Measuring Anyonic Exchange Phases Using Two-Dimensional Coherent Spectroscopy

TL;DR

This work tackles the challenge of identifying anyonic exchange statistics in two-dimensional topologically ordered systems. It introduces two-dimensional coherent spectroscopy (2DCS) as a nonlinear probe and shows that the threshold behavior of a time-ordered four-point function encodes the full exchange phases $R^{ab}_c$ for generic fusion channels. The authors derive and validate (analytically and via exact diagonalization) that the exponents $|\alpha|$ and $|\beta|$ govern threshold scaling along two directions in the $(\omega_{\tau},\omega_t)$ plane, enabling access to both $R^{a\overline{a}}_1$ and $R^{bc}_{\bar{a}}$, including non-Abelian fusion channels. Numerical simulations for toric code (Abelian) and Ising (non-Abelian) anyons corroborate the predictions, suggesting 2DCS as a promising diagnostic for detecting and characterizing anyons in quantum materials.

Abstract

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.

Figures (3)

• Figure 1: Schematics of the two-dimensional coherent spectroscopy setup consisting of three pulses. The time delays between the first and second, second and third, and third pulse and measurement are denoted by $T$, $\tau$, $t$, respectively. For a fixed waiting time $T$, a two-dimensional Fouier transform is performed with respect to $\tau$ and $t$.
• Figure 2: Simulation results for toric code anyons on a $10\times 10$ lattice with $T=7$, $\epsilon_{\tau}=5\cdot 10^{-4}$, $\epsilon_{t}=5\cdot 10^{-2}$, $m_{e}=m_{m}=m_f$, $\Delta_{ff}=1$, and $\Delta_{emf}=2$ for the process in Eq. \ref{['eq:TCpathway']}. (a) Up to broadening effects, $\mathcal{C}$ is only nonzero with the region bounded by the white and red dashed lines corresponding to $\Omega_{t}=0$ and $\xi\Omega_{\tau} = \xi'\Omega_{t}$, respectively. (b) The threshold behavior in $\Omega_{\tau}$ along the white dashed line is consistent with the expected exponent of $\beta=1/2$ associated with mutual semions. (c) The threshold behavior along the red dashed line $\xi \Omega_{\tau}=\xi' \Omega_t$ is consistent with the expected exponent of $\alpha=1$ for fermions.
• Figure 3: Simulation results for Ising anyons on a $10\times 10$ lattice with $T=7$, $\epsilon_{\tau}=5\cdot 10^{-4}$, $\epsilon_{t}=5\cdot 10^{-2}$, $m_{\sigma}=m_{\psi}$, $\Delta_{\psi\psi}=1$, and $\Delta_{\sigma\sigma\psi}=2.5$ for the process in Eq. \ref{['eq:Isingpathway']}. (a) Up to broadening effects, $\mathcal{C}$ is only nonzero with the region bounded by the white and red dashed lines corresponding to $\Omega_{t}=0$ and $\xi\Omega_{\tau} = \xi'\Omega_{t}$, respectively. (b) The threshold behavior in $\Omega_{\tau}$ along the white dashed line is consistent with the expected exponent of $\beta=3/8$ associated with the exchange phase $R^{\sigma\sigma}_{\psi}$. (c) The threshold behavior along the red dashed line $\xi \Omega_{\tau}=\xi' \Omega_t$ is consistent with the expected exponent of $\alpha=1$ for fermions.