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arXiv:2206.15300 · Physical review. B./Physical review. B

Single-magnon excited states of a Heisenberg spin chain using a quantum computer

Shashank Kumar Ranu, Daniel D. Stancil

Published Nov 2, 2022• 3 citations
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Abstract

Excited states of spin chains play an important role in condensed matter physics. We present a method of calculating the single magnon excited states of the Heisenberg spin chain that can be efficiently implemented on a quantum processor for small spin chains. Our method involves finding the stationary points of the energy vs wave-number curve. We implement our method for four-site and eight-site Heisenberg Hamiltonians using numerical techniques as well as using an IBM quantum processor. Finally, we give insight into the circuit complexity and scaling of our proposed method.

DOI
10.1103/physrevb.106.184402

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Excited states of spin chains play an important role in condensed matter physics. We present a method of calculating the single magnon excited states of the Heisenberg spin chain that can be efficiently implemented on a quantum processor for small spin chains. Our method involves finding the stationary points of the energy vs wave-number curve. We implement our method for four-site and eight-site Heisenberg Hamiltonians using numerical techniques as well as using an IBM quantum processor. Finally, we give insight into the circuit complexity and scaling of our proposed method.
Few-magnon physics in the spin-SSS periodic XXZXXZXXZ chain
Few-magnon excitations in Heisenberg-like models play an important role in understanding magnetism and have long been studied by various approaches. However, the quantum dynamics of magnon excitations in a finite-size spin-SSS XXZXXZXXZ chain with single-ion anisotropy remains unsolved. Here, we exactly solve the two-magnon (three-magnon) problem in the spin-SSS XXZXXZXXZ chain by reducing the few-magnons to a fictitious single particle on a one-dimensional (two-dimensional) effective lattice. Such a mapping allows us to obtain both the static and dynamical properties of the model explicitly. The zero-energy-excitation states and various types of multimagnon bound states are manifested, with the latter being interpreted as edge states on the effective lattices. Moreover, we study the real-time multimagnon dynamics by simulating single-particle quantum walks on the effective lattices.
We demonstrate quantum computation of two-point correlation functions for a Heisenberg spin chain. Using the IBM Q 20-qubit quantum machines, we find that, for two sites, the correlation functions produce the exact results reliably. For four sites, results from the IBM Q 20-qubit Tokyo quantum computer are noisy due to read out errors and decoherence. Nevertheless, the correlation functions retain the correct spectral information. This is illustrated in the frequency domain by accurately extracting the magnon energies from peaks in the spectral function.
We present results of a numerical renormalization-group study of the isotropic S=1 Heisenberg chain. The density-matrix renormalization-group techniques used allow us to calculate a variety of properties of the chain with unprecedented accuracy. The ground-state energy per site of the infinite chain is found to be e0{\mathit{e}}_{0}e0​\ensuremath{\simeq}-1.401 484 038 971(4). Open-ended S=1 chains have effective S=1/2 spins on each end, with exponential decay of the local spin moment away from the ends, with decay length \ensuremath{\xi}\ensuremath{\simeq}6.03(1). The spin-spin correlation function also decays exponentially, and although the correlation length cannot be measured as accurately as the open-end decay length, it appears that the two lengths are identical. The string correlation function shows long-range order, with g(\ensuremath{\infty})\ensuremath{\simeq}-0.374 325 096(2). The excitation energy of the first excited state, a state with one magnon with momentum q=\ensuremath{\pi}, is the Haldane gap, which we find to be \ensuremath{\Delta}\ensuremath{\simeq}0.410 50(2). We find many low-lying excited states, including one- and two-magnon states, for several different chain lengths. The magnons have spin S=1, so the two-magnon states are singlets (S=0), triplets (S=1), and quintuplets (S=2). For magnons with momenta near \ensuremath{\pi}, the magnon-magnon interaction in the triplet channel is shown to be attractive, while in the singlet and quintuplet channels it is repulsive.
Abstract An efficient projector Monte Carlo method has been used to calculate the energies of the ground state and the first excited state, and hence the gap between them, of the spin-1 Heisenberg antiferromagnetic chains with linear sizes up to <?CDATA N=40N=40N=40?> . We have computed them on chains up to <?CDATA N=40N=40N=40?> . We have found that the size-dependence of the ground state energy per site has the form <?CDATA [e0(N)−e0]∼1/N5/2s[e_0(N)-e_0]\sim 1/N^{5/2s}[e0​(N)−e0​]∼1/N5/2s?> , which implies that there is a gap between the ground state and the first excited state. The value of the ground state energy per site as <?CDATA N→∞N\rightarrow \inftyN→∞?> is <?CDATA e0=−1.3996e_0=-1.3996e0​=−1.3996?> . The direct extrapolation of <?CDATA ΔE(N)\Delta E(N)ΔE(N)?> also shows that a gap of <?CDATA ΔE=0.36\Delta E=0.36ΔE=0.36?> exists as <?CDATA N→∞N\rightarrow \inftyN→∞?> .
We use the matrix product state formalism to construct stationary scattering states of elementary excitations in generic one-dimensional quantum lattice systems. Our method is applied to the spin-1 Heisenberg antiferromagnet, for which we calculate the full magnon-magnon S matrix for arbitrary momenta and spin, the two-particle contribution to the spectral function, and higher order corrections to the magnetization curve. As our method provides an accurate microscopic representation of the interaction between elementary excitations, we envisage the description of low-energy dynamics of one-dimensional spin chains in terms of these particlelike excitations.
A new method of determining the spectra of simple quantum-mechanical spin Hamiltonians by direct Hamiltonian-matrix diagonalization is presented. The method is illustrated by applying it to the isotropic one-dimensional Heisenberg antiferromagnet, for which calculations were performed for longer spin chains than by previous direct Hamiltonian-matrix diagonalizations. For spin s=1/2 it was possible to accurately determine the ground- and first-excited-state energies for chains of length N=32 and 64. For s=1 the longest chain for which a similar computation was performed was N=32; the singlet-triplet Haldane gap is found to be 0.421\ifmmode\pm\else\textpm\fi{}0.005 which is consistent with earlier Monte Carlo results. Long-spin-chain calculations are performed iteratively by solving smaller-chain problems, and constructing from their eigenvalues, and eigenstates, the longer-spin-chain basis and Hamiltonian matrices. The computations were performed on a small computer, which suggests that much larger problems could be handled with a parallel-processing supercomputer.