Topological phases of matter are purely quantum mechanical and have no classical analogue. Most phases in nature can be classified and studied classically through the concept of symmetry breaking and its theoretical description, Landau-Ginzburg field theory. In contrast to the general wisdom of Landau-Ginzburg field theory, the topological phases share the same symmetry as a trivial insulator and still are different phases. Having gapped spectrum in bulk, they support a metallic edge excition robust against symmetry-respecting perturbation or an emergent fractional excitation in bulk. Following after fractional quantum Hall fluids, many topologically orderd phases, such as spin liquids and topological insulators, have been found and studied. Spin liquids are disordered phases of frustrated antiferromagnets and do not freeze and order even at the lowest temperature. They support an electrically neutral spin-1/2 excitation, which does not exist in a microscopic scale, with emergent dynamical gauge field in bulk and do not have an adiabatic path to a trivial paramagnet phase. The topological insulators are time-reversal symmetric band insulators which cannot evolve smoothly into a trivial insulator with the symmetry, and they have been intensely studied theoretically and experimetally for the last decade. Though the topological insulators are inherently non-interacting systems, they have exotic gapless edge and surface states which demonstrate many interesting quantum phenomena such as fractionalization, axionic electromagnetism, and half quantum Hall effect. In this thesis, we study various quantum phenomena of the topological phases, mainly of the topological insulator and its close relatives in which the physics of spin liquids has been merged into. As they are intrinsically quantum many-body states, the quantum field theory is an invaluable tool to explore the venue of the phases and will be used thorougly in this work. First, we present a topological field theoretic description of the topological insulators and translational symmetric Z2 spin liquid. We demostrate that abelian BF theory is an effective field theory for the two and three dimensional topological insulators. We show that the phenomenologies of BF theory are consistent with those of the topological insulators such as gapless edge and surface excitations and response functions to external gauge field. The same form of the BF theory is also applicable to study the certain classes of Z2 spin liquids with the gapless edge states. We closely examine the possible gapless edge theory of the BF theory and the translational symmetric spin liquids, and we show that the effective BF theories capture the bulk properties as well as the gapless edge spectrum of the spin liquid. Also we discuss the zero-temperature phase diagram of a thin film three dimensional topological insulators with two competing mass terms: time-reversal symmetric exciton condensation and time-reversal symmetry breaking Zeeman effect. There are two phases, quantum spin Hall phase and quantum anomalous Hall phase, and both phases support fractional excitations. We derive an effective topological field theory for the fractional excitations and examine the origin of the fractional excitations. Next we consider the relatives of the topological insulators. First we discuss the proximate symmetry broken phases of topological Mott insulator, a U(1) spin liquid with fermionic spinons in the topological insulator state. The topologically non-trivial band structure of the spinon generates the axion term for the gauge field, and we show that the axion term changes the nature of the confined phase of the spin liquid. Contrast to the conventional confined phase which has only a bond order, the confined phase of the spin liquid is in general a coexisting phase of the two different orders : a current order and a bond order. We then consider another relative of the topological insulator: a bosonic symmetry protected topological phase in two dimensional space, with both PSU(N) and time reversal symmetry. We develop an effective field theory for the phase, which is a SU(N) principal chiral model with a theta-term, and reveal the physics of the topological phase in detail.Finally we consider a topological semimetal, namely a Weyl semimetal. We first demonstrate that the Weyl semimetal can be realized in the magnetically doped topological band insulators. We explicitly derive the low energy theory of Weyl points from the general continuum Hamiltonian and tight-binding model of the topological insulators. Then we study superconducting instabilities of doped Weyl semimetals. We consider a minimal model for a Weyl semimetal and study the superconducting instabilities induced by a short-ranged attractive interaction. With the interaction, we find two competing states: a fully gapped finite-momentum pairing state and a nodal even-parity pairing state. We show that, in a mean field approximateion, the finite-momentum pairing state wins energetically against the usual even-parity paired state. We also show that exotic modes are localized at the full and half quantum vortices of the finite-momentum pairing state.
