Fractional Chern insulator edges: crystalline effects and optical measurements

TL;DR

This work analyzes how crystalline lattice effects alter edge physics of fractional Chern insulators (FCIs), testing the robustness of the chiral Luttinger liquid description beyond the hydrodynamic limit. It combines bosonization with a parton construction for FCIs and employs two-saddle-point analyses alongside matrix-product state simulations to study edge dynamics on lattice models. The authors find that band curvature distorts equal-space edge correlators in the noninteracting limit, but interactions suppress these corrections, while two band-edge saddles produce oscillations and a universal long-time tail; equal-time density correlators retain universal $1/x^2$ scaling fixed by the topological data. They also propose experimental probes, such as time-resolved edge spectroscopy and near-field optical methods, to extract edge velocity and filling in moiré exciton systems and ultracold-atom realizations, enabling direct access to universal edge exponents in realistic platforms.

Abstract

Edge states of chiral topologically ordered phases are commonly described by chiral Luttinger liquids, effective theories that are exact only in the hydrodynamic limit. Motivated by recent bulk observations of fractional Chern insulators (FCIs) in two-dimensional materials and by synthetic realizations in ultracold atoms, we revisit this framework and quantify deviations from the hydrodynamic limit due to lattice effects. Using a combination of analytical arguments and numerical simulations, we disentangle universal from nonuniversal edge properties. We outline experimental probes in excitonic FCIs and in ultracold atom systems, and in particular propose time-resolved edge spectroscopy to directly access the predicted exponents and velocities.

Figures (10)

• Figure 1: (a) Dispersion of the edge states on the strip. (b) Schematic illustration of the checkerboard lattice in a strip geometry and the measurement scheme. Black arrows and solid/dashed lines represent the nearest-neighbor (NN) and next-nearest-neighbor (NNN) hopping terms, respectively; the direction of each arrow indicates the sign of the complex phase in the NN hopping. Thick blue (red) arrows at the bottom (top) edge denote right- (left-) moving edge states. The measurement scheme, which allows extraction of the two point Green's function $\langle a_i^\dagger a_j\rangle$ and the density-density correlator $\langle n_i n_j\rangle$, is depicted schematically as two photodetectors.
• Figure 2: Dynamics of Chern insulators and parton FCIs. (a) Boundary charge density difference $\Delta n(t)$ as a function of time for a Chern insulator with $t_1 = 1$, $t_{2a} = -t_{2b} = 1/\sqrt{2}$, $\phi = -\pi/4$, and $\mu = 1.4 t_1$ on a strip of size $L_x = 100$, $L_y = 50$. (b) Dynamical correlator $G_h(t)$ for the same parameters on a larger strip, $L_x = 600$, $L_y = 120$. A fit to Eq. \ref{['eq:Correlation_Function']} yields $\eta = 0.50$, $\alpha = 0.96$, and ${\mathcal{E}} = 0.12$. Inset: single particle band structure with color indicating boundary weight (darker means larger edge amplitude). (c) Lower part of the parton band structure for the Hamiltonian in Eq. (4) on a strip with $L_x = 600$, $L_y = 100$, $\tilde{t}_1 = t_1$, $\tilde{t}_{2a} = \sqrt{t_{2a}}$, $\tilde{t}_{2b} = i \tilde{t}_{2a}$, and $\mu = -2.0 \tilde{t}_1$. The color scale indicates the parton weight on the boundary; the inset shows the full band structure. (d) Onsite dynamical parton correlator $\Gamma_h(t)$ from Eq. \ref{['eq:Parton_Correlation_Function']}. In the absence of interactions, the correlator initially follows a $t^{-2}$ decay set by the linear edge dispersion (blue), and crosses over to $t^{-1}$ at longer times as band curvature becomes relevant (orange). Inset: $\Gamma_h(t)$ for the same system size at $\mu = -2.4$ (purple), $-2.2$ (red), and $-2.0$ (blue). Note that the oscillation frequency increases with increasing energy difference between $\mu$ and the band bottom of the edge dispersion.
• Figure 3: Dynamics of the bosonic FCI. The particle filling factor is defined by $\nu_b = N_b/(2 L_x L_y)$, corresponding to a lowest band filling of $\nu = 1/2$ with periodic boundary conditions. Time evolution is computed on a strip of $L_x = 40$, $L_y = 4$ with periodic structure along the $x$-direction. In a strip with finite $L_y$, a lower density $\nu_b$ is used to account for the reduced particle density at the edges formed by the hard confinement Supp. (a) Boundary charge density as a function of time for a $-7.5\%$ deviation of the particle number from $\nu_b = 1/4$. (b) Dynamical onsite single particle correlator for several fillings (deviations from $\nu_b = 1/4$) for bond dimension $\chi = 400$; the bulk particle density remains unchanged for these cases Supp. The shaded regions indicate error margins determined by the difference between the $\chi = 400$ and $\chi = 200$ results, which provides a conservative error estimate.
• Figure 4: Static correlators on a strip of width $L_y = 4$ with a filling deviation of $-5\%$. (a) Infinite strip along $x$ (iDMRG) with bond dimensions $\chi = 400, 800, 1600, 3200$ for the single particle correlator $|G_h(x)|$. (b) Finite strips of lengths $L_x = 12, 16, 24, 32, 48, 64$ at fixed $\chi = 1600$ for the connected density--density correlator $|C(x)|$. In both panels, $x$ measures the distance along the lower edge; in (b), $C(x)$ is measured from the center of the lower edge, as indicated in the inset.
• Figure S1: Finite-momentum decay processes that generate a finite decay rate for high-energy excitations: (a) decay process contributing to the particle Green's function; (b) decay process contributing to the hole Green's function.