Dynamics of Anyon Clusters in Fractional Quantum Hall Fluids

TL;DR

This work shows that realistic electron-electron interactions induce nontrivial dynamics and clustering of anyons in Laughlin 1/3 and Moore-Read fractional quantum Hall fluids. By combining Haldane pseudopotentials, Jack polynomial formalism, exact diagonalization on spheres, and Monte Carlo simulations, it reveals bound-state formation and parity-dependent fusion-channel energetics, including a two-anyon molecule with effective charge $e^*=2e/3$ at low temperatures in Laughlin phases and distinct fusion behavior in Moore-Read phases. The low-energy spectra and LDOS fingerprints depend sensitively on the details of the electron interactions, suggesting STM/LDOS as a powerful probe of effective anyon interactions and potentially tunable fusion channels for non-Abelian anyons. These insights bridge microscopic interactions with experimentally accessible signatures and have implications for manipulating topological quasiparticles in quantum information applications.

Abstract

In fractional quantum Hall fluids, the quasiparticle excitations are anyons with fractional charges and statistics. Effective interactions among the anyons can be induced by either model or realistic electron-electron (e-e) interactions. Without losing the generality, we investigate such phenomena for the Laughlin 1/3 and Moore-Read (MR) non-Abelian phases. Anyons display rich internal dynamics in both cases that will lead to interesting experimental consequences. In particular, bound states of two Laughlin anyons are preferred under short-range e-e interactions, leading to 2e/3 -- instead of e/3 -- effective charge carriers at low temperatures, which have been seen in several experiments. MR phases host two topologically distinct fusion channels: 1 and psi. The different effective interactions of e/4 anyons in the two sectors suggest the vanishing of the degeneracy of fusion channels when the e-e interaction is no longer its model Hamiltonian, in which case different bound states could also appear. This indicates the possibility of energetically manipulating the two types of anyons by tuning the bare e-e interactions. We point out that the results of recently developed high-resolution STM measurements will be affected by the effective anyon interactions, where anyons are clustered together after the tunneling of electrons. The low-lying parts of the local density of states affected by various anyon clusters are simulated for both Abelian and non-Abelian systems with (screened) Coulomb interactions.

Figures (14)

• Figure 1: (a) Interaction between two anyons with model electron-electron interaction $\hat{V}_3$ (blue circles) and $\hat{V}_5$ (purple squares) in $\mathcal{H_L}$; (b) Effective interaction between two anyons induced by bare electron-electron interactions: Coulomb (olive squares) and Yukawa with $\lambda = 0.375/l_B$ (green circles), inset: energy differences between the bunching state $E_{\Delta M =2}$ and the most separated state $E_S$ ($\Delta M =12$ for the largest system size $N_e =12$ we computed) against $\lambda$. (c, d) Finite-size scaling of the energy differences between the most separated state $E_S$ (orange crosses), or the bound state $E_B$ when $\Delta M =0$ (yellow pluses), and the bunching state $E_{\Delta M =2}$ with $\hat{V}_3$ (c) and $\hat{V}_5$ (d).
• Figure 2: Energy spectrum of three-quasihole Laughlin 1/3 state with electron-electron (a) Coulomb interaction (b) Yukawa interaction when $\lambda=1/l_B$, both in the LLL; (c) Energy differences between the $\Delta M=6$ state and its competing ones-$\Delta M=8$ (olive circles) and $\Delta M=12$ states (green squares)-plotted as a function of $\lambda$, both exhibiting monotonic behaviours. Red crosses indicate quasihole states, blue pluses denote other excitations outside the QH manifold. The system size is $N_e =10$.
• Figure 3: Effective interaction of two MR quasiholes with different electron-electron interactions. (a) $\hat{V}_1$ and (b) $\hat{V}_3$ within $N_e = 19$ (odd, pink stars) and $N_e = 20$ (even, blue squares) systems, calculated from ED; (c) Coulomb interaction and (d) Yukawa interaction when $\lambda = 0.5 /l_B$ in systems of $N_e = 51$ (odd, pink) and $N_e = 50$ (even, blue), computed by Monte Carlo with error bars shown.
• Figure 4: Spectrum of MR with four quasiholes under various realistic interactions, with odd (here $N_e=13$, first row) and even (here $N_e=14$, second row) numbers of electrons. (a) Coulomb interaction; (b) Yukawa with $\lambda=0.5$; (c) Yukawa with $\lambda = 1$.
• Figure 5: Possible low-energy LDOS for Laughlin 1/3 state with $N_e = 10$ at FWHM = $0.003\text{(first panel)}/0.005\text{(second panel)}\ e^2/\epsilon l_B$. The entire background energies are subtracted in the subfigures since only the energy differences matter here. (a) Coulomb interaction, (b) Yukawa interaction when $\lambda = 1$, both in the LLL.