Ising superconductivity and quantum metal in the two-dimensional transition metal dichalcogenides TaS$_2$ and NbSe$_2$

Atomically-thin transition metal dichalcogenides (TMDs) are of significant contemporary interest, particularly for the strong ``Ising'' spin-orbit coupling (SOC) that results from the broken inversion symmetry in the individual atomic layers. NbSe$_2$, a classic layered superconductor, is the first metallic TMD to have exhibited superconductivity down to this monolayer (1L) limit, and we have recently shown that 1L TaS$_2$, with stronger SOC, also exhibits superconductivity. The atomic-scale thickness of the TMD crystals implies that we can explore the regime where quenching of superconductivity is entirely due to paramagnetic effects. We study this regime, where superconductivity can survive well above 30 T in-plane, and find a strong enhancement of the upper critical field relative to the Pauli paramagnetic limit for both 1L and multilayer samples. I discuss the implications of our results for various spin-orbit coupling effects, local inversion-symmetry breaking in even-layer samples, spin-triplet pairing, and possible topological and other exotic phases of superconductivity. I will also discuss our recent experiments on bilayer and few-layer NbSe$_2$ and TaS$_2$ and our observation of an anomalous metallic phase in the zero-temperature limit induced by a small perpendicular magnetic field. This quantum metal phase has only been observed in highly-disordered thin film superconductors and its observation in a crystalline superconductor, along with a distinct magnetic field scaling, forces reexamination of the diagram of possible electronic phases in two dimensions at zero temperature.