Topological phenomena in physical systems are determined by topological structures and are thus universal and protected against perturbations. We theoretically establish exotic topological phenomena and their consequences for experiments in crystalline systems such as topological insulators and topologically ordered phases. We show that protected one-dimensional fermionic modes may be associated with the line defects such as dislocations in three-dimensional topological insulators, and strong electron-electron repulsion may lead to topological Mott insulators via spontaneous spin-orbit correlations in three dimensions. We also predict anomalous Aharonov Bohm conductance oscillations maximized at half integer multiples of a flux quantum in a topological insulator nanowire with strong surface disorder, arising from surface curvature induced Berry phase. In addition, we classify three-dimensional inversion symmetric insulators and their quantized responses.

Quantum entanglement provides a promising probe to the properties of many-body systems, especially topological phases not captured by local order parameters. We present a characterization of topological insulators using entanglement spectrum based only on bulk ground-state wave function. Further, by studying entanglement of trivial partitions, we establish topological order in candidate Gutzwiller projected wave functions for gapped spin liquids and Laughlin states; and with entanglement's dependence on the ground states for bipartition of a torus into two cylinders, we demonstrate a method to extract the modular matrices and statistics and braiding of quasiparticle excitations. Our method helps to determine the topological order with only the set of ground-state wave functions on a torus. Our variational Monte Carlo calculations of topological entanglement entropy agree well with theory. We also find a violation of the boundary law for a critical spin liquid of Gutzwiller projected Fermi sea on the triangular lattice, where the entanglement entropy's enhancement by a logarithmic factor reflects the presence of emergent fermions in a bosonic wave function.

This dissertation establishes and investigates new phenomena in diverse interacting many-body quantum systems guided by three distinct, but complementary, themes: (i) symmetry and topology, (ii) localization, and (iii) non-Fermi liquids.

The first theme concerns how the interplay of symmetry and topology can offer robust protection for a many-body system. We investigate low-dimensional quantum fermionic models from a general structural perspective. These phases can exhibit fractionalized Majorana zero-energy modes on their boundary. We devise experimentally relevant nonlocal measurements that can be used to detect these topological phases. While our primary focus is on quantum systems, topologically protected behavior can arise in classical mechanical models as well. We extend a recent connection between the topological band theory of electrons and classical physics by proposing a mechanical analogue of a topological nodal semimetal.

The second theme concerns that of many-body localization. We demonstrate that the combination of localization, symmetry, and topology can have radical consequences for quantum systems at high energies, such as the existence of protected gapless boundary modes. We show that, even at these high energies, quantum information can be preserved, and quantum coherence recovered. Quantum coherent dynamics in this regime is unexpected and of great interest for quantum computation.

The third direction in our study of interacting many-body systems concerns non-Fermi liquids. We construct a non-Fermi liquid by bringing together a spin-orbit coupled Fermi surface and fluctuating magnetic order. Using newly developed analytic tools for strongly coupled systems, we demonstrate the stability of the non-Fermi liquid to ordering. This identifies an experimentally accessible candidate for exploring physics that lies beyond Fermi liquid theory.

In this thesis, we will study aspects of two phases close to criticality arising in solid state systems with strong interactions between electrons. In the first part, we study finite temperature transport in a non-Fermi-liquid phase arising from a nodal semimetal with long-range interactions – the so-called Luttinger-Abrikosov-Beneslavskii phase. We are particularly interested in calculating the finite temperature shear viscosity of the phase and find that it is consistent with a bound proposed in the context of gauge-gravity duality. In the second part of the thesis, we study a minimal model of nematic fluctuations in the high-temperature superconductor iron selenide. Nematic fluctuations arising from a quantum critical point have been proposed to explain the phenomenology of several high-temperature superconductors. In a numerical simulation using determinant quantum Monte Carlo methods, we find no direct evidence of a nematic critical point. However, we still observe a wide region of superconductivity correlated with nematic fluctuations as well as an unusual antiferro-quadrupole order.

In this doctoral dissertation, we consider the significance of spin-orbit coupling for the phases of matter which arise for strongly correlated electrons. We explore emergent behavior in quantum many-body systems, including symmetry-breaking orders, quantum spin liquids, and unconventional superconductivity. Our study is cemented by a particular class of Mott-insulating materials, centered around a family of two- and three-dimensional iridium oxides, whose honeycomb-like lattice structure admits peculiar magnetic interactions, the so-called Kitaev exchange. By analyzing recent experiments on these compounds, we show that this unconventional exchange is the key ingredient in describing their magnetism, and then use a combination of numerical and analytical techniques to investigate the implications for the phase diagram as well as the physics of the proximate three-dimensional quantum spin liquid phases. These long-ranged-entangled fractionalized phases should exhibit special features, including finite-temperature stability as well as unconventional high-Tc superconductivity upon charge-doping, which should aid future experimental searches for spin liquid physics. Our study explores the nature of frustration and fractionalization which can arise in quantum systems in the presence of strong spin-orbit coupling.

