Circular Dichroism on the Edge of Quantum Hall Systems: From Many-Body Chern Number to Anisotropy Measurements

TL;DR

This work establishes that in quantum Hall systems, the edge of a finite, confined sample hosts a quantized circular dichroism (CD) response governed by the edge’s low-energy modes, as described by Wen’s chiral Luttinger liquid theory. By isolating the edge contribution, the CD signal becomes a direct, universal probe of the many-body Chern number $C_{ m MB}$, independent of microscopic edge details, and distinguishable from bulk contributions via frequency resolution. The authors derive a general edge sum rule, show robustness to multiple edges and arbitrary boundary shapes, and quantify how edge geometry modulates the CD response through the ratio $S_-/S_+$. Numerical illustrations across integer and fractional Chern insulators, with circular and anisotropic boundaries, demonstrate how edge spectroscopy can extract $|C_{ m MB}|$ and even characterize droplet shapes, offering a practical route for ultracold-atom experiments to measure topological invariants locally at the boundary.

Abstract

Quantum Hall states are characterized by a topological invariant, the many-body Chern number, which determines their quantized Hall conductivity. This invariant also emerges in circular dichroic responses, namely, by applying a circular drive and comparing excitation rates for opposite orientations. This work explores the dichroic response of confined, isolated quantum Hall systems, where bulk and edge contributions cancel exactly:~When the edge response is properly isolated, the circular dichroic signal becomes quantized, serving as a direct and elegant probe of the many-body Chern number encoded in the edge physics. We demonstrate that this quantized edge response is entirely captured by low-energy chiral edge modes, allowing for a universal description of this effect based on Wen's edge theory. Its low-energy nature implies that the quantized edge response can be distinguished from the bulk response in the frequency domain. The edge response is also shown to be a sensitive diagnostic of geometric features. This opens the possibility of characterizing the shape of quantum Hall droplets through edge spectroscopic measurements, without requiring knowledge of the system's boundary profile. We illustrate our findings using realistic models of integer and fractional Chern insulators, with different edge geometries, and propose detection schemes suitable for ultracold atoms.

Figures (14)

• Figure 1: (a) The dichroic response of a QH state is quantized in the bulk according to the many-body Chern number $C_{\rm{MB}}$ [Eq. \ref{['eq:C_dichroism']}]. In a confined system, this bulk response is perfectly cancelled by the edge, as evidenced by the local Chern-marker description [Eq. \ref{['eq:C_dichroism_local']}]. (b) The bulk and edge dichroic responses can be resolved in the frequency domain. (c) The response associated with low-energy edge modes is quantized, reflecting the many-body Chern number $C_{\rm{MB}}$ of the topological insulating state.
• Figure 2:
• Figure 3: (a) Multiple edge modes are introduced in a non-interacting integer quantum Hall state in Harper-Hofstadter bands by including, in addition to a circular hard wall at $r_{\text{wall}}\!=\!14.5$, a ring-shaped potential bump of strength $0.5J$, centered two sites away from the walls. (b) Angular-momentum-resolved spectroscopy: Matrix elements $M_{\ell}^{n,0}\!=\!\left|\int d^2\mathbf{r}\, e^{i \ell \theta}\braket{n|\hat{\rho}(\mathbf{r})|0}\right|^2$, as a function of frequency, $\omega_{n,0}\!=\!(\varepsilon_n-\varepsilon_0)/\hbar$, and transferred (optical) angular momentum $\ell$; $\varepsilon_n$ is the energy of the $n$-th eigenstate. The presence of multiple edge modes is highlighted by the arrows. (c) CD matrix elements $M_{\pm}^{n,0}$ [Eq. \ref{['eq:edge_modes_matrix_elements']}], as a function of $\omega_{n,0}$, with the inset zooming in on the low-energy features. The numerically extracted Chern number contributions, $C_{\text{MB}}^{\rm num} = C_+^{\rm num}-C_-^{\rm num}$, are calculated by isolating the low-energy response ($\hbar\omega\!\leq\!1J$). The matrix elements are averaged over a small frequency window (of width $0.05J/\hbar$) for visualization. (d) $C_{\rm MB}^{\rm num}$ approaches the quantized value for increasing system size, denoted by the total number of occupied sites, $N_{\text{s}}$. We used $\phi\!=\!2/7$ and $\varepsilon_F\!=\!-1.5J$.
• Figure 4: (a) Radial density profile $\rho_A$(/$\rho_B$) of the $A$(/$B$) component of a Halperin $(2,2,1)$ state (yellow(/red) dashes), and the combined density $\rho_A+\rho_B$ (black solid line). The bulk's density approaches $\rho_0=\nu/(2\pi l_B^2)$, with $\nu=|C_{\rm MB}|=2/3$. (b) Low-lying portion of the system's spectrum $E_n$ (black crosses) as a function of its angular momentum $L$; horizontal colored lines display the $M_+^{n,0}$ matrix elements; both edge modes at $L=L_{(2,2,1)}+1$ can be seen to carry a non-zero spectral weight, which together add-up to the many body Chern number $C_{\rm MB}$ (inset).
• Figure 5: (a) Density $\rho(x,y)$ of a Laughlin liquid in an anisotropic (elliptic) anharmonic trap, for a fixed ellipticity parameter $\lambda=0.2$. (b) The two normalized rates $S_\pm$ (black circles) saturate the edge CD sum rule for all the ellipticities $\lambda$ considered, with the expected value $|C_{\rm MB}|=1/2$ (black dashes). (c) The ratio $S_-/S_+$ reveals $\lambda$ via Eq. \ref{['eq:SmSp_ratio_ellipse']}.