We consider universal statistical properties of systems that are characterized by phase states with macroscopic degeneracy of the ground state. A possible topological order in such systems is described by non-linear discrete equations. We focus on the discrete equations which take place in the case of generalized exclusion principle statistics. We show that their exact solutions are quantum dimensions of the irreducible representations of certain quantum group. These solutions provide an example of the point where the generalized exclusion principle statistics and braid statistics meet each other. We propose a procedure to construct the quantum dimer models by means of projection of the knotted field configurations that involved braiding features of one-dimensional topology.
We consider universal statistical properties of systems that are characterized by phase states with macroscopic degeneracy of the ground state. A possible topological order in such systems is described by non-linear discrete equations. We focus on the discrete equations which take place in the case of generalized exclusion principle statistics. We show that their exact solutions are quantum dimensions of the irreducible representations of certain quantum group. These solutions provide an example of the point where the generalized exclusion principle statistics and braid statistics meet each other. We propose a procedure to construct the quantum dimer models by means of projection of the knotted field configurations that involved braiding features of one-dimensional topology.
Recently we conjectured that a certain set of universal topological quantities characterize topological order in any dimension. Those quantities can be extracted from the universal overlap of the ground-state wave functions. For systems with gapped boundaries, these quantities are representations of the mapping class group MCG(M) of the space manifold M on which the systems live. We will here consider simple examples in three dimensions and give physical interpretation of these quantities, related to the fusion algebra and statistics of particles and string excitations. In particular, we will consider dimensional reduction from 3+1D to 2+1D, and show how the induced 2+1D topological data contain information on the fusion and the braiding of non-Abelian string excitations in 3D. These universal quantities generalize the well-known modular S and T matrices to any dimension.
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.
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 particleloop-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 I would like to give my sincere thanks to my advisor Shinsei Ryu.Throughout my PhD study, he taught me a lot about condensed matter physics, particularly on the forefront subjects in topological phases.Under his supervision, I worked on different abstract yet inspiring topics on topological phases.His guide is always clear with mathematical interpretation and physical insight.His advice not only guided me though the research, but also shaped my views and thinking.From my advisor, I learnt the way to convey difficult ideas
While the topological order in two dimensions has been studied extensively since the discovery of the integer and fractional quantum Hall systems, topological states in three spatial dimensions are much less understood.In this paper, we propose a general formalism for constructing a large class of threedimensional topological states by stacking layers of 2D topological states and introducing coupling between them.Using this construction, different types of topological states can be obtained, including those with only surface topological order and no bulk topological quasiparticles, and those with topological order both in the bulk and at the surface.For both classes of states, we study its generic properties and present several explicit examples.As an interesting consequence of this construction, we obtain example systems with nontrivial braiding statistics between string excitations.In addition to studying the string-string braiding in the example system, we propose a topological field-theory description for the layer-constructed systems, which captures not only the string-particle braiding statistics but also the string-string braiding statistics when the coupling is twisted.Last, we provide a proof of a general identity for Abelian string statistics and discuss an example system with non-Abelian strings.
Topologically ordered quantum systems have robust physical properties, such as quasiparticle statistics, and ground-state degeneracy, which do not depend on the microscopic details of the Hamiltonian. We consider topological phase transitions under a deformation such as an effective string tension on a Z3 topological state. This is studied in terms of the gauge-symmetry-preserved quantum state renormalization group, first proposed by He, Moradi, and Wen [Phys. Rev. B 90, 205114 (2014)]. In this approach modular matrices S and T can be obtained and used as order parameters to characterize the topological properties of the phase and determine phase transitions. From a mapping to a classical two-dimensional Potts model on the square lattice, the critical string tension, at which the transition to a topologically trivial phase takes place, can be obtained analytically and agrees with the numerically determined value. Such a transition can be generalized to a ZN topological model under a string tension and determined in the same way. With different deformations, the Z3 topological phase can also be driven to a critical phase, which contains, in the large deformation limit, the wave function analogous to the Rokhsar-Kivelson point in the quantum dimer model in one case and the fully packed loop model in another case.
We investigate transitions between topologically ordered phases in two spatial dimensions induced by the condensation of a bosonic quasiparticle. To this end, we formulate an extension of the theory of symmetry-breaking phase transitions which applies to phases with topological excitations described by quantum groups or modular tensor categories. This enables us to deal with phases whose quasiparticles have noninteger quantum dimensions and obey braid statistics. Many examples of such phases can be constructed from two-dimensional rational conformal field theories, and we find that there is a beautiful connection between quantum group symmetry breaking and certain well-known constructions in conformal field theory, notably the coset construction, the construction of orbifold models, and more general conformal extensions. Besides the general framework, many representative examples are worked out in detail.