Tangent-space methods for uniform matrix product states

Abstract

In these lecture notes we give a technical overview of tangent-space methods for matrix product states in the thermodynamic limit. We introduce the manifold of uniform matrix product states, show how to compute different types of observables, and discuss the concept of a tangent space. We explain how to variationally optimize ground-state approximations, implement real-time evolution and describe elementary excitations for a given model Hamiltonian. Also, we explain how matrix product states approximate fixed points of one-dimensional transfer matrices. We show how all these methods can be translated to the language of continuous matrix product states for one-dimensional field theories. We conclude with some extensions of the tangent-space formalism and with an outlook to new applications.

scipost resubmission

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
• Real time evolution using the density matrix renormalization group
• Matrix Product Density Operators: Simulation of finite-T and dissipative systems
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• Accurate determination of tensor network state of quantum lattice models in two dimensions
• The iTEBD algorithm beyond unitary evolution
• Continuous Matrix Product States for Quantum Fields
• Entanglement entropy for the long range Ising chain
• String order and symmetries in quantum spin lattices
• Prethermalization and Persistent Order in the Absence of a Thermal Phase Transition
• Fermionic Matrix Product States and One-Dimensional Topological Phases
• Holographic quantum states
• Transfer Matrices and Excitations with Matrix Product States
• Error estimates for extrapolations with matrix-product states
• Continuous matrix product states with periodic boundary conditions and an application to atomtronics
• Symmetry Breaking and Convex Set Phase Diagrams for the q-state Potts Model

Cited by in corpus (164)

