Repulsive versus attractive Hubbard model: Transport properties and spin-lattice relaxation rate

We contrast the transport properties (dc resistivity, Seebeck coefficient), optical conductivity, spectral functions, dynamical magnetic susceptibility, and the nuclear magnetic resonance $1/T_1$ spin-lattice relaxation rate of the repulsive and attractive infinite-dimensional Hubbard models in the paramagnetic phase for a generic band filling. The calculations are performed in a wide temperature interval using the dynamical mean-field theory with the numerical renormalization group as the impurity solver. The attractive case exhibits significantly more complex temperature dependencies which can be explained by the behavior of the half-filled Hubbard model in external magnetic field with constant magnetization, to which the attractive Hubbard model maps through the partial particle-hole transformation. The resistivity is nonmonotonous for the strongly attractive case: it peaks significantly above the Mott-Ioffe-Regel value at a temperature $T_{\max }$ where the quasiparticle band disappears. For both signs of $U$ we find particle-hole asymmetry in the self-energy at low energies, but with the opposite kind of excitations having longer lifetime. This leads to a strong suppression of the slope of the Seebeck coefficient in the attractive case rather than an enhancement as in the repulsive case. The spin-lattice relaxation rate in the strongly attractive case has a nonmonotonic temperature dependence, thereby revealing the pairing fluctuations.

Figure 1

Zero-temperature thermodynamic properties of the Hubbard model as a function of the electron-electron interaction parameter $U$. The density is fixed at $n=0.8$. (a) Double occupancy (density of doubly occupied sites) $P_2=\langle n_{\uparrow }{n}_{\downarrow }\rangle$. The inset shows the uniform charge susceptibility ${\chi }_{\mathrm{c}}=\partial \langle n\rangle /\partial \mu$. The temperature dependence of $P_2$ is shown in Fig. 15. (b) Potential, kinetic, and total energy per particle. (c) Chemical potential.

Figure 2

Temperature dependence of entropy per lattice site for a set of $U$ values.

Figure 3

(a) Temperature dependence of the chemical potential (shifted by $-U/2$). The zero-temperature values of the chemical potential (without the shift) are also shown in Fig. 1. (b) Temperature dependence of the renormalized chemical potential ${\mu }_{\mathrm{eff}}\left(T\right)=\mu \left(T\right)-\mathrm{Re}\Sigma (\omega =0,T)$.

Figure 4

Local spectral function $A\left(\omega \right)$ at zero temperature for (a) repulsive and (b) attractive cases. (c) Quasiparticle renormalization factor $Z\equiv Z(T=0)$ as a function of $U$. (d) Low-frequency part of the spectral function rescaled as $A(\omega /Z)$. We plot the results for $U=-2.5,-2,-1.5,-1,-0.5,0,1,2,3,4,5,6D$.

Figure 5

Momentum-resolved spectral functions $A(\epsilon ,\omega )$ for a range of temperatures for attractive interaction with $U/D=-2$.

Figure 6

(a) Imaginary part of the self-energy at $T=0$ for strongly repulsive and strongly attractive interactions reveals particle-hole asymmetry at low-energy scales in both cases. (b) Temperature dependence of $\mathrm{Im}\Sigma \left(\omega \right)$ for the attractive Hubbard model at $U/D=-2.25$.

Figure 7

Resistivity and Seebeck coefficient of the Hubbard model at constant temperature $T=3\times {10}^{-2}D$. The resistivity is expressed in units of the Mott-Ioffe-Regel value ${\rho }_0=(e^2/\hslash )\Phi \left(0\right)/D$. The results for the Seebeck coefficient for the values of $U$ where the calculation is not reliable have been omitted.

