We construct a new model of two-dimensional quantum spin systems that combines both symmetry-protected and intrinsic topological orders. It is referred as the bilayer Doubled Semion model (bDS) and is an instance of symmetry-enriched topological order. The model provides an explicit solution to the problem of combining symmetry-protected topological phases present in the Doubled Semion model with those in the Kitaev model. A bilayer lattice is introduced to this purpose, which allows us to combine a global spin-flip symmetry with a global spin-flavor symmetry. The bDS model exhibits non-trival braiding self-statistics of excitations and novel edge states protected by the $\mathbb{Z}_2\times \mathbb{Z}_2$ global symmetry. This construction makes compatible the methods of topological charge-fractionalization based on dualities and on string-flux mechanisms.
Bilayer Double Semion Model with Z2×Z2 Symmetry-Protected Topological Order
We construct a new model of two-dimensional quantum spin systems that combines both symmetry-protected and intrinsic topological orders. It is referred as the bilayer Doubled Semion model (bDS) and is an instance of symmetry-enriched topological order. The model provides an explicit solution to the problem of combining symmetry-protected topological phases present in the Doubled Semion model with those in the Kitaev model. A bilayer lattice is introduced to this purpose, which allows us to combine a global spin-flip symmetry with a global spin-flavor symmetry. The bDS model exhibits non-trival braiding self-statistics of excitations and novel edge states protected by the Z2×Z2 global symmetry. This construction makes compatible the methods of topological charge-fractionalization based on dualities and on string-flux mechanisms.
We study (2+1)-dimensional phases with topological order, such as fractional quantum Hall states and gapped spin liquids, in the presence of global symmetries. Phases that share the same topological order can then differ depending on the action of symmetry, leading to symmetry-enriched topological (SET) phases. Here, we present a K-matrix Chern-Simons approach to identify distinct phases with Abelian topological order, in the presence of unitary or antiunitary global symmetries. A key step is the identification of a smooth edge sewing condition that is used to check if two putative phases are indeed distinct. We illustrate this method by classifying Z2 topological order (Z2 spin liquids) in the presence of an internal Z2 global symmetry for which we find six distinct phases. These include two phases with an unconventional action of symmetry that permutes anyons leading to symmetry-protected Majorana edge modes. Other routes to realizing protected edge states in SET phases are identified. Symmetry-enriched Laughlin states and double-semion theories are also discussed. Somewhat surprisingly, we observe that (i) gauging the global symmetry of distinct SET phases leads to topological orders with the same total quantum dimension, and (ii) a pair of distinct SET phases can yield the same topological order on gauging the symmetry.
Symmetry protected topological (SPT) phase is a generalization of topological insulator (TI). Different from the intrinsic topological phase, e.g., the fractional quantum hall (FQH) phase, SPT phase is only distinguishable from a trivial disordered phase when certain symmetry is preserved. Indeed, SPT phase has a long history in 1D, and it has been shown that the well known Haldane phase of S=1 Heisenberg chain belongs to this class. However, in higher dimensions, most of the previous studies focus on free electron systems. Until very recently, it was realized that SPT phase also exists in interacting boson/spin systems in higher dimensions. In this talk, I will show that there is an interesting duality map between intrinsic topological phase and SPT phase. I will focus on a simple model that realizes an Ising paramagnet with gapless edge modes protected by Ising symmetry. The duality provides us a simple physical construction that distinguishes this system from a conventional paramagnet: we couple the system to a Z2 gauge field and then show that the flux excitations have different braiding statistics from that of a usual paramagnet. In addition, these braiding statistics directly implies the existence of protected edge modes. The duality map also provides us a complementary way to derive topological terms of SPT phases, which classify different SPT phases in bosonic systems. Finally, I will discuss the possible generalization for interacting fermion/electron systems.
In this thesis, we study the topological phases of quantum spin systems. One project is to investigate a class of anti-ferromagnetic SU(N) Heisenberg models, describing Mott insulators of fermionic ultra-cold alkaline earth atoms on the three-dimensional simple cubic lattice. Our large-N analysis maps a rich phase diagram. One particularly striking state we found spontaneously breaks lattice rotation symmetry, where the cubic lattice breaks up into bilayers, each of which forms a two-dimensional chiral spin liquid state. In the other projects, we study the phenomenon of symmetry fractionalization on anyons as a tool to characterize two-dimensional symmetry enriched topological phases. In particular, we focus on how crystalline symmetries may fractionalize in gapped Z2 spin liquids. If the system has the symmetry of the square lattice, then there are 2080 symmetry fractionalization patterns possible. With exactly solvable models, we realize 487 of these in strictly two-dimensional systems. In addition, we succeed to understand why the remaining patterns cannot be found in the family of models we construct. Some can only appear on the surface of three-dimensional systems with non-trivial point group symmetry protected topological (pgSPT) order, whose boundary degrees of freedom transform non-locally under the symmetries. We construct a simple toy model to show this anomalous crystalline symmetry fractionalization phenomenon associated with a reflection. Moreover, our approach establishes the connection between the pgSPT phases and the topological phases with on-site symmetries in lower dimensions. This insight is very useful for classification of pgSPT orders in general.
We discuss an extension of higher-order topological phases to include bosonic systems. We present two spin models for a second-order topological phase protected by a global ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry. One model is built from layers of an exactly solvable cluster model for a one-dimensional ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological phase, while the other is built from more conventional spin couplings (XY or Heisenberg). These models host gapped, symmetry-protected topological phases on their edges, and corner modes that fall into a projective representation of the symmetry. Using Jordan-Wigner transformations we show that both our models are related to a bilayer of free Majorana fermions that form a fermionic second-order topological phase. We also discuss how our models can be extended to three dimensions to form a third-order topological phase.
We construct a two-dimensional (2D) quantum spin model that realizes an Ising paramagnet with gapless edge modes protected by Ising symmetry. This model provides an example of a ``symmetry-protected topological phase.'' We describe a simple physical construction that distinguishes this system from a conventional paramagnet: We couple the system to a Z2 gauge field and then show that the $\ensuremath{\pi}$-flux excitations have different braiding statistics from that of a usual paramagnet. In addition, we show that these braiding statistics directly imply the existence of protected edge modes. Finally, we analyze a particular microscopic model for the edge and derive a field theoretic description of the low energy excitations. We believe that the braiding statistics approach outlined in this paper can be generalized to a large class of symmetry-protected topological phases.
Higher symmetries can emerge at low energies in a topologically ordered state with no symmetry, when some topological excitations have very high-energy scales while other topological excitations have low energies. The low-energy properties of topological orders in this limit, with the emergent higher symmetries, may be described by higher symmetry-protected topological order. This motivates us, as a simplest example, to study a lattice model of Zn-1 symmetry-protected topological (1-SPT) states in 3+1 dimensions for even n. We write an exactly solvable lattice model and study its boundary transformation. On the boundary, we show the existence of anyons with nontrivial self-statistics. For the n=2 case, where the bulk classification is given by an integer m mod 4, we show that the boundary can be gapped with double-semion topological order for m=1 and toric code for m=2. The bulk ground-state wave-function amplitude is given in terms of the linking numbers of loops in the dual lattice. Our construction can be generalized to arbitrary 1-SPT protected by finite unitary symmetry.