| Summary: | The following thesis is divided in two main parts. Chapters 2, 3 and 4 are devoted to the study of the so called pair-density-wave (PDW) superconducting state and some of its connections to electronic liquid crystal (ELC) phases, its topological aspects in a one dimensional model and its appearance in a quasi-one dimensional system. On the other hand, chapter 5 is focused on the investigation of the classical statistical mechanics properties of dimers, in particular, the dimer model on the Aztec diamond graph and its relation with the octahedron equation. In chapter 2 we present a theory of superconducting states where the Cooper pairs have a nonzero center-of-mass momentum, inhomogeneous superconducting states known as a pair-density-waves (PDWs) states. We show that in a system of spin-1/2 fermions in two dimensions in an electronic nematic spin-triplet phase where rotational symmetry is broken in both real and spin space PDW phases arise naturally in a theory that can be analysed using controlled approximations. We show that several superfluid phases that may arise in this phase can be treated within a controlled BCS mean field theory, with the strength of the spin-triplet nematic order parameter playing the role of the small parameter of this theory. We find that in a spin-triplet nematic phase, in addition to a triplet p-wave and spin-singlet d-wave (or s depending on the nematic phase) uniform superconducting states, it is also possible to have a d-wave (or s) PDW superconductor. The PDW phases found here can be either unidirectional, bidirectional, or tridirectional depending on the spin-triplet nematic phase and which superconducting channel is dominant. In addition, a triple-helix state is found in a particular channel. We show that these PDW phases are present in the weak-coupling limit, in contrast to the usual Fulde-Ferrell-Larkin-Ovchinnikov phases, which require strong coupling physics in addition to a large magnetic field (and often both). In chapter 3 we show that the pair-density-wave (PDW) ...
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