Robert N. C. Pfeifer

Tensor network decompositions in the presence of a global symmetry

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 in the context of tensor network algorithms as well, thus setting the stage for cross-fertilization between these two areas of research.

Entanglement renormalization, scale invariance, and quantum criticality

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 multiscale entanglement renormalization ansatz 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.

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.

Constraining Validity of the Minkowski Energy-Momentum Tensor

There exist two popular energy-momentum tensors for an electromagnetic wave in a dielectric medium. The Abraham expression is robust to experimental verification but more mathematically demanding, while the Minkowski expression is the foundation of a number of simplifications commonly found within the literature, including the relative refractive index transformation often used in modeling optical tweezers. These simplifications are based on neglecting the Minkowski tensor’s material counterpart, a process known to be incompatible with conservation of angular momentum, and in conflict with experimental results, yet they are very successful in a wide range of circumstances. This paper combines existing constraints on their usage with recent theoretical analysis to obtain a list of conditions that must be satisfied to safely use the simplified Minkowski approach. Applying these conditions to an experiment proposed by Padgett et al., we find their prediction in agreement with that obtained using the total energy-momentum tensor.

Boundary quantum critical phenomena with entanglement renormalization

We propose the use of entanglement renormalization techniques to study boundary critical phenomena on a lattice system. The multiscale entanglement renormalization ansatz (MERA), 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 Wilsons 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.

Optical tweezers and paradoxes in electromagnetism

The widespread application of optical forces and torques has contributed to renewed interest in the fundamentals of the electromagnetic force and torque, including long-standing paradoxes such as the Abraham–Minkowski controversy and the angular momentum density of a circularly polarized plane wave. We discuss the relationship between these electromagnetic paradoxes and optical tweezers. In particular, consideration of possible optical tweezers experiments to attempt to resolve these paradoxes strongly suggests that they are beyond experimental resolution, yielding identical observable results in all cases.

Improving the efficiency of variational tensor network algorithms

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 network T, we prove that if the environment of a single tensor from the network can be evaluated with computational cost κ, then the environment of any other tensor from T can be evaluated with identical cost κ. Moreover, we describe how the set of all single tensor environments from T can be simultaneously evaluated with fixed cost 3κ. The usefulness of these results, which are applicable to a variety of tensor network methods, is demonstrated for the optimization of a multiscale entanglement renormalization Ansatz for the ground state of a one-dimensional quantum system, where they are shown to substantially reduce the computation time.

Translation invariance, topology, and protection of criticality in chains of interacting anyons

Using finite-size scaling arguments, the critical properties of a chain of interacting anyons can be extracted from the low-energy spectrum of a finite system. Feiguin showed that an antiferromagnetic chain of Fibonacci anyons on a torus is in the same universality class as the tricritical Ising model and that criticality is protected by a topological symmetry. In the present paper we first review the graphical formalism for the study of anyons on the disk and demonstrate how this formalism may be consistently extended to the study of systems on surfaces of higher genus. We then employ this graphical formalism to study finite rings of interacting anyons on both the disk and the torus and show that analysis on the disk necessarily yields an energy spectrum which is a subset of that which is obtained on the torus. For a critical Hamiltonian, one may extract from this subset the scaling dimensions of the local scaling operators which respect the topological symmetry of the system. Related considerations are also shown to apply for open chains.

Matrix product states for anyonic systems and efficient simulation of dynamics

Matrix product states (MPS) have proven to be a very successful tool to study lattice systems with local degrees of freedom such as spins or bosons. Topologically ordered systems can support anyonic particles which are labeled by conserved topological charges and collectively carry non-local degrees of freedom. In this paper we extend the formalism of MPS to lattice systems of anyons. The anyonic MPS is constructed from tensors that explicitly conserve topological charge. We describe how to adapt the time-evolving block decimation (TEBD) algorithm to the anyonic MPS in order to simulate dynamics under a local and charge-conserving Hamiltonian. To demonstrate the effectiveness of anyonic TEBD algorithm, we used it to simulate (i) the ground state (using imaginary time evolution) of an infinite 1D critical system of (a) Ising anyons and (b) Fibonacci anyons both of which are well studied, and (ii) the real time dynamics of an anyonic Hubbard-like model of a single Ising anyon hopping on a ladder geometry with an anyonic flux threading each island of the ladder. Our results pertaining to (ii) give insight into the transport properties of anyons. The anyonic MPS formalism can be readily adapted to study systems with conserved symmetry charges, as this is equivalent to a specialization of the more general anyonic case.

Two controversies in classical electromagnetism

This paper examines two controversies arising within classical electromagnetism which are relevant to the optical trapping and micromanipulation community. First is the Abraham-Minkowski controversy, a debate relating to the form of the electromagnetic energy-momentum tensor in dielectric materials, with implications for the momentum of a photon in dielectric media. A wide range of alternatives exist, and experiments are frequently proposed to attempt to discriminate between them. We explain the resolution of this controversy and show that regardless of the electromagnetic energy-momentum tensor chosen, when material disturbances are also taken into account the predicted behaviour will always be the same. The second controversy, known as the plane wave angular momentum paradox, relates to the distribution of angular momentum within an electromagnetic wave. The two competing formulations are reviewed, and an experiment is discussed which is capable of distinguishing between the two.