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arXiv:1908.02599 · Physical review. B./Physical review. B

Matrix formulation for non-Abelian families

Tian Lan

Published Dec 5, 2019• 4 citations
arXivHTMLPDF

Abstract

We generalize the $K$ matrix formulation to nontrivial non-Abelian families of ($2+1$)-dimensional topological orders. Given a topological order $\mathcal{C}$, any topological order in the same non-Abelian family as $\mathcal{C}$ can be efficiently described by $\mathbit{a}=({a}_{I})$, where ${a}_{I}$ are Abelian anyons in $\mathcal{C}$, together with a symmetric invertible matrix $K,{K}_{IJ}={k}_{IJ}\ensuremath{-}{t}_{{a}_{I},{a}_{J}}$, where ${k}_{IJ}$ are integers, ${k}_{II}$ are even, and ${t}_{{a}_{I},{a}_{J}}$ are the mutual statistics between ${a}_{I},{a}_{J}$. In particular, when $\mathcal{C}$ is a root whose rank is the smallest in the family, $K$ becomes an integer matrix. Our results make it possible to generate the data of large numbers of topological orders instantly, thus providing a large reservoir of potentially useful topological materials.

DOI
10.1103/physrevb.100.241102

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We generalize the KKK matrix formulation to nontrivial non-Abelian families of (2+12+12+1)-dimensional topological orders. Given a topological order C\mathcal{C}C, any topological order in the same non-Abelian family as C\mathcal{C}C can be efficiently described by \mathbita=(aI)\mathbit{a}=({a}_{I})\mathbita=(aI​), where aI{a}_{I}aI​ are Abelian anyons in C\mathcal{C}C, together with a symmetric invertible matrix K,KIJ=kIJ\ensuremath−taI,aJK,{K}_{IJ}={k}_{IJ}\ensuremath{-}{t}_{{a}_{I},{a}_{J}}K,KIJ​=kIJ​\ensuremath−taI​,aJ​​, where kIJ{k}_{IJ}kIJ​ are integers, kII{k}_{II}kII​ are even, and taI,aJ{t}_{{a}_{I},{a}_{J}}taI​,aJ​​ are the mutual statistics between aI,aJ{a}_{I},{a}_{J}aI​,aJ​. In particular, when C\mathcal{C}C is a root whose rank is the smallest in the family, KKK becomes an integer matrix. Our results make it possible to generate the data of large numbers of topological orders instantly, thus providing a large reservoir of potentially useful topological materials.
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I define quantum loop models whose degrees of freedom are Ising spins on the square lattice as in the toric code, but where the excitations should have non-abelian statistics. The inner product is topological, allowing a direct implementation of the anyonic fusion matrix on the lattice. It also makes deconfined anyons possible for a variety of values of the weight per loop ddd in the ground state. For d=\sqrt{2}, a gapped non-abelian topological phase can occur with only four-spin interactions.
Decohering topological order (TO) is central to the many-body physics of open quantum matter and decoding transitions. We identify statistical mechanical models for decohering non-Abelian TOs, which have been crucial for understanding the error threshold of Abelian stabilizer codes. The decohered density matrix can be described by loop models, whose topological loop weight N is the quantum dimension of the decohering anyon-reducing to the Ising model if N=1. In particular, the Rényi-n moments correspond to n coupled O(N) loop models. Moreover, by diagonalizing the density matrix at maximal error rate, we connect the fidelity between two logically distinct ground states to random O(N) loop and spin models. We find a remarkable stability to quantum channels which proliferate non-Abelian anyons with large quantum dimension, with the possibility of critical phases for smaller dimensions. Intuitively, this stability is due to non-Abelian anyons not admitting finite-depth string operators. We confirm our framework with exact results for Kitaev quantum double models, and with numerical simulations for the non-Abelian phase of the Kitaev honeycomb model. Our work opens up the possibility of non-Abelian TO being robust against maximally proliferating certain anyons, which can inform error-correction studies of these topological memories.
Two-dimensional systems such as quantum spin liquids or fractional quantum Hall systems exhibit anyonic excitations that possess more general statistics than bosons or fermions. This exotic statistics makes it challenging to solve even a many-body system of non-interacting anyons. We introduce an algorithm that allows to simulate anyonic tight-binding Hamiltonians on two-dimensional lattices. The algorithm is directly derived from the low energy topological quantum field theory and is suited for general Abelian and non-Abelian anyon models. As concrete examples, we apply the algorithm to study the energy level spacing statistics, which reveals level repulsion for free semions, Fibonacci anyons, and Ising anyons. Additionally, we simulate nonequilibrium quench dynamics, where we observe that the density distribution becomes homogeneous for large times---indicating thermalization.
Abstract Anyon models are algebraic structures that model universal topological properties in topological phases of matter and can be regarded as mathematical characterization of topological order in two spacial dimensions. It is conjectured that every anyon model, or mathematically unitary modular tensor category, can be realized as the representation category of some chiral conformal field theory, or mathematically vertex operator algebra/local conformal net. This conjecture is known to be true for abelian anyon models providing support for the conjecture. We reexamine abelian anyon models from several different angles. First anyon models are algebraic data for both topological quantum field theories and chiral conformal field theories. While it is known that each abelian anyon model can be realized by a quantum abelian Chern–Simons (CS) theory and chiral conformal field theory, the construction is not algorithmic. Our goal is to provide such an explicit algorithm for a K -matrix in CS theory and a positive definite even one for a lattice conformal field theory. Secondly anyon models and chiral conformal field theories underlie the bulk-edge correspondence for topological phases of matter. But there are interesting subtleties in this correspondence when stability of the edge theory and topological symmetry are taken into consideration. Therefore, our focus is on the algorithmic reconstruction of extremal chiral conformal field theories with small central charges. Finally we conjecture that a much stronger reconstruction holds for abelian anyon models: every abelian anyon model can be realized as the representation category of some non-lattice extremal vertex operator algebra generalizing the moonshine realization of the trivial anyon model.
In the previous chapters we introduced anyons and their properties, we presented how to perform topological quantum computation and studied several examples of topological models. There is a wide variety of research topics concerned with topological quantum computation. Among the many open questions, two have a singular importance. The first natural question is: which physical systems can support non-Abelian anyons? Realising non-Abelian anyons in the laboratory is of fundamental and practical interest. Such exotic statistical behaviour has not yet been encountered in nature. The physical realisation of non-Abelian anyons would be the first step towards the identification of a technological platform for the realisation of topological quantum computation. The second question concerns the efficiency of topological systems in combating errors. It has been proven that the effect of coherent environmental errors in the form of local Hamiltonian perturbations can be suppressed efficiently without degrading the topologically encoded information (Bravyi et al. , 2010). Nevertheless, there is no mechanism that can protect topological order from incoherent probabilistic errors. Topological systems nevertheless constitute a rich and versatile medium that allows imaginative proposals to be developed (Chesi et al. , 2010; Hamma et al. , 2009). Regarding the first question, we can identify two main categories of physical proposals for the realisation of two-dimensional topological systems: systems that are defined on the continuum and discrete systems defined on a lattice. It is natural to ask, which are the most promising architectures to realise in the laboratory?