Effective linear response in non-equilibrium anyonic systems

TL;DR

The paper develops an effective linear-response theory for non-equilibrium anyonic beams in collider geometries, introducing an effective equilibrium defined by $V_\text{eff}$ and $T_\text{eff}$ and deriving four kinetic coefficients that link charge and heat currents to differences between effective and equilibrium channel parameters. By analyzing correlation functions of anyonic vertex operators, it separates time-domain braiding (braiding-only) from real tunneling and collisions, revealing universal Lorenz-number behavior in the collision-free limit that encodes the anyonic statistics via $\nu$. When collisions are included, particle-hole symmetry is broken and finite Seebeck and Peltier coefficients emerge, providing a diagnostic for collisional dynamics and non-equilibrium statistics, with the effective parameters themselves becoming nonuniversal functions of experimental details. The results offer a practical route to probe fractional (and potentially non-Abelian) statistics through transport and thermoelectric measurements, including protocol-oriented guidance for experimental extraction of the effective linear-response coefficients.

Abstract

Linear response theory serves as a fundamental tool in the study of quantum transport, extensively employed to elucidate fundamental mechanisms related to the nature of the particles involved and the underlying symmetries. This framework is, however, limited to equilibrium or near-equilibrium conditions. Here, we develop an effective linear response theory designed to describe charge and thermal quantum transport, where the reference far-from-equilibrium stationary state comprises anyons forming a dilute beam. We apply our theory to study tunnel-coupled anyonic beams in collider geometries, enabling braiding, collisions, and tunneling of anyons at the central collider. Our linear-response transport coefficients directly reflect the fractional charge and statistics of the anyons involved, avoiding the need to measure higher-order current correlations. Moreover, the emergence of finite thermoelectric (Peltier and Seebeck) coefficients signifies the presence of real anyon collisions (as opposed to virtual braiding in the time domain), intimately associated with a broken particle-hole symmetry, specific to anyonic gases.

Figures (10)

