Chiral Edge Excitations of Fractional Chern Insulators

Abstract

Edge excitations are the defining signature of chiral topologically ordered systems. In continuum fractional quantum Hall (FQH) states, these excitations are described by the chiral Luttinger liquid ($χ$LL) theory. Whether this effective description remains valid for fractional Chern insulators (FCIs) on discrete lattices has been a longstanding open question. Here we numerically demonstrate that the charge-one edge spectral function of a $ν=1/2$ FCI on an infinitely long strip with width $L_y=10$ quantitatively follows the predictions of $χ$LL theory. The edge spectrum is gapless, chiral, and linear, with spectral weight increasing linearly with both momentum and energy. We further analyze the influence of lattice size, particle number, trapping potential, and charge sector of excitations on the edge properties. Our results establish a clear correspondence between lattice FCIs and continuum FQH systems and provide guidance for future experimental detection of chiral edge modes.

Figures (5)

• Figure 1: (a) Lattice geometry of a strip with $L_y=10$, in which $\phi$ denotes the magnetic flux per plaquette. The spectral functions of charge-one edge excitations on the top ($n=10$) and bottom ($n=1$) rows are computed using an iMPS with bond dimension $D=2000$ and a Lorentzian broadening factor $\eta=0.005$. (b) Illustration of the path of a local tensor in the iMPS (blue) and its mapped trajectory on the physical lattice (pink). The shaded area marks the region where Stokes’ theorem is applied to evaluate the momentum shift.
• Figure 2: Ground-state particle densities and current distributions of the $\nu=1/2$ bosonic FCI on a strip. Panels (a) and (b) show results of iMPS on an $\infty \times 10$ lattice with one particle per column and trapping strength varied from $V=0$ to $V=0.035$. Panels (c1)–(d2) present data on a finite strip with $L_x=40$, $L_y=10$, and $V=0$. The total particle number is $N_{\mathrm{total}}=L_x$ for (c1-c2) and $N_{\mathrm{total}}=L_x+1$ for (d1-d2). Arrow size and color indicate current strength, while square colors denote local particle densities. Panels (c2) and (d2) zoom in on nine sites near the top boundary, revealing enhanced edge currents $\langle \hat{\mathcal{J}}^x \rangle$ after adding one more particle.
• Figure 3: Spectral weight and average positions of the excited states on an infinite strip with $L_y=10$ and $V=0$. (a) Spectral weight $\mathcal{I}_9(\mathcal{E}_p,p)+\mathcal{I}_{10}(\mathcal{E}_p,p)$ of the lowest chiral excitations near $p=0$. (b) Average positions $\overline{n}$ of the three lowest energy levels of the excited states.
• Figure 4: (a,b) Spectral functions $\mathcal{A}_n(k_x,\omega)$ on an $\infty\times 8$ lattice with iMPS bond dimension $D=1500$, for $n=1,8,4,5$ in (a1,a2,b1,b2), respectively. (c) Particle density and current distributions of the ground state with $L_xL_y+1$ particles on the finite strip with $L_x=30$ and $L_y=8$.
• Figure 5: Average positions and spectral functions on an infinite strip with width $L_y=11$ and harmonic trapping potential $V=0.01$. (a0) and (b0) show the average positions of charge-zero and charge-one excitations, respectively. (a2,a3) display the spectral functions of charge-zero excitations on rows $n=2,10$, while (b2,b3) show the corresponding charge-one spectra. All data are obtained with MPS bond dimension $D=2000$.