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Since their discovery the understanding of unconventional superconductors (USCs) has posed a great challenge in condensed matter physics. One central problem, both of theory and experiment, lies in determining the microscopic origin of electron pairing in such systems. Partly, the difficulty lies in numerically simulating systems hypothesized to represent USCs. In particular, it remains an open question whether the ground state of the two-dimensional Hubbard model realizes an USC. This system epitomizes the combined difficulty of finding both whether electrons form pairs and whether they condense into an USCs phase.

Conversely, the one-dimensional (1D) Hubbard ladder is readily solved numerically using matrix-product state (MPS) methods. Doped away from a half-filled lattice repulsively mediated electron pairing is realized in the system. However, quantum fluctuations hinder ordering in such systems even at zero temperature viz. continuous symmetries cannot be spontaneously broken. Instead, quasi-one-dimensional (Q1D) systems featuring arrays of one-dimensional (1D) chains weakly coupled into a higher-dimensional system can be studied. While pairing is resolved in each 1D system using MPS the condensation of such pairs into a superconductor may be treated using mean-field (MF) theory. The subject of this thesis is the development of the MPS+MF framework: An algorithm utilizing MPS and MF theory capable of solving Q1D systems.

Developing new methods requires comparison with known solutions to learn of their potential inaccuracies. Thus, development is split into three steps: i) Simulation of bosons to test the basic approach, ii) simulation of attractive fermions, iii) simulation of an USC composed of repulsive Hubbard ladders. The first two targets admit comparison to quantum Monte Carlo simulations and, for some parameters, analytical methods. The MPS+MF framework is found to simulate the critical temperature of condensation with a fixed ratio to the true critical temperature, independent of Q1D coupling. Additionally, the framework is found capable of resolving competition of insulating and USC phases.

Utilizing the possibility of evolving states in time using MPS numerics, MPS+MF is extended to perform self-consistent time evolution. We find that this method allows the detection of superconductivity out of equilibrium and, in particular, dynamically induced superconductivity.