• The ITensor Software Library for Tensor Network Calculations
• Time-evolution methods for matrix-product states
• Tensor networks for complex quantum systems
• The Density Matrix Renormalization Group in Chemistry and Molecular Physics: Recent Developments and New Challenges
• Variational quantum simulation of general processes
• Review on novel methods for lattice gauge theories
• Beyond-classical computation in quantum simulation
• Large Classes of Quantum Scarred Hamiltonians from Matrix Product States
• Influence matrix approach to many-body Floquet dynamics
• Witnessing quantum criticality and entanglement in the triangular antiferromagnet KYbSe${}_2$
• Simulating excitation spectra with projected entangled-pair states
• Entanglement generation in $(1+1)D$ QED scattering processes
• Riemannian optimization of isometric tensor networks
• Collisions of false-vacuum bubble walls in a quantum spin chain
• Emergent Symmetry and Conserved Current at a One Dimensional Incarnation of Deconfined Quantum Critical Point
• Symmetry breaking bias and the dynamics of a quantum phase transition
• A scaling hypothesis for matrix product states
• Quasiparticle origin of dynamical quantum phase transitions
• Quantum natural gradient generalised to noisy and non-unitary circuits
• High-Energy Collision of Quarks and Mesons in the Schwinger Model: From Tensor Networks to Circuit QED
• Modelling quantum light-matter interactions in waveguide-QED with retardation and a time-delayed feedback: matrix product states versus a space-discretized waveguide model
• Open Quantum System Dynamics from Infinite Tensor Network Contraction
• Local Matrix Product Operators: Canonical Form, Compression, & Control Theory
• Scaling of temporal entanglement in proximity to integrability
• Approaching the Kosterlitz-Thouless transition for the classical XY model with tensor networks
• Spectral function of the $J_1-J_2$ Heisenberg model on the triangular lattice
• Evidence of Dirac Quantum Spin Liquid in YbZn2GaO5
• Tensor networks for quantum machine learning
• A tensor network view of multilayer multiconfiguration time-dependent Hartree methods
• Efficient MPS methods for extracting spectral information on rings and cylinders
• Dynamical Signatures of Symmetry Broken and Liquid Phases in an $S=1/2$ Heisenberg Antiferromagnet on the Triangular Lattice
• Direct sampling of projected entangled-pair states
• Deformed Symmetry Structures and Quantum Many-body Scar Subspaces
• Automatic differentiation of dominant eigensolver and its applications in quantum physics
• Gauging tensor networks with belief propagation
• Dynamical phase transitions in quantum spin models with antiferromagnetic long-range interactions
• Variational methods for contracting projected entangled-pair states
• Controlled bond expansion for DMRG ground state search at single-site costs
• A scaling hypothesis for projected entangled-pair states
• Approximate analytical gradients and nonadiabatic couplings for the state-average density matrix renormalization group self-consistent field method
• Dynamical quantum phase transitions in spin-$S$ $\mathrm{U}(1)$ quantum link models
• Quantum kernels to learn the phases of quantum matter
• Locally accurate MPS approximations for ground states of one-dimensional gapped local Hamiltonians
• Tensor network approach to the two-dimensional fully frustrated XY model and a chiral ordered phase
• Excitation spectrum of spin-1 Kitaev spin liquids
• Solving frustrated Ising models using tensor networks
• Real-time scattering of interacting quasiparticles in quantum spin chains
• Anatomy of Dynamical Quantum Phase Transitions
• Superfluidity in the 1D Bose-Hubbard Model
• Local quench spectroscopy of many-body quantum systems
• Phase coherence of pairs of Cooper pairs as quasi-long-range order of half-vortex pairs in a two-dimensional bilayer system
• Gaussian time-dependent variational principle for the finite-temperature anharmonic lattice dynamics
• Tensor network approach to phase transitions of a non-Abelian topological phase
• Symmetric cluster expansions with tensor networks
• Spinon confinement and deconfinement in spin-1 chains
• Time-dependent variational principle with controlled bond expansion for matrix product states
• Partial lifting of degeneracy in the $J_1-J_2-J_3$ Ising antiferromagnet on the kagome lattice
• Characterization of topological phase transitions from a non-Abelian topological state and its Galois conjugate through condensation and confinement order parameters
• Complete characterization of non-Abelian topological phase transitions and detection of anyon splitting with projected entangled pair states
• Coherent Many-Body Oscillations Induced by a Superposition of Broken Symmetry States in the Wake of a Quantum Phase Transition
• Generating Function for Tensor Network Diagrammatic Summation
• Efficient variational contraction of two-dimensional tensor networks with a non-trivial unit cell
• Towards a variational Jordan-Lee-Preskill quantum algorithm
• Vaporization dynamics of a dissipative quantum liquid
• Emergent conformal boundaries from finite-entanglement scaling in matrix product states
• Fermionic tensor network methods
• Continuous phase transition between Neel and valence bond solid phases in a J-Q-like spin ladder system
• Fractional quantum Hall states with variational Projected Entangled-Pair States: a study of the bosonic Harper-Hofstadter model
• Non-Hermitian effects of the intrinsic signs in topologically ordered wavefunctions
• Generating function for projected entangled-pair states
• Tensor network approach to the fully frustrated XY model on a kagome lattice with a fractional vortex-antivortex pairing transition
• Quantum Annealing for Neural Network optimization problems: a new approach via Tensor Network simulations
• Relativistic continuous matrix product states for quantum fields without cutoff
• Strong-to-weak spontaneous symmetry breaking meets average symmetry-protected topological order
• Projector formalism for kept and discarded spaces of matrix product states
• Competing orders in the honeycomb lattice $t$-$J$ model
• Divergent nonlinear response from quasiparticle interactions
• Matrix Product States with Backflow correlations
• Symmetry-protected topological phases and competing orders in a spin-1/2 XXZ ladder with a four-spin interaction
• Hybrid Berezinskii-Kosterlitz-Thouless and Ising topological phase transition in the generalized two-dimensional XY model using tensor networks
• Converting long-range entanglement into mixture: tensor-network approach to local equilibration
• Extracting the Speed of Light from Matrix Product States
