Luttinger's Theorem Violation and Green's Function Topological Invariants in a Fractional Chern Insulator

Abstract

Luttinger's theorem constrains the particle density of interacting fermions through global properties of the single-particle Green's function, and its violation signals a breakdown of the identification between the quantized Hall response and the Green-function-based Ishikawa-Matsuyama invariant. This phenomenon becomes especially compelling in strongly correlated topological phases, such as fractional Chern insulators, where fractionalized quasiparticles lack an adiabatic connection to electrons, raising the question of how Green's-function-based topological invariants manifest in such phases. Using exact diagonalization of the fermionic Harper-Hofstadter-Hubbard model, we compute bulk single-particle Green's functions deep inside a fractional Chern insulating phase and directly evaluate the Luttinger count, its possible correction (the Luttinger integral), and the Ishikawa-Matsuyama invariant $N_3[\mathrm{G}]$. We demonstrate a clear violation of Luttinger's theorem and show that the fractional nature of the many-body Chern number is encoded in the Středa response of the Luttinger integral, while the integer invariant $N_3[\mathrm{G}]$ arises from the Středa response of the Luttinger count. We also analytically prove that $N_3[\mathrm{G}]$ is fully determined by the Luttinger count together with the Chern number of the occupied Bloch band, upon neglecting Bloch-band mixing. Finally, we propose an experimental protocol to extract all Green-function-based topological invariants from local density-of-states measurements, experimentally accessible in fractional quantum Hall systems.

Figures (11)

• Figure 1: Schematic illustration of the system, approach and key results.(a) We study the bulk single-particle Green's function $\mathrm{G}(\omega)$ of interacting electrons in a uniform magnetic field realizing a fractional Chern insulator. (b) Schematic spectral function of a FQH state. The electronic spectral density exhibits sharp coherent hole-like and electron-like peaks, corresponding to long-lived charged excitations located well above the neutral many-body gap $\Delta_{\mathrm{MB}}$. The dashed red line indicates the emergence of a zero of $\mathrm{G}$ within the charge gap. (c) In the FCI phase, the particle density and the Luttinger-count density differ, signaling a violation of Luttinger's theorem. (d) The Středa response of the particle density evaluated deep within the bulk yields the fractional many-body Chern number $C^{\mathrm{MB}}$, whereas the Středa response of the Luttinger-count density $n_1$ produces an integer valued $N_3[\mathrm{G}]$ different from $C^{\mathrm{MB}}$.
• Figure 2: $(a)$ Schematic illustration of the diagonal element of a toy single-particle Green's function on the real-frequency axis, with poles at $\varepsilon_\alpha^{(p)} = -2, 2, 4$ (blue circles) and corresponding spectral weights $Z_\alpha^{(p)}=0.7, 0.2, 0.1$. The positions of the zeros $\chi_\alpha^{(s)}$ of $\mathrm{G}_{\alpha\alpha}(\omega)$ are indicated by red crosses. $(b)$ Spectral density $A_{\alpha}(\omega)$ [Eq. \ref{['eq:A_omega']}] and Luttinger counting kernel $n_{1\alpha}(\omega)$ [Eq. \ref{['eq:n1_omega']}].
• Figure 3: Analytical structure of the Green's functions and the self-energy.(a) Local spectral densities of the Green's function, $A(\omega)$, and the self-energy, $S(\omega)$, defined in \ref{['eq:spectral']} computed in the finite box (positive sign) and on the torus (negative sign). Dashed lines denote the electron addition and removal energies, $\mu^{+}=E_0(N+1)-E_0(N)$ and $\mu^{-} = E_0(N) - E_0(N-1)$, extracted from the torus geometry, where $E^{N\pm1}_0$ are the ground state energies in the $N\pm1$-particle sectors. Note that $A(\omega)$ and $S(\omega)$ are plotted in different scales. (b) Momentum-resolved spectral densities $A(k,\omega)$ and $S(k,\omega)$ on the torus geometry. The main panel shows spectra obtained from the thermodynamically averaged Green's function over the degenerate ground-state manifold [\ref{['eq:therm_GF']}], while the inset presents the result of averaging only over a single ground-state [\ref{['eq:signgle_gs_GF']}]. The frequency is measured in units of the fixed-$N$ many-body gap $\Delta_{\mathrm{MB}}$ computed on the torus geometry, with zero frequency set at $(\mu^{+}+\mu^{-})/2$. The momenta $k_x$ and $k_y$ are measured in units of $\frac{2\pi}{N_x}$ and $\frac{2\pi}{q N_y}$
• Figure 4: Středa-response and Luttinger's theorem violation in the FCI state.(a,b) Spatial distributions of the local Luttinger count $n_1(\bm{r})$ and Luttinger integral $\Delta n_1(\bm{r})$, defined in \ref{['eq:localn1n2']}. The presence of a finite, negative $\Delta n_1(\bm{r})$ in the bulk directly signals the breakdown of Luttinger's theorem. (c,d) Středa responses of the Luttinger count and the Luttinger integral densities. The derivative of $n_1$ with respect to magnetic field (averaged over the bulk sites enclosed by the green contour) yields an integer-valued response, corresponding to an Ishikawa-Matsuyama invariant $N_3[\mathrm{G}] \simeq 1$. By contrast, the Středa response of $\Delta n_1$ is fractional and accounts for the difference between $N_3[\mathrm{G}]$ and the many-body Chern number. (e) Středa response of the total bulk electron density as a function of magnetic flux per plaquette. The developing plateau near $1/3$ is consistent with the fractional Hall conductance of the FCI state realized at flux $\varphi = 1/5$. The inset shows the spatially-resolved magnetic response of the total density $n(\bm{r})$ at $\varphi = 1/5$.
• Figure 5: Extraction of the self-energy, the Luttinger count and the Luttinger integral from the spectral functions obtained with ED. (a) The spectral densities $A$ and $S$ defined in \ref{['eq:spectral']} as obtained from the exact diagonalization and extracted from the model \ref{['eq:fit_spec']}. Note, that $A(\omega)$ and $S(\omega)$ are plotted in different scales. (b) and (c) The distribution of the localized versions of the Luttinger count $n_1$ and the Luttinger integral $\Delta n_1$ over the lattice, as defined in \ref{['eq:localn1n2']}. (c) The Středa response of the Luttinger count $n_1$ and the Luttinger integral $\Delta n_1$ densities. The bulk value of the approximate $\frac{\partial n_1}{\partial B}$ allows us to estimate the Ishikawa-Matsuyama invariant by $N_3 = 0.99\pm0.01$ in agreement with our numerical \ref{['fig:fig4']} and analytical results \ref{['sec:N3 analytics']}. (d) The fractional valued Středa response of the Luttinger integral $\Delta N_3 = -0.58\pm0.02$ as before balances the difference between the many-body Chern number and $N_3$.