Algebraic-Dynamical Perturbation Theory of Large-$U$ Hubbard Models. Single Particle Spectrum of Antiferromagnetic Mott Insulating States

Abstract

In this work, we present an analytical framework for studying antiferromagnetic (AFM) Mott insulating states in the Hubbard model. We first derive an analytical solution for the single-particle Green's functions in the atomic limit. Within a second-order perturbation approach, we compute the ground state energy and show that the ground state is antiferromagnetically ordered. Then we derive an analytical solution for single-particle Green's functions when effects of the hopping term are considered in the Néel state. With the analytical solution, we compute the spectral functions and explain various properties of the AFM Mott insulating state as observed both experimentally and numerically: i) magnetic blueshift of the Mott gap; ii) the low energy part in the parental compounds of cuprate high \(T_c\) superconductors, which corresponds to a single band Hubbard model description. This work comprehends the electronic properties of antiferromagnetic Mott states analytically and provides a foundation for future investigations of doped antiferromagnetic Mott insulators, aiming for the mechanism of cuprates high-\(T_c\) superconductivity.

Figures (4)

• Figure 1: \ref{['sfig:1a']} Self-consistent solution to magnetization. Here only single-particle charge fluctuation is taken into account, thus the deviation from full polarization is small. \ref{['sfig:1b']} Spin projected local density of states at different magnetization. The solid line is for spin up and the dashed line is for spin down. To exemplify the blueshift effect and the spectral imbalance, we show results for $m_z$ down to 0.1 beyond the self-consistenct results.
• Figure 2: \ref{['sfig:2']} A direct comparison of $\Delta_{bs}/\Delta_{Param}$ of this work (red line) and data extracted from Fig.(3d) of Ref. fratino-2017-signat-mott(blue dots). We modified $m_z(U)$ by a constant factor as $\rightarrow m_z(U)*1.73/1.94$ to match the staggered magnetization of Ref. fratino-2017-signat-mott in the large-U limit since our computed $m_z(U)$ lacks both spin-wave and thermal fluctuations.
• Figure 3: \ref{['sfig:2a']} MDC for $m_z = 0.45$ of $\mathcal{A}[\hat{c}_{\bm{k} \uparrow}, \hat{c}^\dagger_{\bm{k} \uparrow}]$, i.e. that of a single spin species. \ref{['sfig:2b']} MDC at $m_z = 0.49$. Both are plotted along high symmetry directions $(0,0)\rightarrow (0,\pi) \rightarrow (\pi,\pi) \rightarrow (0,0)$. \ref{['sfig:2c']} Constant energy cuts at $\omega = - U/2 + \delta\omega$ with $\delta\omega \in \{1.5t, t, 0.5t, 0.2t, 0.1t, -0.2t\}$, from left to right, for $m_z = 0.45$. \ref{['sfig:2d']} Constant energy cuts from recent APRES measurements (Fig. (1c) of Ref. hu-2018-eviden-multip) on a cuprate parent compound. \ref{['sfig:2b']} showcase the interesting limit where the fully polarized local moments turn one of the LHB into an exact flat band. \ref{['sfig:2c']} is comparable to ARPES observations of Ref. hu-2018-eviden-multip. Plot for $\delta \omega = -U/2 + 0.2 t$ provides an explanation for the two "Fermi surface sheets" observed.
• Figure 4: Comparison of MDCs between this work and Ref. hu-2018-eviden-multip. \ref{['sfig:4a']} MDCs along the cuts VB1 to VB11 and \ref{['sfig:4b']} HB1 to HB11. \ref{['sfig:4c']} ARPES data of MDC along the same cuts. Cuts are defined the same as in Ref. hu-2018-eviden-multip. Cuts labeled as VB1 to VB11 are given as $\{dk_x, k_y\}$ with $k_y \in (0, 2 \pi)$ and $dk_x \in \{-\pi/2,-\pi/2 + \pi/10, \dots, \pi/2\}$. The HB cuts are given as $\{k_x, dk_y\}$ with $k_x \in (-\pi, \pi)$ and $dk_y \in \{\pi/2, \pi/2 + \pi/10, \dots, 3\pi/2\}$. To make a direct comparison, we note that the current results of LHB have a renormalized bandwidth $\sqrt{1-4 m_z^2} 4t \simeq 0.6$ eV, assuming realistic parameters with $t \simeq 0.4~eV$lebert-2023-param-disper and $m_z \simeq 0.3$(60% polarization) for the 2D square lattice Heisenberg modelsandvik-1997-finit-size. Therefore, our results are directly comparable to the ARPES measurement within an energy window of $0.6~eV$, which is marked by red frames in \ref{['sfig:4c']}.