The non-chiral edge excitations of quantum spin Hall systems and topological insu-lators are described by means of their partition function. The stability of topological phases protected by time-reversal symmetry is rediscussed in this context and put in relation with the existence of discrete anomalies and the lack of modular invariance of the partition function. The Z2 characterization of stable topological insulators is extended to systems with interacting and non-Abelian edge excitations. ar X iv
The non-chiral edge excitations of quantum spin Hall systems and topological insu-lators are described by means of their partition function. The stability of topological phases protected by time-reversal symmetry is rediscussed in this context and put in relation with the existence of discrete anomalies and the lack of modular invariance of the partition function. The Z2 characterization of stable topological insulators is extended to systems with interacting and non-Abelian edge excitations. ar X iv
Chiral-antichiral pairs of non-Abelian Hall states, like the Pfaffian, Read-Rezayi and NASS states, can be used to model two-dimensional time-reversal invariant topological insulators. Their stability was shown to be associated to the presence of a Z2 anomaly and characterized by the same Z2 index introduced for free fermion and Abelian systems. In this work, we continue the stability analysis by providing the form of time-reversal invariant interactions that gap the non-Abelian edge excitations. Our approach is based on the description of non-Abelian states as projections of corresponding “parent ” Abelian states. ar X iv
Chiral–antichiral pairs of non-Abelian Hall states, like the Pfaffian, Read–Rezayi and NASS states, can be used to model two-dimensional time-reversal invariant topological insulators. Their stability was shown to be associated to the presence of a anomaly and characterized by the same index introduced for free fermion and Abelian systems. In this work, we continue the stability analysis by providing the form of time-reversal invariant interactions that gap the non-Abelian edge excitations. Our approach is based on the description of non-Abelian states as projections of corresponding 'parent' Abelian states.
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.
We show that the decorated honeycomb (star) lattice supports a number of topological insulating phases with a nontrivial Z2 invariant and time-reversal symmetry protected gapless edge modes. We investigate the stability of these phases with respect to various symmetry-breaking perturbations and demonstrate the connection to the recent discovery of an exactly solvable S=1/2 chiral spin-liquid model [Yao and Kivelson, Phys. Rev. Lett. 99, 247203 (2007)] with non-Abelian and Abelian excitations on the same lattice at strong interaction strength. Our work highlights the relationship between noninteracting topological band insulators and strongly interacting topologically ordered spin systems, and points to promising avenues for enlarging the number of known examples of both.
Symmetry protected topological (SPT) phases are characterized by robust boundary features, which do not disappear unless passing through a phase transition. These boundary features can be quantified by a topological invariant which, in some cases, is related to a physical quantity, such as the spin conductivity for the quantum spin Hall insulators. In other cases, the boundary features give rise to new physics, such as the Majorana fermion. In all cases the boundary features can be analyzed with the help of an entanglement spectrum and their robustness make them promising candidates for storing quantum information. The topological invariant characterizing SPT phases is strictly only invariant under deformations which respect a certain \nsymmetry. For example, the boundary currents of the quantum spin Hall insulator are only robust against non-magnetic, i.e. time-reversal invariant, impurities. In this thesis we study the SPT phases of spin chains. \nAs a result of our work we find a topological invariant for SPT phases of spin chains which are protected by continuous symmetries. By means of a non-local order parameter we find a way to extract this invariant from the ground state wave function of the system. Using density-matrix-renormalization-group techniques we verify that this invariant is a tool to detect transitions between different topological phases. We find a non-local transformation that maps SPT phases to conventional phases characterized by a local order parameter. This transformation suggests an analogy between topological phases and conventional phases and thus give a deeper understanding of the role of topology in spin systems.
We consider nonchiral symmetry-protected topological phases of matter in two spatial dimensions protected by a discrete symmetry such as ZK or ${\mathbb{Z}}_{K}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{K}$ symmetry. We argue that modular invariance/noninvariance of the partition function of the one-dimensional edge theory can be used to diagnose whether, by adding a suitable potential, the edge theory can be gapped or not without breaking the symmetry. By taking bosonic phases described by Chern-Simons K-matrix theories and fermionic phases relevant to topological superconductors as an example, we demonstrate explicitly that when the modular invariance is achieved, we can construct an interaction potential that is consistent with the symmetry and can completely gap out the edge state.