Universal collective modes from strong electronic correlations: Modified $1/{\mathcal{N}}_f$ theory with application to high-$T_c$ cuprates

A nonzero-temperature technique for strongly correlated electron lattice systems, combining elements of both variational wave function (VWF) approach and expansion in the inverse number of fermionic flavors ($1/{\mathcal{N}}_f$), is developed. The departure point, VWF method, goes beyond the renormalized mean-field theory and provides semiquantitative description of principal equilibrium properties of high-$T_c$ superconducting cuprates. The developed here scheme of $\text{VWF}+1/{\mathcal{N}}_f$, in the leading order provides dynamical spin and charge responses around the VWF solution, generalizing the weak-coupling spin-fluctuation theory to the regime of strong correlations. Thermodynamic corrections to the correlated saddle-point state arise systematically at consecutive orders. Explicitly, $\text{VWF}+1/{\mathcal{N}}_f$ is applied to evaluate dynamical response functions for the hole-doped Hubbard model and compared with available determinant quantum Monte Carlo data, yielding a good overall agreement in the regime of coherent collective-mode dynamics. The emergence of well-defined spin and charge excitations from the incoherent continua is explicitly demonstrated and a nonmonotonic dependence of the charge-excitation energy on the interaction magnitude is found. The charge mode energy saturates slowly when approaching the strong-coupling limit, which calls for a reevaluation of the $t\text{−}J$-model approach to the charge dynamics in favor of more general $t\text{−}J\text{−}U$ and $t\text{−}J\text{−}U\text{−}V$ models. The results are also related to recent inelastic resonant x-ray and neutron-scattering experiments for the high-$T_c$ cuprates.

Figure 1

Static spin susceptibilities along the representative $\mathrm{\Gamma }\text{−}X\text{−}M\text{−}\mathrm{\Gamma }$ contour in the Brillouin zone for the two-dimensional Hubbard model with nonzero nearest- and next-nearest-neighbor hopping ($t<0$ and $t^{\prime }=0.2\left|t\right|$, respectively) at weak coupling ($U/\left|t\right|=2$) and for temperature $k_{B}T=0.2\left|t\right|$. Red solid lines in left panels represent ${\mathrm{SGA}}_{{\lambda }_d}+1/{\mathcal{N}}_f$ calculation, whereas in the right panels the corresponding random-phase approximation (RPA) results are displayed. The hole doping levels are set to $\delta =-0.017$ in panels (a) and (b), $\delta =0.083$ in (c) and (d), $\delta =0.289$ in (e) and (f), and $\delta =0.464$ in (g) and (h). Negative $\delta$ indicates the electron doped system. Blue diamonds are the corresponding DQMC data of Ref. [67] for the same set of model parameters and temperature. The lattice size has been set to $100\times 100$ in the ${\mathrm{SGA}}_{{\lambda }_d}+1/{\mathcal{N}}_f$ calculation and no renormalization factors have been applied to the susceptibilities.

Figure 2

Static spin susceptibilities of the two-dimensional Hubbard model with nearest-neighbor hopping only, and the on-site repulsion $U/\left|t\right|=8$. Hole doping levels are set to $\delta =0.40$ [panel (a)] and to $\delta =0$ at half filling [panel (b)]. Lattice size is taken as $10\times 10$. Blue diamonds represent determinant quantum Monte Carlo data of Ref. [68] at temperature $k_{B}T=\left|t\right|/3$, whereas green circles represent corresponding paramagnetic-phase ${\mathrm{SGA}}_{{\lambda }_d}+1/{\mathcal{N}}_f$ results for $k_{B}T=0.333\left|t\right|$ [panel (a)] and $k_{B}T=0.35\left|t\right|$ [panel (b)]. Red crosses are ${\mathrm{SGA}}_{{\lambda }_d}+1/{\mathcal{N}}_f$ scaled by respective renormalization factors $Z=0.80$ for panel (a) and $Z=0.61$ for panel (b) (see discussion in the text). The blue symbols represent correlated Lindhard susceptibilities, ${\widehat{\chi }}_0(\mathbf{k},\omega =0)$. Lines are guides to the eye. The results correspond to the paradigmatic phase so that the transverse- and longitudinal-susceptibility components are equivalent. The maximum in panel (b) signals the tendency towards antiferromagnetic at the $M$ point.

