Continuum tensor network field states, path integral representations and the encoding of spatial symmetries

Abstract

A natural way to generalise tensor network variational classes to quantum field systems is via a continuous tensor contraction. This approach is first illustrated for the class of quantum field states known as continuous matrix-product states (cMPS). As a simple example of the path-integral representation we show that the state of a dynamically evolving quantum field admits a natural representation as a cMPS. A completeness argument is also provided that shows that all states in Fock space admit a cMPS representation when the number of variational parameters tends to infinity. Beyond this, we obtain a well-behaved field limit of projected entangled pair states (PEPS) in two dimensions that provide an abstract class of quantum field states with natural symmetries. We demonstrate how symmetries of the physical field state are encoded within the dynamics of an auxiliary field system of one dimension less. In particular, the imposition of Euclidean symmetries on the physical system requires that the auxiliary system involved in the class' definition must be Lorentz-invariant. The physical field states automatically inherit entropy area laws from the PEPS class, and are fully described by the dissipative dynamics of a lower dimensional virtual field system. Our results lie at the intersection many-body physics, quantum field theory and quantum information theory, and facilitate future exchanges of ideas and insights between these disciplines.

33 pages, 2 figures, v.2: now includes main points of arXiv:1210.5401, expanded discussion; v2. is published version

References in corpus (18)

• The density-matrix renormalization group in the age of matrix product states
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• A class of quantum many-body states that can be efficiently simulated
• Matrix product states represent ground states faithfully
• Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions
• Continuous Matrix Product States for Quantum Fields
• Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space
• Entanglement renormalization for quantum fields
• Chiral Gauge Theory for Graphene
• The Dynamics of 1D Quantum Spin Systems Can Be Approximated Efficiently
• Gauging quantum states: from global to local symmetries in many-body systems
• Sequential Generation of Matrix-Product States in Cavity QED
• Holographic quantum states
• Transfer Matrices and Excitations with Matrix Product States
• Applying the variational principle to (1+1)-dimensional quantum field theories
• Sequentially generated states for the study of two dimensional systems
• Simulating Dirac fermions with Abelian and non-Abelian gauge fields in optical lattices
• Matrix product states for Hamiltonian lattice gauge theories

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• Efficient Quantum Simulation for Thermodynamics of Infinite-size Many-body Systems in Arbitrary Dimensions
• Holographic Software for Quantum Networks
• Tensor renormalization group in bosonic field theory
• Characterizing the quantum field theory vacuum using temporal Matrix Product states
• Functional Tensor Network Solving Many-body Schrödinger Equation
• Continuum limits of Matrix Product States
• Continuum limits of homogeneous binary trees and the Thompson group
• Approximating Relativistic Quantum Field Theories with Continuous Tensor Networks
• Matrix product approximations to multipoint functions in two-dimensional conformal field theory
• Matrix product approximations to conformal field theories
• State Diagrams to determine Tree Tensor Network Operators
• Topological Defects in Quantum Field Theory with Matrix Product States
• Field Tensor Network States
• The Kibble Zurek Mechanism of Topological Defect Formation in Quantum Field Theory with Matrix Product States
• Continuous matrix-product states in inhomogeneous systems with long-range interactions
• Tensor Renormalization Group for interacting quantum fields
• Spatial distribution of superfluidity and superfluid distillation of Bose liquids
• Generalised ansatz for continuous Matrix Product States
• Symmetries and field tensor network states
• Entanglement renormalization for quantum fields with boundaries and defects
• Holographic Tensor Networks as Tessellations of Geometry