Non-Fermi-liquid signatures in the Hubbard model due to van Hove singularities

When a van Hove singularity is located in the vicinity of the Fermi level, the electronic scattering rate acquires a nonanalytic contribution. This invalidates basic assumptions of Fermi liquid theory and within treatments based on perturbation theory leads to a non-Fermi liquid self-energy and transport properties. Such anomalies are shown to also occur in the strongly correlated metallic state within dynamical mean-field theory. We consider the Hubbard model on a two-dimensional square lattice with nearest- and next-nearest-neighbor hoppings within the single-site dynamical mean-field theory. At temperatures on the order of the low-energy scale $T_0$ an unusual maximum emerges in the imaginary part of the self-energy which is renormalized toward the Fermi level for finite doping. At zero temperature this double-well structure is suppressed but an anomalous energy dependence of the self-energy remains. For the frustrated Hubbard model on the square lattice with next-nearest-neighbor hopping, the presence of the van Hove singularity changes the asymptotic low-temperature behavior of the resistivity from a Fermi liquid to non-Fermi liquid dependency as function of doping. The results of this work are discussed regarding their relevance for high-temperature cuprate superconductors.

Figure 1

(Color online) Imaginary parts of the effective media $\Gamma \left(\omega -i{0}^{+}\right)$ [see Eqs. (9, 10, 11)] as functions of energy $\omega$ for the cases $\alpha =0,1,2$ and two values of the residual interaction $\gamma$. The inset shows the model DOS for the same values of $\alpha$. $W=Z=1$ and $T=\widetilde{ϵ}=0$ was used.

Figure 2

(Color online) (a) Local spectral function $\rho \left(\omega \right)$ for the symmetric Hubbard model on a 2D square lattice for Coulomb repulsion $U=4$ $\left(ϵ=-2\right)$, half bandwidth $W=2$ $\left(t=0.5\right)$, and various temperatures as functions of frequency within DMFT (ENCA). The thin dashed curve represents the noninteracting DOS ${\rho }_0\left(\omega \right)$ and the inset shows the imaginary part of the effective medium $\text{Im}\Gamma \left(\omega -i{0}^{+}\right)$. (b) Imaginary (left) and real (right) part of the self-energy ${\Sigma }^U\left(\omega -i{0}^{+}\right)$ for the same parameters as in (a) and $T$ as indicated. The inset of the left panel shows the imaginary part on a wider energy interval. The short arrows in the right graphs point at the approximate positions of the kinks.

Figure 3

(Color online) Position of the minimum ${\omega }_{\text{min}}$ in $\text{Im}{\Sigma }^U$ in units of the low-energy scale $T_0$ as a function of the rescaled temperature $T/T_0$ for three different Coulomb repulsions. The inset shows the screened local magnetic moment normalized to its high-temperature value ${\mu }_{\infty }^2$ as a function of temperature for the same values of the Coulomb repulsions. The horizontal line indicates the value where ${\mu }_{\text{eff}}^2$ reaches 70% of ${\mu }_{\infty }^2$ and the vertical dashed lines indicate the low-energy scales: $T_0\left(U=3.5\right)\approx 0.082$, $T_0\left(U=4\right)\approx 0.051$, and $T_0\left(U=4.5\right)\approx 0.023$.

Figure 4

(Color online) (a) Local DOS for the asymmetric Hubbard model on a 2D square lattice with $U=6$ and filling $n=0.98$ for various temperatures obtained with the DMFT (ENCA). The inset shows the low-energy region. (b) Imaginary part of the self-energy around the Fermi level for the same parameters as in (a) and temperatures as indicated. The inset shows the imaginary part of the effective medium in the low-energy region.

Figure 5

(Color online) The imaginary parts of the effective medium (upper graphs) and self-energy (lower graphs) of the Hubbard model within DMFT (ENCA) for different noninteracting DOS, which are shown in the insets. The vertical dashed line indicates the position of the Fermi level. The parameters are indicated in the plots.

Figure 6

(Color online) (a) DMFT (NRG) spectral function at $T=0$ for the Hubbard model on a 2D square lattice for $U=4$ and varying hole doping. The inset shows the position of the maximum as a function of doping for $U=4$ and $U=0$. The dashed line is a fit ${\omega }_{\text{max}}=2.2{\delta }^2$. (b) Imaginary part of the self-energy at half filling $\delta =0$ for three values of the Coulomb repulsion $U$. The curves for $U=4.5$ and $U=5$ were rescaled in order to lie on top of the $U=4$ curves at low $\omega$. The dashed line is a fit with a function $a{\left|\omega \right|}^{3/2}$. The upper inset shows the low-energy behavior in a double-logarithmic plot. The lower inset shows the quasiparticle weight for $U=4$ as a function of doping.

Figure 7

(Color online) DMFT (ENCA) spectral functions $\rho \left(\omega \right)$ (left panels) and imaginary parts of the self-energy ${\Sigma }^U\left(\omega -i{0}^{+}\right)$ (right panels) for different filling $n$ for the Hubbard model on the square lattice with additional next-nearest-neighbor hopping $t^{\prime }=-0.2t=-0.1$ and $U=4.25$. The insets in the left panels show the imaginary part of the effective medium while the insets on the right show a closeup of the low-energy region of $\text{Im}{\Sigma }^U$. The noninteracting DOS is shown in the inset for the half-filled case (b).

Figure 8

(Color online) (a) Static quasiparticle scattering rate $\text{Im}{\Sigma }^U\left(-i{0}^{+}\right)$ for the Hubbard with $U=4.25$ on a square lattice with next-nearest-neighbor hopping $t^{\prime }=-0.2t$ as function of temperature in log-log plot. The curves for different filling are multiplied by the factors indicated on the right. (b) Resistivity $\varrho$ (normalized to ${\varrho }_0$) for the same parameters as in (a). The inset shows $\varrho$ without the scaling factors and in a nonlogarithmic plot. The dashed lines represent fits with the indicated functions in both plots.

Figure 9

(Color online) Generalized quasiparticle weight for the Hubbard model on a square lattice with next-nearest-neighbor hopping $t^{\prime }=-0.2t$ and $U=4.25$ as function of temperature for three different filling obtained from DMFT (ENCA) calculations and Eq. (20).