arXiv:2402.06484 · Physical review. B./Physical review. B

Anomalous thermal relaxation and pump-probe spectroscopy of two-dimensional topologically ordered systems

Max McGinley, Michele Fava, S. A. Parameswaran

Published Feb 6, 2024• 6 citations

Abstract

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

DOI: 10.1103/physrevb.109.075108

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arXiv:2402.06484 · Physical review. B./Physical review. B

Anomalous thermal relaxation and pump-probe spectroscopy of two-dimensional topologically ordered systems

Max McGinley, Michele Fava, S. A. Parameswaran

Published Feb 6, 2024• 6 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevb.109.075108

Related papers

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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 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.

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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.

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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.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.

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2d

topological excitations in a

3d

material. Here we show that the characteristic time scale of such experiments is inversely proportional to the initial excitation density, set by the pump intensity. A uniform excitation density created on the surface of a sample spreads subdiffusively into the bulk with a mean depth

\bar z

scaling as

\sim t^{1/3}

when annihilation processes are absent. The propagation becomes logarithmic,

\bar z \sim \log t

, when pair-annihilation is allowed. Furthermore, pair-diffusion between layers leads to a new decay law for the total density,

n(t) \sim (\log^2 t)/t

- slower than in a purely

2d

system. We discuss possible experimental implications for pump-probe experiments in samples of finite width.

t

behavior for the pump-probe signal

χ_\mathit{ZZX}

. By contrast, swapping the pump and probe polarizations produces a nonlinear response

χ_\mathit{XXZ}

that is asymptotically

t

-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.