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Bound states of anyons: a geometric quantization approach

Qingchen Li, Pavel A. Nosov, Taige Wang, Eslam Khalaf

Abstract

The question of anyon interactions and their possible binding plays a key role in the physics of fractional quantum Hall states. Here, we introduce a controlled and scalable approach to study anyon binding by working entirely within the Hilbert space of anyons. The resulting theory is characterized by an effective potential, which captures the electrostatic energy of classical anyon configurations, and a Kähler potential, which simultaneously encodes the anyon Berry phase and the structure of their Hilbert space; both quantities are readily computed using Monte Carlo methods for large systems, enabling reliable extrapolation to the thermodynamic limit. By applying the formalism of geometric quantization on Kähler manifolds, we construct the anyon Hamiltonian, which can be exactly diagonalized in the few-anyon Hilbert space. Applying our approach to the quasiholes of the $ν=1/3$ Laughlin state with screened Coulomb interaction, we find that Laughlin quasiholes form bound states for screening lengths comparable or smaller than the magnetic length. Remarkably, binding occurs despite both the bare electron-electron interaction and the quasihole electrostatic potential being purely repulsive. The bound-state formation is a Berry phase effect, driven by oscillations in the quasihole density profile on the $\ell_B$ scale that are invisible in the quasihole electrostatic potential alone. For multiple quasiholes, we identify a sequence of phases as the screening length is reduced: free $e/3$ anyons, paired $2e/3$ bound states, three-anyon charge-$e$ clusters, and larger composite objects. Finally, we discuss possible signatures in charge imaging experiments on quantum Hall systems and the relevance to the phase diagram of itinerant anyon phases in fractional quantum anomalous Hall materials.

Bound states of anyons: a geometric quantization approach

Abstract

The question of anyon interactions and their possible binding plays a key role in the physics of fractional quantum Hall states. Here, we introduce a controlled and scalable approach to study anyon binding by working entirely within the Hilbert space of anyons. The resulting theory is characterized by an effective potential, which captures the electrostatic energy of classical anyon configurations, and a Kähler potential, which simultaneously encodes the anyon Berry phase and the structure of their Hilbert space; both quantities are readily computed using Monte Carlo methods for large systems, enabling reliable extrapolation to the thermodynamic limit. By applying the formalism of geometric quantization on Kähler manifolds, we construct the anyon Hamiltonian, which can be exactly diagonalized in the few-anyon Hilbert space. Applying our approach to the quasiholes of the Laughlin state with screened Coulomb interaction, we find that Laughlin quasiholes form bound states for screening lengths comparable or smaller than the magnetic length. Remarkably, binding occurs despite both the bare electron-electron interaction and the quasihole electrostatic potential being purely repulsive. The bound-state formation is a Berry phase effect, driven by oscillations in the quasihole density profile on the scale that are invisible in the quasihole electrostatic potential alone. For multiple quasiholes, we identify a sequence of phases as the screening length is reduced: free anyons, paired bound states, three-anyon charge- clusters, and larger composite objects. Finally, we discuss possible signatures in charge imaging experiments on quantum Hall systems and the relevance to the phase diagram of itinerant anyon phases in fractional quantum anomalous Hall materials.

Paper Structure

This paper contains 16 sections, 80 equations, 10 figures, 1 table.

Table of Contents

  1. Review of geometric quantization
  2. Effective potentials in different path integral representations
  3. Bound states from repulsive interaction in the LLL
  4. Bound states in LLL in a uniform field
  5. Simple example: spin 1 Hamiltonians
  6. Connection to plasma limit and Chern-Simons gauge transformation
  7. Basis construction for quasihole Hilbert space
  8. Additional numerical results
  9. Monte Carlo for the fixed center-of-mass and for a fixed angular momentum
  10. Quasihole spectra at 1/5 filling
  11. Quasihole spectra for double-gate screened Coulomb interaction
  12. Error analysis
  13. Finite-size scaling analysis
  14. Ground-state quasihole wavefunctions: an ansatz
  15. Expansion of the binding energy near integer filling
  16. ...and 1 more sections

Figures (10)

  • Figure 1: Results for two quasiholes: (a) Energy spectrum of two Laughlin quasiholes, measured in units of the Haldane pseudopotential $V_3$ relative to well-separated quasiholes, for Yukawa-screened Coulomb interaction $V(r) = \frac{e^{-r/\lambda}}{r}$ for different values of $\lambda$ as a function of the relative angular momentum $L$. (b) Effective electrostatic potential between two quasiholes as a function of the relative position. (c) Effective magnetic field felt by one quasihole as a function of the distance from the other quasihole.
  • Figure 2: Benchmarking Monte Carlo against exact diagonalization: Energy spectra $E_{2qh} - E_{\rm Laughlin} - 2(E_{1qh} - E_{\rm Laughlin})$ for $Ne=7$ at $\nu = 1/3$, comparing MC and ED results for $N_h = 2$ (a,b) and $N_h = 3$ (c,d), with Yukawa-screened Coulomb interaction at $\lambda = 1$ (a,c) and bare Coulomb interaction (b,d). In each panel, ED results are shown both with and without the Trugman-Kivelson pseudopotential term $\alpha\hat{V}_\mathrm{TK}$ (large $\alpha$), together with MC results for maximum $L_z$ and for the fixed center-of-mass.
  • Figure 3: Results for multi-quasiholes: (a,b) Energy spectra of three and four Laughlin quasiholes, respectively, for the Yukawa-screened Coulomb interaction at $\lambda=2$. (c) Density profiles of a single quasihole and multi-quasihole bound states at $L=2,8$ and $18$ corresponding to $N_h=2,3$ and $4$, respectively. (d) Phase diagram for $N_h=2,...,6$ as a function of the screening length $\lambda$. (e) Phase diagram in the thermodynamic limit for different screening lengths $\lambda$, limited to cluster phases with up to six quasiholes. $N_h$=1,2,3,4 results are obtained using the exact interaction evaluated in MC, whereas $N_h=5,6$ results are obtained from pairwise interaction approximation.
  • Figure S1: Comparison of effective potentials and Berry-curvature-induced magnetic fields obtained from different path-integral representations. (a) Effective potentials $U$, $U_{\mathrm{sym}}$ and $U_{\mathrm{dist}}$ as functions of the quasihole separation $\xi$ for $\lambda = 2$. (b) Effective magnetic fields $B$ and $B_{\mathrm{sym}}$ as functions of $\xi$. Dashed lines mark $1/2q$ and $1/q$.
  • Figure S2: MC spectra on the sphere at $\nu=1/3$ for $N_e=100$ with Yukawa-screened Coulomb interaction with $\lambda=1$.(a,b) Spectra for $N_h=2$ and $3$. Each panel shows MC results for the fixed center-of-mass and MC results for maximum $L_z$.
  • ...and 5 more figures