Almost a hundred years ago, two different expressions were proposed for the energy-momentum tensor of an electromagnetic wave in a dielectric. Minkowski's tensor predicted an increase in the linear momentum of the wave on entering a dielectric medium, whereas Abraham's 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.

Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. We discuss how to incorporate a global symmetry, given by a compact, completely reducible group G, in tensor network decompositions and algorithms. This is achieved by considering tensors that are invariant under the action of the group G. Each symmetric tensor decomposes into two types of tensors: degeneracy tensors, containing all the degrees of freedom, and structural tensors, which only depend on the symmetry group. In numerical calculations, the use of symmetric tensors ensures the preservation of the symmetry, allows selection of a specific symmetry sector, and significantly reduces computational costs. On the other hand, the resulting tensor network can be interpreted as a superposition of exponentially many spin networks. Spin networks are used extensively in loop quantum gravity, where they represent states of quantum geometry. Our work highlights their importance also in the context of tensor network algorithms, thus setting the stage for cross-fertilization between these two areas of research.Comment: 4 pages, 4 figures, RevTeX 4-

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 critical 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. The study of quantum critical phenomena through real-space renormalization group (RG) techniques [1,2] has traditionally been obstructed by the accumulation, over successive RG transformations, of short-range entanglement across block boundaries. Entanglement renormalization [3] was recently proposed as a technique to address this problem. By removing short-range entanglement at each iteration of the RG transformation, not only can arbitrarily large lattice systems be considered, but the scale invariance characteristic of critical phenomena is also seen to be restored [3,4].In this paper we explain how to use the multi-scale entanglement renormalization ansatz (MERA) [5] to investigate scale invariant systems [3,4,5,6,7]. It has been showed that the scale invariant MERA can represent the infra-red limit of topologically ordered phases [6]. Here we focus instead on its use at quantum criticality. We present the following results: (i) given a critical Hamiltonian, an adaptation of the algorithm of Ref.[8] to compute a scale invariant MERA for its ground state; then, starting from a scale invariant MERA, (ii) a procedure to identify the scaling operators/dimensions of the theory and (iii) a closed expression for two-point and threepoint correlators; (iv) a connection between the MERA and conformal field theory, which can be used to readily identify the continuum limit of a critical lattice model; finally (v) benchmark calculations for the Ising and Potts models.We note that result (ii) was already discussed by Giovannetti, Montangero and Fazio in Ref. [7] using the binary MERA of Ref. [5]. Our derivations are conducted instead with the ternary MERA of Ref [8] (see Fig. 1), in terms of which results (iii)-(iv) acquire a simple form.We start by considering a finite 1D lattice L made of N sites, each one described by a vector space V of dimension χ. The (ternary) MERA is a tensor network that serves as an ansatz for pure states |Ψ ∈ V ⊗N of the lattice, see Fig. 1. Its tensors, known as disentanglers and isometries, are organized in T ≈ log 3 N layers, each one implementing a RG transformation. Such transformations produce a sequence of lattices,where lattice L τ +1 is a coarse-graining of lattice L τ , and the top lattice L T is sufficiently small to allow exact numerical computations. Let o denote a local observable supported on two contiguous sites of L, and let ρ T be the density matrix that describes the state of the system on two contigu...

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 ͑MERA͒. 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 ansätze for use on anyonic systems opens the door for numerical simulation of large systems of free and interacting anyons in one and two dimensions.

The efficient evaluation of tensor expressions involving sums over multiple indices is of significant importance to many fields of research, including quantum many-body physics, loop quantum gravity, and quantum chemistry. The computational cost of evaluating an expression may depend strongly on the order in which the index sums are evaluated, and determination of the operation-minimizing contraction sequence for a single tensor network (single term, in quantum chemistry) is known to be NP-hard. The current preferred solution is an exhaustive search, using either an iterative depth-first approach with pruning or dynamic programming and memoization, but these approaches are impractical for many of the larger tensor network ansätze encountered in quantum many-body physics. We present a modified search algorithm with enhanced pruning which exhibits a performance increase of several orders of magnitude while still guaranteeing identification of an optimal operation-minimizing contraction sequence for a single tensor network. A reference implementation for MATLAB, compatible with the ncon() and multienv() network contractors of arXiv:1402.0939 and Evenbly and Pfeifer, Phys. Rev. B 89, 245118 (2014), respectively, is supplied.

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Erratum: Colloquium: Momentum of an electromagnetic wave in dielectric media ͓Rev. Mod. Phys. 79,

We propose the use of entanglement renormalization techniques to study boundary critical phenomena on a lattice system. The multiscale entanglement renormalization ansatz ͑MERA͒, in its scale invariant version, offers a very compact approximation to quantum critical ground states. Here we show that, by adding a boundary to the MERA, an accurate approximation to the ground state of a semi-infinite critical chain with an open boundary is obtained, from which one can extract boundary scaling operators and their scaling dimensions. As in Wilson's renormalization-group formulation of the Kondo problem, our construction produces, as a side result, an effective chain displaying explicit separation of energy scales. We present benchmark results for the quantum Ising and quantum XX models with free and fixed boundary conditions.