Exactly soluble 3D lattice models and the braiding statistics of their loop excitations

Ikuo Ichinose, 大林聡之

Chien-Hung Lin

In the past few years, the discovery of topological insulators has resurrected the study of exotic phases of matter beyond the Landau symmetry breaking paradigm. In this thesis, we focus on two particular classes of exotic phases, known as "topological phases" and "symmetry protected topological phases (SPT phases)" in 2D and 3D. Topological phases are generalizations of fractional quantum Hall liquids while SPT phases are generalizations of topological insulators. We investigate topological phases and SPT phases by constructing and analyzing appropriate exactly soluble lattice spin models. We address the following three questions: The first question involves a large class of exactly soluble lattice spin models called string-net models. It is known that string-net models can realize a large class of 2D topological phases, but it is unclear what topological phases can and can not be realized by these models. Thus our first question is: what are the most general topological phases that can be realized by string-net models? We find that a topological phase can be realized by a string-net model if and only if it supports a gapped edge, i.e. the edge can be gapped by suitable local interactions. In the second question we ask: how can we distinguish different 3D topological phases? While it is well known that 3D topological phases can be distinguished by the braiding statistics between particle and loop-like excitations, we construct two exactly soluble lattice models that demonstrate that we also have to consider the braiding statistics of loops with other loops. In the final part of the thesis, we derive a bulk-boundary correspondence for 3D SPT phases. More specifically, we prove that a particular exactly soluble lattice spin model has protected surface states using the fact that the vortex loop excitations in the bulk have non-trivial braiding statistics. We prove the result for a particular 3D exactly soluble lattice model, but our arguments apply more generally.

Jürgen Fuchs, Doron Gepner

Chenjie Wang, Nathanan Tantivasadakarn, Meng Cheng

Chien-Hung Lin, Michael Levin

Motivated by the question of whether or not two gapped Hamiltonians belong to the same phase, the authors compare two exactly solvable lattice models. They show that the only way to see that these models belong to distinct phases is to examine their three-loop braiding statistics.

Tieyan Si, Yue yu

Exactly soluble spin-$\frac{1}2$ models on three-dimensional lattices are proposed by generalizing Kitaev model on honeycomb lattice to three dimensions with proper periodic boundary conditions. The simplest example is spins on a diamond lattice which is exactly soluble. The ground state sector of the model may be mapped into a p-wave paired state on cubic lattice. We observe for the first time a topological phase transition from a gapless phase to a gapped phase in an exactly soluble spin model. Furthermore, the gapless phase can not be gapped by a perturbation breaking the time reversal symmetry. Unknotted and unlinked Wilson loops arise as eigen excitations, which may evolute into linked and knotted loop excitations. We show that these closed string excitations obey abelian statistics in the gapped phase and non-abelian statistics in the gapless phase.