Magnetic Resonance, Index Compression Maps and the Holstein-Primakoff Bosons: Towards a Polynomially Scaling Exact Diagonalization of Isotropic Multispin Hamiltonians

TL;DR

This paper introduces a novel approach combining Holstein-Primakoff bosons and index compression maps to achieve polynomially scaling exact diagonalization of isotropic multispin Hamiltonians, overcoming the exponential growth challenge.

Contribution

The authors develop a systematic method that uses index compression maps and bosonic representations to block-diagonalize multispin Hamiltonians, significantly reducing computational complexity.

Findings

Exact diagonalization of multispin Hamiltonians with reduced computational cost

Analytical characterization of multispin Hilbert space

Method applicable to various finite quantum systems

Abstract

Matrix diagonalization has long been a setback in the numerical simulation of the magnetic resonance spectra of multispin systems since the dimension of the Hilbert space of such systems grows exponentially with the number of spins -- a problem commonly referred to as the "curse of dimensionality". In this paper, we propose two mathematical instruments which, when harmoniously combined, could greatly help surmount to a fair degree and in a systematic manner the curse of dimensionality. These are: 1) the Holstein-Primakoff bosons and 2) what we have termed the "index compression maps". These two allow a bijective mapping of (multi)spin states to integers. Their combination leads to the block diagonalization of the multispin Hamiltonian, thus a computationally exact way of diagonalizing the latter but which also reduces significantly the computational cost. We also show that the eigenvectors and eigenvalues of the Liouvillian operator can be easily determined once those of the related multispin Hamiltonian are known. Interestingly, the method also enables an analytical characterization of the multispin Hilbert space -- a feat hardly attainable with other approaches. We illustrate the method here by showing how a general static isotropic multispin Hamiltonian could be exactly diagonalized with very less computational cost. Nonetheless, we emphasize that the method could be applied to study numerous quantum systems defined on finite Hilbert spaces and embodied with at most pairwise interactions.