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.

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

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Cited by in corpus (95)

• Tensor networks for complex quantum systems
• The iPEPS algorithm, improved: fast full update and gauge fixing
• Surface/State Correspondence as a Generalized Holography
• Lecture Notes of Tensor Network Contractions
• The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems
• Tree Tensor Networks for Generative Modeling
• Tensor Networks in a Nutshell
• Spin-S Kagome quantum antiferromagnets in a field with tensor networks
• Machine Learning by Unitary Tensor Network of Hierarchical Tree Structure
• A measurement-based variational quantum eigensolver
• Quantum critical phase with infinite projected entangled paired states
• Simulation complexity of open quantum dynamics: Connection with tensor networks
• Computing vibrational eigenstates with tree tensor network states (TTNS)
• Loop Quantum Gravity, Exact Holographic Mapping, and Holographic Entanglement Entropy
• T3NS: three-legged tree tensor network states
• Supervised Learning with Projected Entangled Pair States
• X-Symbols for Non-Abelian Symmetries in Tensor Networks
• Systematic construction of spin liquids on the square lattice from tensor networks with SU(2) symmetry
• Exploring the Tensor Networks/AdS Correspondence
• Conformal field theory complexity from Euler-Arnold equations
• qTorch: The Quantum Tensor Contraction Handler
• Investigation of the chiral antiferromagnetic Heisenberg model using PEPS
• Explicit Hamiltonians Inducing Volume Law for Entanglement Entropy in Fermionic Lattices
• Conformal Quasicrystals and Holography
• Tensor network and ($p$-adic) AdS/CFT
• Topological transitions from multipartite entanglement with tensor networks: a procedure for sharper and faster characterization
• Finite-temperature density-matrix renormalization group method for electron-phonon systems: Thermodynamics and Holstein-polaron spectral functions
• Measuring correlations of cold atom systems using multiple quantum probes
• A universal tensor network algorithm for any infinite lattice
• Entanglement and Complexity of Purification in (1+1)-dimensional free Conformal Field Theories
• Tensor network simulation of QED on infinite lattices: learning from (1+1)d, and prospects for (2+1)d
• A programming guide for tensor networks with global $SU(2)$ symmetry
• Entanglement-guided architectures of machine learning by quantum tensor network
• Lectures on Quantum Tensor Networks
• Comparative Study of State-of-the-Art Matrix-Product-State Methods for Lattice Models with Large Local Hilbert Spaces
• Matrix product state algorithms for Gaussian fermionic states
• The spin-1/2 Kagome XXZ model in a field: competition between lattice nematic and solid orders
• Kitaev honeycomb tensor networks: exact unitary circuits and applications
• Magnetization of the spin-1/2 Heisenberg antiferromagnet on the triangular lattice
• A tree tensor network approach to simulating Shor's algorithm
• Quantum Machine Learning Tensor Network States
• QSpace - An open-source tensor library for Abelian and non-Abelian symmetries
• Language Design as Information Renormalization
• All spin-1 topological phases in a single spin-2 chain
• Critical colored-RVB states in the frustrated quantum Heisenberg model on the square lattice
• Thermodynamics of 3D Kitaev quantum spin liquids via tensor networks
• Density of Yang-Lee zeros in the thermodynamic limit from tensor network methods
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• Minimum Cost Loop Nests for Contraction of a Sparse Tensor with a Tensor Network
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• Thermal bosons in 3d optical lattices via tensor networks
• Bosonic and fermionic Gaussian states from Kähler structures
• Quantum criticality on a chiral ladder: an $SU(2)$ iDMRG study
• Efficient calculation of three-dimensional tensor networks
• Distributed-Memory DMRG via Sparse and Dense Parallel Tensor Contractions
• Fermionic criticality is shaped by Fermi surface topology: a case study of the Tomonaga-Luttinger liquid
• Holographic encoding of universality in corner spectra
• The Hubbard model on the Bethe lattice via variational uniform tree states: metal-insulator transition and a Fermi liquid
• Symmetry-protected intermediate trivial phases in quantum spin chains
• 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
• Holographic Foliations: Self-Similar Quasicrystals from Hyperbolic Honeycombs
• Properties of the simplest inhomogeneous and homogeneous Tree-Tensor-States for Long-Ranged Quantum Spin Chains with or without disorder
• Kitaev honeycomb antiferromagnet in a field: quantum phase diagram for general spin
• Note on quantum entanglement and quantum geometry
• Quantum Gauge Networks: A New Kind of Tensor Network
• 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
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• Dimension truncation for open quantum systems in terms of tensor networks
• Construction of Many-Body-Localized Models where all the eigenstates are Matrix-Product-States
• Improved Matrix Product Operator Renormalization Group: application to the N-color random Ashkin-Teller chain
• 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
• 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
• The Software Landscape for the Density Matrix Renormalization Group
• On the simplest scale invariant Tree-Tensor-States preserving the quantum symmetries of the antiferromagnetic XXZ chain
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• Implementing the sine transform of fermionic modes as a tensor network
• Entanglement in simple spin networks with a boundary
• Committor functions via tensor networks
• Representation Theorem for Matrix Product States
• Sparse Hierarchical Tucker Factorization and its Application to Healthcare
• Explicit properties of the simplest inhomogeneous Matrix-Product-State including the Riemann metric of the MPS manifold
• Multi-orbital dynamical mean-field theory with a complex-time solver
• Grassmann tensor networks
• Leaf-to-leaf distances in Catalan trees