Compression of Hamiltonian matrix: Application to spin-1/2 Heisenberg square lattice

Seongsoo Choi,Jongho Kim,Woohyun Kim

doi:10.1063/1.4963834

Abstract

We introduce a simple algorithm providing a compressed representation (∈ℝNorbits×Norbits×ℕNorbits) of an irreducible Hamiltonian matrix (number of magnons M constrained, dimension: Nspins!M!(Nspins−M)!&gt;Norbits) of the spin-1/2 Heisenberg antiferromagnet on the L×L non-periodic lattice, not looking for a good basis. As L increases, the ratio of the matrix dimension to Norbits converges to 8 (order of the symmetry group of square) for the exact ground state computation. The sparsity of the Hamiltonian is retained in the compressed representation. Thus, the computational time and memory consumptions are reduced in proportion to the ratio.

Highlights

IntroductionA matrix is irreducible if it is not similar via a permutation to a block upper triangular matrix.[1] Graph-theoretically, replacing non-zero entries in the matrix by one, and viewing the matrix as the adjacency matrix of a directed graph, the matrix is irreducible if and only if such directed graph is strongly connected.[2] If a symmetric structure is inherent in such an irreducible matrix, there is a block-diagonalized similar matrix having the same eigenvalues.[3] Once we construct the bases of various irreducible subspaces, low-rank matrices can be obtained, which allow us to compute all the eigenpairs
In mathematics, a matrix is irreducible if it is not similar via a permutation to a block upper triangular matrix.[1]
We show how to obtain a compressed representation of an irreducible Hamiltonian of the spin-1/2 Heisenberg antiferromagnet on the L × L lattice.[6]

Summary

Introduction

A matrix is irreducible if it is not similar via a permutation to a block upper triangular matrix.[1] Graph-theoretically, replacing non-zero entries in the matrix by one, and viewing the matrix as the adjacency matrix of a directed graph, the matrix is irreducible if and only if such directed graph is strongly connected.[2] If a symmetric structure is inherent in such an irreducible matrix, there is a block-diagonalized similar matrix having the same eigenvalues.[3] Once we construct the bases of various irreducible subspaces, low-rank matrices can be obtained, which allow us to compute all the eigenpairs It takes a great deal of effort and time to search for a good basis blockdiagonalizing such an irreducible matrix and calculate new matrix elements. In contrast to the original Hamiltonian and the irreducible matrix (sparse matrices), the low-rank block is the following dense matrix (numerically represented):[5]

Methods

Results

Conclusion