Lifshitz transition in the phase diagram of two-leg $t$-$J$ ladder systems at low filling

TL;DR

This study combines numerical and analytical methods to explore the phase diagram of a two-leg $t$-$J$ ladder at low fillings, revealing a Lifshitz transition, a crossover in superconducting order, and precursor Wigner crystal behavior.

Contribution

It identifies a Lifshitz-like transition and crossover in the phase diagram of the $t$-$J$ ladder, incorporating effects of nearest-neighbor repulsion and low filling.

Findings

Lifshitz-like band filling transition observed

Crossover from s-wave to d-wave superconducting order

Precursor to Wigner crystal at low fillings

Abstract

We use a combination of numerical matrix product states (MPS) and analytical approaches to investigate the phase diagram of the two-leg $t$-$J$ ladder in the region of low to intermediate fillings. We choose the same coupling strength along the leg- and rung-directions, but study the effect of adding a nearest-neighbor repulsion $V$. We observe a rich phase diagram and analytically identify a Lifshitz-like band filling transition, which can be associated to a numerically observed crossover from s-wave to d-wave like superconducting quasi-long range order (QLRO). Due to the strong interactions, the Lifshitz transition is smeared into a crossover region which separates two distinct Luttinger theories with unequal physical meaning of the Luttinger parameter. Our numerically exact MPS results spotlight deviations from standard Luttinger theory in this crossover region and is consistent with Luttinger theory sufficiently far away from the Lifshitz transition. At very low fillings, studying the Friedel-like oscillations of the local density identifies a precursor region to a Wigner crystal at small values of the magnetic exchange interaction $J/t$. We discuss analytically how tuning parameters at these fillings modifies the phase diagram, and find good agreement with MPS results.