In a recent paper Coldea et al (2010 Science 327 177) report observations of the weak confinement of kinks in the Ising spin chain ferromagnet CoNb 2 O 6 at low temperatures. To interpret the entire spectra of magnetic excitations measured via neutron scattering, they introduce a phenomenological model, which takes into account only the two-kink configurations of the spin chain. We present the exact solution of this model. The explicit expressions for the two-kink bound-state energy spectra and for the relative intensities of neutron scattering on these magnetic modes are obtained in terms of the Bessel function.
In a recent paper Coldea et al (2010 Science 327 177) report observations of the weak confinement of kinks in the Ising spin chain ferromagnet CoNb 2 O 6 at low temperatures. To interpret the entire spectra of magnetic excitations measured via neutron scattering, they introduce a phenomenological model, which takes into account only the two-kink configurations of the spin chain. We present the exact solution of this model. The explicit expressions for the two-kink bound-state energy spectra and for the relative intensities of neutron scattering on these magnetic modes are obtained in terms of the Bessel function.
Kink bound states in the one-dimensional ferromagnetic Ising chain compound CoNb2O6 have been studied using high-resolution time-domain terahertz spectroscopy in zero applied magnetic field. When magnetic order develops at low temperature, nine bound states of kinks become visible. Their energies can be modeled exceedingly well by the Airy function solutions to a 1D Schrödinger equation with a linear confining potential. This sequence of bound states terminates at a threshold energy near 2 times the energy of the lowest bound state. Above this energy scale we observe a broad feature consistent with the onset of the two particle continuum. At energies just below this threshold we observe a prominent excitation that we interpret as a novel bound state of bound states--two pairs of kinks on neighboring chains.
Abstract With recent advances in terahertz (THz) sources and detection, two-dimensional coherent spectroscopy (2DCS), which allows one to probe nonlinear responses, now reaches the meV regime relevant for quasiparticle excitations in magnetic materials. This opens a promising route to reveal many-body phenomena that evade linear-response probes. To date, most experimental applications have focused on classical magnets, and a solid demonstration in a quantum magnet has yet to be established. Here we present a theoretical study of 2DCS in CoNb 2 O 6 , a quasi-one-dimensional Ising magnet that is believed to host fractionalized spinons which at low temperatures are confined by weak interchain coupling. Our analysis, which builds on an experimentally constrained effective S = 1/2 Hamiltonian, is found to reveal unambiguous 2DCS signatures of spinon deconfinement above the low-temperature ordered phase. Using a four-spinon approximation, we track these 2DCS signatures by sequentially building a faithful microscopic model for CoNb 2 O 6 , starting from the exactly solvable one-dimensional transverse-field Ising model (1 d TFIM) and successively adding additional interactions. In particular, adding a bond-dependent staggered YZ interaction to the 1 d -TFIM already reproduces many key spectral features of the full material Hamiltonian. Within this TFIM+YZ model, we find a series of bound states, including a four-spinon bound state that is distinct from the familiar two-spinon bound states. We further find that introducing a confinement potential suppresses sharp spinon-echo features, which are thought to reflect an underlying continuum of fractionalized excitations. Our results provide concrete predictions for future THz 2DCS experiments on CoNb 2 O 6 and related quasi-one-dimensional quantum magnets.
The Heisenberg XXZ spin-(1/2) chain is considered in the massive antiferromagnetic regime in the presence of a staggered longitudinal magnetic field. The Hamiltonian of the model is characterised by the anisotropy parameter and by the magnetic-field strength h . At zero magnetic field, the model is exactly solvable. In the thermodynamic limit, it has two degenerate vacua and the kinks (which are also called spinons) interpolating between these vacua, as elementary excitations. Application of the staggered magnetic field breaks integrability of the model and induces the long-range attractive potential between two adjacent kinks leading to their confinement into the bound states. The energy spectra of the resulting two-kink bound states are perturbatively calculated in the extreme anisotropic (Ising) limit to the first order in the inverse anisotropy constant , and also for any to the first order in the weak magnetic field h .
We present experimental and theoretical evidence of novel bound state formation in the low transverse field ordered phase of the quasi-one-dimensional Ising-like material CoNb2O6. High-resolution single-crystal inelastic neutron scattering measurements observe that small transverse fields lead to a breakup of the spectrum into three parts, each evolving very differently upon increasing field. This can be naturally understood starting from the excitations of the ordered phase of the transverse field Ising model, domain wall quasiparticles (solitons). Here, the transverse field and a staggered off-diagonal exchange create one-soliton hopping terms with opposite signs. We show that this leads to a rich spectrum and a special field, when the strengths of the off-diagonal exchange and transverse field match, at which solitons become localized; the highest field investigated is very close to this special regime. We solve this case analytically and find three two-soliton continua, along with three novel bound states. Perturbing away from this novel localized limit, we find very good qualitative agreement with the experimental data. We also present calculations using exact diagonalization of a recently refined Hamiltonian model for CoNb2O6 and using diagonalization of the two-soliton subspace, both of which provide a quantitative agreement with the observed spectrum. The theoretical models qualitatively and quantitatively capture a variety of nontrivial features in the observed spectrum, providing insight into the underlying physics of bound state formation.
For many years, CoNb_{2}O_{6} has served as an exemplar of the one-dimensional Ising model. However, recent experimental and theoretical analyses challenge its applicability to this material. Prior to that, a tailored spin model for 3d^{7} systems such as Co^{2+}, known as the JKΓ model, has emerged, featuring Heisenberg (J), Kitaev (K), and Gamma (Γ) interactions. While these interactions are permitted by the symmetry of the system, their role in CoNb_{2}O_{6} remains enigmatic. We present a microscopic theory based on spin-orbit entangled J_{eff}=1/2 states, aimed at elucidating the roles of Kitaev and Gamma interactions in shaping Ising anisotropy. Leveraging strong coupling theory, we identify a dominant ferromagnetic Kitaev interaction. Furthermore, by comparing dynamical structure factors obtained via exact diagonalization with those from inelastic neutron scattering experiments, we find an antiferromagnetic Γ interaction, which dictates the Ising axis and explains the mechanism behind moment pinning. Our theory provides a microscopic origin for Ising behavior in spin-orbit coupled one-dimensional chains and posits CoNb_{2}O_{6} as a rare Kitaev chain.
The infinite Heisenberg XXZ spin-21 chain in the gapped antiferromagnetic regime has two degenerate vacua and kink topological excitations (which are also called spinons) interpolating between these vacua as elementary excitations. Application of an arbitrary weak staggered longitudinal magnetic field h induces a long-range attractive potential between two adjacent spinons leading to their confinement into ``meson'' bound states. Exploiting the integrability of the XXZ model in the deconfined phase h=0, we perform perturbative calculations of the energy spectra of the two-spinon bound states in the weak confinement regime at $h\ensuremath{\rightarrow}+0$, using the strength of the staggered magnetic field h as a small parameter. Both transverse and longitudinal dynamical structure factors of the local spin operators are calculated as well in the two-spinon approximation in the weak confinement regime to the leading order in h.