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arXiv:2212.03442 · Physical review. B./Physical review. B

Confinement of many-body Bethe strings

Jiahao Yang, Tao Xie, Stanislav E. Nikitin, Jianda Wu, Andrey A. Podlesnyak

Published Jul 18, 2023• 5 citations
arXivHTMLPDF

Abstract

Based on the Bethe ansatz approach and inelastic neutron scattering experiments, we reveal the evolution of confinement of many-body Bethe strings in ordered regions of the quasi-one-dimensional antiferromagnet ${\mathrm{YbAlO}}_{3}$. In the antiferromagnetic phase, the spin dynamics is dominated by confined length-1 Bethe strings, whose dominancy in the high-energy branch of the excitation spectrum yields to confined length-2 Bethe strings when the material is tuned to the spin-density-wave phase. In the thermal-induced disordered region, the confinement effect disappears, and the system restores the conventional quantum integrable physics of the one-dimensional Heisenberg model. Our results establish a unified picture based on a Bethe string for the spin dynamics in different magnetic phases of ${\mathrm{YbAlO}}_{3}$, and thus provide profound insight into many-body quantum magnetism.

DOI
10.1103/physrevb.108.l020402

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By circumventing the difficulty of obtaining exact string state solutions to Bethe ansatz equations, we devise a truncated string state space approach for investigating spin dynamics in a nonintegrable spin-12\frac{1}{2}21​ Heisenberg chain subjected to a staggered field at various magnetizations. The obtained dynamical spectra reveal a series of elastic peaks at integer multiples of the ordering wave vector QQQ, indicating the presence of multi-QQQ Bethe string states within the ground state. The spectrum exhibits a separation between different string continua as the strength of the staggered field increases at low magnetization, reflecting the confinement of the Bethe strings. This approach provides a unified string-state-based framework for understanding spin dynamics in low-dimensional nonintegrable Heisenberg models, which has a successful application to observations across various phases of the quasi-one-dimensional antiferromagnet YbAlO3{\mathrm{YbAlO}}_{3}YbAlO3​.
Quantum dynamics of strongly correlated systems is a challenging problem. Although the low energy fractional excitations of one-dimensional integrable models are often well understood, exploring quantum dynamics in these systems remains challenging in the gapless regime, especially at intermediate and high energies. Based on the algebraic Bethe ansatz formalism, we study spin dynamics in a representative one-dimensional strongly correlated model, i.e., the antiferromagnetic spin-12\frac{1}{2}21​ XXZ chain with the Ising anisotropy, via the form-factor formulas. Various excitations at different energy scales are identified crucial to the dynamic spin structure factors under the guidance of sum rules. At small magnetic polarizations, gapless excitations dominate the low energy spin dynamics arising from the magnetic-field-induced incommensurability. In contrast, spin dynamics at intermediate and high energies is characterized by the two- and three-string states, which are multiparticle excitations based on the commensurate N\'eel ordered background. Our work is helpful for experimental studies on spin dynamics in both condensed matter and cold atom systems beyond the low energy effective Luttinger liquid theory. Based on an intuitive physical picture, we speculate that the dynamic feature at high energies due to the multiparticle antibound state excitations can be generalized to nonintegrable spin systems.
The spin fluctuations parallel to the external magnetic field in the ground state of the one-dimensional (1D) s=12s=\frac{1}{2}s=21​ Heisenberg antiferromagnet are dominated by a two-parameter set of collective excitations. In a cyclic chain of N sites and magnetization 0<Mz<N/2,0<{M}_{z}<N/2,0<Mz​<N/2, the ground state, which contains 2Mz{2M}_{z}2Mz​ spinons, is reconfigured as the physical vacuum for a different species of quasiparticles, identifiable in the framework of the coordinate Bethe ansatz by characteristic configurations of Bethe quantum numbers. The dynamically dominant excitations are found to be scattering states of two such quasiparticles. For $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty},$ these collective excitations form a continuum in $(q,\ensuremath{\omega})$ space with an incommensurate soft mode. Their matrix elements in the dynamic spin structure factor ${S}_{\mathrm{zz}}(q,\ensuremath{\omega})$ are calculated directly from the Bethe wave functions for finite N. The resulting line- shape predictions for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ complement the exact results previously derived via algebraic analysis for the exact two-spinon part of ${S}_{\mathrm{zz}}(q,\ensuremath{\omega})$ in the zero-field limit. They are relevant for neutron-scattering experiments on quasi-1D antiferromagnetic compounds in a strong magnetic field.
Despite being predicted almost a century ago as the excitations of the spin-1/2 antiferromagnetic chain, Bethe strings were overshadowed by the more dominant multi-spinon-pair excitations and were only observed recently. Here we confirm their existence in the quasi-one-dimensional quantum antiferromagnet BaCo 2 V 2 O 8 using a magnetic field to separate them from the spinons. We find excellent quantitative agreement over an extended field range between these experimental results and our theoretical predictions based on the algebraic Bethe ansatz formalism. This work proves the existence of Bethe strings (multimagnon bound states) up to length-3 in BaCo 2 V 2 O 8 .
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We report a high-resolution terahertz spectroscopic study of quantum spin dynamics in the antiferromagnetic Heisenberg-Ising spin-chain compound BaCo_{2}V_{2}O_{8} as a function of temperature and longitudinal magnetic field. Confined spinon excitations are observed in an antiferromagnetic phase below T_{N}≃5.5 K. In a field-induced gapless phase above B_{c}=3.8 T, we identify many-body string excitations as well as low-energy fractional psinon or antipsinon excitations by comparing to Bethe ansatz calculations. In the vicinity of B_{c}, the high-energy string excitations are found to have a dominant contribution to the spin dynamics as compared with the fractional excitations.