A large class of topological orders can be understood and classified using the string-net condensation picture. These topological orders can be characterized by a set of data (N, di, F ijk lmn , δ ijk ). We describe a way to detect this kind of topological order using only the ground state wave function. The method involves computing a quantity called the "topological entropy" which directly measures the quantum dimension D = P i d 2 i . [3,4], and edge excitations [1]. In this paper, we demonstrate that topological order is manifest not only in these dynamical properties but also in the basic entanglement of the ground state wave function. We hope that this characterization of topological order can be used as a theoretical tool to classify trial wave functions -such as resonating dimer wave functions [5], Gutzwiller projected states, [6][7][8][9][10] or quantum loop gas wave functions [11]. In addition, it may be useful as a numerical test for topological order. Finally, it demonstrates definitively that topological order is a property of a wave function, not a Hamiltonian. The classification of topologically ordered states is nothing but a classification of complex functions of thermodynamically large numbers of variables.Main Result: We focus on the (2 + 1) dimensional case (though the result can be generalized to any dimension). Let Ψ be an arbitrary wave function for some two dimensional lattice model. For any subset A of the lattice, one can compute the associated quantum entanglement entropy S A .[12] The main result of this paper is that one can determine the "quantum dimension" D of Ψ by computing the entanglement entropy S A of particular regions

We show that quantum systems of extended objects naturally give rise to a large class of exotic phases -namely topological phases. These phases occur when the extended objects, called "string-nets", become highly fluctuating and condense. We derive exactly soluble Hamiltonians for 2D local bosonic models whose ground states are string-net condensed states. Those ground states correspond to 2D parity invariant topological phases. These models reveal the mathematical framework underlying topological phases: tensor category theory. One of the Hamiltoniansa spin-1/2 system on the honeycomb lattice -is a simple theoretical realization of a fault tolerant quantum computer. The higher dimensional case also yields an interesting result: we find that 3D string-net condensation naturally gives rise to both emergent gauge bosons and emergent fermions. Thus, string-net condensation provides a mechanism for unifying gauge bosons and fermions in 3 and higher dimensions.

Recently, several authors have investigated topological phenomena in periodically driven systems of noninteracting particles. These phenomena are identified through analogies between the Floquet spectra of driven systems and the band structures of static Hamiltonians. Intriguingly, these works have revealed phenomena that cannot be characterized by analogy to the topological classification framework for static systems. In particular, in driven systems in two dimensions (2D), robust chiral edge states can appear even though the Chern numbers of all the bulk Floquet bands are zero. Here, we elucidate the crucial distinctions between static and driven 2D systems, and construct a new topological invariant that yields the correct edge-state structure in the driven case. We provide formulations in both the time and frequency domains, which afford additional insight into the origins of the ''anomalous'' spectra that arise in driven systems. Possibilities for realizing these phenomena in solid-state and cold-atomic systems are discussed.

We describe a simple real space renormalization group technique for two dimensional classical lattice models. The approach is similar in spirit to block spin methods, but at the same time it is fundamentally based on the theory of quantum entanglement. In this sense, the technique can be thought of as a classical analogue of DMRG. We demonstrate the method -which we call the tensor renormalization group method -by computing the magnetization of the triangular lattice Ising model. Introduction:The density matrix renormalization group (DMRG) technique has proved extraordinarily powerful in the analysis of one dimensional quantum systems. [1,2] Thus it is natural to try to develop an analogous renormalization group method in higher dimensions. Such a method could solve many currently intractable problems (such as the 2D Hubbard model).Recent work has focused on generalizing DMRG to higher dimensional quantum systems.[3] But it is also natural to try to generalize to higher dimensional classical lattice models. While classical real space renormalization group methods (such as block spin methods [4]) have been around for many years, they have never achieved the generality or precision of DMRG.In this paper, we address this problem in the two dimensional case. We use ideas from quantum information theory to develop a numerical renormalization group method that can effectively solve any two dimensional classical lattice model. The technique -which we call the tensor renormalization group (TRG) method -has no sign problem and works equally well for models with complex weights.Accurate numerical methods based on transfer matrices [5,6] have already been developed for 2D classical systems. The advantage of the approach described here is that it is a fully isotropic coarse graining procedure, similar in spirit to block spin methods. It is thus naturally suited to investigating universal long distance physics. Also, on a more theoretical level, the method reveals the relationship between classical RG and quantum entanglement. Finally, if only for its simplicity, we feel that the method is a useful numerical tool in two dimensions as well as a natural candidate for higher dimensional generalizations.Tensor network models: The tensor renormalization group method applies to a set of classical lattice models called "tensor network models." [7] Many well known statistical mechanical models, such as the Ising model, Potts model, and the six vertex model, can be written naturally as tensor network models. In fact, as we show later, all classical lattice models with local interactions can be written as tensor network models.To describe a tensor network model on the honeycomb lattice, one must specify a (cyclically symmetric) tensor T ijk with indices i, j, k running from 1 to D for some D.The corresponding tensor network model has a degree of freedom i = 1, ..., D on each bond of the honeycomb lattice. The weight for a configuration (i, j, k, ...) is given

We construct a 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 π-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.

We show that the particle-hole conjugate of the Pfaffian state-or "anti-Pfaffian" state-is in a different universality class from the Pfaffian state, with different topological order. The two states can be distinguished easily by their edge physics: their edges differ in both their thermal Hall conductance and their tunneling exponents. At the same time, the two states are exactly degenerate in energy for a nu=5/2 quantum Hall system in the idealized limit of zero Landau level mixing. Thus, both are good candidates for the observed sigma_{xy}=5/2(e;{2}/h) quantum Hall plateau.

We analyze generalizations of two-dimensional topological insulators which can be realized in interacting, time reversal invariant electron systems. These states, which we call fractional topological insulators, contain excitations with fractional charge and statistics in addition to protected edge modes. In the case of s(z) conserving toy models, we show that a system is a fractional topological insulator if and only if sigma(sH)/e* is odd, where sigma(sH) is the spin-Hall conductance in units of e/2pi, and e* is the elementary charge in units of e.

A simple physical realization of an integer quantum Hall state of interacting two dimensional bosons is provided. This is an example of a "symmetry-protected topological" (SPT) phase which is a generalization of the concept of topological insulators to systems of interacting bosons or fermions. Universal physical properties of the boson integer quantum Hall state are described and shown to correspond to those expected from general classifications of SPT phases.