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arXiv:2210.16249 · Physical Review Letters

Signatures of Fractional Statistics in Nonlinear Pump-Probe Spectroscopy

Max McGinley, Michele Fava, S. A. Parameswaran

Published Feb 6, 2024• 20 citations
arXivHTMLPDF

Abstract

We show that the presence of anyons in the excitation spectrum of a two-dimensional system can be inferred from nonlinear spectroscopic quantities. In particular, we consider pump-probe spectroscopy, where a sample is irradiated by two light pulses with an adjustable time delay between them. The relevant response coefficient exhibits a universal form that originates from the statistical phase acquired when anyons created by the first pulse braid around those created by the second. This behavior is shown to be qualitatively unchanged by nonuniversal physics including nonstatistical interactions and small nonzero temperatures. In magnetic systems, the signal of interest can be measured using currently available terahertz-domain probes, highlighting the potential usefulness of nonlinear spectroscopic techniques in the search for quantum spin liquids.

DOI
10.1103/physrevlett.132.066702

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We show that the presence of anyons in the excitation spectrum of a two-dimensional system can be inferred from nonlinear spectroscopic quantities. In particular, we consider pump-probe spectroscopy, where a sample is irradiated by two light pulses with an adjustable time delay between them. The relevant response coefficient exhibits a universal form that originates from the statistical phase acquired when anyons created by the first pulse braid around those created by the second. This behavior is shown to be qualitatively unchanged by nonuniversal physics including nonstatistical interactions and small nonzero temperatures. In magnetic systems, the signal of interest can be measured using currently available terahertz-domain probes, highlighting the potential usefulness of nonlinear spectroscopic techniques in the search for quantum spin liquids.
We study the behavior of linear and nonlinear spectroscopic quantities in two-dimensional topologically ordered systems, which host anyonic excitations exhibiting fractional statistics. We highlight the role that braiding phases between anyons have on the dynamics of such quasiparticles, which as we show dictates the behavior of both linear response coefficients at finite temperatures, as well as nonlinear pump-probe response coefficients. These quantities, which act as probes of temporal correlations in the system, are shown to obey distinctive universal forms at sufficiently long timescales. As well as providing an experimentally measurable fingerprint of anyonic statistics, the universal behavior that we find also demonstrates anomalously fast thermal relaxation: correlation functions decay as a “squished exponential” C ( t ) ∼ exp ( − [ t / τ ] 3 / 2 ) at long times. We attribute this unusual asymptotic form to the nonlocal nature of interactions between anyons, which allows relaxation to occur much faster than in systems with quasiparticles interacting via local, nonstatistical interactions. While our results apply to any Abelian or non-Abelian topological phase in two-dimensions, we discuss in particular the implications for candidate quantum spin liquid materials, wherein the relevant quantities can be measured using pre-existing time-resolved terahertz-domain spectroscopic techniques. Published by the American Physical Society 2024
We show that the braiding of anyons in a quantum spin liquid leaves a distinct dynamical signature in the nonlinear pump–probe response. Using a combination of exact diagonalization and matrix product state techniques, we study the nonlinear pump–probe response of the toric code in a magnetic field, a model with mobile electric e and magnetic m anyonic excitations. While the linear response signal oscillates and decays with time like ∼ t − 1.3 , the amplitude of the nonlinear signal for χ XZZ ( 3 ) features a linear-in-time enhancement at early times and a stronger enhancement t α with α > 1 at later times. The comparison between χ XZZ ( 3 ) , which involves nontrivial braiding of e and m anyons, and χ XXX ( 3 ) that involves trivial braiding of the same types of anyons, distinguishes the braiding statistics of anyons. We support our analysis with a hard-core anyon model with statistical gauge fields to develop further insights into the time dependence of the pump–probe response. Pump–probe spectroscopy provides a distinctive new probe of quantum spin liquid states, beyond the inconclusive broad features observed in single spin-flip inelastic neutron scattering.
Identifying experimental signatures of anyons, which exhibit fractional exchange statistics, remains a central challenge in the study of two-dimensional topologically ordered systems. Previous theoretical work has shown that the threshold behavior in linear response spectroscopy can reveal the fractional exchange statistics between an anyon and its antiparticle. In this work, we extend this framework to nonlinear, two-dimensional coherent spectroscopy. We demonstrate by analyzing time-ordered four-point correlation functions that the threshold behavior of nonlinear response functions encodes the fractional statistics between general pairs of anyons that can combine to any composite topological charge. This feature in particular provides a powerful probe for unambiguously distinguishing non-Abelian anyons, which can form multiple composite charges with distinct nontrivial braid statistics. Our approach is validated using numerical simulations that are consistent with the correct fractional exchange statistics for both the Abelian anyons in the toric code and non-Abelian Ising anyons.
Identifying experimental signatures of anyons, which exhibit fractional exchange statistics, remains a central challenge in the study of two-dimensional topologically ordered systems. Previous theoretical work has shown that the threshold behavior in linear response spectroscopy can reveal the fractional exchange statistics between an anyon and its antiparticle. In this work, we extend this framework to nonlinear, two-dimensional coherent spectroscopy. We demonstrate by analyzing time-ordered four-point correlation functions that the threshold behavior of nonlinear response functions encodes the fractional statistics between general pairs of anyons that can combine to any composite topological charge. This feature in particular provides a powerful probe for unambiguously distinguishing non-Abelian anyons, which can form multiple composite charges with distinct nontrivial braid statistics. Our approach is validated using numerical simulations that are consistent with the correct fractional exchange statistics for both the Abelian anyons in the toric code and non-Abelian Ising anyons.
We demonstrate how pump-probe techniques enable specific spectroscopic diagnostics of fracton phases of matter by studying lineon-planon braiding in the paradigmatic X-cube model. Our discussion builds on previous works probing anyonic exchange statistics in conventional spin liquids, but the extension to fracton phases reveals qualitatively new phenomena due to the restricted mobility of fractionalized excitations. A key feature is that nearby planons can form an emergent bound state by accessing different planes via alternative pairing configurations. We show that this bound state qualitatively modifies the long-time linear and nonlinear responses, leading to an asymptotic linear-in-ttt behavior for the pump-probe signal χZZXχ_\mathit{ZZX}χZZX​. By contrast, swapping the pump and probe polarizations produces a nonlinear response χXXZχ_\mathit{XXZ}χXXZ​ that is asymptotically ttt-independent. This asymmetry reflects the fact that the two species of fractionalized excitations live in different spatial dimensions. Thus, the pump-probe signals studied here are sensitive to (i) nontrivial braiding statistics in three dimensions, (ii) the existence of bound states among fractionalized excitations, and (iii) the one-dimensional mobility of lineons. Our results therefore provide spectroscopic signatures that distinguish fracton phases from conventional topologically ordered spin liquids.
Unambiguous identification of fractionalized excitations in quantum spin liquids has been a long-standing issue in correlated topological phases. Conventional spectroscopic probes, such as the dynamical spin structure factor, can only detect composites of fractionalized excitations, leading to a broad continuum in energy. Lacking a clear signature in conventional probes has been the biggest obstacle in the field. In this work, we theoretically investigate what kinds of distinctive signatures of fractionalized excitations can be probed in two-dimensional nonlinear spectroscopy by considering the exactly solvable Kitaev spin liquids. We demonstrate the existence of a number of salient features of the Majorana fermions and fluxes in two-dimensional nonlinear spectroscopy, which provide crucial information about such excitations.