Two-dimensional coherent spectroscopy (2DCS) is a nonlinear spectroscopy technique capable of identifying whether apparent continua in linear response are made out of multiplets of sharp deconfined quasiparticles. This makes it a potent tool for experimental identification of fractionalized phases. Previous discussions have focused on limits where the quasiparticles in question are infinitely long lived. In this paper we discuss 2DCS in the regime where the fractionalized quasiparticles can themselves decay. We introduce a powerful path-integral-based approach, whereby the computation of nonlinear susceptibilities reduces to an efficient exercise in diagrammatic perturbation theory. We apply this method to compute the 2DCS response of the one-dimensional transverse field Ising model, in the presence of integrability-breaking perturbations. We discuss aspects of the self-energy of the fractionalized quasiparticles that may be extracted via 2DCS, such as the momentum-dependent decay rate.
Two-dimensional coherent spectroscopy (2DCS) is a nonlinear spectroscopy technique capable of identifying whether apparent continua in linear response are made out of multiplets of sharp deconfined quasiparticles. This makes it a potent tool for experimental identification of fractionalized phases. Previous discussions have focused on limits where the quasiparticles in question are infinitely long lived. In this paper we discuss 2DCS in the regime where the fractionalized quasiparticles can themselves decay. We introduce a powerful path-integral-based approach, whereby the computation of nonlinear susceptibilities reduces to an efficient exercise in diagrammatic perturbation theory. We apply this method to compute the 2DCS response of the one-dimensional transverse field Ising model, in the presence of integrability-breaking perturbations. We discuss aspects of the self-energy of the fractionalized quasiparticles that may be extracted via 2DCS, such as the momentum-dependent decay rate.
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.
Two-dimensional coherent spectroscopy (2DCS) with terahertz radiation offers a promising new avenue for the exploration of many-body phenomena in quantum magnets. This includes the potential diagnosis of fractionalized excitations, for which linear response often struggles due to the indistinguishability of a continuum of fractional excitations from that caused by disorders or impurities. However, the interpretation of the complex results produced by 2DCS remains a challenge. Here, we explore a numerical approach based on exact diagonalization (ED) to help push forward our understanding of 2DCS and, as an example, use it to study the impact of confinement in the one-dimensional transverse field Ising (TFIM) model. We first validate our approach by comparing numerical ED and exact analytical results for the spectroscopic signatures of fractionalized spinons in the model, which exhibits a sharp spinon-echo signal in 2DCS, and show how to understand the finite-size effects inevitable with the inherently small system sizes in ED calculations. By augmenting the model with a longitudinal field, which breaks integrability and introduces confinement of spinon pairs, we observe significant changes to the 2DCS spectrum, such as nonsymmetric broadening of the spinon-echo signal. To further elucidate these changes, we introduce a ``four-kink'' approximation, an effective model that captures the interactions between multiple (confined) spinon pairs. Comparing ED data to these four-kink results provides understanding of the multiple features of the 2DCS spectra, allowing us to interpolate between finite-system size data and the thermodynamic limit. One advantage of our ED approach is its possible extension to finite temperatures, which we explore using thermally pure quantum states and demonstrate how the intensity and spectroscopic patterns of 2DCS change when going beyond the integrable model. Our numerically exact results provide a benchmark for future experiments and theoretical studies relying on approximation methods and pave the way for the exploration of fractionalized excitations in quantum magnets.
We show that the new technique of terahertz 2D coherent spectroscopy is capable of giving qualitatively new information about fractionalized spin systems. For the prototypical example of the transverse field Ising chain, we demonstrate theoretically that, despite the broad continuum of excitations in linear response, the 2D spectrum contains sharp features that are a coherent signature of a "spinon echo," which gives previously inaccessible information such as the lifetime of the two-spinon excited state. The effects of disorder and finite lifetime, which are practically indistinguishable in the linear optical or neutron response, manifest in dramatically different fashion in the 2D spectra. Our results may be directly applicable to model quasi-1D transverse field Ising chain systems such as CoNb_{2}O_{6}, but the concept can be applied to fractionalized spin systems in general.
Identifying quantum spin liquids, magnon breakdown, or fractionalized excitations in quantum magnets is an ongoing challenge due to the ambiguity in interpreting excitation continua occurring in linear-response probes. Recently, it was proposed that techniques measuring higher-order response, such as two-dimensional coherent spectroscopy (2DCS), could resolve such ambiguities. Numerically simulating nonlinear response functions can, however, be computationally very demanding. We present an efficient Lanczos-based method to compute second-order susceptibilities χ 2 ( ω t , ω τ ) directly in the frequency domain. Applying this to extended Kitaev models describing α − RuCl 3 , we find qualitatively different nonlinear responses between intermediate magnetic field strengths and the high-field regime. To put these results into context, we derive the general 2DCS response of partially polarized magnets within the linear spin-wave approximation, establishing that χ 2 ( ω t , ω τ ) is restricted to a distinct universal form if the excitations are conventional magnons. Deviations from this form, as predicted in our Lanczos-based simulations for α − RuCl 3 , can hence serve in 2DCS experiments as direct criteria to determine whether an observed excitation continuum is of conventional two-magnon type or of different nature.
Two-dimensional coherent spectroscopy (2DCS) is an established method for characterizing molecules and has been proposed in the terahertz regime as a tool for probing exotic excitations of quantum magnets; however, the precise nature of the coupling between pump field and spin degrees of freedom has remained unclear. Here, we develop a general response theory of 2DCS and show how magnetoelectric as well as polarization couplings contribute to 2DCS in addition to the typically assumed magnetization. We propose experimental protocols to probe the presence of polarization couplings, for instance, from exchange-striction or spin-current mechanism, when the electric field couples to quadratic terms in spin operators. We provide example calculations for the paradigmatic twisted Kitaev chain material CoNb 2 O 6 and highlight the crucial role of contributions from cross-coupling between polarization and magnetic nonlinear susceptibilities. Our work paves the way for systematic studies of light-matter couplings in quantum magnets and for establishing 2DCS as a versatile tool for probing fractional excitations of exotic magnetic quantum phases.
We study electric field induced two-dimensional coherent nonlinear optical spectroscopy (2DCS) in a Kitaev magnet at finite temperature. We show that 2DCS is susceptible to both types of fractional quasiparticles of this quantum spin-liquid, i.e., fermions and flux visons. Focusing on the second order response, we find a strong antidiagonal feature in the two-dimensional frequency plane, related to the galvanoelectric effect of the fractional fermions. Perpendicular to the antidiagonal, the width of this feature is set by quasiparticle relaxation rates beyond the bare Kitaev magnet, thereby providing access to single-particle characteristics within the multi-particle 2DCS continuum. While the structure of the 2DCS susceptibility stems from the fermionic quasiparticles and displays Fermi blocking versus temperature, the emergent bond randomness which arises due to thermally populated visons strongly modifies the fermionic spectrum. Therefore also the presence of gauge excitations is manifest in the 2DCS susceptibility as the temperature is increased beyond the flux proliferation crossover. Our results are consistent with and extend previous findings on second harmonic generation in Kitaev magnets.