We study unusual gapped topological phases where they admit \mathbb{Z}_N ℤ N fractional excitations in the same manner as topologically ordered phases, yet their ground state degeneracy depends on the local geometry of the system. Placing such phases on 2D lattice, composed of an arbitrary connected graph and 1D line, we find that the fusion rules of quasiparticle excitations are described by the Laplacian of the graph and that the number of superselection sectors is related to the kernel of the Laplacian. Based on this analysis, we further show that the ground state degeneracy is given by \bigl[N\times \prod_{i}\text{gcd}(N, p_i)\bigr]^2 [ N × ∏ i gcd ( N , p i ) ] 2 , where p_i p i ’s are invariant factors of the Laplacian that are greater than one and gcd stands for the greatest common divisor. We also discuss braiding statistics between quasiparticle excitations.
Anisotropic higher rank ZN topological phases on graphs
We study unusual gapped topological phases where they admit \mathbb{Z}N ℤ N fractional excitations in the same manner as topologically ordered phases, yet their ground state degeneracy depends on the local geometry of the system. Placing such phases on 2D lattice, composed of an arbitrary connected graph and 1D line, we find that the fusion rules of quasiparticle excitations are described by the Laplacian of the graph and that the number of superselection sectors is related to the kernel of the Laplacian. Based on this analysis, we further show that the ground state degeneracy is given by \bigl[N\times \prod{i}\text{gcd}(N, p_i)\bigr]^2 [ N × ∏ i gcd ( N , p i ) ] 2 , where p_i p i ’s are invariant factors of the Laplacian that are greater than one and gcd stands for the greatest common divisor. We also discuss braiding statistics between quasiparticle excitations.
Motivated by recent interest in fracton topological phases, we explore the interplay between gapped 2D ZN topological phases which admit fractional excitations with restricted mobility and geometry of the lattice on which such phases are placed. We investigate the properties of the phases in a geometric context---graph theory. By placing the phases on a 2D lattice consisting of two arbitrary connected graphs, ${G}_{x}\ensuremath{\boxtimes}{G}_{y}$, we study the behavior of fractional excitations of the phases. We derive the formula of the ground-state degeneracy of the phases, which depends on invariant factors of the Laplacian.
Topological order in two-dimensions can be described in terms of deconfined quasiparticle excitations—anyons—and their braiding statistics. However, it has recently been realized that this data does not completely describe the situation in the presence of an unbroken global symmetry. In this case, there can be multiple distinct quantum phases with the same anyons and statistics, but with different patterns of symmetry fractionalization—termed symmetry enriched topological order. When the global symmetry group G , which we take to be discrete, does not change topological superselection sectors—i.e. does not change one type of anyon into a different type of anyon—one can imagine a local version of the action of G around each anyon. This leads to projective representations and a group cohomology description of symmetry fractionalization, with the second cohomology group H 2 ( G , A a b e l i a n ) being the relevant group. In this paper, we treat the general case of a symmetry group G possibly permuting anyon types. We show that despite the lack of a local action of G , one can still make sense of a so-called twisted group cohomology description of symmetry fractionalization, and show how this data is encoded in the associativity of fusion rules of the extrinsic ‘twist’ defects of the symmetry. Furthermore, building on work of Hermele (2014 Phys. Rev. B 90 184418 ), we construct a wide class of exactly-solvable models which exhibit this twisted symmetry fractionalization, and connect them to our formal framework.
2+1d topological phases are well characterized by the fusion rules and braiding/exchange statistics of fractional point excitations. In 4+1d, some topological phases contain only fractional loop excitations. What kind of loop statistics exist? We study the 4+1d gauge theory with 2-form \mathbb{Z}_2 ℤ 2 gauge field (the loop-only toric code) and find that while braiding statistics between two different types of loops can be nontrivial, the self “exchange” statistics are all trivial. In particular, we show that the electric, magnetic, and dyonic loop excitations in the 4+1d toric code are not distinguished by their self-statistics. They tunnel into each other across 3+1d invertible domain walls which in turn give explicit unitary circuits that map the loop excitations into each other. The SL(2, \mathbb{Z}_2 ℤ 2 ) symmetry that permutes the loops, however, cannot be consistently gauged and we discuss the associated obstruction in the process. Moreover, we discuss a gapless boundary condition dubbed the “fractional Maxwell theory” and show how it can be Higgsed into gapped boundary conditions. We also discuss the generalization of these results from the \mathbb{Z}_2 ℤ 2 gauge group to \mathbb{Z}_N ℤ N .