Figure 3

Comparison of imaginary parts of the dynamical charge and spin susceptibilities (top and bottom panels, respectively) obtained within the ${\mathrm{SGA}}_{{\lambda }_d}+1/{\mathcal{N}}_f$ and DQMC methods for undoped ($\delta \to 0$), near-optimally-doped ($\delta =0.15$), and overdoped ($\delta =0.40$) $300\times 300$ square-lattice Hubbard models, each along two lines in $\mathbf{k}$ space. The model parameters are $t<0, t^{\prime }=0.3\left|t\right|, U=8\left|t\right|$, temperature $k_{B}T=\left|t\right|/3$ (DQMC), $k_{B}T=0.35\left|t\right|$ (${\mathrm{SGA}}_{{\lambda }_d}+1/{\mathcal{N}}_f$ at $\delta =0$), and $k_{B}T=0.333\left|t\right|$ (${\mathrm{SGA}}_{{\lambda }_d}+1/{\mathcal{N}}_f$ at $\delta >0$). Color maps illustrate imaginary parts of ${\mathrm{SGA}}_{{\lambda }_d}+1/{\mathcal{N}}_f$ dynamical susceptibilities, whereas red lines follow maximal values of intensities for a given wave vector. Yellow symbols are the corresponding maximum-intensity frequencies for analytically continued 64-site DQMC data of Ref. [69]. For definition of ${\mathrm{SGA}}_{{\lambda }_d}+1/{\mathcal{N}}_f$, see Table 1. Analytic continuation was performed as $i{\omega }_n\to w+i0.02\left|t\right|$.

Figure 4

Comparison of imaginary parts of the dynamical longitudinal spin structure factor, $S^{zz}(\mathbf{k},\omega )$, obtained within the ${\mathrm{SGA}}_{{\lambda }_d}+1/{\mathcal{N}}_f$ and DQMC methods for undoped ($\delta =0$), near-optimally-doped ($\delta =0.20$), and overdoped ($\delta =0.40$) square-lattice Hubbard models. The model parameters are $t<0, t^{\prime }=0, U=8\left|t\right|$, and temperature $k_{B}T=\left|t\right|/3$ [within ${\mathrm{SGA}}_{{\lambda }_d}+1/{\mathcal{N}}_f$ we actually set $k_{B}T=0.35\left|t\right|$ for panels (a) and (b) and $k_{B}T=0.333\left|t\right|$ for panels (c)–(f); see discussion in the text]. Color maps represent imaginary parts of ${\mathrm{SGA}}_{{\lambda }_d}+1/{\mathcal{N}}_f$ dynamical susceptibilities, evaluated for the $500\times 500$ lattice, whereas red lines follow maximal values of $S^{zz}(\mathbf{k},\omega )$. Yellow symbols are the maximum-intensity frequencies for analytically continued $10\times 10$ lattice DQMC data of Ref. [68]. The analytic continuation of ${\mathrm{SGA}}_{{\lambda }_d}+1/{\mathcal{N}}_f$ data was performed as $i{\omega }_n\to \omega +i\epsilon$ with $\epsilon =0.04\left|t\right|$.

Figure 5

Imaginary parts of (a) spin and (b) charge dynamical susceptibilities as a function of the on-site Coulomb repulsion, $U$, for the Hubbard model with nonzero next- and next-nearest neighbor hopping, taken at the $X$ point in the Brillouin zone. The employed model parameters are $t<0, t^{\prime }=0.3\left|t\right|, k_{B}T=0.35\left|t\right|$, and hole doping $\delta =0.2$. The calculations have been performed for a $300\times 300$ square lattice using ${\mathrm{SGA}}_{{\lambda }_d}+1/{\mathcal{N}}_f$ approximation (see Table 1).

