Diagonalizing transfer matrices and matrix product operators: a medley of exact and computational methods

Abstract

Transfer matrices and matrix product operators play an ubiquitous role in the field of many body physics. This paper gives an ideosyncratic overview of applications, exact results and computational aspects of diagonalizing transfer matrices and matrix product operators. The results in this paper are a mixture of classic results, presented from the point of view of tensor networks, and of new results. Topics discussed are exact solutions of transfer matrices in equilibrium and non-equilibrium statistical physics, tensor network states, matrix product operator algebras, and numerical matrix product state methods for finding extremal eigenvectors of matrix product operators.

arXiv commentLecture notes from a course at Vienna University

References in corpus (89)

• Quantum entanglement
• Fault-tolerant quantum computation by anyons
• Non-Abelian Anyons and Topological Quantum Computation
• The density-matrix renormalization group in the age of matrix product states
• The density-matrix renormalization group
• Area laws for the entanglement entropy - a review
• Topological entanglement entropy
• Efficient classical simulation of slightly entangled quantum computations
• Detecting topological order in a ground state wave function
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Efficient simulation of one-dimensional quantum many-body systems
• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• String-net condensation: A physical mechanism for topological phases
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations
• Matrix Product Density Operators: Simulation of finite-T and dissipative systems
• Entanglement renormalization
• Thermodynamic limit of the density matrix renormalization for the spin-1 Heisenberg chain
• An Area Law for One Dimensional Quantum Systems
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• Time-dependent variational principle for quantum lattices
• Unifying time evolution and optimization with matrix product states
• Matrix product states represent ground states faithfully
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• Area laws in quantum systems: mutual information and correlations
• Nonequilibrium Steady States of Matrix Product Form: A Solver's Guide
• Mixed-state dynamics in one-dimensional quantum lattice systems: a time-dependent superoperator renormalization algorithm
• Tensor renormalization group approach to 2D classical lattice models
• DMRG and periodic boundary conditions: a quantum information perspective
• Criticality, the area law, and the computational power of PEPS
• Classical simulation of quantum many-body systems with a tree tensor network
• The iTEBD algorithm beyond unitary evolution
• Corner Transfer Matrix Renormalization Group Method
• From density-matrix renormalization group to matrix product states
• Simulation of two dimensional quantum systems on an infinite lattice revisited: corner transfer matrix for tensor contraction
• Matrix product operator representations
• Tensor Network Renormalization
• Algorithms for entanglement renormalization
• The computational complexity of PEPS
• Symmetry-protected topological orders for interacting fermions -- Fermionic topological nonlinear $\sigma$ models and a special group supercohomology theory
• Entanglement spectrum and boundary theories with projected entangled-pair states
