Kinks in the electronic dispersion of the Hubbard model away from half filling

We study kinks in the electronic dispersion of a generic strongly correlated system by dynamic mean-field theory (DMFT). The focus is on doped systems away from particle-hole symmetry where valence fluctuations matter potentially. Three different algorithms are compared to asses their strengths and weaknesses, as well as to clearly distinguish physical features from algorithmic artifacts. Our findings extend a view previously established for half-filled systems where kinks reflect the coupling of the fermionic quasiparticles to emergent collective modes, which are identified here as spin fluctuations. Kinks are observed when strong spin fluctuations are present and, additionally, a separation of energy scales for spin and charge excitations exists. Both criteria are met by strongly correlated systems close to a Mott-insulator transition. The energies of the kinks and their doping dependence fit well to the kinks in the cuprates, which is surprising in view of the spatial correlations neglected by DMFT.

Figure 1

Spectral function for $U=D$ and $\mu =0$ (left panel) and $\mu =0.25D$ (right panel). While the NRG and DMRG results are for $T=0$, the ENCA curves are computed for $T=0.17D$ (left panel) and for $T=0.22D$ (right panel). The case $\mu =0$ implies half-filling, $n=0.5$, while for $\mu =0.25D$ the fillings are $n_{\mathrm{NRG}}\approx 0.583$, $n_{\mathrm{DMRG}}\approx 0.577$, and $n_{\mathrm{ENCA}}=0.573$.

Figure 2

Imaginary (top row) and real (bottom row) parts of the self-energy for $\mu =0$ (left column) and $\mu =0.25D$ (right column) and $U=D$. Other parameters are as in Fig. 1.

Figure 3

Imaginary part of the magnetic (top row) and charge (bottom row) susceptibility for $\mu =0$ (left column) and $\mu =0.25D$ (right column) and $U=D$. Other parameters are as in Fig. 1.

Figure 4

Spectral densities at $U=2D$ and chemical potentials $\mu =0$ [panel (a)] and $\mu =0.5D$ [panel (b)]. For NRG and DMRG $T=0$ holds, while for the ENCA $T=0.027D\approx 0.15T^{*}$ ($\mu =0$) and $T=0.1D\approx 0.6T^{*}$ ($\mu =0.5D$). For $\mu =0.5D$, the fillings are $n_{\mathrm{NRG}}\approx 0.580$, $n_{\mathrm{DMRG}}\approx 0.592$, and $n_{\mathrm{ENCA}}=0.567$.

Figure 5

Fermi liquid scales $T_Z^{*}$ and $T_{\chi }^{*}$ extracted from the quasiparticle weight and from the maximum in the spin susceptibility, respectively, as a function of $U$ for two values of the chemical potential. The magnetic scale $T_{\chi }^{*}$ is rescaled by a single factor $T_Z^{*}=a{T}_{\chi }^{*}=3.125·T_{\chi }^{*}$ to obtain coinciding energies.

Figure 6

Rescaled self-energy $T^{*}\Sigma \left(\omega \right)$ for $U=2D$ and (a) $\mu =0$ (b) $\mu =0.5D$ as a function of energy measured in units of the low energy scale $T^{*}$. Within the RG approaches we find $T^{*}\approx 0.31D$ and $T^{*}\approx 0.33D$ for $\mu =0$ and $\mu =0.5D$, respectively. In ENCA we find $T^{*}\approx 0.18D$ for and $T^{*}\approx 0.2D$ for $\mu =0$ and $\mu =0.5D$, respectively. The temperatures are as in Fig. 4.

Figure 7

(a) Rescaled imaginary part of the magnetic susceptibility $T^{*}\mathrm{Im}\chi (\omega /T^{*})$ for $\mu =0$ (left panel) and $\mu =0.5D$ (right panel) at $U=2D$ as a function of frequency. Parameters are as in Figs. 4. (b) $T^{*}\mathrm{Im}\chi (\omega /T^{*})$ vs $\omega /T^{*}$ calculated with DMFT(NRG) for $\mu =0$ (left panel) and $\mu =0.25D$ (right panel) at various values of $U$.

Figure 8

Imaginary part of the charge susceptibility for $\mu =0$ (left panel) and $\mu =0.5D$ (right panel) at $U=2D$ as function of frequency in units of the half bandwidth $D$. Parameters are as in Fig. 4.

Figure 9

Spectral density $\rho (\omega ,{\epsilon }_k)$ obtained within DMFT (ENCA) as function of frequency $\omega$ and bare electronic energy ${\epsilon }_k$ for $U=2D$ at (a) half filling ($\mu =0$) and (b) $\mu =0.5D$.

Figure 10

Positions of the minima between the many-body resonance and the Hubbard bands calculated by NRG as function of the quasiparticle weight $Z$. The inset shows the same data in a double-logarithmic plot. The dashed line depicts a power law $Z^{0.25}$ for comparison.

Figure 11

Comparison of the kink position from a numerical analyis of $\mathrm{Re}\Sigma$ and from Eq. (11) for $\mu =0.25D$ for the two RG impurity solvers as a function of $U$.

Figure 12

Illustration of the KK transforms of two test functions $f_1$ and $f_2$, which are depicted in the inset. The first ($f_1$) displays a kink while the second ($f_2$) does not, but has the same average slope for small frequencies. The KK transforms are shifted by $a_i,b_i$ such that KK$\left[f_i\right](-a_i)+b_i=0$.

Figure 13

(a) $\mathrm{Re}\Sigma \left(\omega \right)$ (top panel) and $\mathrm{Im}\Sigma \left(\omega \right)$ (bottom panel) at finite doping as function of $\omega /T^{*}$ obtained from DMFT (NRG). The dashed lines in the top panel indicate linear fits used to determine kink positions. In the bottom panel the parabola ${(\omega /T^{*})}^2$ expected from Fermi liquid theory is included for comparison. The coherence scale $T^{*}\approx 0.16D$ is the low energy scale for $U=2.6D$ (cf. Fig. 5). (b) Same as in panel (b), but for $\mu =0.5D$ at $U=2D$. Note that the kink at positive of frequencies is very weak and concomitantly $\mathrm{Im}\Sigma \left(\omega \right)$ displays parabolic behavior up to $\omega \approx T^{*}$ except for a very shallow hump.

Figure 14

Kink energies ${\omega }_{☆}^{\pm }$ as function of the frequency of the maximum $T_{\chi }^{*}/D$ in the imaginary part of the spin susceptibility for (a) $\mu =0$ and (b) $\mu =0.25D$.

Figure 15

Dependence of the kink positions ${\omega }_{☆}^{\pm }$ and of the scale of the magnetic modes on hole doping for $U=2D$ (top panel) and $U=2.8D$ (bottom panel) obtained from DMFT (NRG).