Topological phases of matter are characterised by long-range entanglement between their constituent degrees of freedom, which allows them to host excitations with non-trivial exchange statistics. While topological phases in 2+1d are relatively well studied, less is known about their 3+1d counterparts. In this thesis, we examine features of these higher-dimensional phases in a tractable example model, based on a generalisation of lattice gauge theory called higher lattice gauge theory. As well as hosting point-like excitations, the model supports loop-like excitations with non-trivial loop-loop and point-loop braiding statistics. We explicitly construct operators to produce and move these excitations, and use these to find the loop-loop and point-loop braiding relations. These creation operators also reveal that some of the excitations are confined, costing energy to separate, while others are condensed and can be produced locally. This is discussed in the context of condensation-confinement transitions between different cases of this model. We also explain a method for measuring topological charge and use explicit measurement operators to re-derive a relationship between the number of charges measured by a 2-torus and the ground-state degeneracy of the higher lattice gauge theory model on the 3-torus. From these measurement operators, we can see that the ground state degeneracy on the 3-torus is related to the number of types of linked loop-like excitations.
Topological phases are gapped quantum phases of matter classified beyond the paradigm of Landau's symmetry breaking theory. In particular, topologically ordered phases are topological phases with topological order. Unlike the conventional symmetry-breaking order described by local order parameter, topological order is characterized by the excitations. For example, in (2+1)D topologically ordered phases, there are particle-like excitations called anyons. In contrast, in (3+1)D topologically ordered phases, there are both particle-like and loop-like excitations. These excitations exhibit braiding statistics, meaning that carrying one excitation adiabatically around another transform the ground state by a phase (or generally a unitary transformation). The braiding statistics of these excitations serves as a crucial ingredient in characterizing the topological order in these phases. One important aspect is to understand the braiding statistics in topologically ordered phases. While the anyonic braiding statistics in (2+1)D topologically ordered phases is well established, the braiding statistics in (3+1)D topologically ordered phases is far from clear. In (3+1)D, the most well-understood braiding statistics is the particle-loop braiding statistics, where a particle winds around a loop excitation acquires an Aharonov Bohm phase. In the thesis, we introduce a new braiding statistics, which we called the Borromean-Rings braiding statistics. It is a particle-loop-loop braiding statistics where a particle travels around two unlinked loops acquires a braiding phase. Unlike the particle-loop braiding, the particle trajectory in the Borromean-Rings braiding is not linked with any of the two loops. Another important aspect is to probe the topological order in topologically ordered phases. One usual way to probe the topological order is to examine the properties of the phases on various orientable surfaces. For example, for (2+1)D topologically ordered phases, the number of degenerate ground states on Torus counts the number of anyons. Also, ground state transformation under twisting the torus encodes the braiding data. In the thesis, we attempt to probe the topological order by putting the phases on non-orientable surfaces. We find that the ground state degeneracy and the ground state transformation reveals a special kind of anyons, called the parity symmetric anyons, which show a lost of orientation in their braiding statistics. They in turn distinguish different topological order. Even without topological order, there can be non-trivial topological phases in the presence of global symmetry. These topological phases are called the symmetry protected topological phases. For example, (3+1)D fermionic matter with time reversal symmetry and charge conservation symmetry has a non-trivial phase, namely the topological insulators. While the bulk of the topological insulators behaves like trivial insulators, the surface is anomalous with odd number of massless dirac fermions. While most of the understanding of topological insulators is developed upon the free fermion picture, such phase survives under interaction. In the thesis, we derive hydrodynamic field theory for the topological insulators by functional bosonization. Such formulation is robust against interaction. In addition, by suitably incorporating electron interactions, we predict new topological phases, called the fractional topological insulators. Besides, the surface theory of topological insulators also justify the famous duality between free massless dirac fermion and QED_3.
We construct generalized Hofstadter models that possess ``color-entangled'' flat bands and study interacting many-body states in such bands. For a system with periodic boundary conditions and appropriate interactions, there exist gapped states at certain filling factors for which the ground-state degeneracy depends on the number of unit cells along one particular direction. This puzzling observation can be understood intuitively by mapping our model to a single-layer or a multilayer system for a given lattice configuration. We discuss the relation between these results and the previously proposed ``topological nematic states,'' in which lattice dislocations have non-Abelian braiding statistics. Our study also provides a systematic way of stabilizing various fractional topological states in C>1 flat bands and provides some hints on how to realize such states in experiments.