Charge Susceptibility and Kubo Response in Hatsugai-Kohmoto-related Models

TL;DR

The paper addresses charge response in Mottness-relevant Hatsugai-Kohmoto (HK) and orbital HK (OHK) models, where non-rigid lower and upper Hubbard bands create a multi-pole density response, a nontrivial particle-hole continuum, and potential deviations from standard plasmon and f-sum behavior. It derives the exact density response $\chi^0_{\text{HK}}(\mathbf q,\omega)$, decomposing it into direct and cross Hubbard-band contributions, and treats Coulomb coupling via RPA to reveal a plasmon dispersion $\omega_{\mathbf p}(q) \sim 1/q$ and a modified f-sum rule arising from a nonlocal, long-range diamagnetic current. The work clarifies how the Kubo current operator must be defined in HK by deriving it from gauge-invariant minimal coupling, demonstrating current conservation for finite systems and highlighting the noncommutativity of the limits $\mathbf q\to 0$ and $L\to\infty$; it reconciles these findings with the continuity equation and resolves prior disputes related to DMFT interpretations. It also connects HK physics to DMFT in the $d\to\infty$ limit, where momentum sectors decouple and Mottness is captured by fixed-point structure, suggesting that the observed anomalies are intrinsic to the long-range HK interaction rather than ground-state degeneracy. Overall, the results illuminate Mottness-driven charge dynamics, plasmon anomalies, and the precise role of boundary conditions in strongly correlated band models, with implications for DMFT-based descriptions of correlated metals.

Abstract

We study in depth the charge susceptibility for the band Hatsugai-Kohmoto (HK) and orbital (OHK) models. As either of these models describes a Mott insulator, the charge susceptibility takes on the form of a modified density response function with lower and upper Hubbard bands, thereby giving rise to a multi-pole structure. The particle-hole continuum consists of hot spots along the $ω$ vs $q$ axis arising from inter-band transitions. Such transitions, which are strongly suppressed in non-interacting systems, obtain here because of the non-rigidity of the Hubbard bands. This modified density response function gives rise to a plasmon dispersion that is inversely dependent on the momentum, resulting in an additional contribution to the conventional f-sum rule. This extra contribution originates from a long-range diamagnetic contribution to the current. This results in a non-commutativity of the long-wavelength ($q\rightarrow 0$) and thermodynamic ($L\rightarrow\infty$) limits. When the correct limits are taken, we find that the Kubo response computed with either open or periodic boundary conditions yields identical results that are consistent with the continuity equation contrary to recent claims. We also show that the long wavelength pathology of the current noted previously also plagues the Anderson impurity model interpretation of dynamical mean-field theory (DMFT).

Figures (4)

• Figure 1: Schematic diagram of filling surface momentum for Hubbard bands $P_L$ and $P_U$. $P_L/P_U$ are the intersections of the chemical potential with the lower/upper Hubble band. The colored arrows represent processes in the particle-hole continuum. Green and blue arrows indicate finite momentum ($q\ne0$) transitions, while the red arrow corresponds to an zero momentum transition which is forbidden in rigid bands. The processes drawn here generalize to the insulating case in which neither the upper nor the lower bands cross the chemical potential.
• Figure 2: Plasmon dispersion in the $(\mathbf{q},\omega)$ plane, as extracted from the pole of $\mathrm{Im} \chi^{\mathrm{RPA}}_{\mathrm{HK}}$. We observe a $1/q$ momentum dependence, suggesting a modification of the f-sum rule from the conventional form in the long wavelength limit. Here, r.l.u. stands for reciprocal lattice units. We work with a square-lattice tight-binding dispersion for our model, $\xi(\mathbf{k}) = -2t\left(\cos(k_x) + \cos(k_y)\right) - 4t_p\cos(k_x)\cos(k_y)$, where $t=0.1$eV, $t_p = 0.03$eV, and $U = 0.14$eV, where $U$ is the interaction strength as defined in Eq. (\ref{['eq:bHK_hamiltonian']}).
• Figure 3: The $f$-sum rule for band HK and Fermi liquid systems, plotted as a function of $\mathbf{q}$. The blue line represents the sum rule numerically calculated from the HK model, which clearly does not approach zero as $q \to 0$. For comparison, the orange line represents the sum rule from a generic Fermi liquid, which scales as $q^2$ for small $q$. Note that this plot has the point $q=0$ excluded. The parameters involved in calculating the result for band HK are same as those provided in the caption of Fig. (\ref{['fig:1/q_plasmon']}).
• Figure 4: The imaginary part of the susceptibility for (a) HK model and (b) Fermi Liquid $\text{Sr}_2\text{Ru}\text{O}_4$. The HK charge susceptibility exhibits a prominent peak along the $\omega$-axis, arising precisely at $\omega=U$, where $U$ is the interaction strength as defined in Eq. (\ref{['eq:bHK_hamiltonian']}). The high intensity at small momenta indicates that the HK model adheres to a different $f$-sum rule as compared to a Fermi liquid. For the the Fermi Liquid $\text{Sr}_2\text{Ru}\text{O}_4$, the prominent intensities corresponds to the intra-band PHC. The inter-band energy spacing is roughly $\omega = 100$ meV, where the PHC originates on the $\omega$-axis.