Abstract
We combine the Density Matrix Renormalization Group (DMRG) with Matrix Product State tangent space concepts to construct a variational algorithm for finding ground states of one dimensional quantum lattices in the thermodynamic limit. A careful comparison of this variational uniform Matrix Product State algorithm (VUMPS) with infinite Density Matrix Renormalization Group (IDMRG) and with infinite Time Evolving Block Decimation (ITEBD) reveals substantial gains in convergence speed and precision. We also demonstrate that VUMPS works very efficiently for Hamiltonians with long range interactions and also for the simulation of two dimensional models on infinite cylinders. The new algorithm can be conveniently implemented as an extension of an already existing DMRG implementation.
Highlights
The strategy of renormalization group (RG) techniques to successively reduce a large number of microscopic degrees of freedom to a smaller set of effective degrees of freedom has led to powerful numerical and analytical methods to probe and understand the effective macroscopic behavior of both classical and quantum many-body systems [1,2,3,4]
For (d) in particular, the final energy error obtained by infinite density matrix renormalization group (IDMRG) is still almost 10% higher than the value obtained by variational uniform matrix product state algorithm (VUMPS)
We have introduced a novel algorithm for calculating matrix product states (MPS) ground-state approximations of strongly correlated onedimensional quantum lattice models with nearest-neighbor or long-range interactions, in the thermodynamic limit
Summary
The strategy of renormalization group (RG) techniques to successively reduce a large number of microscopic degrees of freedom to a smaller set of effective degrees of freedom has led to powerful numerical and analytical methods to probe and understand the effective macroscopic behavior of both classical and quantum many-body systems [1,2,3,4]. The underlying variational Ansatz of matrix product states (MPS) [7,8,9,10,11,12,13] belongs to a class of Ansätze known as tensor network states [11,14,15] These variational classes encode the many-body wave function in terms of virtual entanglement degrees of freedom living on the boundary and satisfy an area law scaling of entanglement entropy per construction. We present a new variational algorithm, inspired by tangent space ideas [13,27,28], that combines the advantages of IDMRG and ITEBD and addresses some of their shortcomings As such it is directly formulated in the thermodynamic limit, but at the same time optimizes the state by solving effective eigenvalue problems, rather than employing imaginary-time evolution. These involve infinite geometric sums of the transfer matrix, which are further studied in Appendix D
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