Flat-Band Stoner Instability and Peierls-Phase Origin of the Transdimensional Anomalous Hall Effect in Rhombohedral Graphite

Abstract

A ``transdimensional'' anomalous Hall effect (TDAHE), where both in-plane ($\Bpar$) and out-of-plane magnetic fields produce hysteretic Hall signals, was recently observed in nine-layer rhombohedral graphite~\cite{Li2026Nature}. We present a microscopic theory attributing the TDAHE to a flat-band Stoner instability coupled to Peierls-phase gap modulation. The flat-band density of states satisfies the Stoner criterion $Uρ(\varepsilon_F) > 1$~\cite{Bultinck2020}, driving a spin-valley-locked ferromagnet whose valley polarization $η$ breaks time-reversal symmetry and generates an intrinsic anomalous Hall conductivity (AHC). The orbital $g$-factor $g_{\mathrm{orb}} = e d_0 v_F (N{-}1)/2 \propto (N{-}1)$ then lets $\Bpar$ modulate the gap, producing the transdimensional response $\propto η$. A self-consistent $2N$-band Hartree-Fock calculation yields complete valley polarization ($η\to 1$) below a mean-field transition $T_c^{\mathrm{MF}} \approx 2.2$~K, reduced by 2D-Ising critical fluctuations to the experimental $T_c \approx 1.6$~K, with $R_{xy} \approx 1.5$~k$Ω$. Because both Hall responses are carried by one order parameter $η$, they share a single $T_c$, as observed; $T_c$ is governed by the Stoner product $Uρ$ and is insensitive to the intervalley exchange, which only gates whether the valley-polarized phase forms, so the exchange strength is not a fitted parameter. Beyond reproducing $R_{xy}$, $T_c$, and the phase window, the theory predicts a sharp onset of valley polarization between $N = 7$ and $N = 9$, a symmetry selection rule fixing the crescent Fermi surface to the $m{=}1$ nematic channel, and a transdimensional-to-conventional Hall ratio $\sigmaPHE/\sigmaAHE^{\mathrm{tot}} = g_{\mathrm{orb}}\Bpar/m$ independent of $η$ and $U$. The $Z$-independence of the intrinsic AHC is verified within dynamical mean-field theory.