13,133 research outputs found
Detecting Topological Order with Ribbon Operators
We introduce a numerical method for identifying topological order in
two-dimensional models based on one-dimensional bulk operators. The idea is to
identify approximate symmetries supported on thin strips through the bulk that
behave as string operators associated to an anyon model. We can express these
ribbon operators in matrix product form and define a cost function that allows
us to efficiently optimize over this ansatz class. We test this method on spin
models with abelian topological order by finding ribbon operators for
quantum double models with local fields and Ising-like terms. In
addition, we identify ribbons in the abelian phase of Kitaev's honeycomb model
which serve as the logical operators of the encoded qubit for the quantum
error-correcting code. We further identify the topologically encoded qubit in
the quantum compass model, and show that despite this qubit, the model does not
support topological order. Finally, we discuss how the method supports
generalizations for detecting nonabelian topological order.Comment: 15 pages, 8 figures, comments welcom
The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems
We present a compendium of numerical simulation techniques, based on tensor
network methods, aiming to address problems of many-body quantum mechanics on a
classical computer. The core setting of this anthology are lattice problems in
low spatial dimension at finite size, a physical scenario where tensor network
methods, both Density Matrix Renormalization Group and beyond, have long proven
to be winning strategies. Here we explore in detail the numerical frameworks
and methods employed to deal with low-dimension physical setups, from a
computational physics perspective. We focus on symmetries and closed-system
simulations in arbitrary boundary conditions, while discussing the numerical
data structures and linear algebra manipulation routines involved, which form
the core libraries of any tensor network code. At a higher level, we put the
spotlight on loop-free network geometries, discussing their advantages, and
presenting in detail algorithms to simulate low-energy equilibrium states.
Accompanied by discussions of data structures, numerical techniques and
performance, this anthology serves as a programmer's companion, as well as a
self-contained introduction and review of the basic and selected advanced
concepts in tensor networks, including examples of their applications.Comment: 115 pages, 56 figure
Small Black holes vs horizonless solutions in AdS
It is argued that the appropriate macroscopic description of half-BPS mesonic
chiral operators in generic toric gauge theories is in terms
of the geometric quantization of smooth horizonless configurations. The
relevance of different ensemble macroscopic descriptions is emphasized :
lorentzian vs euclidean configurations as (semiclassical) microstates vs saddle
points in an euclidean path integral.Comment: 10 pages; v2 improved entropy discussion and new references include
Simulation of anyons with tensor network algorithms
Interacting systems of anyons pose a unique challenge to condensed matter
simulations due to their non-trivial 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 1-D Multi-scale Entanglement
Renormalisation 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 ansaetze for use on
anyonic systems opens the door for numerical simulation of large systems of
free and interacting anyons in one and two dimensions.Comment: Fixed typos, matches published version. 16 pages, 21 figures, 4
tables, RevTeX 4-1. For a related work, see arXiv:1006.247
Forecasts of non-Gaussian parameter spaces using Box-Cox transformations
Forecasts of statistical constraints on model parameters using the Fisher
matrix abound in many fields of astrophysics. The Fisher matrix formalism
involves the assumption of Gaussianity in parameter space and hence fails to
predict complex features of posterior probability distributions. Combining the
standard Fisher matrix with Box-Cox transformations, we propose a novel method
that accurately predicts arbitrary posterior shapes. The Box-Cox
transformations are applied to parameter space to render it approximately
multivariate Gaussian, performing the Fisher matrix calculation on the
transformed parameters. We demonstrate that, after the Box-Cox parameters have
been determined from an initial likelihood evaluation, the method correctly
predicts changes in the posterior when varying various parameters of the
experimental setup and the data analysis, with marginally higher computational
cost than a standard Fisher matrix calculation. We apply the Box-Cox-Fisher
formalism to forecast cosmological parameter constraints by future weak
gravitational lensing surveys. The characteristic non-linear degeneracy between
matter density parameter and normalisation of matter density fluctuations is
reproduced for several cases, and the capabilities of breaking this degeneracy
by weak lensing three-point statistics is investigated. Possible applications
of Box-Cox transformations of posterior distributions are discussed, including
the prospects for performing statistical data analysis steps in the transformed
Gaussianised parameter space.Comment: 14 pages, 7 figures; minor changes to match version published in
MNRA
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