In this thesis, I will first derive and study the effective field theories of isotropic-nematic quantum phase transitions of Quantum states. The low-energy theory of the nematic field has z=2 dynamics due to a Berry phase of the order parameter, which is related to the Hall viscosity in parity and time-reversal-symmetry (TRS) broken states. The vortex of the nematic field, which is physically a disclination, creates a nonzero geometry curvature in the disclination core. The leading coupling between the nematic field and gauge field includes a Wen-Zee term which links the geometry curvature with the gauge theory. In the second part of this thesis, I investigate the geometry related issues in Weyl semimetals and SPT states, and explore the novel character of geometry defect in SPT states inherited from the topological nature of manybody system. In addition, I would introduce a general way to induce topological phase transition via decorated defect condensate. In the final part of this thesis, I begin with the bilayer Half-filled Landau Level system where the two composite Fermi surface acquires interlayer coherence and forms bonding/anti-bonding composite fermi sea. The corresponding interlayer coherent composite Fermi liquid(ICCFL) phase provides a straightforward landscape to verify the Dirac nature in Son's theory and extract the hidden Berry phase structure of the composite Fermi surface. The ICCFL phase contains two Fermi surfaces which are detached in most regions but adhesive at two hot spots. Such nematic structure is a consequence of the Berry phase encoded in the Dirac Fermi surface which is absent in HLR theory. Due to the nematicity in ICCFL, the system supports half-quantum vortex with deconfined $\frac{\pi}{2}$ gauge flux and the phase transition toward ICCFL contains a Lifshitz criticality with $z=3$ dynamical exponent. In addition, the exciton order parameter carries topological spin number so the ICCFL contains a unique Wen-Zee term which connects EM response with the background geometry curvature.
In this thesis, I will first derive and study the effective field theories of isotropic-nematic quantum phase transitions of Quantum states. The low-energy theory of the nematic field has z=2 dynamics due to a Berry phase of the order parameter, which is related to the Hall viscosity in parity and time-reversal-symmetry (TRS) broken states. The vortex of the nematic field, which is physically a disclination, creates a nonzero geometry curvature in the disclination core. The leading coupling between the nematic field and gauge field includes a Wen-Zee term which links the geometry curvature with the gauge theory. In the second part of this thesis, I investigate the geometry related issues in Weyl semimetals and SPT states, and explore the novel character of geometry defect in SPT states inherited from the topological nature of manybody system. In addition, I would introduce a general way to induce topological phase transition via decorated defect condensate. In the final part of this thesis, I begin with the bilayer Half-filled Landau Level system where the two composite Fermi surface acquires interlayer coherence and forms bonding/anti-bonding composite fermi sea. The corresponding interlayer coherent composite Fermi liquid(ICCFL) phase provides a straightforward landscape to verify the Dirac nature in Son's theory and extract the hidden Berry phase structure of the composite Fermi surface. The ICCFL phase contains two Fermi surfaces which are detached in most regions but adhesive at two hot spots. Such nematic structure is a consequence of the Berry phase encoded in the Dirac Fermi surface which is absent in HLR theory. Due to the nematicity in ICCFL, the system supports half-quantum vortex with deconfined 2π gauge flux and the phase transition toward ICCFL contains a Lifshitz criticality with z=3 dynamical exponent. In addition, the exciton order parameter carries topological spin number so the ICCFL contains a unique Wen-Zee term which connects EM response with the background geometry curvature.
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.
