Hybrid Monte Carlo for Fractional Quantum Hall States

TL;DR

This work introduces a hybrid Monte Carlo approach with global updates and a double stereographic projection on the sphere to efficiently sample Laughlin and Moore-Read FQH wave functions at system sizes $N>1000$. The method yields high-quality thermodynamic-limit data, enabling precise measurements of the topological shift from edge dipoles and non-Abelian braiding matrices for MR quasiholes, with two rotational schemes yielding braiding matrices consistent with Ising anyon theory. By overcoming autocorrelation and sampling limitations of Metropolis MC, the authors demonstrate improved convergence and scalability, and discuss applications to FQH/FQAH stability under perturbations and decoherence. The combination of accurate edge physics, robust non-Abelian statistics, and scalable computation provides a powerful tool for exploring topological order in large systems and in more complex geometries or perturbative settings.

Abstract

We develop a hybrid Monte Carlo method to efficiently compute the physical observables from the samplings of the Laughlin and the Moore-Read wave functions of fractional quantum Hall (FQH) systems. With the advancements in methodology, including global updates and double stereographic projection on spherical geometry, our hybrid Monte Carlo simulation is significantly faster than the widely used Metropolis Monte Carlo scheme. As a result, we can readily simulate systems with electron numbers $N > 1000$ on both disk and sphere geometries. We apply this method to investigating the topological shift obtained from the edge dipole moment, computed from the density of the wave function on the disk. We also numerically computed the non-Abelian braiding matrices for different braiding schemes of the Moore-Read quasiholes on the sphere. Results with much better quality compared with previous works have been achieved. With the thermodynamic limit results obtained at ease, we also discuss the future usage of our method to clarify the questions on the instability of fractional quantum Hall states in an ideal Chern band setting or under quantum decoherence.

Figures (5)

• Figure 1: Electron density of Integer and Laughlin states at different fillings on disk. (a)-(d) The electron density relative to $\rho_0=\frac{1}{m}$ against the spatial distance to the origin $r$ (in unit of $l_B$) for $m=1,\,3,\,5,\,7$. The four panels share the same legend in the top-right corner, indicating the number of electrons $N$. (e) The edge dipole moment computed from the density profile using Eq. \ref{['eq:edge_dipole']}. The gray horizontal lines are the theoretical prediction $-\frac{1}{4\pi}\frac{m-1}{m}$.
• Figure 2: Electron density of Moore-Read state on disk geometry. (a) Electron density against the spatial distance to the origin $r$ in a disk geometry, with number of electrons $N=40,80,120,160,200, 400$. (b) The analyzed edge dipole moment for the MR state against $1/N$. The horizontal line is the theoretical prediction $-\frac{3}{16\pi}$ (see main text).
• Figure 3: Berry phase and braiding matrix of Moore-Read state on sphere geometry. (a) The braiding scheme used to compute the Berry phase of two quasiholes in the Moore-Read state. One electron at a fixed latitude is brought around the other sitting at the north pole, subtending a solid angle $\Omega$ at the center (area of the blue shaded segment). (b) The scheme used to compute the braiding matrix of swapping two quasiholes among 4 quasiholes, each with polar angle $\pm\theta$, forming a tetrahedron. (c) Numerical result of the statistical part of the braiding phase with $N=100,101,200,201$ electrons. The horizontal lines are the theoretical prediction for even and odd sectors, which are 0 and $\pi$, respectively. (d) Result of the braiding matrix using the parameterization in Eq. \ref{['eq:parameterization']}. The blue, orange, and yellow markers denote the numerical results for the angles $\eta$, $\beta$, and $\alpha$, respectively, with $N=20,60,100$ electrons. The dashed lines are $\frac{\pi}{4}$, $0$, and $-\frac{\pi}{2}$ respectively. (e) and (f) shows the $\alpha$ angle for two rotation schemes (shown in insets) comparing to analytic results in Eqs. \ref{['braiding matrix rotation scheme 1']} and \ref{['braiding matrix rotation scheme 2']} respectively.
• Figure S1: Each row from left to right: A schematic of the braiding process on the sphere where quasihole trajectories are generated by rotation of the entire sphere, the corresponding braiding scheme on the disk obtained from the stereographic projection (note that the north pole maps to the center of the disk and the south pole maps to infinity), the world-lines of each quasihole in this braiding scheme in 2+1 dimension, the corresponding braided fusion tree, and the braiding matrix obtained from resolving the fusion tree diagram.
• Figure S2: Double stereographic projection for HMC.$P_1$, $P_2$ are two points on the sphere, and $z_1$, $z_2$ are points on the complex plane from the usual stereographic mapping, that are the intersections of the straight line connecting the south pole and points on the sphere with the complex plane. Since $P_2$ is located in the southern hemisphere, one has $|z_2|>1$. The double stereographic projection for $P_2$ starts from the north pole instead and maps $P_2$ to $w_2=1/z_2^*$, which is inside the unit disk.