Matrix product state renormalization

Abstract

The truncation or compression of the spectrum of Schmidt values is inherent to the matrix product state (MPS) approximation of one-dimensional quantum ground states. We provide a renormalization group picture by interpreting this compression as an application of Wilson's numerical renormalization group along the imaginary time direction appearing in the path integral representation of the state. The location of the physical index is considered as an impurity in the transfer matrix and static MPS correlation functions are reinterpreted as dynamical impurity correlations. Coarse-graining the transfer matrix is performed using a hybrid variational ansatz based on matrix product operators, combining ideas of MPS and the multi-scale entanglement renormalization ansatz. Through numerical comparison with conventional MPS algorithms, we explicitly verify the impurity interpretation of MPS compression, as put forward by [V. Zauner et al., New J. Phys. 17, 053002 (2015)] for the transverse-field Ising model. Additionally, we motivate the conceptual usefulness of endowing MPS with an internal layered structure by studying restricted variational subspaces to describe elementary excitations on top of the ground state, which serves to elucidate a transparent renormalization group structure ingrained in MPS descriptions of ground states.

15 pages, 10 figures, published version

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Cited by in corpus (15)

• Loop optimization for tensor network renormalization
• Isometric Tensor Network States in Two Dimensions
• Diagonalizing transfer matrices and matrix product operators: a medley of exact and computational methods
• Renormalization of tensor networks using graph independent local truncations
• Probing the anomalous dynamical phase in long-range quantum spin chains through Fisher-zero lines
• Precise extrapolation of the correlation function asymptotics in uniform tensor network states with application to the Bose-Hubbard and XXZ models
• A scaling hypothesis for matrix product states
• Local scale transformations on the lattice with tensor network renormalization
• Condensation-Driven Phase Transitions in Perturbed String Nets
• Holographic quantum simulation of entanglement renormalization circuits
• Transport theory for femtosecond laser-induced spin-transfer torques
• Exact holographic tensor networks for the Motzkin spin chain
• Extracting the Speed of Light from Matrix Product States
• Entanglement compression in scale space: from the multiscale entanglement renormalization ansatz to matrix product operators
• Self-congruent point in critical matrix product states: An effective field theory for finite-entanglement scaling