Two-dimensional coherent spectroscopy of CoNb$_2$O$_6$

TL;DR

The paper develops terahertz two-dimensional coherent spectroscopy (2DCS) as a tool to probe nonlinear spin dynamics in the quasi-one-dimensional quantum magnet CoNb$_2$O$_6$, focusing on spinon deconfinement and multi-particle bound states. Starting from the exactly solvable $1d$-TFIM and progressively incorporating realistic exchange terms to form $\mathcal{H}=\mathcal{H}_1+\mathcal{H}_2+\mathcal{H}_3$, the authors compute the third-order response $\chi^{(3)}_{yyyy}$ using a four-kink approximation to reveal signatures of spinon interactions and a four-kink bound state (tetraquark) in the high- and low-temperature regimes. They identify characteristic 2DCS features such as spinon-echo lines, off-diagonal continuums, and a robust tetraquark signal in the $I(\omega)$ spectrum, and show how interchain confinement discretizes continua, reshapes signals, and preserves tetraquark signatures near the localized limit. The results provide concrete, experimentally testable predictions for THz 2DCS on CoNb$_2$O$_6$ and illustrate the power of nonlinear spectroscopy to resolve interaction effects and bound-state formation in quantum magnets.

Abstract

With recent advances in terahertz (THz) sources and detection, two-dimensional coherent spectroscopy (2DCS), which allows to probe nonlinear responses in a two-frequency plane, 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 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 interactions to capture its key low-energy physics. 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 in the two-frequency space, which are thought to reflect an underlying continuum of fractionalized excitations. Our results provide concrete predictions and clear targets for future THz 2DCS experiments on CoNb$_2$O$_6$ and related quasi-one-dimensional quantum magnets.

Figures (13)

• Figure 1: Quasi-1D Ising compound CoNb$_2$O$_6$. A schematic of the crystal structure is shown, with Nb$^{5+}$ ions omitted for clarity. The magnetic Co$^{2+}$ ions, each surrounded by six O$^{2-}$ ions forming octahedra, build zigzag chains along the $c$ axis through edge sharing. The intra-chain Hamiltonian $\mathcal{H}_1 + \mathcal{H}_2$ is dominated by a ferromagnetic Ising interaction. The effect of the much weaker interchain coupling is represented by the mean-field term $\mathcal{H}_3$, which becomes relevant only below the Néel temperature $T_\mathrm{N} \approx 2.9$ K. The structure is drawn using VESTA Momma2008Weitzel1976.
• Figure 2: Probing the 4-kink sector with nonlinear spectroscopy. (a) Schematic of a two-pulse THz magnetic-field protocol used to measure the nonlinear magnetization response. The nonlinear magnetization $m_{\mathrm{NL}}$ contains contributions from susceptibilities $\chi^{(n)}$ with $n\ge 2$. (b) An example excitation pathway driven by $M^y$, involving the ground state (GS) as the initial and final states and intermediate states $\ket{P}, \ket{Q}, \ket{R}$ that contribute to the third-order susceptibility $\chi^{(3)}_{yyyy}$. The third-order process involves four insertions of $M^y$ (three from the field pulses and one at detection), indicated by orange arrows. Spins are shown schematically as arrows (white/red denote up/down). Starting from the fully polarized ferromagnetic (FM) state, one application of $M^y$ can create a reversed-spin domain bounded by two kinks (two spinons). A second application can reach four-kink intermediate states $\ket{Q}$.
• Figure 3: Construction of the full Hamiltonian. Starting from the exactly solvable 1$d$-TFIM Hamiltonian, we gradually introduce additional interaction terms present in CoNb$_2$O$_6$. The most dominant terms are included in $\mathcal{H}_1$, which consists of the Ising interaction along with subdominant XY and YZ terms. $\mathcal{H}_2$ is introduced to achieve quantitative agreement with inelastic neutron scattering experiments. The interchain coupling is represented by the mean-field term $\mathcal{H}_3$, which becomes relevant only below $T_\mathrm{N}$.
• Figure 4: 2DCS spectra of transverse-field Ising chain. Imaginary part of third-order response $\mathrm{Im}[\chi^{(3;2)}_{yyyy}(\omega_t, \omega_\tau)]$ for a chain with $L=100$ sites at transverse fields $B_y = 0,\, 1$, and $3$ T, shown from (a) to (c). At $B_y = 0$, all non-rephasing (NR), rephasing (R), and terahertz-rectification (TR) signals appear as point-like features. A finite $B_y$ introduces dispersion to the spinons, leading to a finite energy bandwidth of $4h_y = 4g\mu_\mathrm{B}B_y$ centered at $J$. As a result, all point signals stretch into continuous line structures, as indicated by the purple arrows, with the NR feature barely visible due to the phase-twisting effect.
• Figure 5: Linear response spectra of TFIM + XY. (a--e) Dynamical structure factor $S(\omega)=\mathcal{FT}\,[\langle M_y(t)M_y(0)\rangle]$ and (f--j) its higher-order counterpart $I(\omega)=\mathcal{FT}\,[\langle M_y^{2}(t)M_y^{2}(0)\rangle]$. Linear scales are shown in panels (a--e), and logarithmic scales in panels (f--j). Both spectra were obtained using a four-kink calculation with $L = 100$ sites. (a, f) Transverse-field dependence of the Ising-model spectra. (b, g) Transverse-field dependence of the Ising + XY model. Purple boxes highlight the transverse-field-induced hybridization of the (confined) single spin-flip state with the spinon continuum. (c--e, h--j) Interpolation between the pure Ising limit and the Ising model with additional XY terms ($\alpha_{\mathrm{S}} = 0.251$) for $B_y = 0,\, 1,\, \text{and}\, 3~\text{T}$, parameterised by $\lambda \in [0,1]$.