Symmetry-protected topological (SPT) phases host gapless surface states that are robust against weak, symmetry-preserving interactions. However, strong interactions can drive spontaneous breaking of the protecting symmetries, leading to gapped ordered surfaces. The phase boundary separating the gapless and ordered phases can host exotic conformal field theory states/universality classes typically not observed in conventional quantum phase transitions. We refer to these exotic quantum critical states as surface topological quantum critical points (sTQCPs). In this dissertation, we study attractive interaction-driven quantum criticality on the surface of three-dimensional topological insulators, a class of SPT phases hosting Dirac fermion surface states. When multiple Dirac cones are present, the interaction parameter space becomes intrinsically multi-dimensional, and the phase boundary separating the gapless and ordered phases forms a continuous manifold. Using renormalization group (RG) analysis, we show that in the limit of suppressed quantum fluctuations, the universality of this phase boundary is governed by conformal manifolds: continuous families of interacting conformal field theories characterized by exactly marginal operators. These manifolds are featureless in the sense that they host a single universality class. Their emergence is traced to continuous symmetries of the RG equations that are otherwise absent in the interaction Hamiltonian. We further demonstrate that higher-order quantum fluctuations break these emergent symmetries, fragmenting the conformal manifolds into isolated fixed points of varying infrared stabilities. Remarkably, we find that along the RG flow within the manifold, an EPR-like entanglement entropy in fermion flavor space always increases. Infrared stable conformal field theories correspond to maximally entangled interaction operators, while weakly entangled fixed points are unstable. These results establish the central role of entangled conformal operators and their entropy in shaping the universality classes of sTQCPs. Finally, we propose an experimentally accessible route to realizing surface quantum criticality by coupling a topological insulator surface to a metallic thin film. Strong electron–phonon interactions in the metal mediate effective attractive interactions between surface Dirac fermions, enhanced via a quantum-well resonance mechanism, providing a concrete pathway for accessing sTQCP physics in real materials.
Symmetry-protected topological (SPT) phases host gapless surface states that are robust against weak, symmetry-preserving interactions. However, strong interactions can drive spontaneous breaking of the protecting symmetries, leading to gapped ordered surfaces. The phase boundary separating the gapless and ordered phases can host exotic conformal field theory states/universality classes typically not observed in conventional quantum phase transitions. We refer to these exotic quantum critical states as surface topological quantum critical points (sTQCPs). In this dissertation, we study attractive interaction-driven quantum criticality on the surface of three-dimensional topological insulators, a class of SPT phases hosting Dirac fermion surface states. When multiple Dirac cones are present, the interaction parameter space becomes intrinsically multi-dimensional, and the phase boundary separating the gapless and ordered phases forms a continuous manifold. Using renormalization group (RG) analysis, we show that in the limit of suppressed quantum fluctuations, the universality of this phase boundary is governed by conformal manifolds: continuous families of interacting conformal field theories characterized by exactly marginal operators. These manifolds are featureless in the sense that they host a single universality class. Their emergence is traced to continuous symmetries of the RG equations that are otherwise absent in the interaction Hamiltonian. We further demonstrate that higher-order quantum fluctuations break these emergent symmetries, fragmenting the conformal manifolds into isolated fixed points of varying infrared stabilities. Remarkably, we find that along the RG flow within the manifold, an EPR-like entanglement entropy in fermion flavor space always increases. Infrared stable conformal field theories correspond to maximally entangled interaction operators, while weakly entangled fixed points are unstable. These results establish the central role of entangled conformal operators and their entropy in shaping the universality classes of sTQCPs. Finally, we propose an experimentally accessible route to realizing surface quantum criticality by coupling a topological insulator surface to a metallic thin film. Strong electron–phonon interactions in the metal mediate effective attractive interactions between surface Dirac fermions, enhanced via a quantum-well resonance mechanism, providing a concrete pathway for accessing sTQCP physics in real materials.
