Exploring two-dimensional coherent spectroscopy with exact diagonalization: Spinons and confinement in one-dimensional quantum magnets
Yoshito Watanabe, Simon Trebst, Ciarán Hickey
Abstract
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.