gauss's law (38) serves as the dual formulation of the definition of disclination density (27). We thereby see that the duality maps E ij to a rotated strain tensor:…”

Detecting subsystem symmetry protected topological order via entanglement entropy

“…6 without a prior understanding of the relevant symmetry. The fact that a symmetry-agnostic quantity like the SPEE can characterize SSPT order reflects its fundamental differences from standard SPT order [23].…”

“…Hence we have |G A | = 2 |A|−|∂A|+1 , so S A = N − 1. We see that the SPEE takes the value γ = 1 (= log 2 2) for the cluster state, due to the subsystem symmetries forming non-local constraints lying along the boundary of A [19,22,23]. As a first venture away from the cluster state, we consider a family of states |C(θ) = U (θ)|C , where the circuit U (θ) is defined by acting with the two-body unitary (H ⊗ H)Cθ(H ⊗ H) on every pair of neighbouring sites, with…”

“…Recently, however, it has been observed that γ may deviate from the expected value due to the presence of symmetry-protected topological (SPT) order localized around ∂A [18][19][20][21][22][23]. One setting in which this occurs is for states with subsystem SPT (SSPT) order [21,[24][25][26].…”

Compactifying fracton stabilizer models

“…The exploration of topological order in three dimensions led to the surprising discovery of fracton models, whose topological excitations have emergent mobility restrictions [1][2][3][4][5][6][7][8][9][10][11][12][13]. A large class of these models can be realized by local commuting projector Hamiltonians, and obey lattice definitions of topological order [14].…”