We define an index for 2D G-invariant invertible states of bosonic lattice systems in the thermodynamic limit for a finite symmetry group G with a unitary action. We show that this index is an invariant of the symmetry protected phase.
Abstract We introduce an index for symmetry-protected topological (SPT) phases of infinite fermionic chains with an on-site symmetry given by a finite group G. This index takes values in Z2×H1(G,Z2)×H2(G,U(1)p) with a generalised Wall group law under stacking. We show that this index is an invariant of the classification of SPT phases. When the ground state is translation invariant and has reduced density matrices with uniformly bounded rank on finite intervals, we derive a fermionic matrix product representative of this state with on-site symmetry.
We study invertible states of 1D bosonic quantum lattice systems. We show that every invertible 1D state is in a trivial phase: after tensoring with some unentangled ancillas, it can be disentangled by a fuzzy analog of a finite-depth quantum circuit. If an invertible state has symmetries, it may be impossible to disentangle it in a way that preserves the symmetries, even after adding unentagled ancillas. We show that in the case of a finite unitary symmetry G, the only obstruction is an index valued in degree-2 cohomology of G. We show that two invertible G-invariant states are in the same phase if and only if their indices coincide.
According to a recent theoretical conjecture for a classification of symmetry protected topological (SPT) phases which can arise in a collection of lattice boson models called group cohomology models, each model belongs to a distinct SPT phase, and each SPT phase can be realized by a model. Here the authors show that for SPT phases with symmetry groups that are finite, Abelian, and unitary, each group cohomology model can be characterized by a unique set of topological invariants and therefore belongs to a distinct SPT phase. In two dimensions it is further shown that the complete set of group cohomology models accounts for all SPT phases, while in three dimensions evidence pointing toward such a proof is found.
Abstract We consider symmetry-protected topological phases with on-site finite groupGsymmetry $\beta $ for two-dimensional quantum spin systems. We show that they have H3(G,T) -valued invariant.
A H3(G,T)-valued index of symmetry protected topological phases with on-site finite group symmetry for two-dimensional quantum spin systems
We consider SPT-phases with on-site finite group G symmetry β for two-dimensional quantum spin systems. We show that they have H3(G,T)-valued invariant.