Infinite temperature spin dynamics in the asymmetric Hatsugai-Kohmoto model

TL;DR

This work addresses the infinite-temperature spin dynamics of the asymmetric Hatsugai-Kohmoto model, a solvable cousin of the Hubbard model, to understand how spin excitations reflect Mott physics and hopping asymmetry. The authors derive the high-temperature single-particle Green's function with weights $(1-\nu)$ and $\nu$, obtain the self-energy $\Sigma_\sigma(k,\omega_n)$, and compute the dynamical spin structure factor via exact Kubo formulas, focusing on one dimension with generalization to higher dimensions. They find a rich spectrum: the longitudinal spin structure factor supports multiple sound-like modes, while the transverse response exhibits a Lifshitz-like transition at $U=|t_\uparrow-t_\downarrow|$ with a quadratic dispersion $\omega=U(qa)^2$ at small $q$; in the Falicov-Kimball/flat-band limit, the response becomes momentum-independent with characteristic singularities and Dirac-delta features. The total spin structure factor is momentum-independent, determined solely by the filling $\nu$, providing a clean analytic benchmark for high-temperature spin dynamics in strongly correlated itinerant systems.

Abstract

We focus on the infinite temperature dynamical spin structure factor of the asymmetric Hatsugai-Kohmoto model, the relative of the asymmetric Hubbard model. It is characterized by distinct single particle energies for the two spin species, which interact with each other through a contact interaction in momentum space. We evaluate its spin structure factor exactly and follow the evolution of its excitation spectrum for all fillings and interactions, identify signatures of the Mott transition and fingerprints of the asymmetric hoppings. The longitudinal spin structure factor exhibits sound like and interaction induced gapped excitations, whose number gets doubled in the presence of hopping asymmetry. The transverse response displays the competition of interaction and asymmetry induced gaps and results in a quadratic excitation branch at their transition. The complete asymmetric case features momentum-independent dynamical structure factor, characteristic to transitions involving a flat band.

Figures (2)

• Figure 1: The possible transitions for finite $q$ and $\omega$ are visualized schematically between various bands. The blue and red lines denote the lower, $\varepsilon_\sigma(k)$ and upper, $\varepsilon_{\sigma'}(k)+U$ Hubbard band, respectively. With increasing $U$, the last transition becomes impossible due to energy conservation when a clean gap appears between these band. Further increasing the interaction pushes the third process to very high energies, therefore in the large $U$ limit, only the first 2 transitions are possible, characterized by $w_0$.
• Figure 2: The evolution of the transverse and longitudinal dynamical spin structure factors are visualized for half filling, $\nu=1/2$ for both spin orientations. The interaction varies as $U/t_\uparrow=0$, 0.5 and 1 from top to bottom and the results are insensitive to the sign of $U$.. Column (c) shows the transverse spin structure factor for the symmetric case, the longitudinal one is equivalent to it up to an overall factor of 2. Columns (b) and (d) correspond to the transverse and longitudinal spin structure factor for $t_\downarrow=t_\uparrow/2$, while columns (a) and (e) depicts the same quantities for $t_\downarrow=0$, respectively. The center panel in column (b) corresponds to $t_\uparrow-t_\downarrow=U$, and displays a Lifshitz transition when the $\omega\sim |q|$ sound like excitation changes to $\omega=U (qa)^2$ for small $\omega$ and $q$, denoted by black dashed line. Column (a) corresponds to the transverse spin structure factor in the fully asymmetric case with no momentum dependence due to the flat band, therefore the explicit frequency dependence is displayed, revealing the characteristic inverse square root singularities from Eq. \ref{['squareroot']}. Small energy but finite momentum excitations are also present in, e.g., the bottom panel of column (c), denoted by red arrow. The color encoding is the same for the 12 right panels.