Developments in the Tensor Network -- from Statistical Mechanics to Quantum Entanglement

Abstract

Tensor networks (TNs) have become one of the most essential building blocks for various fields of theoretical physics such as condensed matter theory, statistical mechanics, quantum information, and quantum gravity. This review provides a unified description of a series of developments in the TN from the statistical mechanics side. In particular, we begin with the variational principle for the transfer matrix of the 2D Ising model, which naturally leads us to the matrix product state (MPS) and the corner transfer matrix (CTM). We then explain how the CTM can be evolved to such MPS-based approaches as density matrix renormalization group (DMRG) and infinite time-evolved block decimation. We also elucidate that the finite-size DMRG played an intrinsic role for incorporating various quantum information concepts in subsequent developments in the TN. After surveying higher-dimensional generalizations like tensor product states or projected entangled pair states, we describe tensor renormalization groups (TRGs), which are a fusion of TNs and Kadanoff-Wilson type real-space renormalization groups, focusing on their fixed point structures. We then discuss how the difficulty in TRGs for critical systems can be overcome in the tensor network renormalization and the multi-scale entanglement renormalization ansatz.

34 pages, 20 figures, typos and bugs are fixed

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Cited by in corpus (64)

• Tensor Network Algorithms: a Route Map
• Nonstabilizerness via matrix product states in the Pauli basis
• Automatic structural optimization of tree tensor networks
• Nonstabilizerness in U(1) lattice gauge theory
• Real-time scattering in the lattice Schwinger model
• Optimal sampling of dynamical large deviations in two dimensions via tensor networks
• Partial lifting of degeneracy in the $J_1-J_2-J_3$ Ising antiferromagnet on the kagome lattice
• Critical endpoint of (3+1)-dimensional finite density $Z_3$ gauge-Higgs model with tensor renormalization group
• Entanglement bipartitioning and tree tensor networks
• Classical algorithms for many-body quantum systems at finite energies
• Bond-weighting method for the Grassmann tensor renormalization group
• Ground-state phase diagram of a spin-1/2 frustrated XXZ ladder
• Converting long-range entanglement into mixture: tensor-network approach to local equilibration
• Corner transfer matrix renormalization group approach in the zoo of Archimedean lattices
• Entanglement and confinement in lattice gauge theory tensor networks
• Critical line of the triangular Ising antiferromagnet in a field from a $C_3$-symmetric corner transfer matrix algorithm
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• Meson content of entanglement spectra after integrable and nonintegrable quantum quenches
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• Loschmidt echo, emerging dual unitarity and scaling of generalized temporal entropies after quenches to the critical point
• Parton Distribution Functions in the Schwinger model from Tensor Network States
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• The Software Landscape for the Density Matrix Renormalization Group
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