Miguel Aguado

Entanglement Renormalization and Topological Order

The multiscale entanglement renormalization ansatz (MERA) is argued to provide a natural description for topological states of matter. The case of Kitaevs toric code is analyzed in detail and shown to possess a remarkably simple MERA description leading to distillation of the topological degrees of freedom at the top of the tensor network. Kitaev states on an infinite lattice are also shown to be a fixed point of the renormalization group flow associated with entanglement renormalization. All of these results generalize to arbitrary quantum double models.

Creation, Manipulation, and Detection of Abelian and Non-Abelian Anyons in Optical Lattices

Anyons are particlelike excitations of strongly correlated phases of matter with fractional statistics, characterized by nontrivial changes in the wave function, generalizing Bose and Fermi statistics, when two of them are interchanged. This can be used to perform quantum computations [A. Yu. Kitaev, Ann. Phys. (N.Y.) 303, 2 (2003)]. We show how to simulate the creation and manipulation of Abelian and non-Abelian anyons in topological lattice models using trapped atoms in optical lattices. Our proposal, feasible with present technology, requires an ancilla particle which can undergo single-particle gates, be moved close to each constituent of the lattice and undergo a simple quantum gate, and be detected.

A hierarchy of topological tensor network states

We present a hierarchy of quantum many-body states among which many examples of topological order can be identified by construction. We define these states in terms of a general, basis-independent framework of tensor networks based on the algebraic setting of finite-dimensional Hopf C*-algebras. At the top of the hierarchy we identify ground states of new topological lattice models extending Kitaevs quantum double models [Ann. Phys. 303, 2 (2003)10.1016/S0003-4916(02)00018-0]. For these states we exhibit the mechanism responsible for their non-zero topological entanglement entropy by constructing an entanglement renormalization flow. Furthermore, we argue that the hierarchy states are related to each other by the condensation of topological charges.

Explicit tensor network representation for the ground states of string-net models

We provide a simple expression for the ground states of arbitrary string-net models in the form of local tensor networks. These tensor networks encode the data of the fusion category underlying a string-net model and thus represent all doubled topological phases of matter in the infrared limit according to Levin and Wen [Phys. Rev. B 71, 045110 (2005)]. Furthermore, our construction highlights the importance of the fat lattice equivalence between lattice and continuum descriptions of string-net models.

Mapping Kitaev’s quantum double lattice models to Levin and Wen’s string-net models

We exhibit a mapping identifying Kitaevs quantum double lattice models explicitly as a subclass of Levin and Wens string net models via a completion of the local Hilbert spaces with auxiliary degrees of freedom. This identification allows to carry over to these string net models the representation-theoretic classification of the excitations in quantum double models, as well as define them in arbitrary lattices, and provides an illustration of the abstract notion of Morita equivalence. The possibility of generalising the map to broader classes of string nets is considered.

Scaling law for topologically ordered systems at finite temperature

Understanding the behavior of topologically ordered lattice systems at finite temperature is a way of assessing their potential as fault-tolerant quantum memories. We compute the natural extension of the topological entanglement entropy for T>0, namely, the subleading correction I{sub topo} to the area law for mutual information. Its dependence on T can be written, for Abelian Kitaev models, in terms of information-theoretical functions and readily identifiable scaling behavior, from which the interplay between volume, temperature, and topological order, can be read. These arguments are extended to non-Abelian quantum double models, and numerical results are given for the D(S{sub 3}) model, showing qualitative agreement with the Abelian case.

Electric–magnetic duality of lattice systems with topological order

Abstract We investigate the duality structure of quantum lattice systems with topological order, a collective order also appearing in fractional quantum Hall systems. We define electromagnetic (EM) duality for all of Kitaevʼs quantum double models based on discrete gauge theories with Abelian and non-Abelian groups, and identify its natural habitat as a new class of topological models based on Hopf algebras. We interpret these as extended string-net models, whereupon Levin and Wenʼs string-nets, which describe all intrinsic topological orders on the lattice with parity and time-reversal invariance, arise as magnetic and electric projections of the extended models. We conjecture that all string-net models can be extended in an analogous way, using more general algebraic and tensor-categorical structures, such that EM duality continues to hold. We also identify this EM duality with an invertible domain wall. Physical applications include topology measurements in the form of pairs of dual tensor networks.

Simulation of anyons with tensor network algorithms

Interacting systems of anyons pose a unique challenge to condensed-matter simulations due to their nontrivial exchange statistics. These systems are of great interest as they have the potential for robust universal quantum computation but numerical tools for studying them are as yet limited. We show how existing tensor network algorithms may be adapted for use with systems of anyons and demonstrate this process for the one-dimensional multiscale entanglement renormalization ansatz (MERA). We apply the MERA to infinite chains of interacting Fibonacci anyons, computing their scaling dimensions and local scaling operators. The scaling dimensions obtained are seen to be in agreement with conformal field theory. The techniques developed are applicable to any tensor network algorithm, and the ability to adapt these ansatze for use on anyonic systems opens the door for numerical simulation of large systems of free and interacting anyons in one and two dimensions.

Clash of positivities in topological density correlators

We discuss the apparent conflict between reflection positivity and positivity of the topological susceptibility in two-dimensional nonlinear sigma models and in four-dimensional gauge theories. We pay special attention to the fact that this apparent conflict is already present on the lattice; its resolution puts some nontrivial restrictions on the short-distance behavior of the lattice correlator. It is found that these restrictions can be satisfied both in the case of asymptotic freedom and the dissident scenario of a critical point at finite coupling.

From entanglement renormalisation to the disentanglement of quantum double models

We describe how the entanglement renormalisation approach to topological lattice systems leads to a general procedure for treating the whole spectrum of these models, in which the Hamiltonian is gradually simplified along a parallel simplification of the connectivity of the lattice. We consider the case of Kitaevs quantum double models, both Abelian and non-Abelian, and we obtain a rederivation of the known map of the toric code to two Ising chains; we pay particular attention to the non-Abelian models and discuss their space of states on the torus. Ultimately, the construction is universal for such models and its essential feature, the lattice simplification, may point towards a renormalisation of the metric in continuum theories.