Advances on Tensor Network Theory: Symmetries, Fermions, Entanglement, and Holography

Abstract

This is a short review on selected theory developments on Tensor Network (TN) states for strongly correlated systems. Specifically, we briefly review the effect of symmetries in TN states, fermionic TNs, the calculation of entanglement Hamiltonians from Projected Entangled Pair States (PEPS), and the relation between the Multi-scale Entanglement Renormalization Ansatz (MERA) and the AdS/CFT or gauge/gravity duality. We stress the role played by entanglement in the emergence of several physical properties and objects through the TN language. Some recent results along these lines are also discussed.

Invited Colloquium for EPJB, 31 pages, 13 figures; revised version, accepted for publication

References in corpus (36)

• 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
• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• 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
• Accurate determination of tensor network state of quantum lattice models in two dimensions
• The iTEBD algorithm beyond unitary evolution
• From density-matrix renormalization group to matrix product states
• Tensor network states and algorithms in the presence of a global U(1) symmetry
• Continuous Matrix Product States for Quantum Fields
• String order and symmetries in quantum spin lattices
• Simulating Strongly Correlated Quantum Systems with Tree Tensor Networks
• Robustness of a perturbed topological phase
• Variational quantum Monte Carlo simulations with tensor-network states
• Efficient Tree Tensor Network States (TTNS) for Quantum Chemistry: Generalizations of the Density Matrix Renormalization Group Algorithm
• Resonating valence bond states in the PEPS formalism
• Strings, Projected Entangled Pair States, and variational Monte Carlo methods
• Implementing global Abelian symmetries in projected entangled-pair state algorithms
• Fermionic Quantum Critical Point of Spinless Fermions on a Honeycomb Lattice
• Entanglement renormalization and gauge symmetry
• Matrix Product States: Symmetries and Two-Body Hamiltonians
• Complete-Graph Tensor Network States: A New Fermionic Wave Function Ansatz for Molecules
• Unconstrained Tree Tensor Network: An adaptive gauge picture for enhanced performance
• Simulation of anyons with tensor network algorithms
• Fermionic Implementation of Projected Entangled Pair States Algorithm
• Area law and vacuum reordering in harmonic networks
• Exploiting translational invariance in Matrix Product State simulations of spin chains with periodic boundary conditions
• Symmetries and boundary theories for chiral Projected Entangled Pair States
• Fate of the cluster state on the square lattice in a magnetic field
• Time evolution of projected entangled pair states in the single layer picture
• RVB superconductors with fermionic projected entangled pair states
• Entanglement, Tensor Networks and Black Hole Horizons
• Topological Minimally Entangled States via Geometric Measure

Cited by in corpus (53)

• Surface/State Correspondence as a Generalized Holography
• Tree Tensor Networks for Generative Modeling
• Tensor Networks in a Nutshell
• A measurement-based variational quantum eigensolver
• Loop Quantum Gravity, Exact Holographic Mapping, and Holographic Entanglement Entropy
• Supervised Learning with Projected Entangled Pair States
• Systematic construction of spin liquids on the square lattice from tensor networks with SU(2) symmetry
• Conformal field theory complexity from Euler-Arnold equations
• Investigation of the chiral antiferromagnetic Heisenberg model using PEPS
• Explicit Hamiltonians Inducing Volume Law for Entanglement Entropy in Fermionic Lattices
• Topological transitions from multipartite entanglement with tensor networks: a procedure for sharper and faster characterization
• Measuring correlations of cold atom systems using multiple quantum probes
• Finite-temperature density-matrix renormalization group method for electron-phonon systems: Thermodynamics and Holstein-polaron spectral functions
• Entanglement and Complexity of Purification in (1+1)-dimensional free Conformal Field Theories
• Lectures on Quantum Tensor Networks
• Comparative Study of State-of-the-Art Matrix-Product-State Methods for Lattice Models with Large Local Hilbert Spaces
• The spin-1/2 Kagome XXZ model in a field: competition between lattice nematic and solid orders
• Magnetization of the spin-1/2 Heisenberg antiferromagnet on the triangular lattice
• A tree tensor network approach to simulating Shor's algorithm
• QSpace - An open-source tensor library for Abelian and non-Abelian symmetries
• Thermodynamics of 3D Kitaev quantum spin liquids via tensor networks
• All spin-1 topological phases in a single spin-2 chain
• Bosonic and fermionic Gaussian states from Kähler structures
• Holographic unitary renormalization group for correlated electrons -- II: insights on fermionic criticality
• Minimum Cost Loop Nests for Contraction of a Sparse Tensor with a Tensor Network
• Thermal bosons in 3d optical lattices via tensor networks
• Efficient calculation of three-dimensional tensor networks
• Fermionic criticality is shaped by Fermi surface topology: a case study of the Tomonaga-Luttinger liquid
• Distributed-Memory DMRG via Sparse and Dense Parallel Tensor Contractions
• The Hubbard model on the Bethe lattice via variational uniform tree states: metal-insulator transition and a Fermi liquid
• The Green's function Monte Carlo combined with projected entangled pair state approach to the frustrated $J_1$-$J_2$ Heisenberg model
• Entanglement properties of the two-dimensional SU(3) AKLT state
• Variational quantum eigensolver with embedded entanglement using a tensor-network ansatz
• Quantum-Classical Machine learning by Hybrid Tensor Networks
• Probabilistic Inference in the Era of Tensor Networks and Differential Programming
• Properties of the simplest inhomogeneous and homogeneous Tree-Tensor-States for Long-Ranged Quantum Spin Chains with or without disorder
• Holographic Foliations: Self-Similar Quasicrystals from Hyperbolic Honeycombs
• Kitaev honeycomb antiferromagnet in a field: quantum phase diagram for general spin
• Long-range spin-orbital order in the spin-orbital SU(2)$\times$SU(2)$\times$U(1) model
• Automatic Structural Search of Tensor Network States including Entanglement Renormalization
• Dynamical quantum phase transition and thermal equilibrium in the lattice Thirring model
• Quantum Gauge Networks: A New Kind of Tensor Network
• Dimension truncation for open quantum systems in terms of tensor networks
• The Software Landscape for the Density Matrix Renormalization Group
• Representation Theorem for Matrix Product States
• Committor functions via tensor networks
• A hybrid method integrating Green's function Monte Carlo and projected entangled pair states
• Revealing quantum phase string effect in doped Mott-insulator: a tensor network state approach
• Complexity for deep neural networks and other characteristics of deep feature representations
• Competing pair density wave orders in the square lattice $t$-$J$ model
• Quantum phase diagram of the spin-$\frac{1}{2}$ Heisenberg antiferromagnet on the square-kagome lattice: a tensor network study
• Grassmann tensor networks
• Multi-orbital dynamical mean-field theory with a complex-time solver