We study the topological classification of parafermionic chains in the presence of a modified time-reversal symmetry that satisfies
T2=1. Such chains can be realized in one-dimensional structures embedded in fractionalized two-dimensional states of matter, e.g., at the edges of a fractional quantum spin Hall system, where counterpropagating modes may be gapped either by backscattering or by coupling to a superconductor. In the absence of any additional symmetries, a chain of
Zm parafermions can belong to one of several distinct phases. We find that when the modified time-reversal symmetry is imposed, the classification becomes richer. If
m is odd, each of the phases splits into two subclasses. We identify the symmetry-protected phase as a Haldane phase that carries a Kramers doublet at each end. When
m is even, each phase splits into four subclasses. The origin of this split is in the emergent Majorana fermions associated with even values of
m. We demonstrate the appearance of such emergent Majorana zero modes in a system where the constituent particles are either fermions or bosons.
Based on the recently established parafermionic matrix product states, we study the classification of one-dimensional gapped phases of parafermions with time-reversal (TR) symmetry satisfying
T2=1. Without extra symmetry, it has been found that
Zp parafermionic gapped phases can be classified as topological phases, spontaneous symmetry breaking (SSB) phases, and a trivial phase, which are uniquely labeled by the divisors
n of
p. In the presence of TR symmetry, however, the enriched classification is characterized by three indices
n, $\ensuremath{\kappa}$, and $\ensuremath{\mu}$, where $\ensuremath{\kappa}\ensuremath{\in}{\mathbb{Z}}_{2}$ denotes the linear or projective TR actions on the edges, and $\ensuremath{\mu}\ensuremath{\in}{\mathbb{Z}}_{2}$ indicates the commutation relations between the TR and (fractionalized) charge operator. For the
Zr-symmetric parafermionic ground states, where
r=p for trivial or topological phases, and
r=p/n for SSB phases, each original gapped phase with odd
r is divided into two phases, while each phase with even
r is further separated into four phases. The gapped parafermionic phases with the TR symmetry include the symmetry protected topological phases, symmetry enriched topological phases, and the SSB coexisting symmetry protected topological phases. From analyzing the structures and symmetries of their reduced density matrices of these resulting topological phases, we can obtain the topologically protected degeneracies of their entanglement spectra.
The concept of topology in condensed matter physics has led to the discovery of rich and exotic physics in recent years. Especially when strong correlations are included, phenomenons such as fractionalization and anyonic particle statistics can arise. In this thesis, we study several systems hosting topological phases of interacting fermions. In the first part, we consider one-dimensional systems of parafermions, which are generalizations of Majorana fermions, in the presence of a Z_N charge symmetry. We classify the symmetry-protected topological (SPT) phases that can occur in these systems using the projective representations of the symmetries and find a finite number of distinct phases depending on the prime factorization of N. The different phases exhibit characteristic degeneracies in their entanglement spectrum (ES). Apart from these SPT phases, we report the occurrence of parafermion condensate phases for certain values of N. When including an additional Z_N symmetry, we find a non-Abelian group structure under the addition of phases. In the second part of the thesis, we focus on two-dimensional lattice models of spinless fermions. First, we demonstrate the detection of a fractional Chern insulator (FCI) phase in the Haldane honeycomb model on an infinite cylinder by means of the density-matrix renormalization group (DMRG). We report the calculation of several quantities characterizing the topological order of the state, i.e., (i)~the Hall conductivity, (ii)~the spectral flow and level counting in the ES, (iii)~the topological entanglement entropy, and (iv)~the charge and topological spin of the quasiparticles. Since we have access to sufficiently large system sizes without band projection with DMRG, we are in addition able to investigate the transition from a metal to the FCI at small interactions which we find to be of first order. In a further study, we consider a time-reversal symmetric model on the honeycomb lattice where a Chern insulator (CI) induced by next-nearest neighbor interactions has been predicted by mean field theory. However, various subsequent studies challenged this picture and it was still unclear whether the CI would survive quantum fluctuations. We therefore map out the phase diagram of the model as a function of the interactions on an infinite cylinder with DMRG and find evidence for the absence of the CI phase. However, we report the detection of two novel charge-ordered phases and corroborate the existence of the remaining phases that had been predicted in mean field theory. Furthermore, we characterize the transitions between the various phases by studying the behavior of correlation length and entanglement entropy at the phase boundaries. Finally, we develop an improvement to the DMRG algorithm for fermionic lattice models on cylinders. By using a real space representation in the direction along the cylinder and a real space representation in the perpendicular direction, we are able to use the momentum around the cylinder as conserved quantity to reduce computational costs. We benchmark the method by studying the interacting Hofstadter model and report a considerable speedup in computation time and a severely reduced memory usage.
