Matrix product state renormalization

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.Comment: 15 pages, 10 figures, published versio

Faster Methods for Contracting Infinite 2D Tensor Networks

We revisit the corner transfer matrix renormalization group (CTMRG) method of Nishino and Okunishi for contracting two-dimensional (2D) tensor networks and demonstrate that its performance can be substantially improved by determining the tensors using an eigenvalue solver as opposed to the power method used in CTMRG. We also generalize the variational uniform matrix product state (VUMPS) ansatz for diagonalizing 1D quantum Hamiltonians to the case of 2D transfer matrices and discuss similarities with the corner methods. These two new algorithms will be crucial to improving the performance of variational infinite projected entangled pair state (PEPS) methods.Comment: 20 pages, 5 figures, V. Zauner-Stauber previously also published under the name V. Zaune

Transfer Matrices and Excitations with Matrix Product States

We investigate the relation between static correlation functions in the ground state of local quantum many-body Hamiltonians and the dispersion relations of the corresponding low energy excitations using the formalism of tensor network states. In particular, we show that the Matrix Product State Transfer Matrix (MPS-TM) - a central object in the computation of static correlation functions - provides important information about the location and magnitude of the minima of the low energy dispersion relation(s) and present supporting numerical data for one-dimensional lattice and continuum models as well as two-dimensional lattice models on a cylinder. We elaborate on the peculiar structure of the MPS-TM's eigenspectrum and give several arguments for the close relation between the structure of the low energy spectrum of the system and the form of static correlation functions. Finally, we discuss how the MPS-TM connects to the exact Quantum Transfer Matrix (QTM) of the model at zero temperature. We present a renormalization group argument for obtaining finite bond dimension approximations of MPS, which allows to reinterpret variational MPS techniques (such as the Density Matrix Renormalization Group) as an application of Wilson's Numerical Renormalization Group along the virtual (imaginary time) dimension of the system.Comment: 39 pages (+8 pages appendix), 14 figure

Transfer matrices and excitations with matrix product states

We use the formalism of tensor network states to investigate the relation between static correlation functions in the ground state of local quantum many-body Hamiltonians and the dispersion relations of the corresponding low-energy excitations. In particular, we show that the matrix product state transfer matrix (MPS-TM)—a central object in the computation of static correlation functions—provides important information about the location and magnitude of the minima of the low-energy dispersion relation(s), and we present supporting numerical data for one-dimensional lattice and continuum models as well as two-dimensional lattice models on a cylinder. We elaborate on the peculiar structure of the MPS-TM's eigenspectrum and give several arguments for the close relation between the structure of the low-energy spectrum of the system and the form of the static correlation functions. Finally, we discuss how the MPS-TM connects to the exact quantum transfer matrix of the model at zero temperature. We present a renormalization group argument for obtaining finite bond dimension approximations of the MPS, which allows one to reinterpret variational MPS techniques (such as the density matrix renormalization group) as an application of Wilson's numerical renormalization group along the virtual (imaginary time) dimension of the system