A classification of invertible phases of bosonic quantum lattice systems in one dimension

Abstract

We study the entanglement properties of quantum phases of bosonic 1d lattice systems in infinite volume. We show that the ground state of any gapped local Hamiltonian is Short-Range Entangled: it can be disentangled by a fuzzy analog of a finite-depth quantum circuit. We characterize Short-Range Entangled states in terms of decay properties of their Schmidt coefficients. If a Short-Range Entangled state has symmetries, it may be impossible to disentangle it in a way that preserves the symmetries. We show that in the case of a finite unitary symmetry G the only obstruction for the existence of a symmetry-preserving disentangler is an index valued in degree-2 cohomology of G. We show that two Short-Range Entangled states are in the same phase if and only if their indices coincide.

Classification of (2+1)D invertible fermionic topological phases with symmetry

“…The anyons v, v −1 are permuted by a Z 2 0-form symmetry (the charge conjugation symmetry of Spin(2c − )). Thus the theory has a Z 4 one-form symmetry generated by (say) the v line, with the anomaly described by the SPT phase 2π 2c − 16 4d P(Y 2 ) , (108) 16 The modified cocycle condition dB…”

“…This obscures the physical distinction between different invertible phases, and also removes us from the setting of topological phases of matter, which rely on the notion of a gapped Hamiltonian acting on a many-body Hilbert space. A more direct approach to topological phases of matter, in terms of operator algebras, is also under development, although the results are so far less comprehensive than the TQFT approach [16][17][18][19].…”

An index for two-dimensional SPT states

“…A are SRE defect states.Figure 2: A is a cone-like region at p with boundary components (∂A) − and (∂A) + . All invertible defect states at a point are SRE defects states.The last corollary is also a direct consequence of Lemma 4.1 from[9]. Let ψ be a 2d SRE state, and let α…”

“…In the presence of a symmetry group G we can consider a class of G-invariant states and define the same notions using G-invariant LGA equivalence and G-invariant factorized state. The latter is defined to be a factorized state ψ 0 with G-invariant vectors |v j ∈ V j , such that A j ψ = v j |A j |v j for any A j ∈ A j (see [9] for more details). Similarly, we have a monoid (Φ G , •, τ G ) of G-invariant phases with abelian group Φ * G of G-invertible phases.…”

“…Our approach is motivated by the work [16]. The absence of non-trivial invertible one-dimensional states needed for the definition of the index was proved recently in the work [9], to which this paper is complementary.…”