Figure 6

Comparison of the peak energies for spin [top panels, (a) and (b)] and charge [bottom panels, (c) and (d)] dynamical susceptibilities for the Hubbard model as a function of the Hubbard $U$. The remaining parameters are $t<0, t^{\prime }=0.3\left|t\right|, k_{B}T=0.35\left|t\right|$, hole doping $\delta =0.2$, and lattice size is taken as $300\times 300$. All susceptibilities are evaluated at the $X$ point, and four symbols described inside the panels (a)–(d) correspond to six schemes listed in Table 1. Lines are guides to the eye.

Figure 7

A simplified flowchart of $\text{VWF}+1/{\mathcal{N}}_f$ approach to order $\mathcal{O}\left(1\right)$. Pink ellipses are entry and end points of the algorithm and orange blocks represent the preparation stage. The light-blue decision tree on the bottom left represents the self-consistent diagrammatic VWF solution (see Sec. 2); the light-green block represents evaluation of dynamical corrections in the correlated state. The results, presented in this paper, have been obtained following blue lines. Red dashed lines reflect the $\mathcal{O}\left(1\right)$ contributions to thermodynamics and are yet to be implemented, which will allow us to study the impact of collective modes on static quantities (see discussion of Sec. 7).

Figure 8

Stability analysis of the paramagnetic state for the case of zero next-nearest hopping. The panels show calculated ${\mathrm{SGA}}_{{\lambda }_d}+1/{\mathcal{N}}_f$ static spin [panels (a)–(f)] and charge [panels (g)–(l)] susceptibility [${\chi }_s^{\prime }(\mathbf{k},\omega =0)$ and ${\chi }_c^{\prime }(\mathbf{k},\omega =0)$, respectively] for the Hubbard model with the same parameters as those used to generate Fig. 4, i.e., $t<0, t^{\prime }=0, U=8\left|t\right|$, lattice size $500\times 500$, and doping levels $\delta =0,0.2$, and 0.4. The temperature has been set to $k_{B}T=0.35\left|t\right|$ for $\delta =0$, and $k_{B}T=0.333\left|t\right|$ for $\delta =0.2$ and $\delta =0.4$. Hole doping levels are provided inside the panels.

Figure 9

Stability analysis of the paramagnetic state for the case of nonzero next-nearest hopping. The panels show calculated ${\mathrm{SGA}}_{{\lambda }_d}+1/{\mathcal{N}}_f$ static spin [panels (a)–(f)] and charge [panels (g)–(l)] susceptibility (${\chi }_s^{\prime }(\mathbf{k},\omega =0)$ and ${\chi }_c^{\prime }(\mathbf{k},\omega =0)$, respectively) for the Hubbard model with the same parameters as those used to generate Fig. 3, i.e., $t<0, t^{\prime }=0.3\left|t\right|, U=8\left|t\right|$, lattice size $300\times 300$, and hole doping $\delta =0,0.15$, and 0.4. The temperature has been set to $k_{B}T=0.35\left|t\right|$ for $\delta =0$ and to $k_{B}T=0.333\left|t\right|$ otherwise. Hole doping levels are marked inside the panes.

Figure 10

Calculated static spin susceptibility at the $M$ point for the Hubbard model ($t<0, t^{\prime }=0.3\left|t\right|, k_{B}T=0.35\left|t\right|$, hole doping $\delta =0.2$, and lattice $300\times 300$) as a function of the on-site Coulomb repulsion $U$. Blue and yellow points correspond to ${\mathrm{SGA}}_{{\lambda }_f}+1/{\mathcal{N}}_f$ and ${\mathrm{LD}}_{{\lambda }_f}+1/{\mathcal{N}}_f$, respectively. Lines are guides to the eye. The maximum is located in the regime of metal-insulator transition $U\approx 8\left|t\right|$.