Reducing Memory Cost of Exact Diagonalization using Singular Value Decomposition

TL;DR

This paper introduces a modified Lanczos algorithm that employs singular value decomposition to significantly reduce memory usage in diagonalizing lattice Hamiltonians, especially effective for low entanglement systems, without relying on variational ansatzes.

Contribution

The authors develop a Lanczos-SVD algorithm that partitions the lattice and applies SVD at each iteration, enabling exact diagonalization with much lower memory requirements for certain quantum systems.

Findings

Successfully tested on Kagomé lattice Heisenberg models with up to 36 sites.

Uses less than 15GB of memory, outperforming traditional methods.

Convergence improves with low entanglement entropy, as predicted.

Abstract

We present a modified Lanczos algorithm to diagonalize lattice Hamiltonians with dramatically reduced memory requirements, {\em without restricting to variational ansatzes}. The lattice of size $N$ is partitioned into two subclusters. At each iteration the Lanczos vector is projected into two sets of $n_{{\rm svd}}$ smaller subcluster vectors using singular value decomposition. For low entanglement entropy $S_{ee}$, (satisfied by short range Hamiltonians), the truncation error is expected to vanish as $\exp(-n_{{\rm svd}}^{1/S_{ee}})$. Convergence is tested for the Heisenberg model on Kagom\'e clusters of 24, 30 and 36 sites, with no lattice symmetries exploited, using less than 15GB of dynamical memory. Generalization of the Lanczos-SVD algorithm to multiple partitioning is discussed, and comparisons to other techniques are given.