Pair-Density-Wave in the Strong Coupling Limit of the Holstein-Hubbard model

TL;DR

This study uses DMRG to demonstrate the emergence of a quasi-long-range pair-density-wave order in a strongly coupled Holstein-Hubbard model, revealing novel superconducting states with broken symmetries and valley polarization.

Contribution

It provides the first controlled calculation showing how PDW states can arise in the strong-coupling limit of the Holstein-Hubbard model, linking microscopic mechanisms to emergent order.

Findings

Strong quasi-long-range PDW order observed

Spontaneous breaking of time-reversal and inversion symmetries

Identification of valley-polarized intra-pocket pairing

Abstract

A pair-density-wave (PDW) is a novel superconducting state with an oscillating order parameter. A microscopic mechanism that can give rise to it has been long sought but has not yet been established by any controlled calculation. Here we report a density-matrix renormalization group (DMRG) study of an effective $t$-$J$-$V$ model, which is equivalent to the Holstein-Hubbard model in a strong-coupling limit, on long two-, four- and six-leg triangular cylinders. While a state with long-range PDW order is precluded in one dimension, we find strong quasi-long-range PDW order with a divergent PDW susceptibility as well as the spontaneous breaking of time-reversal and inversion symmetries. Despite the strong interactions, the underlying Fermi surfaces and electron pockets around the $K$ and $K^\prime$ points in the Brillouin zone can be identified. We conclude that the state is valley-polarized and that the PDW arises from intra-pocket pairing with an incommensurate center of mass momentum. In the two-leg case, the exponential decay of spin correlations and the measured central charge $c\approx 1$ are consistent with an unusual realization of a Luther-Emery liquid.