"This thesis is devoted to the investigation of the quantum liquid crystal phases in strongly correlated electronic systems. Such phases are characterized by their partially broken spatial symmetries and are observed in various strongly correlated systems as being summarized in Chapter 1. In quantum liquid crystal phases formed by fermions, behaviors beyond Landau's theory of the Fermi liquids appear naturally, signed by the absence of well-defined quasiparticle excitations. Although quantum liquid crystal phases often involve strong-coupling physics, the associated symmetry analysis can often be carried out also in a relatively weak coupling regime and this is the main approach taken in this thesis. Two different quantum liquid crystal phases, the nematic state and the smectic state, are studied in Chapters 2 and 3 respectively, by investigating various instabilities of a Landau Fermi liquid. A nematic phase is a state that breaks the rotational symmetry spontaneously but preserves the translational symmetries. It can be reached by the Pomeranchuk instability of a Landau Fermi liquid. The Pomeranchuk instability in the charge (spin-singlet) sector of a Fermi liquid was studied previously within a random phase approximation and by a high-dimensional bosonization method. In Chapter 2, we concentrate on the Pomeranchuk instability in the spin-triplet sector, and two different classes of ordered phases are discovered, dubbed the a and ,8-phases by analogy to the superfluid 3He-A and B-phases. The Fermi surfaces in the a-phases exhibit spontaneous anisotropic distortions, while in the ,8-phases the Fermi surfaces remain circular or spherical with topologically non-trivial spin configurations in momentum space. The behaviors of the low energy bosonic excitations are studied both in the order phases (a and .B) and at the quantum critical point separating the ordered phases from the Fermi liquid phase. In Chapter 3, the smectic phase and the quantum phase transition between a metallic nematic state and a metallic smectic phase are investigated using the random phase approximation, by considering the formation of a unidirectional charge density wave in an electronic nematic state. The associated quantum critical point is investigated within the Hertz-Millis approach. Non-Fermi liquid behaviors are discovered both at the quantum phase transition and in the smectic ordered phase. The theory we studied here provides a simple description for the so-called ""fluctuating stripe"" phase in cuprates. Similar to its classical counterpart, the phase transition between the nematic and smectic phases are described by a ""gauge-like"" theory as required by the rotational invariance in close analogy with the McMillan-deGennes theory of the smectic-nematic phase transition in classical liquid crystals. However, unlike the classical McMillandeGennes theory, in which the coupling between the ""gauge-like"" nematic fluctuations and the ""matter-like"" smectic fields is relevant, in the sense of renormalization group, this coupling is irrelevant at the quantum phase transition. This property enables the study of low energy fluctuations and the fate of fermions near this quantum critical point. In Chapter 4, quantum liquid crystal phases with a broken time-reversal symmetry are investigated. We start by studying the general scenario of timereversal symmetry breaking phases in the absence of magnetic ordering in systems with long-lived low-energy fermionic quasiparticle excitations in two dimensions. Using a Berry phase approach, we classified possible time-reversal symmetry breaking phases by the accompanying spatial symmetry breaking patterns. Two of the simplest situations are referred to as the type I state, which breaks simultaneously the space-inversion symmetry and the time-reversal symmetry, and the type I I state, in which the chiral symmetry is broken in addition to the breaking of the time-reversal symmetry. Although the starting point is general and has little relation with the quantum liquid crystal phases, both type I and type II states defined above are realized by (generalized) Pomeranchuk instabilities. In fact , a type I I state is usually a nematic state which also breaks the time-reversal symmetry. Interestingly, the type II states also shows spontaneously generated Hall effect even in the absence of a magnetic field, which is known as the spontaneous anomalous Hall effect. In Chapter 5, we investigate the stability of a quadratic band-crossing point in two-dimensional fermionic systems that provide an explicit example, within well controlled approximations, for the nematic ordering discussed in Chapter 2 and spontaneous anomalous Hall effect discussed in Chapter 4. At the non-interacting level, we show that a quadratic band-crossing point exists and is topologically stable when the Berry flux of this crossing point is ±27r and the point symmetry group at this point has either fourfold or sixfold rotational symmetries. Strikingly, we find that this would-be topologically stable quadratic band-crossing is marginally unstable against short-range interactions: a quadratic band-crossing point in two dimensions is unstable against shortrange repulsion, no matter how weak! Four ordered phases are found in this system, the quantum anomalous Hall state, the quantum spin Hall phase, the nematic phase, and the nematic-spin-nematic phase."
