The transition metal dichalcogenides are layered material and have been successfully exfoliated to
atomically thin layers in experiment. Recently extensive study are focused on the atomically thin
transition metal dichalcogenides due to the various novel properties the material exhibits. In the
monolayer 2H-transition metal dichaocogenides, the intrinsic inversion symmetry breaking in the
crystal structure generates the strong spin-orbit coupling. As the crystal structure has the out of
plane mirror symmetry, the electrons’ spin are all pinned to the z direction, forming the Ising type
spin texture in the momentum space. As the Ising spin-orbit coupling dramatically modifies the
electronic properties, especially, it strongly influences the superconducting 2H type atomically thin
transition metal dichalcogenides. In addition to the 2H type, 1T type transition metal dichalcogenides
maintain its inversion symmetry and have different crystal structure, so the physical properties
of 1T-TaS

_{2} are different from its 2H counterpart. In this thesis, we will focus on the effect of
Ising spin-orbit coupling on the 2H superconducting transition metal dichalcogenides of NbSe

_{2} and MoS

_{2}, and study the Mott physics in 1T-TaS

_{2}. As the spin-orbit coupling plays the key role in the novel properties shown in transition metal dichalcogenides, at last we study the novel topological
phase in the spin-orbit coupled ultra cold atoms in the cubic optical lattice.

In the chapter 2, 3, we focus on the superconducting atomic NbSe

_{2} layers and study the effect of in-plane magnetic field on the superconducting properties. We will show that the Ising spin-orbit
coupling strongly enhance in-plane upper critical field in the 2D NbSe

_{2}. In the low temperature, the in-plane magnetic field induced continuous phase transition is observed from the superconductivity
to the normal metal, unlike the abrupt first order transition in the conventional BCS superconductor.
Since the Ising enhanced in-plane H

_{c2} goes much beyond the Pauli limit H

_{P} , we further study the superconductivity state in the presence of in-plane magnetic field between H

_{c2} and H

_{P}. We found the in-plane magnetic field can drive the superconducting monolayer NbSe

_{2} to a nodal topological superconductor, with a large number of Majorana zero modes at the edge forming the Majorana flat bands connecting the nodes. The tunnelling spectroscopy is proposed to detect the topological phase transition.

In the chapter 4, we discuss the superconductivity pairing symmetry in the strong gated MoS

_{2}. A recent tunnelling spectroscopy shows the density of states for strong gated superconducting MoS

_{2} has the “V ” shape instead of the standard “U” shape. Combined with the group theory, we systematically analyze the possible pairing in MoS

_{2} and show that a spin-triplet nodal pairing with the d vector parallel to the Ising spin-orbit coupling can consistently explain the experimental tunneling spectroscopy measurement. In order to further confirm the spin-triplet pairing order parameter, the phase sensitive measurement is further suggested to directly measure the parity of the superconductivity.

In the chapter 5, we study the Mott insulating state in the 1T-TaS

_{2}. The 1T-TaS

_{2} is a cluster Mott insulator with 13 Ta atoms forming the star of David as the unit cell. The first principle calculation shows a narrow band is isolated from other bands in the star of David superlattice. As the isolated band is half filled, the Mott insulating state is described by the half filled Hubbard model. We derive the effective spin model up to the ring exchange term from the Hubbard model and use the density matrix renormalisation group to numerically solve the spin model. There is no spin order observed but a spinon Fermi surface emerges in the spin-spin correlation function. We conclude
that the 1T-TaS

_{2} is possible a gapless spin liquid candidate with spinon Fermi surface.

In the chapter 6, we systematically study the superfluid phase in the spin-orbit coupled cold
atoms in the cubic optical lattice. In the presence of the 1D synthetic spin-orbit coupling, the nodal
ring superfluid emerges with a drumhead like Majorana pockets at the surface. With 2D spin-orbit
coupling, most part of the nodal ring degeneracies are gapped out and only point nodes are left,
the nodal ring superfluid evolves into the Weyl superfluid with Majorana flat bands connecting
the Weyl nodes. In the 3D spin-orbit coupling, the Weyl nodes in the Weyl superfluid will have
energy shift so that the Majorana flat bands gain finite slope. It gives the Majorana zero modes
at the surface finite group velocity and makes Majorana zero modes spiral forward at the surface.
The large numbers of Majorana zero modes at the surface are suggested to use spatial resolved
radio-frequency spectroscopy to detect.

Appendix A, B, C, D, E contain the details of the theoretical derivation and numerically calculation
for the previous chapters.

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