# Metallic and insulating stripes and their relation with superconductivity in the doped Hubbard model

## TL;DR

This study uses variational quantum Monte Carlo methods to analyze stripe order and its relation to superconductivity in the doped Hubbard model, revealing doping-dependent phases and the importance of spin modulation.

## Contribution

It introduces a detailed variational approach with backflow correlations to investigate stripe and superconducting phases across various doping levels in the Hubbard model.

## Key findings

- Stripe order with specific periodicities is stabilized at certain dopings.
- Insulating and non-superconducting states are found at doping levels 1/8 and 1/6.
- Metallic and superconducting correlations emerge at lower dopings like 1/12 and 1/10.

## Abstract

The dualism between superconductivity and charge/spin modulations (the so-called stripes) dominates the phase diagram of many strongly-correlated systems. A prominent example is given by the Hubbard model, where these phases compete and possibly coexist in a wide regime of electron dopings for both weak and strong couplings. Here, we investigate this antagonism within a variational approach that is based upon Jastrow-Slater wave functions, including backflow correlations, which can be treated within a quantum Monte Carlo procedure. We focus on clusters having a ladder geometry with $M$ legs (with $M$ ranging from $2$ to $10$) and a relatively large number of rungs, thus allowing us a detailed analysis in terms of the stripe length. We find that stripe order with periodicity $\lambda=8$ in the charge and $2\lambda=16$ in the spin can be stabilized at doping $\delta=1/8$. Here, there are no sizable superconducting correlations and the ground state has an insulating character. A similar situation, with $\lambda=6$, appears at $\delta=1/6$. Instead, for smaller values of dopings, stripes can be still stabilized, but they are weakly metallic at $\delta=1/12$ and metallic with strong superconducting correlations at $\delta=1/10$, as well as for intermediate (incommensurate) dopings. Remarkably, we observe that spin modulation plays a major role in stripe formation, since it is crucial to obtain a stable striped state upon optimization. The relevance of our calculations for previous density-matrix renormalization group results and for the two-dimensional case is also discussed.