• Figure 1: Basic setup and effective anyonic distributions. (a) The setup consists of two anyonic channels, channel 1 and channel 2, that communicate through the central collider, where anyons are allowed to tunnel with the probability $\mathcal{T}_C$. This tunneling induces the charge and heat tunneling currents between the channels, $I_\text{charge}$ and $J_\text{heat}$, respectively. Here, channel 2 is at equilibrium, being characterized by the chemical potential $V_{2,\text{eq}}$ and temperature $T_{2,\text{eq}}$. Channel 1 is out of equilibrium, driven by receiving anyons from the equilibrium source channel $S1$ (biased by voltage $V_{S1}$) through the diluter that has the transmission probability $\mathcal{T}_1$ [cf. Eq. \ref{['eq:t1_expressions']} in Materials and Methods]. Downstream of the diluter, channel 1 is characterized by the effective chemical potential $V_{1,\text{eff}}$ and effective temperature $T_\text{1,eff}$, introduced in Ref. LandscapePRL25, as well as by charge current $I_1$. The directions of currents in the setup are shown by red arrows; the drains of the channels are marked with $D_{1,2}$. (b) Effective particle (red curves) and hole (blue curves) distributions $n^\text{neq}_{p,1}$ and $n^\text{neq}_{h,1}$ [defined by Eq. \ref{['eq:distribution_definition']}] of the non-equilibrium channel 1. In the collision-free limit $\mathcal{T}_1 \ll 1$ (solid curves), effective particle and hole distributions are symmetric with respect to the effective Laughlin surface (gray vertical dashed line) corresponding to $\nu e V_\text{1,eff}^0$ from Eq. \ref{['eq:veff_collision_free']}]. This symmetry is sabotaged, when $\mathcal{T}_1$ is increased (dashed curves are plotted for $\mathcal{T}_1 \approx 0.1$).
• Figure 2: Two extra HOM setups. (a) The central collider bridges two non-equilibrium channels 1 and 2, which are both in the collision-free limit. (b) The two channels are both at equilibrium upstream of the central collider.
• Figure 3: Charge and heat conductances in the effective linear response regime. The magenta, red, and blue symbols correspond to setups shown in Figs. \ref{['fig:model']}(a), \ref{['fig:extra_setups']}(a) and \ref{['fig:extra_setups']}(a), respectively. For non-equilibrium channels, we exclude collisions of non-equilibrium anyons. The normalized charge conductance (a), normalized heat conductance (b), and the Lorenz number (c) are shown as functions of the filling fraction. Panel (d) shows the effective particle distribution functions, $n_\text{p}^s$, for equilibrium ($s$=eq, blue curve) and non-equilibrium ($s$=neq, red curve) anyonic channels that have the same (effective or real) potential and temperature.
• Figure 4: Effect of anyonic collisions on charge and heat currents and thermoelectric coefficients. (a) and (b): Diagrams that present the sign (positive in blue areas and negative in red areas) of the product of charge and heat currents, $I_\text{charge}^\text{coll} J_\text{heat}^\text{coll}$, where the subscript "coll" highlights the inclusion of anyonic collisions. The crossing point (the white star) between different-color areas denotes the effective equilibrium point. White dashed arrows indicate the directions of boundaries between the blue and red areas at the effective-equilibrium point. The equilibrium potential and temperature in channel 2 are given in units of $V_{1,\text{eff}}^0 = I_1 \sin (2\pi\nu)/\nu^2 e^2$ [Eq. \ref{['eq:veff_collision_free']}] and $T_\text{1,eff}^0 \approx 17.4 I_1/e$ [Eq. (S54) of the SM Sec. S4], which are the effective chemical potential and effective temperature, respectively, of the non-equilibrium channel 1 in the collision-free limit (hence, superscript "0"). Panel (a) describes the collision-free limit with a vanishingly small $\mathcal{T}_1$, where the white arrows are parallel to either the temperature or bias axis, indicating vanishing Seebeck and Peltier coefficients. In panel (b), where $\mathcal{T}_1=0.1$, the boundaries between the areas are tilted, implying non-zero values of Seebeck and Peltier coefficients. (c) The Seebeck coefficient as a function of $\mathcal{T}_1$ for $T_{S1} \ll V_{S1}$ (when $S_1 = \nu I_1$) increases linearly as a function of $\mathcal{T}_1$. (d) When $\mathcal{T}_1$ is kept fixed, the dependence of the Seebeck coefficient on the ratio $S_1/(\nu I_1)$ is comparatively much less significant, both for relatively large, $\mathcal{T}_1 = 0.1$ (squares), and small, $\mathcal{T}_1 = 0.012$ (circles), transmission.
• Figure S1: All regions, $\mathcal{R}_1$, $\mathcal{R}_2$, $\mathcal{R}_3$ and $\mathcal{R}_4$ that are relevant to the integral of Eq. \ref{['eq:second_order_correlation']}. The former three regions have finite contributions to the non-equilibrium correlation function, $\langle e^{-i \sqrt{\nu}\phi_1 (t^-,L)} e^{i \sqrt{\nu}\phi_1 (0^+, L)} \rangle_\text{neq,2}$, to the leading order of dilution $|\xi_1|^2$. The phase factor of $\mathcal{R}_{\eta_1\eta_2} (s_1,s_2)$, cf. Eq. \ref{['eq:r_eta_value']}, has been marked out for each corresponding region. Regions $\mathcal{R}_1$, $\mathcal{R}_2$ and $\mathcal{R}_3$ are colored in blue and red, to highlight the fact that $\mathcal{R}_{\eta_1\eta_2} (s_1,s_2)$ has a finite phase in these regions. Within these regions, areas with most significant contribution to the integral are highlighted in red. The focal points that dominate integrals in Regions $\mathcal{R}_2$ and $\mathcal{R}_3$ are highlighted by orange and green stars, respectively.