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arXiv:2209.00720 · Physical review. B./Physical review. B

Nonlinear spectroscopy of bound states in perturbed Ising spin chains

GiBaik Sim, Johannes Knolle, Frank Pollmann

Published Mar 7, 2023• 22 citations
arXivHTMLPDF

Abstract

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-$1\mathrm{D}$ Ising spin chain materials, such as ${\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}$.

DOI
10.1103/physrevb.107.l100404

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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​.
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 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.
Mesonic bound states of domain walls (DWs) can be stabilized in quasi-one-dimensional magnetic compounds. Here, we theoretically study the Raman light scattering response of a twisted Kitaev chain with tilted magnetic fields as a minimal model for confinement in CoNb2O6{\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}CoNb2​O6​. By both numerical matrix product states and few-DW variational states, we show that confinement-induced bound states directly manifest themselves as sharp peaks in the Raman response. Remarkably, by tuning the polarization of the incident light field, we demonstrate that the Raman response offers insights into the intrinsic symmetry of the bound-state wave function.
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.
In a recent experiment on CoNb2O6{\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}CoNb2​O6​, R. Coldea et al. [Science 327, 177 (2010)] found experimental evidence of the exceptional Lie algebra E8{E}_{8}E8​. The emergence of this symmetry was theoretically predicted long ago for the transverse quantum Ising chain in the presence of a weak longitudinal field. We consider an accurate microscopic model of CoNb2O6{\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}CoNb2​O6​ incorporating additional couplings and calculate numerically the dynamical structure function using a recently developed matrix-product-state method. The excitation spectra show bound states characteristic of the weakly broken E8{E}_{8}E8​ symmetry. We compare the observed bound-state signatures in this model to those found in the transverse Ising chain in a longitudinal field and to experimental data.
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.