Out of equilibrium dynamics with Matrix Product States

Abstract

Theoretical understanding of strongly correlated systems in one spatial dimension (1D) has been greatly advanced by the density-matrix renormalization group (DMRG) algorithm, which is a variational approach using a class of entanglement-restricted states called Matrix Product States (MPSs). However, DRMG suffers from inherent accuracy restrictions when multiple states are involved due to multi-state targeting and also the approximate representation of the Hamiltonian and other operators. By formulating the variational approach of DMRG explicitly for MPSs one can avoid errors inherent in the multi-state targeting approach. Furthermore, by using the Matrix Product Operator (MPO) formalism, one can exactly represent the Hamiltonian and other operators. The MPO approach allows 1D Hamiltonians to be templated using a small set of finite state automaton rules without reference to the particular microscopic degrees of freedom. We present two algorithms which take advantage of these properties: eMPS to find excited states of 1D Hamiltonians and tMPS for the time evolution of a generic time-dependent 1D Hamiltonian. We properly account for time-ordering of the propagator such that the error does not depend on the rate of change of the Hamiltonian. Our algorithms use only the MPO form of the Hamiltonian, and so are applicable to microscopic degrees of freedom of any variety, and do not require Hamiltonian-specialized implementation. We benchmark our algorithms with a case study of the Ising model, where the critical point is located using entanglement measures. We then study the dynamics of this model under a time-dependent quench of the transverse field through the critical point. Finally, we present studies of a dipolar, or long-range Ising model, again using entanglement measures to find the critical point and study the dynamics of a time-dependent quench through the critical point.

35 pages, 10 figures, submitted to NJP for the focus issue "Out of equilibrium dynamics in strongly interacting 1D systems"

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Cited by in corpus (68)

• Time-evolution methods for matrix-product states
• Time-evolving a matrix product state with long-ranged interactions
• Kitaev chains with long-range pairing
• The mass spectrum of the Schwinger model with Matrix Product States
• Many-body localization and delocalization in large quantum chains
• Probing real-space and time resolved correlation functions with many-body Ramsey interferometry
• Time Dependent Variational Principle with Ancillary Krylov Subspace
• Tensor Network Algorithms: a Route Map
• Quantum dynamics of thermalizing systems
• Topological phases in ultracold polar-molecule quantum magnets
• Few-body Bose gases in low dimensions -- a laboratory for quantum dynamics
• Many-body delocalization dynamics in long Aubry-André quasiperiodic chains
• Numerical Assessment for Accuracy and GPU Acceleration of TD-DMRG Time Evolution Schemes
• Semi-synthetic zigzag optical lattice for ultracold bosons
• Topological phases with long-range interactions
• Slow many-body delocalization beyond one dimension
• Open source Matrix Product States: Opening ways to simulate entangled many-body quantum systems in one dimension
• Simulating generic spin-boson models with matrix product states
• A differentiable programming method for quantum control
• Dynamics of a quantum phase transition in the 1D Bose-Hubbard model: excitations and correlations induced by a quench
• Performance of the time-dependent variational principle for matrix product states in long-time evolution of a pure state
• Matrix Product Representation of Locality Preserving Unitaries
• Parallel time-dependent variational principle algorithm for matrix product states
• Optimized Lie-Trotter-Suzuki decompositions for two and three non-commuting terms
• Topological Persistence Machine of Phase Transitions
• Weak Versus Strong Disorder Superfluid-Bose Glass Transition in One Dimension
• Non-equilibrium phase transitions in $(1+1)$-dimensional quantum cellular automata with controllable quantum correlations
• Chebyshev matrix product state approach for time evolution
• Disorderless quasi-localization of polar gases in one-dimensional lattices
• Stochastic Adaptive Single-Site Time-Dependent Variational Principle
• Long-range random transverse-field Ising model in three dimensions
• Entanglement and Nonlocality in Infinite 1D Systems
• Macroscopic Quantum Tunneling Escape of Bose-Einstein Condensates
• Dynamics of interacting fermions in spin-dependent potentials
• Comparing numerical and analytical approaches to strongly interacting two-component mixtures in one dimensional traps
• Chiral Light--Matter Interaction Beyond the Rotating-Wave Approximation
• The Molecular Hubbard Hamiltonian: Field Regimes and Molecular Species
• Atomic Quantum Technologies for Quantum Matter and Fundamental Physics Applications
• Entanglement in the Quantum Game of Life
• Ultra-fast control of magnetic relaxation in a periodically driven Hubbard model
• Fast trimers in one-dimensional extended Fermi-Hubbard model
• Many-body Quantum Chaos and Entanglement in a Quantum Ratchet
• Signatures of non-trivial pairing in the quantum walk of two-component bosons
• Entangled Dynamics in Macroscopic Quantum Tunneling of Bose-Einstein Condensates
• Spin-models, dynamics and criticality with atoms in tilted optical superlattices
• Symmetries and entanglement in the one-dimensional spin-1/2 XXZ model
• Dynamical correlation functions from complex time evolution
• Magnetic phase transitions of insulating spin-orbit coupled Bose atoms in one-dimensional optical lattices
• Controllable quantum spin glasses with magnetic impurities embedded in quantum solids
• Phase-transitions of the random bond Potts chain with long-range interactions
• Random Sampling Neural Network for Quantum Many-Body Problems
• Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double-Well
• Topological Defects in Quantum Field Theory with Matrix Product States
• Simulating fermions in spin-dependent potentials with spin models on an energy lattice
• Simulating real-time molecular electron dynamics efficiently using the time-dependent density matrix renormalization group
• Bosonic molecules in a lattice: unusual fluid phase from multichannel interactions
• The Kibble Zurek Mechanism of Topological Defect Formation in Quantum Field Theory with Matrix Product States
• Efficient computations of quantum canonical Gibbs state in phase space
• Cooling Fermions in an Optical Lattice by Adiabatic Demagnetization
• Quantum tensor network algorithms for evaluation of spectral functions on quantum computers
• Realizing non-trivial doublon formation using a quantum computer
• Efficient approximation for global functions of matrix product operators
• The Software Landscape for the Density Matrix Renormalization Group
• Time evolution of controlled many-body quantum systems with matrix product operators
• Quantum batteries at the verge of a phase transition
• Locally accurate tensor networks for thermal states and time evolution
• Optimized Observable Readout from Single-shot Images of Ultracold Atoms via Machine Learning
• Manipulating multimer propagation using lattice modulation