Non-universal Thermal Hall Responses in Fractional Quantum Hall Droplets

Abstract

We analytically compute the thermal Hall conductance (THC) of fractional quantum Hall droplets under realistic conditions that go beyond the idealized linear edge theory with conformal symmetry. Specifically, we consider finite-size effects at low temperature, nonzero self-energies of quasiholes, and general edge dispersions. We derive measurable corrections in THC that are consistent with the experimental observables. Although the quantized THC is commonly regarded as a topological invariant that is independent of edge confinement, our results show that this quantization remains robust only for arbitrary edge dispersion in the thermodynamic limit. Furthermore, the THC contributed by Abelian modes can become extremely sensitive to finite-size effects and irregular confining potentials in any realistic experimental system. In contrast, non-Abelian modes show robust THC signatures under perturbations, indicating an intrinsic stability of non-Abelian anyons.

Figures (4)

• Figure 1: Non-universal thermal Hall responses of different edge modes.(a) Thermal Hall conductance $\kappa/(\kappa_0 T)$ versus $\beta\alpha_1$ for Abelian $U(1)$ (blue), Majorana (orange, different line styles for distinct sectors), and the non-Abelian (NA) component of the Gaffnian (green) edge modes. At $\beta\alpha_1=0$, $\kappa/(\kappa_0 T)$ equals the central charge. The $U(1)$ mode decreases nearly linearly with $\beta\alpha_1$, while the Majorana and NA modes remain almost constant, revealing the robustness of non-Abelian excitations against finite-size effects. The inset shows the experimental regime (gray area), where the error in available $U(1)$ data is proportional to $\alpha \beta$, consistent with theory banerjee2017observedbanerjee2018observationmelcer2022absent. (b) Dependence of $\kappa$ on quasihole creation energy and edge-mode velocity: the $U(1)$ mode (left) varies linearly with these parameters, whereas the Majorana mode (right) remains essentially unchanged, underscoring its insensitivity to microscopic details.
• Figure 2: Thermodynamic observables under nonlinear dispersions. (a) Thermal Hall conductance (THC) of Abelian modes for $\epsilon_m=\alpha_1\Delta m+\alpha_2(\Delta m)^2$ with $\alpha_2=10^{-4} \ll \alpha_1$. The dashed line shows the analytical result from Eq. \ref{['eqn:1+2']} in the limit $\beta\alpha_1\to 0$. (b,c) Normalized specific heat $C'=(k_B/\alpha_n^2)C_L$ versus temperature under quadratic and cubic dispersions, with $\alpha_2,\alpha_3 \sim 10^{-26}$. The dashed lines indicate high-temperature fits. For quadratic (cubic) dispersions, $C_L \propto T^{1/3}$ ($T^{1/5}$).
• Figure A.1: Contour plot with poles for the U(1) modes specific heat. The $\zeta(s)$ contributes the pole at $s=1$, $\zeta(s+1)$ contributes pole at $s=0$, and $\Gamma(s+2)$ contribute poles at $s=-2,-3,-4...$
• Figure A.2: Mapping from cylinder to disk geometry. On a cylinder with two chiral edges $\gamma_{1,2}$, inserting a flux through the hole is equivalent to nucleating a conjugate anyon pair $(a,\bar{a})$ in the bulk, and one can drag them to opposite boundaries, represented by an open Wilson line stretching between $\gamma_1$ and $\gamma_2$. Here, the bulk-edge correspondence appears as a gluing condition that originates from electron locality and enforces conjugate anyon charges on the two edges. Shrinking $\gamma_2$ to a point in the bulk maps the cylinder to a disk, where the Wilson line now terminates at the single boundary $\gamma_1$. In this limit, the cylinder gluing condition gives the bulk-edge correspondence on the disk, i.e., the bulk fixes the topological sector of the remaining edge, thereby determining the Majorana boundary condition (NS or R), the parity, and the $U(1)_2$ charge sector of the Moore-Read edge CFT. The resulting disk partition functions are the corresponding character combinations sohal2020entanglement.