Anyon Dispersion in Aharonov-Casher Bands and Implications for Twisted MoTe${}_2$

TL;DR

The work provides a controlled framework to compute and interpret anyon dispersion in FQAH states realized in AC bands by projecting interactions onto Laughlin quasihole states and constructing momentum-resolved quasihole wavefunctions. Dispersion arises from the interplay between a periodic potential generated by quantum geometry inhomogeneity and a noncommutative quasihole guiding-center coordinate governed by the many-body Berry phase; this yields a finite bandwidth ε_{oldsymbol{ appa}} and a characteristic q^2-fold degeneracy in the Brillouin zone. Using Monte Carlo sampling, the authors show that the quasihole bandwidth increases with geometric inhomogeneity and interaction screening length, predicting ≈1 meV dispersions for twisting MoTe_2, suggesting itinerant anyon physics can be relevant in clean samples. They further formulate a microscopic Lagrangian description of quasihole dynamics via geometric quantization, paving the way for a many-anyon theory that retains only the anyon degrees of freedom and connects to long-distance Chern-Simons physics. Overall, the results establish a scalable, analytically controlled route to connect microscopic band geometry and interactions to emergent itinerant anyon behavior in realistic materials.

Abstract

The discovery of fractional quantum anomalous Hall (FQAH) states in two-dimensional heterostructures has opened the door to realizing phases of dispersing anyons. Here, we develop an analytically controlled theory of anyon dispersion in FQAH states realized in ideal or Aharonov-Casher (AC) bands by projecting interactions onto the space of Laughlin quasiholes. Constructing quasihole momentum eigenstates allows efficient evaluation of the single quasihole dispersion using Monte Carlo. We find that the quasihole bandwidth grows with increasing quantum-geometry inhomogeneity of the AC band and with increasing interaction screening length. For realistic parameters relevant to the bands of twisted MoTe${}_2$, the quasihole bandwidth is of order 1 meV, suggesting that itinerant-anyon physics may play an important role in sufficiently clean samples. Furthermore, we develop a microscopic Lagrangian framework in terms of a quasihole guiding-center coordinate, which reproduces the momentum-space formula for the dispersion. This approach reveals that quasihole dispersion originates from the combined effects of an interaction-generated periodic potential, arising from non-uniform quantum geometry of the single particle bands, and the quasihole many-body Berry phase arising from the background magnetic field. The latter endows the guiding-center coordinate with a noncommutative structure, converting the periodic potential into a finite dispersion. Finally, we outline how this framework generalizes to multiple quasiholes, enabling a microscopic theory of charged excitations in FQAH systems that retains only the anyon degrees of freedom.

Figures (11)

• Figure 1: Summary of the numerical results for Laughlin quasiholes in first harmonic approximated AC bands and tMoTe${}_2$ AC bands with realistic parameters. a-c: The dispersion of quasihole wavefunction $\psi^{AC}_{{\bm k}}(\{{\bm z}_i\})$ defined using Eq. \ref{['PsiAC']} and Eq. \ref{['eq:momentum-state']}, in an AC band with first harmonic approximation, $Q(z)=-2K\sum_{{\bm b}_i}\cos({\bm b}_i\cdot{\bm z})$ where ${\bm b}_i,i=1,2,3$ are the reciprocal lattice vector of $\Lambda^{*}$ with ${\bm b}_3=-{\bm b}_1-{\bm b}_2$, under screened Coulomb interaction Eq. \ref{['V:sCoul-real']}, measured in natural unit pseudopotential $V_1 = \int \frac{d^2 {\bm q}}{(2\pi)^2} V({\bm q}) e^{-{\bm q}^2}L_1({\bm q}^2)$. $K,d$ controls the non-uniformity and gate distance respectively. $l_B$ is set to 1. a. Dispersion plotted in the original unfolded Brillouin Zone for $N_e=56,\ K=0.4,\ d=1.0$, the $q^2=9$ fold degeneracy can be observed clearly, with minor deviations due to finite-size effects and MC noise. Average energy has been subtracted. b. Dependence of the quasihole bandwidth $w$ for $N_e=16$ system on different $K$ and $d$. c. Bandwidth as a function of $k$ for $d=1.0$: $w$ grows linearly with $K$ in the small $K$ regime and starts to saturate at large $K$. d. AC quasiholes bandwidth dependence of the twist angle $\theta$ in tMoTe${}_2$ with gate screened Coulomb interaction in the mBZ, here we take $N_e=40,\ d=10 \ l_B\approx 20\ \text{nm}$. The twist angle dependence is obtained from a single MC dataset with $\theta$ controlling the length scale and thus the corresponding energy scale. Shades around the curve represent error bar.
• Figure 2: Illustration of quasiholes obtaining dispersion in nonuniform field. a. Quaisholes with exchange phase $e^{i\frac{\pi}{q}}$ moving in uniform field with continuous magnetic translation symmetry and moving in non-uniform magnetic field with discrete magnetic translation symmetry. b. Emergence of the $q^2$ fold degeneracy of quasiholes, taking $q=3$ as an example: the reduced BZ has folding direction either ${\bm b}_1$ or ${\bm b}_2$, unfolding procedure maps adjacent states in the reduced BZ to $q$ sectors in that direction and thus gives rise to $q$ fold degeneracy along the folding direction. Since the folding direction is chosen to be either ${\bm b}_1$ or ${\bm b}_2$, the physical system has to be $q^2$ fold degenerate to be consistent with the unfolding procedure.
• Figure 3: (a) Single particle quantum geometry, i.e. real space kähler potential $Q({\bm r})$, for twisted MoTe${}_2$. The hexagon outlines Wigner-Seitz moiré unit cell. (b) Dispersion of tMoTe${}_2$ with gate screened Coulomb interaction in the mBZ, here we take $N_e=40,\ d=10 \ l_B\approx 20\ \text{nm},\ \theta=3^\circ$
• Figure 4: Quasihole quantum geometry in AC band with first harmonic approximation, $K=0.25$. Reconstructed from weak field approximation and finite size scaling, $N_{\Phi}=2500$. a. Momentum space Kähler potential ${\mathcal{R}}({\bm k})$. b. Real space Kähler potential ${\mathcal{Q}}({\bm z})$. c. Effective magnetic field ${\mathcal{B}}({\bm z})$. In b-c, the hexagon outlines Wigner-Seitz moiré unit cell.
• Figure S1: Consistency check with $1/3$ Laughlin state. a. Guiding center structure factor taken for system with $N_e=147$. The dashed line represents analytical small $|{\bm q}|$ expansion. b. Finite size scaling for Coulomb energy calculated from physical structure factor. Thermodynamic value $E_{\rm Coulomb}$ is extracted from scaling.