On ℤ2-indices for ground states of fermionic chains

Abstract

For parity-conserving fermionic chains, we review how to associate Z 2 -indices to ground states in finite systems with quadratic and higher-order interactions as well as to quasifree ground states on the infinite CAR algebra. It is shown that the Z 2 -valued spectral flow provides a topological obstruction for two systems to have the same Z 2 -index. A rudimentary definition of a Z 2 -phase label for a class of parity-invariant and pure ground states of the one-dimensional infinite CAR algebra is also provided. Ground states with differing phase labels cannot be connected without a closing of the spectral gap of the infinite GNS Hamiltonian.

The classification of symmetry protected topological phases of one-dimensional fermion systems

“…This index takes value in Z 2 × 1 ( , Z 2 ) × 2 ( , (1) ), which is analogous to the indices introduced in [19] in the context of spin-topological quantum field theory (spin-TQFT) and [11,20,39] for the fermionic MPS setting. When is trivial, the index is Z 2 -valued and recovers the index studied in [4,24]. The key ingredient for the definition is again the split property of unique gapped ground states for fermionic systems proven recently in [24].…”

“…In contrast to quantum spin chains, for paritysymmetric gapped ground states without additional symmetries, there are two distinct phases. A Z 2index to distinguish these phases in infinite systems was introduced in [4] and independently in [24]. It was outlined in [4] that this Z 2 -index is an invariant of the classification of unique parity-invariant gapped ground state phases using techniques from [29] and [28].…”

“…A Z 2index to distinguish these phases in infinite systems was introduced in [4] and independently in [24]. It was outlined in [4] that this Z 2 -index is an invariant of the classification of unique parity-invariant gapped ground state phases using techniques from [29] and [28]. The aim of this article is to extend the analysis of fermionic gapped ground states to the case with an on-site symmetry; namely, a classification of one-dimensional fermionic SPT phases.…”

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

“…Taken together, these results imply that the index of an invertible G-invariant state on A depends only on its G-invariant stable phase. Then it is easy to see that the index we defined above is the same as defined in [15].…”

Rigorous Index Theory for One-Dimensional Interacting Topological Insulators

“…Topological insulators [3] for non-interacting fermions are completely classified according to the "periodic table" [4,5], and are characterized by the indices that are written as an integral over the Brillouin zone when the model has translation invariance (see, e.g., [3,6]), or, in more general, by the indices for projections defined by methods of noncommutative geometry (see, e.g., [7,8]). Although similar classification and characterization of interacting topological insulators have been investigated intensively (see, e.g, [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]), mathematically rigorous results are still limited [24][25][26][27][28][29][30]. See [31][32][33][34][35][36][37] for closely related rigorous index theorems for bosonic (or quantum spin) systems.…”