Resolving competition of charge density wave and superconducting phases using the matrix product state plus mean field algorithm

Materials with strong electronic correlations often exhibit a superconducting phase in close competition with other insulating phases, which is outstandingly difficult to resolve, e.g., for a group of quasi-two-dimensional (Q2D) materials such as the cuprates, even for the simplified minimal model of these materials, the doped 2D Hubbard model. The present work shows how quasi-one-dimensional (Q1D) systems, 2D and three-dimensional (3D) arrays of weakly coupled 1D correlated electrons, are much more amenable to resolving such competition, treating both instabilities on equal footing. Using the recently established matrix product state plus mean field <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mo>(</a:mo><a:mi>MPS</a:mi><a:mo>+</a:mo><a:mi>MF</a:mi><a:mo>)</a:mo></a:math> approach for fermions [Bollmark , ], we demonstrate that large systems can be reached readily in these systems, which opens up the thermodynamic regime via extrapolation. Focusing on basic model systems, 3D arrays of negative-<b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>U</b:mi></b:math> Hubbard chains with additional nearest-neighbor interaction <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>V</c:mi></c:math>, we show that despite the MF component of the <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>MPS</d:mi><d:mo>+</d:mo><d:mi>MF</d:mi></d:math> technique, we can reproduce the expected coexistence of the superconductivity and charge density wave at <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi>V</e:mi><e:mo>=</e:mo><e:mn>0</e:mn></e:mrow></e:math> for density <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mi>n</f:mi><f:mo>=</f:mo><f:mn>1</f:mn></f:mrow></f:math>. We then show how we can tune away from coexistence by both tuning <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>V</g:mi></g:math> and doping the system. This work thus paves the way to deploy two-channel <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mi>MPS</h:mi><h:mo>+</h:mo><h:mi>MF</h:mi></h:math> theory on some highly demanding high-<i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:msub><i:mi>T</i:mi><i:mi>c</i:mi></i:msub></i:math> superconducting systems, such as 3D arrays of repulsive-<j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mi>U</j:mi></j:math> doped Hubbard ladders; we recently characterized the properties of such arrays in single-channel <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mi>MPS</k:mi><k:mo>+</k:mo><k:mi>MF</k:mi></k:math> calculations [Bollmark , ].