The topological defects of Spin(n+1) nematics in two spatial dimensions, known as disclinations, are characterized by the ${\ensuremath{\pi}}_{1}(\mathbb{R}{P}^{n})={\text{Z}}_{2}$ homotopy group for $n\ensuremath{\ge}2$. We argue that incompressible quantum liquids of disclinations can exist as stable low-temperature phases and host composite quasiparticles which combine a fractional amount of fundamental Z2 charge with a unit of topological charge. The fourfold topological ground state degeneracy on a torus admits a fermionic or semionic quasiparticle exchange statistics. The topological nontriviality of these states is visible in the existence of protected gapless edge modes. While the fermionic nematic and gapped Z2 spin liquids have equivalent topological orders, they are still thermodynamically distinct due to having different edge modes, in analogy to the topologically nontrivial and trivial states of quantum spin-Hall systems. The analysis proceeds by recasting the Z2 gauge theory of spin nematics as a continuum limit theory with a larger gauge structure. Nematic fractionalization parallels that of a quantum Hall liquid, but the large gauge symmetry restores the time-reversal symmetry and restricts the quasiparticle fusion rules and statistics. The conclusions from field theory analysis are complemented with the construction of plausible host microscopic models.
In this thesis, we investigate several topics pertaining to emergent collective quantum phenomena in the domain of correlated fermions, using the quantum Monte Carlo method. They display exotic low temperature phases as well as phase transitions which are beyond the Landau–Ginzburg theory. The interplay between three key points is crucial for us: fermion statistics, many body effects and topology. We highlight the following several achievements: 1. Successful modeling of continuum field theories with lattice Hamiltonians, 2. their sign-problem-free Monte Carlo simulations of these models, 3. and numerical results beyond mean field descriptions. First, we consider a model of Dirac fermions with a spin rotational invariant inter- action term that dynamically generates a quantum spin Hall insulator. Surprisingly, an s-wave superconducting phase emerges due to the condensation of topological de- fects of the spin Hall order parameter. When particle-hole symmetry is present, the phase transition between the topological insulator and the superconducting phase is an example of a deconfined quantum critical point(DQCP). Although its low energy effec- tive field theory is purely bosonic, the exact conservation law of the skyrmion number operator rules out the possibility of realizing this critical point in lattice boson models. This work is published in Ref. [1]. Second, we dope the dynamically generated quantum spin Hall insulator mentioned above. Hence it is described by a field theory without Lorentz invariance due to the lack of particle-hole symmetry. This sheds light on the extremely hot topic of twisted bilayergraphene: Why is superconductivity generated when the repulsive Coulomb interaction is much stronger than the electron-phonon coupling energy scale? In our case, Cooper pairs come from the topological skyrmion defects of the spin current order parameter, which are charged. Remarkably, the nature of the phase transition is highly non-mean-field-like: one is not allowed to simply view pairs of electrons as single bosons in a superfluid-Mott insulator transition, since the spin-current order parameter can not be ignored. Again, due to the aforementioned skyrmions, the two order parameters are intertwined: One phase transition occurs between the two symmetry breaking states. This work is summarized in Ref. [2]. Third, we investigate the 2 + 1 dimensional O(5) nonlinear sigma model with a topological Wess-Zumino-Witten term. Remarkably, we are able to perform Monte Carlo calculations with a UV cutoff given by the Dirac Landau level quantization. It is a successful example of simulating a continuous field theory without lattice regularization which leads to an additional symmetry breaking. The Dirac background and the five anti-commuting Dirac mass terms naturally introduce the picture of a non-trivial Berry phase contribution in the parameter space of the five component order parameter. Using the finite size scaling method given by the flux quantization, we find a stable critical phase in the low stiffness region of the sigma model. This is a candidate ground state of DQCP when the O(5) symmetry breaking terms are irrelevant at the critical point. Again, it has a bosonic low energy field theory which is seemingly unable to be realized in pure boson Hamiltonians. This work is summarized in Ref. [3].
