Topological properties of gapless phases in an interacting spinful wire

Abstract

We study topology in gapless phases of an interacting spinful model with spin-charge separation. We focus on the gapless boundaries between $\mathbb{Z}_2$ symmetry-breaking phases. We find two topologically non-trivial gapless states that occur at the boundary between a non-trivial and a trivial insulator. They correspond to topological Luther-Emery liquid and topological Mott insulator. The Luther-Emery liquid is characterized by gapless charge excitations and features topological edge modes that carry fractional spin, while the topological Mott insulator has gapless spin sector and features edge states that carry fractional charge. Surprisingly, even though there is no mean-field description of the interacting gapless phases, as there is no local order parameter, we show that they can be adiabatically connected to a non-interacting topological metal. This non-interacting state is a phase boundary between decoupled Su-Schrieffer-Heeger chains with the winding number $ν=2$ and chains with $ν=1$.

Figures (15)

• Figure 1: Transition between topological phases with different winding numbers in the Su-Schrieffer-Heeger chains. The critical phase with $w_2=v_2$ represents a topological metal, the phase where one of the chains is gapless while the second one is gapped and has a zero mode at the boundary.
• Figure 2: Phase diagram of the model \ref{['full_model_main']}. Each of the phases is characterized by an order parameter \ref{['order_parameters_ferm']}. The green dots and lines schematically illustrate the distribution of electronic density. The topologically non-trivial gapless phases are shown in red, the trivial ones are shown in blue.
• Figure 3: Bosonic ground state of gapped sectors of the gapless phases. a) topologically trivial case with no edge modes. It represents a trivial Mott insulator for $\nu=c$ and a trivial Luther-Emery liquid for $\nu=s$ b) topologically non-trivial state, that has fourfold degenerate edge states. For $\nu=c$ it corresponds to the edge states in topological Mott insulator and the $\nu=s$ case describes edge modes in topological Luther-Emery liquid.
• Figure 4: A snapshot of the ground state in a trivial Luther-Emery liquid. Red dots schematically illustrate positions of electrons and empty black dots correspond to empty sites.
• Figure 5: Fluctuations of charges in the ground state of a trivial Luther-Emery liquid. The ground state can be thought as $\text{SSH}_+$ state with fluctuating charge positions.