Studies of free particles in low-dimensional quantum systems such as two-leg ladders provide insight into the influence of statistics on collective behaviour. The behaviours of bosons and fermions are well understood, but two-dimensional systems also
admit excitations with alternative statistics known as anyons. Numerical analysis of hard-core $mathbb{Z}_3$ anyons on the ladder reveals qualitatively distinct behaviour, including a novel phase transition associated with crystallisation of hole degrees of freedom into a periodic foam. Qualitative predictions are extrapolated for all Abelian $mathbb{Z}_q$ anyon models.
The infinite Density Matrix Renormalisation Group (iDMRG) algorithm is a highly successful numerical algorithm for the study of low-dimensional quantum systems, and is also frequently used to initialise the more popular finite DMRG algorithm. Impleme
ntations of both finite and infinite DMRG frequently incorporate support for the protection and exploitation of symmetries of the Hamiltonian. In common with other variational tensor network algorithms, convergence of iDMRG to the ground state is not guaranteed, with the risk that the algorithm may become stuck in a local minimum. In this paper I demonstrate the existence of a particularly harmful class of physically irrelevant local minima affecting both iDMRG and to a lesser extent also infinite Time-Evolving Block Decimation (iTEBD), for which the ground state is compatible with the protected symmetries of the Hamiltonian but cannot be reached using the conventional iDMRG or iTEBD algorithms. I describe a modified iDMRG algorithm which evades these local minima, and which also admits a natural interpretation on topologically ordered systems with a boundary.
The numerical study of anyonic systems is known to be highly challenging due to their non-bosonic, non-fermionic particle exchange statistics, and with the exception of certain models for which analytical solutions exist, very little is known about t
heir collective behaviour as a result. Meanwhile, the density matrix renormalisation group (DMRG) algorithm is an exceptionally powerful numerical technique for calculating the ground state of a low-dimensional lattice Hamiltonian, and has been applied to the study of bosonic, fermionic, and group-symmetric systems. The recent development of a tensor network formulation for anyonic systems opened up the possibility of studying these systems using algorithms such as DMRG, though this has proved challenging both in terms of programming complexity and computational cost. This paper presents the implementation of DMRG for finite anyonic systems, including a detailed scheme for the implementation of anyonic tensors with optimal scaling of computational cost. The anyonic DMRG algorithm is demonstrated by calculating the ground state energy of the Golden Chain, which has become the benchmark system for the numerical study of anyons, and is shown to produce results comparable to those of the anyonic TEBD algorithm and superior to the variationally optimised anyonic MERA, at far lesser computational cost.
A fundamental process in the implementation of any numerical tensor network algorithm is that of contracting a tensor network. In this process, a network made up of multiple tensors connected by summed indices is reduced to a single tensor or a numbe
r by evaluating the index sums. This article presents a MATLAB function ncon(), or Network CONtractor, which accepts as its input a tensor network and a contraction sequence describing how this network may be reduced to a single tensor or number. As its output it returns that single tensor or number. The function ncon() may be obtained by downloading the source of this preprint.
We present several results relating to the contraction of generic tensor networks and discuss their application to the simulation of quantum many-body systems using variational approaches based upon tensor network states. Given a closed tensor networ
k $mathcal{T}$, we prove that if the environment of a single tensor from the network can be evaluated with computational cost $kappa$, then the environment of any other tensor from $mathcal{T}$ can be evaluated with identical cost $kappa$. Moreover, we describe how the set of all single tensor environments from $mathcal{T}$ can be simultaneously evaluated with fixed cost $3kappa$. The usefulness of these results, which are applicable to a variety of tensor network methods, is demonstrated for the optimization of a Multi-scale Entanglement Renormalization Ansatz (MERA) for the ground state of a 1D quantum system, where they are shown to substantially reduce the computation time.
Bipartite entanglement entropies, calculated from the reduced density matrix of a subsystem, provide a description of the resources available within a system for performing quantum information processing. However, these quantities are not uniquely de
fined on a system of non-Abelian anyons. This paper describes how reduced density matrices and bipartite entanglement entropies (such as the von Neumann and Renyi entropies) may be constructed for non-Abelian anyonic systems, in ways which reduce to the conventional definitions for systems with only local degrees of freedom.
This article describes a single species of non-interacting massless dust on $mathbb{R}^{0|18}$, whose behaviour in the low-energy limit is equivalent to an interacting family of massive particles resembling the Standard Model plus WIMPs on a curved 3
+1D space--time manifold (though with some liberties taken with gravity). The coupling between mass and curvature is not strictly equivalent to general relativity, but reproduces the usual metrics for large uncharged spherically symmetric sources at reasonable distances from the event horizon. Tunable parameters may be chosen so that electroweak particle masses and force couplings calculated to tree level lie within a few percent of their Standard Model values. This model is consequently of interest as a novel approximation to the Standard Model and gravitation. Extensive new physics, including a tripartite coloured preon substructure for fermions, is predicted at energies beyond the strong nuclear scale.
The use of entanglement renormalization in the presence of scale invariance is investigated. We explain how to compute an accurate approximation of the critical ground state of a lattice model, and how to evaluate local observables, correlators and c
ritical exponents. Our results unveil a precise connection between the multi-scale entanglement renormalization ansatz (MERA) and conformal field theory (CFT). Given a critical Hamiltonian on the lattice, this connection can be exploited to extract most of the conformal data of the CFT that describes the model in the continuum limit.
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Robert N. C. Pfeifer
, Timo A. Nieminen
, Norman R. Heckenberg andn Halina Rubinsztein-Dunlop
2009
Almost a hundred years ago, two different expressions were proposed for the energy--momentum tensor of an electromagnetic wave in a dielectric. Minkowskis tensor predicted an increase in the linear momentum of the wave on entering a dielectric medium
, whereas Abrahams tensor predicted its decrease. Theoretical arguments were advanced in favour of both sides, and experiments proved incapable of distinguishing between the two. Yet more forms were proposed, each with their advocates who considered the form that they were proposing to be the one true tensor. This paper reviews the debate and its eventual conclusion: that no electromagnetic wave energy--momentum tensor is complete on its own. When the appropriate accompanying energy--momentum tensor for the material medium is also considered, experimental predictions of all the various proposed tensors will always be the same, and the preferred form is therefore effectively a matter of personal choice.
