SO(4) symmetry and off-diagonal long-range order in the Hubbard bilayer

TL;DR

This paper generalizes Yang's η pairing to the Hubbard bilayer, constructing eigenstates with multiple types of off-diagonal long-range order and revealing an SO(4) symmetry that underpins these phenomena.

Contribution

It introduces a generalized η pairing operator for the Hubbard bilayer and explicitly constructs eigenstates with multiple off-diagonal long-range orders, highlighting the SO(4) symmetry.

Findings

Eigenstates with three types of off-diagonal long-range order constructed.

SO(4) symmetry identified and related to order parameters.

Conditions for spontaneous symmetry breaking and collective modes derived.

Abstract

Yang's $\eta$ pairing operator is generalized to explore off-diagonal long-range order in the Hubbard bilayer with an arbitrary chemical potential. With this operator and a constraint condition on annihilation and creation operators, we construct explicitly eigenstates which possess simultaneously three kinds of off-diagonal long-range order, i.e., the intralayer one and the interlayer one for on-site pairing, and that for interlayer nearest-neighbor pairing. As in the simple Hubbard model there is also an SO(4) symmetry, with the the generators properly defined. A sufficient condition leads to at least one of the above three kinds of off-diagonal long-range order. A constraint relation among different kinds of off-diagonal long-range order is also given. There exists a triplet of collective modes if the U(1) symmetry of a subgroup is spontaneously broken.