Transfer Matrices and Excitations with Matrix Product States

Abstract

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.

arXiv comment39 pages (+8 pages appendix), 14 figures

References in corpus (40)

• 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 renormalization
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• Time-dependent variational principle for quantum lattices
• Matrix product states represent ground states faithfully
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• Corner Transfer Matrix Renormalization Group Method
• Lieb-Robinson Bounds and the Exponential Clustering Theorem
• Matrix product operator representations
• Algorithms for entanglement renormalization
• Continuous Matrix Product States for Quantum Fields
• Density Matrix Renormalization Group Method for 2D Classical Models
• String order and symmetries in quantum spin lattices
• N-representability is QMA-complete
• Locality in Quantum and Markov Dynamics on Lattices and Networks
• Entanglement renormalization, scale invariance, and quantum criticality
• Physical Meaning of the Affleck-Kennedy-Lieb-Tasaki S=1 Quantum spin chain
• Density matrix renormalisation group for a quantum spin chain at non-zero temperature
• Quantum phase transitions in matrix product systems
• Thermodynamics of the anisotropic Heisenberg chain calculated by the density matrix renormalization group method
• Variational matrix product ansatz for dispersion relations
• Matrix product states for critical spin chains: finite size scaling versus finite entanglement scaling
• A Density Matrix Algorithm for 3D Classical Models
• Effect of disorder on quantum phase transitions in anisotropic XY spin chains in a transverse field
• Elementary excitations in gapped quantum spin systems
• Calculus of continuous matrix product states
• Equal Time Correlations in Haldane Gap Antiferromagnets
• Holographic quantum states
• Real-time dynamics at finite temperature by DMRG: A path-integral approach
• Particles, holes and solitons: a matrix product state approach
• Algorithms for entanglement renormalization: boundaries, impurities and interfaces
• Boundary quantum critical phenomena with entanglement renormalization
• Temperature driven crossover phenomena in the correlation lengths of the one-dimensional t-J model
• Spectral gaps of AKLT Hamiltonians using Tensor Network methods
• S matrix from matrix product states
• Onset of incommensurability in quantum spin chains
• Incommensurability and edge states in the one-dimensional S=1 bilinear-biquadratic model
• Long-time dynamics of quantum chains: transfer-matrix renormalization group and entanglement of the maximal eigenvector
• Thermodynamic properties and thermal correlation lengths of a Hubbard model with bond-charge interaction

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