[Paper Review] Extracting spinon self-energies from two-dimensional coherent spectroscopy

Oliver Hart, Rahul Nandkishore|arXiv (Cornell University)|Aug 26, 2022

Spectroscopy and Quantum Chemical Studies4 citations

TL;DR

This paper develops a real-time path integral framework for computing nonlinear susceptibilities in interacting quantum many-body systems, enabling the extraction of spinon self-energies—particularly momentum-dependent decay rates—from two-dimensional coherent spectroscopy (2DCS) responses. Applied to the one-dimensional transverse field Ising model with integrability-breaking interactions, the method reveals how quasiparticle decay via domain-wall pair emission or thermal scattering imprints distinct spectral features in 2DCS, including anomalous broadening and new peak structures at $\pm 2\delta_p$ in the frequency domain.

ABSTRACT

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 manuscript 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.

Motivation & Objective

• To extend 2DCS theory beyond noninteracting or two-level systems to describe interacting many-body systems with finite quasiparticle lifetimes.
• To develop a real-time path integral approach that enables diagrammatic many-body perturbation theory for nonlinear response functions.
• To identify experimentally accessible signatures of quasiparticle self-energy—especially momentum-dependent decay rates—via 2DCS in fractionalized phases.
• To analyze how interactions induce domain-wall decay in the 1D transverse field Ising model and how this affects 2DCS spectra.
• To distinguish between homogeneous and inhomogeneous broadening in 2DCS, enabling experimental detection of fractionalization beyond linear response.

Proposed method

• Formulates the nonlinear response via a generating functional $ Z[h^{\text{cl}}, h^{\text{q}}] $ in real-time path integral formalism, mapping the problem to a diagrammatic perturbation theory framework.
• Represents the system in terms of Majorana fermions and uses the path integral to compute free Green's functions and magnetization operators.
• Derives third-order response functions by expanding the generating functional to cubic order in the external fields, capturing nonlinear susceptibilities.
• Computes self-energy corrections perturbatively up to second order, with vertex factors involving interaction terms and spinor weights $ A_{ab}^{\sigma}(p) $.
• Evaluates the self-energy components using trace identities and matrix products, ensuring Hermiticity and physical consistency.
• Fourier transforms the time-domain 2DCS response to extract frequency-resolved spectra, identifying rephasing streaks and anomalous peaks at $ \pm 2\delta_p $.

Experimental results

Research questions

• RQ1 How can 2DCS be used to extract the momentum-dependent self-energy of fractionalized quasiparticles in interacting quantum systems?
• RQ2 What spectral signatures in 2DCS arise from quasiparticle decay due to interactions, such as domain-wall pair emission in the 1D transverse field Ising model?
• RQ3 How do inhomogeneous broadening (multi-particle continuum) and homogeneous broadening (finite lifetime) manifest differently in 2DCS spectra?
• RQ4 Can the path integral approach accurately describe nonlinear responses in systems with realistic interactions beyond noninteracting or effective two-level models?
• RQ5 What role does the splitting $ \delta_p $ between doublet levels play in generating additional peaks in the 2DCS spectrum, and how does it relate to anomalous broadening?

Key findings

• The 2DCS response exhibits a rephasing signal proportional to $ \sin[2E_p(t - \tau)] $, which generates additional peaks at $ (\pm 2\delta_p, 0) $, $ (0, \pm 2\delta_p) $, and $ (\pm 2\delta_p, \mp 2\delta_p) $ in the $ (\omega_t, \omega_\tau) $ plane.
• Anomalous broadening becomes visible when intrinsic broadening $ \gamma_p $ is reduced and $ \delta_p $ is increased, allowing the splitting to dominate spectral features.
• The second-order self-energy correction involves nontrivial momentum-dependent vertex contractions, with components expressed through traces and matrix products of $ \check{A}^\sigma(p) $ matrices.
• The self-energy components $ B_{ab}(1|2,3) $ and $ C_{ab}(1,2|3) $ are explicitly computed using trace identities, yielding $ \operatorname{Tr}[\check{A}^{\sigma_j}(p_j)(\check{A}^{\sigma_k}(p_k))^\top] = 2[\sigma_j\sigma_k - \cos(\vartheta_{p_j} + \vartheta_{p_k})] $.
• The method successfully captures decay via three-domain-wall continuum when the single-domain-wall dispersion overlaps with the continuum, enabling spontaneous decay without thermal bath.
• In the absence of such overlap, thermal scattering provides a decay mechanism, and the resulting self-energy leads to observable spectral distortions in the 2DCS signal.