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arXiv:1312.4514 · Physical Review Letters

Hierarchy of Bound States in the One-Dimensional Ferromagnetic Ising Chain CoNb 2 O 6 Investigated by High-Resolution Time-Domain Terahertz Spectroscopy

C. M. Morris, Rolando Valdés Aguilar, Asim Kumar Ghosh, S. M. Koohpayeh, Jason W. Krizan, R. J. Cava, Oleg Tchernyshyov, T. M. McQueen, N. P. Armitage

Published Apr 2, 2014• 97 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.112.137403

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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.
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.
Close to the quantum critical point of the transverse-field Ising spin-chain model, an exotic dynamic spectrum was predicted to emerge upon a perturbative longitudinal field. The dynamic spectrum consists of eight particles and is governed by the symmetry of the E8{E}_{8}E8​ Lie algebra. Here we report on high-resolution terahertz spectroscopy of quantum spin dynamics in the ferromagnetic Ising-chain material CoNb2O6{\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}CoNb2​O6​. At 0.25 K in the magnetically ordered phase we identify characteristics of the first six E8{E}_{8}E8​ particles, m1{\mathrm{m}}_{1}m1​ to m6{\mathrm{m}}_{6}m6​, and the two-particle (m1+m2)({\mathrm{m}}_{1}+{\mathrm{m}}_{2})(m1​+m2​) continuum in an applied transverse magnetic field of Bc1D=4.75T{B}_{c}^{\text{1D}}=4.75 \mathrm{T}Bc1D​=4.75T, before the three-dimensional magnetic order is suppressed above ${B}_{c}^{\text{3D}}\ensuremath{\approx}5.3 \mathrm{T}$. The observation of the higher-energy particles (m3tom6)({\mathrm{m}}_{3} \mathrm{to} {\mathrm{m}}_{6})(m3​tom6​) above the low-energy two-particle continua features quantum many-body effects in the exotic dynamic spectrum.
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.
Experimental Observation of High-Energy E8E_8E8​ Particles in the Ising Chain Ferromagnet CoNb2_22​O6_66​
Close to the quantum critical point of the transverse-field Ising spin-chain model, an exotic dynamic spectrum was predicted to emerge upon a perturbative longitudinal field. The dynamic spectrum consists of eight particles and is governed by the symmetry of the E8E_8E8​ Lie algebra. Here we report on high-resolution terahertz spectroscopy of quantum spin dynamics in the ferromagnetic Ising-chain material CoNb2_22​O6_66​. At 0.25 K in the magnetically ordered phase we identify characteristics of the first six E8E_8E8​ particles, m1\mathbf{m}_1m1​ to m6\mathbf{m}_6m6​, and the two-particle (m1+m2\mathbf{m}_1+\mathbf{m}_2m1​+m2​) continuum in an applied transverse magnetic field of Bc1D=4.75B_c^{1D}=4.75Bc1D​=4.75 T, before the three-dimensional magnetic order is suppressed above Bc3D≈5.3B_c^{3D}\approx 5.3Bc3D​≈5.3 T. The observation of the higher-energy particles (m3\mathbf{m}_3m3​ to m6\mathbf{m}_6m6​) above the low-energy two-particle continua features quantum many-body effects in the exotic dynamic spectrum.
We study the nonlinear response of nonintegrable one-dimensional (1D) spin models using infinite matrix-product state techniques. As a benchmark and demonstration of the method, we first calculate the two-dimensional (2D) coherent spectroscopy for the exactly soluble ferromagnetic transverse field Ising model where excitations are freely moving domain walls. We then investigate the distinct signatures of confined bound states by introducing a longitudinal field and observe the emergence of strong nonrephasinglike signals. To interpret the observed phenomena, we use a two-kink approximation to perturbatively compute the 2D spectra. We find good agreement in comparison with the exact results of the infinite matrix-product state method in the strongly confined regime. We discuss the relevance of our results for quasi-1D1\mathrm{D}1D Ising spin chain materials, such as CoNb2O6{\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}CoNb2​O6​.
Weakly coupled Ising chains provide a condensed-matter realization of confinement. In these systems, kinks and antikinks bind into mesons due to an attractive interaction potential that increases linearly with the distance between the particles. While single mesons have been directly observed in experiments, the role of the multiparticle continuum and bound states of mesons in the excitation spectrum is far less clear. Using time-dependent density-matrix renormalization group methods, we study the dynamical structure factors of one- and two-spin operators in a transverse-field two-leg Ising ladder in the ferromagnetic phase. The propagation of time-dependent correlations and the two-spin excitation spectrum reveal the existence of interchain bound states, which are absent in the one-spin dynamical structure factor. We also identify two-meson bound states that appear at higher energies, above the thresholds of several two-meson continua.