We use a combination of numerical matrix product states (MPS) and analytical approaches to investigate the phase diagram of the two-leg $t\text{−}J$ 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-type band-filling transition, which can be associated to a numerically observed crossover from $s$-wave- to $d$-wave-like superconducting quasi-long-range order. 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 are consistent with Luttinger theory sufficiently far away from the Lifshitz transition. At very low fillings, studying the Friedel-type 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 modify the phase diagram, and find good agreement with MPS results.

• Received 15 December 2022
• Revised 4 September 2023
• Accepted 2 October 2023

DOI:https://doi.org/10.1103/PhysRevB.108.155148

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society