The subject of condensed matter physics is very rich — there are an infinite number of parameters producing a diversity of exciting phenomena. As a theorist, my goal is to distill general principles out of this complexity — to construct theories that can coherently explain many known examples altogether.

This thesis is composed of several attempts to develop such theories in topics related to spontaneously symmetry breaking. A remarkable feature of many-body interacting systems is that although they are described by equations respecting various symmetries, they may spontaneously organize into a state that explicitly breaks symmetries. Examples are numerous: various types of crystalline and magnetic orders, Bose-Einstein condensates of cold atoms, superfluids of liquid helium, chiral symmetry in QCD, neutron stars, and cosmic inflation.

These systems with spontaneously broken continuous symmetries have gapless excitations, so called Nambu-Goldstone bosons (NGBs). Although the properties of NGBs are well understood in Lorentz-invariant systems, surprisingly, some basic properties of NGBs such as their number and dispersion in nonrelativistic systems have not been discussed from a general perspective. In the first part of this thesis, we solve this issue by developing and analyzing an effective Lagrangian that coherently captures the low-energy, long-distance physics of many different symmetry-breaking states all at once.

Next, we examine whether these NGBs originating from spontaneous symmetry breaking remain to be well-defined excitations inside a metal, where low-energy electrons near Fermi surface can collide with them. Our result is a one equation criterion that specifies whether the interactions between electrons and NGBs can be ignored, or whether it completely changes their character. In the latter case, unusual phases of matter such as non-Fermi liquids may arise; in that case, NGBs are overdamped and cannot form particle-like excitations in spite of the assumed symmetry breaking.

In the last part of this thesis, we investigate the possibility of spontaneously breaking of Hamiltonian itself. The homogeneity of time is one of the most fundamental symmetries of nature, underlying the conservation of the energy. The question is whether this symmetry can be spontaneously broken, as suggested recently by Wilczek, in analogy with ordinary crystals. Contrary to his proposal that attracted a significant attention and stimulated many further studies, we prove a no-go theorem that rules out spontaneously breaking of time translation, in the ground state or in the canonical ensemble of a general Hamiltonian.

The commonly adopted classification of the electronic phases of matter as metals, insulators, semimetals, semiconductors and superconductors can be refined by studying the topological properties of the band structure in these phases. This unveils a rich and diverse substructure, and helps to conclude the existence of topological phases -- phases whose properties are robust against perturbations, possibly with the requirement that the perturbations preserve certain symmetries. In this dissertation, several topological phases are studied, and predictions are made for some unusual physical phenomena involving these phases. Some of these phenomena may be observable using current experimental techniques. Topological phases often carry unconventional surface states; experimental signatures for the surface states of a strong topological insulator are presented as well.

We begin by systematically discussing the gapped phases, including some topological insulators and topological superconductors, proximate to a three-dimensional Dirac node. Studying topological defects in these phases reveals interesting duality relations between them, which provide potential routes for unconventional or non-Landau-Ginzburg-Wilson continuous phase transitions between them. Topological textures, on the other hand, can lead to very different physics, and we show that a bulk superconductor vortex in a topological insulator -- a texture with a non-trivial Hopf index -- acts like a fermion. As a corollary, we show that if the vortex string is terminated by a surface, the resulting dangling end traps a Majorana zero mode. The Majorana mode is found to be stable against doping the topological insulator, provided the doping is below a critical value set by Berry phase properties of the Fermi surface of the doped insulator. Importantly, several existing superconductors such as Cu-doped Bi₂Se₃ and TlBiTe₂ are predicted to host surface Majorana modes. In the absence of superconductivity, the surface of a topological insulator has gapless, spin-momentum locked states. This locking leads to an interesting response to circularly polarized radiation, with the dominant photocurrent being tied to the Berry curvature of the surface bands.

Having studied several theoretical and experimental implications of gapped topological phases, we turn our attention towards a gapless topological phase -- the Weyl semimetal. We investigate transport in this phase in the presence of Coulomb interactions or disorder at finite temperature, and compare our results to experimental data on the candidate Weyl semimetal Y₂Ir₂O₇ , finding encouraging agreement.