Quantum phases of matter can be classified into two kinds: gapped or gapless, depending on whether the equilibrium ground state is separated from excited states by a finite energy window or not in the thermodynamic limit. Topology has played an important role in the understanding of both gapped and gapless phases of matter. This thesis uncovers several robust physical properties of both kinds of systems which are largely unexplored before. \\ For gapped systems, we study topological orders, which are characterized by the existence of anyonic quasiparticles with fractional braiding statistics. In the first part, we introduce a family of Abelian quantum Hall insulators, at filling factors
ν=p/2q for bosonic systems and
ν=p/(p+2q) for fermionic systems, with
p and
q being two coprime integers. These states are termed the \textit{nondiagonal} quantum Hall states, which are constructed in a coupled wire model that exhibits an intimate relation to the nondiagonal conformal field theory and has a constrained pattern of motion for anyons. We unveil a nontrivial interplay between charge symmetry and translation symmetry in this system, and establish the nondiagonal quantum Hall states as symmetry-enriched topological orders. In the second part, we introduce a related model of interacting electrons, but in the absence of a strong quantizing magnetic field, known as the \textit{toric code insulator}. It is constructed from a two-dimensional array of strongly coupled one-dimensional topological superconductors, which feature electron fractionalization and possesses anyonic excitations described by the
Z2 topological order. The motion of anyons is again constrained, and different types of anyons can be related by the translation symmetry. Nondiagonal quantum Hall states and the toric code insulator are convincing examples that itinerant electronic systems can realize interesting symmetry-enriched topological orders.\\ For gapless quantum matters we focus on a particularly familiar class, the metals, but we uncover their topological aspects which were much less appreciated. Every metal is characterized by a Fermi surface of gapless excitations, and the Fermi surface surrounds a manifold in the momentum space, known as the Fermi sea. The ground state of a non-interacting metal, or a Fermi gas, consists of all single-electron states within the Fermi sea. The shape of the Fermi sea has a natural topological characterization in terms of the Euler characteristic
χF, and a fundamental question that we address in this thesis is: \textbf{does
χF imply any robust physical properties in metals, and how can they be measured?} In the third part of the thesis,
χF is connected to physically measurable quantities, such as equal-time density-density correlation functions, and multipartite entanglement measures, such as the mutual information. Particularly, for three dimensions, we establish that the 4-partite mutual information is a robust probe of the Fermi sea topology even in the presence of Fermi-liquid interactions. In the last part, we focus on two dimensions, and propose an experimentally accessible platform to probe
χF for any two-dimensional electron gas (2DEG) that respects time-reversal symmetry. The proposed experimental setup is a Josephson
π-junction, and we predict that the electrical transport in the direction \textit{along} the junction exhibits a current-rectification effect that is controlled by the underlying Fermi sea topology. Due to the relation to Andreev state transport, we term this effect \textit{topological Andreev rectification}. The size of rectification is quantified by a \textit{nonlocal rectified conductance}, which is established to attain an integer quantized value that equals
χF (in unit of
e2/h). Extensive numerical simulations are performed to support our theoretical proposal, and material platforms such as InAs heterostructures and graphene are identified as promising candidates to demonstrate this effect.
Topological phases of matter are usually realized in deconfined phases of gauge theories. In this context, confined phases with strongly fluctuating gauge fields seem to be irrelevant to the physics of topological phases. For example, the low-energy theory of the two-dimensional (2D) toric code model (i.e., the deconfined phase of
Z2 gauge theory) is a $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ Chern-Simons theory in which gauge charges (i.e.,
e and
m particles) are deconfined and the gauge fields are gapped, while the confined phase is topologically trivial. In this paper, we point out a route to constructing exotic three-dimensional (3D) gapped fermionic phases in a confining phase of a gauge theory. Starting from a parton construction with strongly fluctuating compact $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ gauge fields, we construct gapped phases of interacting fermions by condensing two linearly independent bosonic composite particles consisting of partons and $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ magnetic monopoles. This can be regarded as a 3D generalization of the 2D Bais-Slingerland condensation mechanism. Charge fractionalization results from a Debye-H\"uckel--type screening cloud formed by the condensed composite particles. Within our general framework, we explore two aspects of symmetry-enriched 3D Abelian topological phases. First, we construct a new fermionic state of matter with time-reversal symmetry and $\mathrm{\ensuremath{\Theta}}\ensuremath{\ne}\ensuremath{\pi}$, the fractional topological insulator. Second, we generalize the notion of anyonic symmetry of 2D Abelian topological phases to the charge-loop excitation symmetry (
Charles) of 3D Abelian topological phases. We show that line twist defects, which realize
Charles transformations, exhibit non-Abelian fusion properties.
The low energy effective field theories of (2 + 1) dimensional topological phases of matter provide powerful avenues for investigating entanglement in their ground states. In [1] the entanglement between distinct Abelian topological phases was investigated through Abelian Chern-Simons theories equipped with a set of topological boundary conditions (TBCs). In the present paper we extend the notion of a TBC to non-Abelian Chern-Simons theories, providing an effective description for a class of gapped interfaces across non-Abelian topological phases. These boundary conditions furnish a defining relation for the extended Hilbert space of the quantum theory and allow the calculation of entanglement directly in the gauge theory. Because we allow for trivial interfaces, this includes a generic construction of the extended Hilbert space in any (compact) Chern-Simons theory quantized on a Riemann surface. Additionally, this provides a constructive and principled definition for the Hilbert space of effective ground states of gapped phases of matter glued along gapped interfaces. Lastly, we describe a generalized notion of surgery, adding a powerful tool from topological field theory to the gapped interface toolbox.