• PEPS as ground states: degeneracy and topology
• Bethe Ansatz Solution of the Asymmetric Exclusion Process with Open Boundaries
• Tensor-entanglement renormalization group approach to 2D quantum systems
• Matrix-product-groundstates for one-dimensional spin-1 quantum antiferromagnets
• The Loop Algorithm
• Density Matrix Renormalization Group Method for 2D Classical Models
• String order and symmetries in quantum spin lattices
• Variational study of hard-core bosons in a 2-D optical lattice using Projected Entangled Pair States (PEPS)
• Simulating Strongly Correlated Quantum Systems with Tree Tensor Networks
• Stripes in the two-dimensional t-J model with infinite projected entangled-pair states
• Corner Transfer Matrix Algorithm for Classical Renormalization Group
• Representations of the quadratic Algebra and Partially Asymmetric Diffusion with Open Boundaries
• Equivalence of the Variational Matrix Product Method and the Density Matrix Renormalization Group applied to Spin Chains
• Quantum circuits for strongly correlated quantum systems
• Exact solution of a cellular automaton for traffic
• Density matrix renormalisation group for a quantum spin chain at non-zero temperature
• Two-Dimensional Tensor Product Variational Formulation
• Thermodynamics of the anisotropic Heisenberg chain calculated by the density matrix renormalization group method
• Variational matrix product ansatz for dispersion relations
• Explicit tensor network representation for the ground states of string-net models
• Implementing global Abelian symmetries in projected entangled-pair state algorithms
• Topological order in PEPS: Transfer operator and boundary Hamiltonians
• Exact Stationary State for an ASEP with Fully Parallel Dynamics
• Tensor network renormalization yields the multi-scale entanglement renormalization ansatz
• Exploring corner transfer matrices and corner tensors for the classical simulation of quantum lattice systems
• Density Matrix Renormalization Group and Reaction-Diffusion Processes
• Excitations and the tangent space of projected entangled-pair states
• Matrix Product Density Operators: Renormalization Fixed Points and Boundary Theories
• Planar lattice gases with nearest-neighbour exclusion
• Critical properties of the reaction - diffusion model 2A -> 3A, 2A ->0
• Geometry of Matrix Product States: metric, parallel transport and curvature
• Shadows of Anyons
• Matrix product operators for symmetry-protected topological phases: Gauging and edge theories
• Twisted Injectivity in PEPS and the Classification of Quantum Phases
• Real-time dynamics at finite temperature by DMRG: A path-integral approach
• Chiral topological spin liquids with projected entangled pair states
• Particles, holes and solitons: a matrix product state approach
• Scattering particles in quantum spin chains
• The Bethe ansatz as a matrix product ansatz
• Numerical Contraction of the Tensor Network generated by the Algebraic Bethe Ansatz
• The Algebraic Bethe Ansatz and Tensor Networks
• Efficient evaluation of partition functions of frustrated and inhomogeneous spin systems
• Generalization of the matrix product ansatz for integrable chains
• S matrix from matrix product states
• Derivation of Matrix Product Ansatz for the Heisenberg Chain from Algebraic Bethe Ansatz
• Stochastic Matrix Product States
• Matrix product state renormalization
• 2D Potts Model Correlation Lengths: Numerical Evidence for ${\xi }_o={\xi }_d$ at ${\beta }_t$