This dissertation discusses some applications of conformal field theories (CFTs) in topological phases of matter. The first part is devoted to a discussion of coupled wire constructions of some novel quantum Hall systems. Through a theoretical coupled wire model, we construct strongly correlated electronic integer quantum Hall states with filling factor 16. The edge state is a bosonic chiral (E8)1 CFT, which is closely related to topological paramagnets in (3+1)d. As a distinguishing feature, these states support electric and thermal Hall transport violating the Wiedemann-Franz law. We further construct two descendant non-Abelian quantum Hall states at filling 8, each carrying bosonic chiral (G2)1 or (F4)1 edge theories, and hosting Fibonacci anyonic excitations in the bulk. Finally, we discover a new notion of particle-hole conjugation based on the E8 state that relates the G2 and F4 Fibonacci states, which is reminiscent of similar physics in half-filled Landau level. The second part is focused on the surface topological orders of 3D bulk topological systems. Symmetry protected and symmetry-enriched topological (SPT/SET) phases in three dimensions are quantum systems that support non-trivial two-dimensional surface states. These surface states develop finite excitation energy gaps when the relevant symmetries are broken. On the other hand, one-dimensional gapless modes can populate along interfaces that separate adjacent gapped surface domains with distinct symmetry-breaking orders. A surface strip pattern in general reduces the low-energy SPT/SET surface degrees of freedom onto a 2D array of gapless 1D channels. These channels can be coupled to one another by quasiparticle tunneling, and these inter-wire interactions collectively provide an effective description of the surface state. In this part, we study a general class of symmetry-preserving or breaking SPT/SET surface states that admit finite excitation energy gaps and Abelian topological orders via the coupled wire construction. In particular, we focus on the prototype Abelian surface topological orders that fall under the ADE classification of simply-laced Lie algebras. We also elaborate on the emergent symmetry and duality properties of the coupled wire models. The third part is to discuss the relation between the conformal boundary state and (2+1)d SPT phases. We propose a diagnostic tool for detecting nontrivial symmetry-protected topological (SPT) phases protected by a symmetry group G in 2 + 1 dimensions. Our method is based on directly studying the 1 + 1-dimensional anomalous edge conformal field theory (CFT) of SPT phases. We claim that if the CFT is the edge theory of an SPT phase, then there must be an obstruction to cutting it open. This obstruction manifests as the non-existence of boundary states in the CFT that preserves both the conformal symmetry and the global symmetry G. We discuss the relation between edgeability and gappability in the presence of G. We study several examples including time-reversal symmetric topological insulators, ZN symmetric bosonic SPT phases, and Z2 × Z2 symmetric topological superconductors.
The Nobel Prize in physics of year 2016 has been awarded to David J. Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz for their theoretical discoveries of topological phase transitions and topological phase of matter. This paper starts with a brief introduction to their groundbreaking works made in 1970s–1980s, including the KT phase transition due to topological defects, vortex and antivortex pairs, in two-dimensional superfluid or superconductor, the TKNN number identifying the integer quantum Hall state, as well as the Haldane model to realize quantum Hall effect without external magnetic field in crystal lattice. Their insightful findings have revealed a new quantum state of matter, which cannot be described by local order parameter within Landau symmetry-breaking theory but by some global topological invariant of electronic states. In topology, a branch of mathematics, topological invariant describes the preserved properties of an object under continuous deformations. The combination of topology with the state of matter is a breakthrough in condensed matter physics. The band topology theory based on Berry phase is introduced in order to understand the origin of topology of the electronic states in solids. But only in last ten years, their pioneering works have been rediscovered and greatly extended. Around 2005, the time-reversal symmetry protected topological state identified by Z2 number has been discovered theoretically and soon it is realized in material called as Z2 topological insulator. Quickly, the concept of symmetry protected topological states has been generalized from time-reversal symmetry to all the space group symmetries, from electronic states to those in photonic crystal, phonon spectrum and cold atoms, and even from the insulators to metals. The first experimentally realized topological metal is Dirac semimetal, a three-dimensional version of graphene, with massless Dirac equation describing the low energy excitation around Fermi level. The discovery of Weyl semimetal in TaAs family has stimulated lots of studies on the chiral Weyl fermions, which have not been found in vacuum but observed as quasiparticles in Weyl semimetals. More topological semimetals, such as node-line semimetal and multiple-degenerate nodal point semimetal, have been also proposed. The distinguished property of topological states of matter is its stability in both bulk and boundary states against the perturbation preserving the protecting symmetry. This is expected for potential applications in electronic devices. Therefore, three Laureates of Nobel Prize in physics of 2016 revealed the secrets of exotic matter and opened the door to topological states.