UC Riverside

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.