We study symmetries and defects of a wide class of two-dimensional Abelian topological phases characterized by Lie algebras. We formulate the symmetry group of all Abelian topological field theories. The symmetries relabel quasiparticles (or anyons) but leave exchange and braiding statistics unchanged. Within the class of
ADE phases, in particular, these anyonic symmetries have a natural origin from the Lie algebra. We classify one-dimensional gapped phases along the interface between identical topological states according to symmetries. This classification also applies to gapped edges of a wide range of fractional quantum spin Hall (QSH) states. We show that the edge states of the
ADE QSH systems can be gapped even in the presence of time-reversal and charge-conservation symmetries. We distinguish topological point defects according to anyonic symmetries and bound quasiparticles. Although in an Abelian system, they surprisingly exhibit non-Abelian fractional Majorana-like characteristics from their fusion behavior.
Time-reversal invariant three-dimensional topological insulators can be defined fundamentally by a topological field theory with a quantized axion angle $\ensuremath{\theta}$ of 0 or $\ensuremath{\pi}$. It was recently shown that fractional quantized values of $\ensuremath{\theta}$ are consistent with time-reversal invariance if deconfined, gapped, fractionally charged bulk excitations appear in the low-energy spectrum due to strong correlation effects, leading to the concept of a fractional topological insulator. These fractionally charged excitations are coupled to emergent gauge fields, which ensure that the microscopic degrees of freedom, the original electrons, are gauge-invariant objects. A first step towards the construction of microscopic models of fractional topological insulators is to understand the nature of these emergent gauge theories and their corresponding phases. In this work, we show that low-energy effective gauge theories of both Abelian or non-Abelian type are consistent with a fractional quantized axion angle if they admit a Coulomb phase or a Higgs phase with gauge group broken down to a discrete subgroup. The Coulomb phases support gapless but electrically neutral bulk excitations while the Higgs phases are fully gapped. The Higgs and non-Abelian Coulomb phases exhibit multiple ground states on boundaryless spatial three-manifolds with nontrivial first homology, while the Abelian Coulomb phase has a unique ground state. The ground-state degeneracy receives an additional contribution on manifolds with boundary due to the induced boundary Chern-Simons term.
We explore a scenario where local interactions form one-dimensional gapped interfaces between a pair of distinct chiral two-dimensional topological states-referred to as phases 1 and 2-such that each gapped region terminates at a domain wall separating the chiral gapless edge states of these phases. We show that this type of T junction supports pointlike fractionalized excitations obeying parafermion statistics, thus implying that the one-dimensional gapped interface forms an effective topological parafermionic wire possessing a nontrivial ground state degeneracy. The physical properties of the anyon condensate that gives rise to the gapped interface are investigated. Remarkably, this condensate causes the gapped interface to behave as a type of anyon "Andreev reflector" in the bulk, whereby anyons from one phase, upon hitting the interface, can be transformed into a combination of reflected anyons and outgoing anyons from the other phase. Thus, we conclude that while different topological orders can be connected via gapped interfaces, the interfaces are themselves topological.