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The dualism between superconductivity and charge/spin modulations (theso-called stripes) dominates the phase diagram of many strongly-correlatedsystems. A prominent example is given by the Hubbard model, where these phasescompete and possibly coexist in a wide regime of electron dopings for both weakand strong couplings. Here, we investigate this antagonism within a variationalapproach that is based upon Jastrow-Slater wave functions, including backflowcorrelations, which can be treated within a quantum Monte Carlo procedure. Wefocus 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 adetailed analysis in terms of the stripe length. We find that stripe order withperiodicity $\lambda=8$ in the charge and $2\lambda=16$ in the spin can bestabilized at doping $\delta=1/8$. Here, there are no sizable superconductingcorrelations and the ground state has an insulating character. A similarsituation, with $\lambda=6$, appears at $\delta=1/6$. Instead, for smallervalues of dopings, stripes can be still stabilized, but they are weaklymetallic at $\delta=1/12$ and metallic with strong superconducting correlationsat $\delta=1/10$, as well as for intermediate (incommensurate) dopings.Remarkably, we observe that spin modulation plays a major role in stripeformation, since it is crucial to obtain a stable striped state uponoptimization. The relevance of our calculations for previous density-matrixrenormalization group results and for the two-dimensional case is alsodiscussed.

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