Symmetry protected topological phases of 1D interacting fermions with spin-charge separation

TL;DR

This paper demonstrates that various gapped phases in 1D interacting fermionic systems with spin-charge separation possess nontrivial topological properties, including protected edge modes, and classifies them as symmetry-protected topological phases.

Contribution

It establishes a correspondence between the eight gapped phases and SPT phases classified by group cohomology, and introduces measurable nonlocal order parameters for these phases.

Findings

Identification of topologically nontrivial gapped phases with protected edge modes

Classification of phases as SPT phases using group cohomology

Proposal of nonlocal order parameters to detect SPT phases

Abstract

The low energy behavior of a huge variety of one-dimensional interacting spinful fermionic systems exhibits spin-charge separation, described in the continuum limit by two sine-Gordon models decoupled in the charge and spin channels. Interaction is known to induce, besides the gapless Luttinger liquid phase, eight possible gapped phases, among which are the Mott, Haldane, charge-/spin-density, and bond-ordered wave insulators, and the Luther Emery liquid. Here we prove that some of these physically distinct phases have nontrivial topological properties, notably the presence of degenerate protected edge modes with fractionalized charge/spin. Moreover, we show that the eight gapped phases are in one-to-one correspondence with the symmetry-protected topological (SPT) phases classified by group cohomology theory in the presence of particle-hole symmetry P. The latter result is also exploited to characterize SPT phases by measurable nonlocal order parameters which follow the system evolution to the quantum phase transition. The implications on the appearance of exotic orders in the class of microscopic Hubbard Hamiltonians, possibly without P symmetry at higher energies, are discussed.