# $\eta$-pairing in correlated fermion models with spin-orbit coupling

## TL;DR

This paper extends $ta$-pairing theory to Hubbard models with spin-orbit coupling, revealing exact pseudospin symmetry and conditions for $ta$-pairing states in various topological and semimetal systems.

## Contribution

It generalizes $ta$-pairing to spin-orbit coupled Hubbard models, identifying exact symmetry conditions and analyzing phase stability in topological semimetals.

## Key findings

- Exact $ta$-pairing conditions in SOC Hubbard models.
- Identification of stable $ta$-pairing and charge-density-wave phases.
- Existence of an exactly solvable multicritical line.

## Abstract

We generalize the $\eta$-pairing theory in Hubbard models to the ones with spin-orbit coupling (SOC) and obtain the conditions under which the $\eta$-pairing operator is an eigenoperator of the Hamiltonian. The $\eta$ pairing thus reveals an exact $SU(2)$ pseudospin symmetry in our spin-orbit coupled Hubbard model, even though the $SU(2)$ spin symmetry is explicitly broken by the SOC. In particular, these exact results can be applied to a variety of Hubbard models with SOC on either bipartite or non-bipartite lattices, whose noninteracting limit can be a Dirac semimetal, a Weyl semimetal, a nodal-line semimetal, and a Chern insulator. The $\eta$ pairing conditions also impose constraints on the band topology of these systems. We then construct and focus on an interacting Dirac-semimetal model, which exhibits an exact pseudospin symmetry with fine-tuned parameters. The stability regions for the exact $\eta$-pairing ground states (with momentum $\bm{\pi}$ or $\bm{0}$) and the exact charge-density-wave ground states are established. Between these distinct symmetry-breaking phases, there exists an exactly solvable multicritical line. In the end, we discuss possible experimental realizations of our results.