UC San Diego

Topology and dynamics are among central topics in contemporary condensed matter physics. In this thesis, I present our works on topological properties of topological superconductors with spin-$\frac{3}{2}$ fermions, and real-time dynamics of one-dimensional ($1D$) integrable antiferromagnetic spin-$\frac{1}{2}$ chains.

We systematically generalize the exotic $^3$He-B phase, which not only exhibits unconventional symmetry but is also isotropic and topologically non-trivial, to arbitrary partial-wave channels with multi-component fermions. The concrete example with four-component fermions is illustrated including the isotropic $f$, $p$ and $d$-wave pairings in the spin septet, triplet, and quintet channels, respectively. The odd partial-wave channel pairings are topologically non-trivial, while pairings in even partial-wave channels are topologically trivial. The topological index reaches the largest value of $N^2$ in the $p$-wave channel ($N$ is half of the fermion component number). The surface spectra exhibit multiple linear and even high order Dirac cones. Applications to multi-orbital condensed matter systems and multi-component ultra-cold large spin fermion systems are discussed.

Besides fully gapped unconventional superconductors, nodal superconducting systems can also exhibit nontrivial topological properties. Recent experiments provide evidence to unconventional superconductivity in the YPtBi material with nodal spin-septet pairing. We systematically study topological pairing structures in spin-$\frac{3}{2}$ systems with the cubic group symmetries and calculate the surface Majorana spectra, which exhibit both the zero energy flat band and the cubic dispersion. The signatures of these surface states in the quasi-particle interference patterns are studied, which can be tested in future tunneling experiments.

Although the low energy fractional excitations of one dimensional integrable models are often well-understood, exploring quantum dynamics in these systems remains challenging in the gapless regime, especially at intermediate and high energies. Based on the algebraic Bethe ansatz formalism, we study spin dynamics in the antiferromagnetic spin-$\frac{1}{2}$ XXZ chain with the Ising anisotropy via the form-factor formulae. Various excitations at different energy scales are identified crucial to the dynamic spin structure factors under the guidance of sum rules. At small magnetic polarizations, gapless excitations dominate the low energy spin dynamics arising from the magnetic-field-induced incommensurability. In contrast, spin dynamics at intermediate and high energies is characterized by the two- and three-string states, which are multi-particle excitations based on the commensurate N eel ordered background. The dynamic spectra of the identified dominant excitations evolve with clear energy separations when tuning the magnetic field, conveying a simple and straightforward way to clearly identify the novel string excitations in proper condensed matter systems.