# Quantum spin Hall density wave insulator of correlated fermions

## TL;DR

This paper introduces a novel topological quantum order called the quantum spin-Hall density wave (QSHDW) state, characterized by a $Z_2$ invariant, spatial modulation, and weak topological insulator properties in correlated fermion systems.

## Contribution

It proposes a new topological phase driven by spin-orbit density wave order, with unique symmetry and boundary state features, extending quantum spin-Hall concepts to density wave systems.

## Key findings

- Identifies a $Z_2$ topological invariant in density wave order
- Predicts weak topological insulator behavior with spin-polarized boundary states
- Suggests realizations in quantum wires and quasi-2D systems

## Abstract

We present the theory of a new type of topological quantum order which is driven by the spin-orbit density wave order parameter, and distinguished by $Z_2$ topological invariant. We show that when two oppositely polarized chiral bands [resulting from the Rashba-type spin-orbit coupling $\alpha_k$, $k$ is crystal momentum] are significantly nested by a special wavevector ${\bf Q}\sim(\pi,0)/(0,\pi)$, it induces a spatially modulated inversion of the chirality ($\alpha_{k+Q}=\alpha_k^*$) between different sublattices. The resulting quantum order parameters break translational symmetry, but preserve time-reversal symmetry. It is inherently associated with a $Z_2$-topological invariant along each density wave propagation direction. Hence it gives a weak topological insulator in two dimensions, with even number of spin-polarized boundary states. This phase is analogous to the quantum spin-Hall state, except here the time-reversal polarization is spatially modulated, and thus it is dubbed quantum spin-Hall density wave (QSHDW) state. This order parameter can be realized or engineered in quantum wires, or quasi-2D systems, by tuning the spin-orbit couping strength and chemical potential to achieve the special nesting condition.