• Unified tensor network theory for frustrated classical spin models in two dimensions
• Quantifying Rare Events in Stochastic Reaction-Diffusion Dynamics Using Tensor Networks
• Reliability of lattice gauge theories in the thermodynamic limit
• Charge-4$e$ Superconductivity in a Hubbard model
• Symmetry-protected Topological Phases in Spinful Bosons with a Flat Band
• Theory of charge-6e condensed phase in Kagome lattice superconductors
• Dynamical Spectral Response of Fractonic Quantum Matter
• Efficient Representation of Minimally Entangled Typical Thermal States in two dimensions via Projected Entangled Pair States
• Infinite Grassmann Time-Evolving Matrix Product Operator Method in the Steady State
• Differentiable programming tensor networks for Kitaev magnets
• Large-Scale $2+1$D $\mathrm{U}(1)$ Gauge Theory with Dynamical Matter in a Cold-Atom Quantum Simulator
• Matrix product state fixed points of non-Hermitian transfer matrices
• Multi-impurity method for the bond-weighted tensor renormalization group
• Dynamical correlation functions from complex time evolution
• Quantum spin spiral ground state of the ferrimagnetic sawtooth chain
• Real-Time Scattering in Ising Field Theory using Matrix Product States
• Rényi free energy and variational approximations to thermal states
• Time-dependent variational principle of mixed matrix product states in the thermodynamic limit
• Contrasting pseudo-criticality in the classical two-dimensional Heisenberg and ${\mathrm{RP}}^2$ models: zero-temperature phase transition versus finite-temperature crossover
• Higher Berry Curvature from the Wave Function I: Schmidt Decomposition and Matrix Product States
• Momentum-resolved time evolution with matrix product states
• Variational method in relativistic quantum field theory without cutoff
• Detecting emergent continuous symmetries at quantum criticality
• Double-bracket quantum algorithms for quantum imaginary-time evolution
• Tensor-network approach to thermalization in open quantum many-body systems
• Magnon Spectra of Cuprates beyond Spin Wave Theory
• 2D excitation information by MPS method on infinite helixes
• The Haldane gap in the SU(3) [3 0 0] Heisenberg chain
• Chiral spin liquid in a ${\mathbb{Z}}_3$ Kitaev model
• Triggering Boundary Phase Transitions through Bulk Measurements in 2D Cluster States
• Tangent space generators of matrix product states and exact Floquet quantum scars
• Ashkin-Teller phase transition and multicritical behavior in a classical monomer-dimer model
• Lattice regularisation and entanglement structure of the Gross-Neveu model
• Continuous matrix-product states in inhomogeneous systems with long-range interactions
• Simulating chiral spin liquids with projected entangled-pair states
• Extracting Off-Diagonal Order from Diagonal Basis Measurements
• Spectral gaps of two- and three-dimensional many-body quantum systems in the thermodynamic limit
• Continuous Wigner-Mott transition at $\nu =1/5$
• Efficient and systematic calculation of arbitrary observables for the matrix product state excitation ansatz
• Two Dimensional Isometric Tensor Networks on an Infinite Strip
• Efficient optimization and conceptual barriers in variational finite Projected Entangled-Pair States
• Dynamics of two-dimensional open quantum lattice models with tensor networks
• Bang-bang preparation of quantum many-body ground states in two dimensions: optimization of the algorithm with a two-dimensional tensor network
• A New Framework for Quantum Phases in Open Systems: Steady State of Imaginary-Time Lindbladian Evolution
• Equivalence checking of quantum circuits via intermediary matrix product operator
• Efficient iPEPS Simulation on the Honeycomb Lattice via QR-based CTMRG
• Entanglement Renormalization of the class of Continuous Matrix Product States
• Efficient optimization of variational tensor-network approach to three-dimensional statistical systems
• Macroproperties vs. Microstates in the Classical Simulation of Critical Phenomena in Quench Dynamics of 1D Ising Models
• Non-Landau quantum phase transition in modulated SU(N) Heisenberg spin chains
• Cost of Locally Approximating High-Dimensional Ground States of Contextual Quantum Models
• Nonlocal order parameter of pair superfluids
• Accelerating two-dimensional tensor network optimization by preconditioning
• Prospects for NMR Spectral Prediction on Fault-Tolerant Quantum Computers
• Tangent Space Excitation Ansatz for Quantum Circuits
• Searching for superconductivity in doped triangular lattice Kitaev magnets
• Achieving the quantum field theory limit in far-from-equilibrium quantum link models
• Two-stage melting of an inter-component Potts long-range order in two dimensions
• Universal quasi-Fermi liquid physics of one-dimensional interacting fermions
• Persistent oscillation of a Cooper-pair condensate of topological defects in a nonintegrable quantum Ising chain
• Pseudospectral method for solving PDEs using Matrix Product States
• Quasi-Fermi liquid behavior in a one-dimensional system of interacting spinless fermions
• Density Matrix Renormalization Group simulations of the SU(N) Fermi-Hubbard chain implementing the full SU(N) symmetry via Semi-Standard Young Tableaux and Unitary Group Subduction Coefficients
• Emergent Intermediate Phase in the $J_1$-$J_2$ XY model from Tensor Network Approaches
• Towards Excitations and Dynamical Quantities in Correlated Lattices with Density Matrix Embedding Theory
• Stabilizer Rényi Entropy for Translation-Invariant Matrix Product States
• Variational boundary based tensor network renormalization group
• Les Houches Lecture Notes on Tensor Networks
• Superfluid dome in the spatially modulated two-dimensional XY model
• Protecting Hilbert space fragmentation through quantum Zeno dynamics
• Unconventional Spin Dynamics and Supersolid Excitations in the Triangular-Lattice XXZ Model
• Chiral Edge Excitations of $\nu =1/2$ Fractional Chern Insulators in the Bosonic Hofstadter Model
• Finding periodic orbits in projected quantum many-body dynamics
• Equivalence of residual entropy of hexagonal and cubic ices from tensor network methods
• Multi-orbital dynamical mean-field theory with a complex-time solver
• Quantum Dimension Reduction of Hidden Markov Models
• Matrix product states and first quantization
• Grassmann tensor networks
• Triangular lattice models of the Kalmeyer-Laughlin spin liquid from coupled wires
• Phases of Interacting Fibonacci Anyons on a Ladder at Half-Filling
• Quantitative Description of Strongly Correlated Materials by Combining Downfolding Techniques and Tensor Networks
• Explicit properties of the simplest inhomogeneous Matrix-Product-State including the Riemann metric of the MPS manifold