Monte Carlo Sampling for Wave Functions Requiring (Anti)Symmetrization

TL;DR

The paper introduces a scalable Monte Carlo framework to compute observables for wave functions that require explicit (anti)symmetrization across particle clusters, such as Read-Rezayi and Moore–Read states in two dimensions. By organizing permutations into equivalence classes using set partitions for $k=2$ and doubly stochastic $k\times k$ matrices for general $k$, it avoids factorial growth and enables MC evaluation of energies and correlators in large systems and standard geometries. It provides detailed error analyses comparing full symmetrization, naive sampling, and a refined sampling strategy that adaptively includes more representations to control bias, with demonstrated accuracy against Pfaffian/ED benchmarks for RR$k$ states. The method extends to higher cluster numbers and offers insights into competitive energetics (e.g., RR3 vs Jain states) and potential topological properties (e.g., Fibonacci anyons) in larger systems, representing a practical route beyond exact diagonalization. Overall, the approach significantly broadens the computational reach for topologically ordered quantum states by leveraging cluster-based decompositions and controlled MC sampling.

Abstract

Many strongly correlated states, such as those arising in the fractional quantum Hall effect and spin liquids, are described by wave functions obtained by dividing particles into multiple clusters, constructing a readily evaluable wave function in each cluster, and (anti)symmetrizing across these clusters. We introduce a method to compute quantities such as energies and correlators, using Monte Carlo simulations for these states. Our framework overcomes the factorial scaling of explicit (anti)symmetrization, allowing for studies of systems beyond the reach of exact diagonalization.

Figures (5)

• Figure 1: (a) Pair-correlation function for the $\nu{=}1$ Moore-Read state of $N{=}20$ bosons on the sphere obtained with different methods keeping the same computational resources and runtime. (b) Same as (a), except, for clarity, results from only the refined and naive methods are shown.
• Figure 2: Pair-correlation function $g(r)$ for the Moore-Read state of (a) bosons at $\nu{=}1$ and (c) fermions at $\nu{=}1/2$ for $N{=}30$ particles, and the $3$-cluster Read-Rezayi state of (b) bosons at $\nu{=}3/2$ and (d) fermions at $\nu{=}3/5$ for $N{=}24$ particles obtained in the spherical geometry by different methods.
• Figure 3: Competition between the bosonic $k{=}3$ Read-Rezayi and Jain states at $\nu{=}3/2$ for the hard-core $V_{0}$ interaction.
• Figure S1: The bias $\mathcal{B}_{\rm reduced}$ [see Eq. \ref{['eq: bias_reduced']}] as a function of number of particles $N$ [left panel], as well as $\mathcal{E}/\mathcal{S}$ [right panel], for system sizes $N{=}\{ 8,10,12,14,16 \}$ for the bosonic Moore-Read state at $\nu{=}1$ for the naive and refined methods with the same number of Monte Carlo iterations.
• Figure S2: The CPU time per core required to converge the static structure factor computed in spherical geometry to within a standard deviation of $0.01$ from the exact result for the $\nu{=}1$ bosonic Moore-Read state for various systems of $N$ bosons on the sphere using the refined method with the number of extra terms $\mathcal{E}{=}\lceil 0.4e^{0.3912N} \rceil$. The red squares represent the numerically obtained results for $N{=}\{8,10,12,14,16\}$, while the red line represents the best-fit exponential, ${-}108.83{+}1.33e^{0.46N}$. The inset shows the curve fitting extrapolated to a system size of up to $40$ particles to give a rough estimate of its convergence time (it has the same $x$-axis and $y$-axis labels as the main plot).