# Symmetry and topology in condensed matter physics

## Abstract

Recently there has been a surging interest in the topological phases of matter, including the symmetry-protected topological phases, symmetry-enriched topological phases, and topological semimetals. This thesis is aiming at finding new ways of searching and probing these topological phases of matter in order to deepen our understanding of them. The body of the thesis consists of three parts. In the first part, we study the search of filling-enforced topological phases of matter in materials. It shows the existence of symmetry-protected topological phases enforced by special electron fillings or fractional spin per unit-cell. This is an extension of the famous Lieb-Schultz-Mattis theorem. The original LSM theorem states that the symmetric gapped ground state of the system must exhibit topological order when there's fractional spin or fractional electron filling per unit-cell. However, the LSM theorem can be circumvented when commensurate magnetic flux is present in the system, which enlarge the unit-cells to accommodate integer numbers of electrons. We utilize this point to prove that the ground state of the system must be a symmetry-protected topological phase when magnetic translation symmetry is satisfied, which we coin the name “generalized LSM theorem”. The theorem is proved using two different methods. The first proof is to use the tensor network representation of the ground state wave-function. The second proof consists of a physical argument based on the idea of entanglement pumping. As a byproduct of this theorem, a large class of decorated quantum dimer models are introduced, which satisfy the condition of the generalized LSM theorem and exhibit SPT phases as their ground states. In part II, we switch to the nonlinear response study of Weyl semimetals. Weyl semimetals (WSM) have been discovered in time-reversal symmetric materials, featuring monopoles of Berry’s curvature in momentum space. WSM have been distinguished between Type-I and II where the velocity tilting of the cone in the later ensures a finite area Fermi surface.To date it has not been clear whether the two types results in any qualitatively new phenomena. In this part we focus on the shift-current response ($\sigma_{shift}(\omega)$), a second order optical effect generating photocurrents. We find that up to an order unity constant, $\sigma_{shift}(\omega)\sim \frac{e^3}{h^2}\frac{1}{\omega}$ in Type-II WSM, diverging in the low frequency $\omega\rightarrow 0$ limit. This is in stark contrast to the vanishing behavior ($\sigma_{shift}(\omega)\propto \omega$) in Type-I WSM. In addition, in both Type-I and Type-II WSM, a nonzero chemical potential $\mu$ relative to nodes leads to a large peak of shift-current response with a width $\sim |\mu|/\hbar$ and a height $\sim \frac{e^3}{h}\frac{1}{|\mu|}$, the latter diverging in the low doping limit. We show that the origin of these divergences is the singular Berry’s connections and the Pauli-blocking mechanism. Similar results hold for the real part of the second harmonic generation, a closely related nonlinear optical response. In part III, we propose a new kind of thermo-optical experiment: the nonreciprocal directional dichroism induced by a temperature gradient. The nonreciprocal directional dichroism effect, which measures the difference in the optical absorption coefficient between counterpropagating lights, occurs only in systems lacking inversion symmetry. The introduction of temperature-gradient in an inversion-symmetric system will also yield nonreciprocal directional dichroism effect. This effect is then applied to quantum magnetism, where conventional experimental techniques have difficulty detecting magnetic mobile excitations such as magnons or spinons exclusively due to the interference of phonons and local magnetic impurities. A model calculation is presented to further demonstrate this phenomenon.