Construction of Variational Matrix Product States for the Heisenberg Spin-1 Chain

TL;DR

This paper introduces a simple variational matrix product state that accurately approximates the ground state of the spin-1 Heisenberg chain, generalizing the AKLT state with added dimer and trimer features.

Contribution

The authors develop a new variational MPS with bond dimension 8 that captures the ground state energy and properties of the Haldane phase more efficiently than previous methods.

Findings

Achieves ground state energy within 0.04% of exact results.

Replicates the entanglement spectrum degeneracy structure.

Matches DMRG results for correlation functions.

Abstract

We propose a simple variational wave function that captures the correct ground state energy of the spin-1 Heisenberg chain model to within 0.04\%. The wave function is written in the matrix product state (MPS) form with the bond dimension $D=8$, and characterized by three fugacity parameters. The proposed MPS generalizes the Affleck-Kennedy-Lieb-Tasaki (AKLT) state by dressing it with dimers, trimers, and general $q$-dimers. The fugacity parameters control the number and the average size of the $q$-mers. Furthermore, the $D=8$ variational MPS state captures the ground states of the entire family of bilinear-biquadratic Hamiltonian belonging to the Haldane phase to high accuracy. The 2-4-2 degeneracy structure in the entanglement spectrum of our MPS state is found to match well with the results of density matrix renormalization group (DMRG) calculation, which is computationally much heavier. Spin-spin correlation functions also find excellent fit with those obtained by DMRG.