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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>M</mml:mi> </mml:math> legs (with M <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>M</mml:mi> </mml:math> ranging from 2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mn>2</mml:mn> </mml:math> to 10 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mn>10</mml:mn> </mml:math> ) 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> </mml:math> in the charge and 2\lambda=16 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>16</mml:mn> </mml:mrow> </mml:math> in the spin can be stabilized at doping \delta=1/8 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:mn>8</mml:mn> </mml:mrow> </mml:math> . Here, there are no sizable superconducting correlations and the ground state has an insulating character. A similar situation, with \lambda=6 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:math> , appears at \delta=1/6 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:mn>6</mml:mn> </mml:mrow> </mml:math> . Instead, for smaller values of dopings, stripes can be still stabilized, but they are weakly metallic at \delta=1/12 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:mn>12</mml:mn> </mml:mrow> </mml:math> and metallic with strong superconducting correlations at \delta=1/10 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:mn>10</mml:mn> </mml:mrow> </mml:math> , 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.

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