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arXiv:1905.11420 · Physical Review Letters

Resolving Continua of Fractional Excitations by Spinon Echo in THz 2D Coherent Spectroscopy

Yuan Wan, N. P. Armitage

Published Jun 25, 2019• 99 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.122.257401

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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.
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.
This work explores the effectiveness of two-dimensional coherent spectroscopy for understanding the microscopic properties of quantum magnets. Motivated by experiments on the Ising chain magnet CoNb2{}_{2}2​O6{}_{6}6​, it is shown that the 2D spectra are markedly different for the two proposed microscopic Hamiltonians - the transverse field Ising model and the twisted Kitaev spin chain - even though the linear response is similar. The findings showcase the power of 2D spectroscopy for quantifying microscopic descriptions and understanding the dynamics of fractionalized excitations of quantum magnets.
The identification of microscopic models describing the low-energy properties of correlated materials has been a central goal of spectroscopic measurements. We demonstrate how 2D non-linear spectroscopy can be used to distinguish effective spin models whose linear responses show similar behavior. Motivated by recent experiments on the quasi-1D Ising magnet CoNb2_22​O6_66​, we focus on two proposed models, the ferromagnetic twisted Kitaev chain with bond dependent interactions and the transverse field Ising model. The dynamical spin structure factor probed in linear response displays similar broad spectra for both models from their fermionic domain wall excitations. In sharp contrast, the 2D non-linear spectra of the two models show clear qualitative differences: those of the twisted Kitaev model contain off-diagonal peaks originating from the bond dependent interactions and transitions between different fermion bands absent in the transverse field Ising model. We discuss the different signatures of spin fractionalization in integrable and non-integrable regimes of the models and their connection to experiments.
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.
Inelastic neutron scattering (INS) has traditionally been one of the primary methods for investigating quantum magnets, particularly in identifying a continuum of excitations as a hallmark of spin fractionalization in quantum spin liquids (QSLs). However, INS faces severe limitations due to its inability to distinguish between such QSL signatures and similar excitation continua arising from highly frustrated magnetic orders with large unit cells or classical spin liquids. In contrast, two-dimensional coherent spectroscopy (2DCS) has emerged as a powerful tool to probe nonlinear excitation dynamics, offering insights into the underlying mechanisms behind these broad spectral features. In this paper, we utilize classical molecular dynamics (MD) techniques to explore the 2DCS responses of frustrated magnets with dominant Kitaev interactions. Comparing the classical and quantum versions of the pure Kitaev model our results indicate both clear similarities, in the form of sharp line features, and clear distinctions, in the locations of these features and in selection rules. Moreover, in the extended KΓΓ′KΓΓ'KΓΓ′ model, we show that the 2DCS response of the Kitaev spin liquid is completely distinct from that of large unit cell magnetic orders, despite both generating a broad continuum in INS. Additionally, we demonstrate the extreme sensitivity of classical 2DCS to thermal fluctuations and discuss the potential significance of quantum coherence in experimental settings. Overall, our work illustrates the potential of 2DCS in resolving the complex physics underlying ambiguous spin excitation continua, thereby enhancing our understanding of the dynamics in these frustrated systems.