Generalized Lanczos method for systematic optimization of tensor network states

Abstract

We propose a generalized Lanczos method to generate the many-body basis states of quantum lattice models using tensor-network states (TNS). The ground-state wave function is represented as a linear superposition composed from a set of TNS generated by Lanczos iteration. This method improves significantly both the accuracy and the efficiency of the tensor-network algorithm and allows the ground state to be determined accurately using TNS with very small virtual bond dimensions. This state contains significantly more entanglement than each individual TNS, reproducing correctly the logarithmic size dependence of the entanglement entropy in a critical system. The method can be generalized to non-Hamiltonian systems and to the calculation of low-lying excited states, dynamical correlation functions, and other physical properties of strongly correlated systems.

Emergent symmetry and conserved current at a one-dimensional incarnation of deconfined quantum critical point

“…The MPS [72][73][74][75] represents the quantum many-body state in a chain of D × D matrices multiplied together, whose entanglement is bounded by the bond dimension D. The MPS can provide very efficient representations for gapped quantum many-body states in 1D, due to the area-law scaling of entanglement. However for gapless states at quantum critical points, the entanglement scales logarithmically, which generally requires tensor network [76,77] with more complicated structure but more powerful expressing capability to represent the state, but the computational cost for those tensor networks will also be much higher.…”

Direct solution of multiple excitations in a matrix product state with block Lanczos

“…The strategy used in Ref. 19 is to attach a single index to the orthogonality center which labels the wavefunctions. This effectively bundles states together to make a bundled MPS.…”

“…Many tens or hundreds of excitations have been solved in the models we investigate with only a moderately large bond dimension. The advantage of directly infusing the DMRG algorithm with block Lanczos will avoid the issues with solving a generic cost function to obtain the excitations [16,17,19].…”

Reorthonormalization of Chebyshev matrix product states for dynamical correlation functions

“…In this method, the Chebyshev expansion is applied twice. We first carry out a CheMPS calculation to obtain a set of Chebyshev vectors represented using MPS, and then reorthonormalize these MPS to obtain a set of many-body basis states 25 . Within the truncated Hilbert space spanned by this set of basis states, we re-diagonalize the Hamiltonian and evaluate the spectral function, using the Chebyshev expansion again.…”