Symmetry-Protected Local Minima in Infinite DMRG

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. Implementations 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.Comment: 13 pages, 9 figures, 1 table, RevTeX 4.1. New title, greatly expanded explanations, fixed some typos (incl. reference to equation in caption of Fig.3). Reversed orientation convention for arrow on accessory site to match arrows on physical sites: all site arrows are now inboun

Theory and Practice of Transactional Method Caching

Nowadays, tiered architectures are widely accepted for constructing large scale information systems. In this context application servers often form the bottleneck for a system's efficiency. An application server exposes an object oriented interface consisting of set of methods which are accessed by potentially remote clients. The idea of method caching is to store results of read-only method invocations with respect to the application server's interface on the client side. If the client invokes the same method with the same arguments again, the corresponding result can be taken from the cache without contacting the server. It has been shown that this approach can considerably improve a real world system's efficiency. This paper extends the concept of method caching by addressing the case where clients wrap related method invocations in ACID transactions. Demarcating sequences of method calls in this way is supported by many important application server standards. In this context the paper presents an architecture, a theory and an efficient protocol for maintaining full transactional consistency and in particular serializability when using a method cache on the client side. In order to create a protocol for scheduling cached method results, the paper extends a classical transaction formalism. Based on this extension, a recovery protocol and an optimistic serializability protocol are derived. The latter one differs from traditional transactional cache protocols in many essential ways. An efficiency experiment validates the approach: Using the cache a system's performance and scalability are considerably improved

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 $\mathcal{T}$, 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 $3\kappa$. 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.Comment: 12 pages, 8 figures, RevTex 4.1, includes reference implementation. Software updated to v1.02: Resolved two scenarios in which multienv would generate errors for valid input

Finite Density Matrix Renormalisation Group Algorithm for Anyonic Systems

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 their 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.Comment: 24 pages, 37 figure files (25 floating figures). RevTeX 4.1. Minor changes for clarity in Figs. 9 & 11, matching published versio

Visualisation of Cherenkov Radiation and the Fields of a Moving Charge

For some physics students, the concept of a particle travelling faster than the speed of light holds endless fascination, and Cherenkov radiation is a visible consequence of a charged particle travelling through a medium at locally superluminal velocities. The Heaviside--Feynman equations for calculating the magnetic and electric fields of a moving charge have been known for many decades, but it is only recently that the computing power to plot the fields of such a particle has become readily available for student use. This article investigates and illustrates the calculation of Maxwell's D field in homogeneous isotropic media for arbitrary, including superluminal, constant velocity, and uses the results as a basis for discussing energy transfer in the electromagnetic field.Comment: 18 pages, 8 figures, 2 MATLAB listings. Version 2: Corrected display for letter paper format. Added publication info. Version 3: Corrected typos in Eqs. 5, 8, 1

Faster identification of optimal contraction sequences for tensor networks

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 upon the order in which the index sums are evaluated, and determination of the operation-minimising 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 memoisation, but these approaches are impractical for many of the larger tensor network Ansaetze 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-minimising 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 arXiv:1310.8023 respectively, is supplied.Comment: 25 pages, 12 figs, 2 tables, includes reference implementation of algorithm, v2.01. Update corrects the display of contraction sequences involving single-tensor traces (i.e. where an index in the input appears twice on the same tensor

Tensor network states and algorithms in the presence of a global U(1) 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. In a recent paper [arXiv:0907.2994v1] we discussed how to incorporate a global internal symmetry, given by a compact, completely reducible group G, into tensor network decompositions and algorithms. Here we specialize to the case of Abelian groups and, for concreteness, to a U(1) symmetry, often associated with particle number conservation. We consider tensor networks made of tensors that are invariant (or covariant) under the symmetry, and explain how to decompose and manipulate such tensors in order to exploit their symmetry. In numerical calculations, the use of U(1) symmetric tensors allows selection of a specific number of particles, ensures the exact preservation of particle number, and significantly reduces computational costs. We illustrate all these points in the context of the multi-scale entanglement renormalization ansatz.Comment: 22 pages, 25 figures, RevTeX