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• Review on novel methods for lattice gauge theories
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• Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners
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• Probing the anomalous dynamical phase in long-range quantum spin chains through Fisher-zero lines
• Non-Abelian chiral spin liquid in a quantum antiferromagnet revealed by an iPEPS study
• Quantum critical phase with infinite projected entangled paired states
• Tensor lattice field theory with applications to the renormalization group and quantum computing
• Temporal entanglement, quasiparticles and the role of interactions
• Renormalization group flows of Hamiltonians using tensor networks
• Dualities in one-dimensional quantum lattice models: topological sectors
• Conformal Boundary Conditions of Symmetry-Enriched Quantum Critical Spin Chains
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• Residual entropies for three-dimensional frustrated spin systems with tensor networks
• A scaling hypothesis for matrix product states
• Matrix product operator symmetries and intertwiners in string-nets with domain walls
• Entanglement structure of current-driven diffusive fermion systems
• Matrix Product Representation of Locality Preserving Unitaries
• Approaching the Kosterlitz-Thouless transition for the classical XY model with tensor networks
• Accurate simulation for finite projected entangled pair states in two dimensions
• Efficient MPS methods for extracting spectral information on rings and cylinders
• Study of anyon condensation and topological phase transitions from a ${\mathbb{Z}}_4$ topological phase using Projected Entangled Pair States
• Open System Tensor Networks and Kramers' Crossover for Quantum Transport
• Gauging tensor networks with belief propagation
• Variational methods for contracting projected entangled-pair states
• Entanglement order parameters and critical behavior for topological phase transitions and beyond
• Topological aspects of the critical three-state Potts model
• Tensor network approach to the two-dimensional fully frustrated XY model and a chiral ordered phase
• Solving frustrated Ising models using tensor networks
• Phase coherence of pairs of Cooper pairs as quasi-long-range order of half-vortex pairs in a two-dimensional bilayer system
• Tensor Networks with a Twist: Anyon-permuting domain walls and defects in PEPS
• Non-equilibrium dynamics of symmetry-resolved entanglement and entanglement asymmetry: Exact asymptotics in Rule 54
• Generating Function for Tensor Network Diagrammatic Summation
• Experimental realization of a quantum image classifier via tensor-network-based machine learning
• Artificial out-of-plane Ising antiferromagnet on the kagome lattice with very small further neighbour couplings
• Approximate optimization, sampling and spin-glass droplets discovery with tensor networks
• Fractional quantum Hall states with variational Projected Entangled-Pair States: a study of the bosonic Harper-Hofstadter model
• Quantum Compressed Sensing with Unsupervised Tensor-Network Machine Learning
• Kibble-Zurek mechanism in a one-dimensional incarnation of deconfined quantum critical point
• Atomic Quantum Technologies for Quantum Matter and Fundamental Physics Applications
• Efficient Quantum Simulation for Thermodynamics of Infinite-size Many-body Systems in Arbitrary Dimensions
• Tensor network approach to the fully frustrated XY model on a kagome lattice with a fractional vortex-antivortex pairing transition
• A gapped SU(3) spin liquid with Z_3 topological order
• The minimal canonical form of a tensor network
• Unified tensor network theory for frustrated classical spin models in two dimensions
• Extracting the Speed of Light from Matrix Product States
• Hybrid Berezinskii-Kosterlitz-Thouless and Ising topological phase transition in the generalized two-dimensional XY model using tensor networks
• Gapped $Z_2$ spin liquid in the breathing kagome Heisenberg antiferromagnet
• Corner transfer matrix renormalization group approach in the zoo of Archimedean lattices
• Nonequilibrium critical dynamics in the quantum chiral clock model
• Charge-4$e$ Superconductivity in a Hubbard model
• Entanglement and the density matrix renormalisation group in the generalised Landau paradigm
• Matrix product state fixed points of non-Hermitian transfer matrices
• Exact Hidden Markovian Dynamics in Quantum Circuits
• Contrasting pseudo-criticality in the classical two-dimensional Heisenberg and ${\mathrm{RP}}^2$ models: zero-temperature phase transition versus finite-temperature crossover
• AKLT-states as ZX-diagrams: diagrammatic reasoning for quantum states
• A tensor network approach to sensing quantum light-matter interactions
• Generalized Gibbs Ensemble Description of Real Space Entanglement Spectra of (2+1)-dimensional Chiral Topological Systems with $SU(2)$ Symmetry
• Topological Spin Liquids: Robustness under perturbations
• Dual-isometric Projected Entangled Pair States
• Ashkin-Teller phase transition and multicritical behavior in a classical monomer-dimer model
• Tangent space generators of matrix product states and exact Floquet quantum scars
• String order parameters for symmetry fractionalization in an enriched toric code
• Simulating chiral spin liquids with projected entangled-pair states
• A Quantum-Inspired Algorithm for Wave Simulation Using Tensor Networks
• A Simple and General Equation for Matrix Product Unitary Generation
• Influence functional of many-body systems: temporal entanglement and matrix-product state representation
• Efficient optimization of variational tensor-network approach to three-dimensional statistical systems
• Entanglement Spectrum as a diagnostic of chirality of Topological Spin Liquids: Analysis of an $\mathrm{SU}(3)$ PEPS
• From gauging to duality in one-dimensional quantum lattice models
• Emergent Intermediate Phase in the $J_1$-$J_2$ XY model from Tensor Network Approaches
• Les Houches Lecture Notes on Tensor Networks
• Two-stage melting of an inter-component Potts long-range order in two dimensions
• Variational boundary based tensor network renormalization group
• Efficient numerical evaluation of thermodynamic quantities on infinite (semi-)classical chains
• Superfluid dome in the spatially modulated two-dimensional XY model
• Equivalence of residual entropy of hexagonal and cubic ices from tensor network methods
• Multipartite Entanglement in Crossing the Quantum Critical Point