Suppression of the charge fluctuations by nonlocal correlations close to the Mott transition

TL;DR

This work tackles how nonlocal correlations affect charge fluctuations near the Mott transition in the 2D Hubbard model by applying the ladder dynamical vertex approximation (lDΓA) to go beyond DMFT. The study finds that nonlocal spin fluctuations strongly suppress the large uniform charge susceptibility predicted by DMFT near half filling, while increasing doping eventually enhances charge fluctuations above DMFT predictions, signaling a metallization tendency. The analysis combines charge and spin susceptibilities, correlation lengths, self-energy, and thermodynamic energies to reveal a consistent rebalancing of fluctuations and a doping-dependent shift in electronic mobility. The results underscore the importance of nonlocal correlations for accurately describing charge dynamics in low-dimensional correlated systems and provide a framework for investigating related phenomena, including possible effects on superconductivity at optimal doping.

Abstract

In this paper, we investigate the impact of nonlocal correlations on charge fluctuations in the two-dimensional single-band Hubbard model close to the Mott metal-to-insulator transition, employing the ladder dynamical vertex approximation. At half filling and for interaction strengths and temperatures where the system is in the Mott insulating phase, charge fluctuations are strongly suppressed. Under these conditions, dynamical mean-field theory (DMFT) calculations predict a strong enhancement of the charge susceptibility at small (electron or hole) doping. However, these DMFT results include only the effects of purely local correlations despite the importance of nonlocal correlations in two-dimensional systems. We have, hence, carried out ladder dynamical vertex approximation (lD$Γ$A) simulations which allow for the inclusion of such nonlocal correlation effects while retaining the local ones of DMFT. Our lD$Γ$A numerical data show that close to half filling the large uniform charge susceptibility of DMFT is strongly suppressed by nonlocal fluctuations but gradually increases with (electron) doping. At a certain doping value, charge fluctuations eventually become larger in lD$Γ$A with respect to DMFT, indicating that the absence of nonlocal correlations underestimates the mobility of the charge carriers in this parameter regime. This metallization effect is also reflected in an enhancement of the lD$Γ$A kinetic and potential energies and a corresponding reduction of the (absolute value of the) lD$Γ$A Matsubara self-energy with respect to DMFT.

Figures (11)

• Figure 1: Difference between the potential energies obtained from two- and one-particle correlation functions on the left- and right-hand sides of Eq. \ref{['equ:lambdacondition2']}, respectively, as a function of $\lambda_{\mathrm{d}}$ where $\lambda_{\mathrm{m}}$ is determined (for a given $\lambda_{\mathrm{d}}$) from Eq. \ref{['equ:lambdacondition1']}. Crossing with the $x$ axes corresponds to the solution of Eqs. \ref{['equ:lambdacondition']}. Data are shown for $U=2.64$ and $\beta=44.0$ (see Fig. \ref{['fig:PD']} and discussion in Sec. \ref{['sec:charge_susceptibility']}) for four different values of the filling $n$.
• Figure 2: DMFT paramagnetic phase diagram of the two-dimensional Hubbard model on a square lattice with nearest ($t$), next-nearest ($t'$), and next-to-next-nearest ($t"$) neighbor hopping, plotted as a function of interaction strength $U$ and temperature $T$ at half filling. Hysteresis is visualized via color coding, representing the difference in double occupancy $d$ obtained from the metallic and insulating solutions, respectively. $U_{c1}$ (blue line) and $U_{c2}$ (green line) are the boundaries between the coexistence region and the paramagnetic metallic (PM) and paramagnetic insulating (PI) phases, respectively. The red dot marks the parameter set ($U=2.64$ and $\beta=44$) used to explore the system away from half filling. Model parameters: $t = 0.25$, $t' = -0.075$, and $t" = 0.05$.
• Figure 3: Uniform charge susceptibility $\chi_{\mathrm{d},\mathbf{q_0}=\mathbf{0}}^{\omega_0=0}$ for $U\!=\!2.64$ and $\beta\!=\!44.0$ (see red square in Fig. \ref{['fig:PD']}) as a function of filling $n$ obtained from DMFT (red diamonds) and $\mathrm{lD}\Gamma\mathrm{A}_\text{dm}\,$ (green squares). For comparison also $\frac{1}{2}\frac{dn}{d\mu}$ is also depicted (blue circles), which is equivalent to the uniform charge susceptibility within DMFT. The orange curve is obtained by fitting $\mathrm{lD}\Gamma\mathrm{A}_\text{dm}\,$ data using Eq. \ref{['equ:fitting_chid']} in Appendix \ref{['sec:fitting']} with the fitting parameters are $A = 0.6307$, $B = 0.0658$, and $C = 0.1473$. Inset: Enlargement of the $\mathrm{lD}\Gamma\mathrm{A}_\text{dm}\,$ uniform charge susceptibility around its maximum value.
• Figure 4: Spin susceptibility $\chi_{\mathrm{m},{\mathbf{q}}_\pi}^{\omega_0}$ as a function of filling $n$ for $\mathrm{lD}\Gamma\mathrm{A}_\text{m}\,$ where only the magnetic channel is corrected ($\lambda_{\mathrm{m}} \neq 0$, $\lambda_{\mathrm{d}} = 0$, brown hexagons) and for $\mathrm{lD}\Gamma\mathrm{A}_\text{dm}\,$, where we include both magnetic and charge corrections ($\lambda_{\mathrm{m}} \neq 0$, $\lambda_{\mathrm{d}} \neq 0$, green squares). Here, $\mathbf{q}_\pi = (\pi,\pi)$ denotes the antiferromagnetic ordering vector. Data are shown on a logarithmic scale on the $y$-axis for the same parameters $U$ and $T$ as in Fig. \ref{['fig:charge_susceptibility']}.
• Figure 5: Upper row: Local (momentum summed) charge susceptibility $\sum_{\mathbf{q}} \chi_{\mathrm{d},{\mathbf{q}}}^{\lambda_{\mathrm{d}},\omega}$ of DMFT ($\lambda_{\mathrm{d}}\!=\!0$, red diamonds) vs. $\mathrm{lD}\Gamma\mathrm{A}_\text{dm}\,$ ($\lambda_{\mathrm{d}}\!\ne\!0$, green squares) as a function of the bosonic Matsubara frequency $\omega$ for four different values of filling $n$ gradually increasing from (a) to (d); Lower rows: Static $\mathrm{lD}\Gamma\mathrm{A}_\text{dm}\,$ charge susceptibility $\chi_{\mathrm{d},{\mathbf{q}}}^{\lambda_{\mathrm{d}},\omega=0}$ as a function of momentum ${\mathbf{q}}$ for eight different values of $n$ gradually increasing from (e) to (l). Interaction strength and temperature are the same as in Fig. \ref{['fig:charge_susceptibility']}.