# Ground state phase diagram of the doped Hubbard model on the 4-leg cylinder

## TL;DR

This study maps the ground state phases of the doped Hubbard model on a 4-leg cylinder, revealing how next-nearest-neighbor hopping influences phases like stripe order, superconductivity, and Luther-Emery liquids.

## Contribution

It demonstrates the sensitivity of the Hubbard model's ground state to next-nearest-neighbor hopping and identifies distinct phases with coexisting orders and unique correlation properties.

## Key findings

- Without $t'$ the system exhibits an insulating stripe phase with charge order.
- Small negative $t'$ induces a phase with algebraic superconducting and charge correlations.
- Small positive $t'$ results in a phase with infinite fermion correlation length and antiferromagnetic spin correlations.

## Abstract

We study the ground state properties of the Hubbard model on a 4-leg cylinder with doped hole concentration per site $\delta\leq 12.5\%$ using density-matrix renormalization group. By keeping a large number of states for long system sizes, we find that the nature of the ground state is remarkably sensitive to the presence of next-nearest-neighbor hopping $t'$. Without $t'$ the ground state of the system corresponds with the insulating filled stripe phase with long-range charge-density-wave (CDW) order and short-range incommensurate spin correlations appears. However, for a small negative $t'$ a phase characterized by coexisting algebraic d-wave superconducting (SC)- and algebraic CDW correlations. In addition, it shows short range spin- and fermion correlations consistent with a canonical Luther-Emery (LE) liquid, except that the charge- and spin periodicities are consistent with half-filled stripes instead of the $4 k_F$ and $2 k_F$ wavevectors generic for one dimensional chains. For a small positive $t'$ yet another phase takes over showing similar SC and CDW correlations. However, the fermions are now characterized by a (near) infinite correlation length while the gapped spin system is characterized by simple staggered antiferromagnetic correlations. We will show that this is consistent with a LE formed from a weakly coupled (BCS like) d-wave superconductor on the ladder where the interactions have only the effect to stabilize a cuprate style magnetic resonance.