Figure 8

Temperature dependence of resistivity and Seebeck coefficient. ${\rho }_0$ is the MIR resistivity. We note that for even higher temperatures, not shown in the plot, the resistivity for $U/D=-2$ and $U/D=-2.25$ starts to increase, i.e., there appears to be no saturation of resistivity for either sign of $U$. At very low temperatures (for $T/D\lesssim 0.01$), the results for the Seebeck coefficient become unreliable due to increasing error in dividing two small values of the transport integrals $L_{12}$ and $L_{11}$, but also due to the intrinsic problems of the NRG method in the calculations of the self-energy function at very small energies and temperatures (causality violations).

Figure 9

Resistivity on the log-log scale for attractive $U$ near the localization transition. The dashed line has slope 2, expected for the Fermi-liquid regime. The dotted horizontal line indicates the Mott-Ioffe-Regel limit. For $U/D=-2.25$, two characteristic energy scales can be defined: the Fermi-liquid temperature $T_{\mathrm{FL}}$ and the resistivity peak temperature $T_{\max }$. Inset: rescaled quasiparticle renormalization factor $Z\left(T\right)/Z(T=0)$. Deviation from 1 indicates the end of the Landau Fermi-liquid regime.

Figure 10

Resistivity at constant attractive $U$ for a range of electron densities $\langle n\rangle$.

Figure 11

Resistivity and Seebeck coefficient at $T/D=0.03$ for three choices of the transport function $\Phi$. The data for the Seebeck coefficient in the shaded rectangle are not reliable for reasons explained in the text.

Figure 12

Seebeck coefficient vs temperature for two values of $U$, one positive (top panel) and one negative (bottom panel), for three choices of the transport function $\Phi$. The two values are chosen so that they correspond to a comparable value of the quasiparticle renormalization factor $Z$. The results for $T/D\lesssim 0.01$ are not correct (note, for example, the downturn of $S$ for $U/D=4$ instead of linear low-temperature behavior).

Figure 13

Optical conductivity for attractive $U/D=-2.25$ for a range of temperatures. The arrows indicate the evolution with increasing temperature in different frequency regions. Upper panel roughly corresponds to $T\lesssim T_{\max }$, lower to $T\gtrsim T_{\max }$.

Figure 14

Optical conductivity for $U/D=-2$ (left) and $U/D=4$ (right) for a range of band fillings $n$.

Figure 15

(a) Zero-frequency slope of the imaginary part of the local dynamical spin susceptibility, i.e., $1/T_{1}T$ up to constant prefactor. (b) Double occupancy as a function of temperature. (c) Zero-frequency slope multiplied by the temperature, i.e., $1/T_1$. (d) Zero-temperature spin relaxation rate vs Hubbard parameter $U$. The legends in (a) and (b) show the value of U/D.

Figure 16

Imaginary part of the dynamical spin susceptibility for a range of temperatures for (a) repulsive interaction and (b) attractive interaction.

Figure 17

Dynamical charge susceptibility of the attractive Hubbard model for a range of temperatures. The insets shows the frequency of the peak as a function of the temperature.

Figure 18

Properties of the single-impurity Anderson model at half-filling as a function of the temperature $T$ and the magnetic field $B$. (a) Magnetization, (b) impurity spin susceptibility, (c) impurity charge susceptibility, and (d) impurity entropy. (e) Temperature dependence of the impurity spin susceptibility evaluated along the constant-magnetization contours (top to bottom: $\langle S_z\rangle =0.05,0.1,0.1,0.2$).

Figure 19

Conductivity $F(T,B)$ for a single-spin species in the single-impurity Anderson model at half-filling in external magnetic field (see the text for the exact definition). (a) Contour plot in the $(T,B)$ plane. (b) Temperature dependence of the conductivity along the constant-magnetization contours (top to bottom: $\langle S_z\rangle =0.05,0.1,0.15,0.2$).

Figure 20

Optical conductivity for $n=0.8$ for a range of repulsion parameters $U$. The finite width of the Drude peak for $U=0$ is due to artificial broadening in the calculation. The absence of the Drude peak for $U/D=-2.5$ at $T/D=0.1$ shows that the system is in the (bad) insulator regime.