Weyl Semi-Metals are materials whose properties are strongly

influenced by spin-orbit couplings. Their description, at low

energies, is in terms of non-relativistic linearly dispersing

massless fermions. In this thesis, we explore possible of new

correlated phases. In particular we focus on excitonic phases due to

particle-hole instabilities in chapters 2 and 3, and on

superconducting phases from particle-particle instabilities in

chapter 4.

The range of the interaction plays a crucial role in determining the

most stable phase. For particle-hole instabilities, short-range

interactions yield eight phases. At stoichiometry, they all

require minimum interaction strengths to ensure their emergence.

Only one of them, the chiral excitonic insulator(EI) phase,

opens a gap at the nodes. It is energetically most favored and is

characterized by a complex vectorial order parameter. Also, it is

ferromagnetic with the phase of the order parameter determining the

direction of the induced net spin polarization's. In contrast

long-range interactions can condense a second gapped state namely

the Charge Density Wave (CDW). To highlight the physics, we employ

the multipole expansion and analyze to leading order. Expanding the

interaction potential and the

order parameter in spherical harmonics, the gap

equation is obtained and analyzed to obtain the minimum interaction

strengths linked to phases. We end that the critical coupling for

CDW phase is half that of the EI phase. Thus, under the Coulomb

interaction, CDW phase is more energetically favorable.

In chapter 4, we turn to possible superconducting states induced

by particle-particle instabilities. As the energy spectrum has even

nodes in the Brillouin zone, both intra-nodal finite-momentum

pairing and inter-nodal zero-momentum BCS pairing are allowed. For

local attractive interaction the finite momentum pairing state with

chiral p-wave symmetry is the most favorable phase at finite carrier

density. For chemical potential at the node the state is preempted

by a fully gapped CDW phase. On the other hand, for long-range

attractive interactions, the p-wave BCS superconducting state wins

out for all values of the chemical potential.

In quantum materials, the interactions between the bulk lattice, free electrons, and lattice vibrations give rise to interesting phases of matter. In recent years, new materials have been discovered that have nontrivial topological physics, allowing them to host topologically protected surface states. Much work has been done in exploring whether or not such materials can be rendered superconducting via the proximity effect. In this work, I will explore interesting phases of two topological materials: Weyl semimetals and transition metal dichalcogenides. In the first section, I explore the properties of proximity induced superconductivity within these materials. Using a numerical technique known as the spectral method, I determine what, if any, types of superconductivity can be induced within these materials, and whether or not the superconducting state retains their topological properties. Ultimately, I show that, in the absence of a non-trivial tunneling interaction at the interface, no interesting superconducting properties can be induced. In the second section, I will explore the optical absorption properties of a semiconducting materials in the presence of electron-phonon interactions, which typically give rise to phonon side-bands in optical spectra. I demonstrate how, with an appropriate transformation, an expression for the optical spectra to arbitrary phonon side-band order can be calculated for a general interacting system. To demonstrate its validity, I fit the model to the measured optical absorption spectrum of a device consisting of layered transition metal dichalcogenides.

The explosion of recent work suggests that the next generation of solid state devices will be built upon an understanding of topological phases and phenomena. Key to the development and application of such devices are the notions of tunability, the manipulation of device properties with some external stimulus, and interactions, the additional influences on electrons.

Tunable devices form the basis for transistors and computer technology, but tunable topological devices have not been greatly explored. Thus, here we extend a previous proposal[2], which realized a two-node Weyl semimetal with a heterostructure of alternating topological/normal insulator layers and magnetic coupling in the direction perpendicular to the layers, with the coupling placed parallel to the layers. The magnetic coupling, arising, for example, from ferromagnetic insulators, here creates a line-node semimetal and it turns out to allow for tunable features in the device which can, in principle, be measured in future experimental studies. Interestingly, the Fermi surface can be tuned to have the topology of either a sphere or a torus, a unique aspect of line nodes.

Interactions in topological devices provide additional routes for further development. A good example is the problem of dilute magnetic impurities, providing a window into the structure of topological states. Here we use monolayer transition metal group-VI dichalcogenides for a simple model of topological bands in a semiconductor. The system is hexagonal but lacks an inversion center and includes strong spin-orbit coupling from the heavy transition metal, resulting in spin-split bands in separate valleys around the K points, with finite Berry curvature, and consequently a contrasted optical circular dichroism. The hole-doped regime possesses separate Fermi surfaces, with opposite spins on opposite sides of the Brillouin zone, producing an interesting spin structure in the Kondo ground state. Furthermore, the selective absorption of circularly polarized light according to valley/spin leads to the manipulation of the spin structure directly. We extensively study the Kondo ground state resulting from the quasi-equilibrium configuration inferred from the application of circularly polarized light, a situation which involves topology, spin-orbit interactions, hybridization with a magnetic impurity, and tunability of the spin state with light.

Recent focus on two dimensional materials and spin-coupled phenomena holds future potential for fast, efficient, flexible, and transparent devices.

The fundamental operation of a spintronic device depends on the injection, transmission, and detection of spins in a conducting channel. Long spin lifetimes during transit are critical for realizing this technology. An attractive platform for this purpose is graphene, which has high mobilities and low spin-orbit coupling. Unfortunately, measured spin lifetimes are orders of magnitude smaller than theoretically expected. A source of spin loss is the resistance mismatch between the ferromagnetic electrodes and graphene. While this has been studied numerically, here we provide a closed form expression for Hanle spin precession which is the standard method of measuring spin lifetimes. This allows for a detailed characterization of the nonlocal spin valve device.

Strong spin-orbit interaction has the potential to engender unconventional superconducting states. A cousin to graphene, two dimensional transition metal dichalcogenides entwine interaction, spin-orbit coupling, and topology. The noninteracting electronic states have multiple valleys in the energy dispersion and are topologically nontrivial. We report on the possible superconducting states of hole-doped systems, and analyze to what extent the correlated phase inherits the topological aspects of the parent crystal. We find that local attractive interactions or proximal coupling to $s$-wave superconductors lead to a pairing which is an equal mixture of a spin singlet and the $m = 0$ spin triplet. Its topology allows quasiparticle excitations of net nonzero Berry curvature via pair-breaking by circularly polarized light. The valley contrasting optical response, where oppositely circularly polarized light couples to different valleys, is present even in the superconducting state, though with smaller magnitude.