Transport and optical conductivity in the Hubbard model: A high-temperature expansion perspective

We derive analytical expressions for the spectral moments of the dynamical response functions of the Hubbard model using the high-temperature series expansion. We consider generic dimension $d$ as well as the infinite-$d$ limit, arbitrary electron density $n$, and both finite and infinite repulsion $U$. We use moment-reconstruction methods to obtain the one-electron spectral function, the self-energy, and the optical conductivity. They are all smooth functions at high temperature and, at large $U$, they are featureless with characteristic widths of the order of the lattice hopping parameter $t$. In the infinite-$d$ limit, we compare the series expansion results with accurate numerical renormalization group and interaction expansion quantum Monte Carlo results. We find excellent agreement down to surprisingly low temperatures, throughout most of the bad-metal regime, which applies for $T\gtrsim (1-n)D$, the Brinkman-Rice scale. The resistivity increases linearly in $T$ at high temperature without saturation. This results from the $1/T$ behavior of the compressibility or kinetic energy, which play the role of the effective carrier number. In contrast, the scattering time (or diffusion constant) saturates at high $T$. We find that $\sigma (n,T)\approx (1-n)\sigma (n=0,T)$ to a very good approximation for all $n$, with $\sigma (n=0,T)\propto t/T$ at high temperatures. The saturation at small $n$ occurs due to a compensation between the density dependence of the effective number of carriers and that of the scattering time. The $T$ dependence of the resistivity displays a kneelike feature which signals a crossover to the intermediate-temperature regime where the diffusion constant (or scattering time) starts increasing with decreasing $T$. At high temperatures, the thermopower obeys the Heikes formula, while the Wiedemann-Franz law is violated with the Lorenz number vanishing as $1/T^2$. The relevance of our calculations to experiments probing high-temperature transport in materials with strong electronic correlations or ultracold atomic gases in an optical lattice is briefly discussed.

Figure 1

The optical conductivity for the Bethe lattice at $U=\infty$ as calculated within NRG-DMFT for $n=0.9$ at several high temperatures. When multiplied by $\frac{T}{D}$ (with $D$ the half-bandwidth), the curves collapse onto each other, in accordance with the first term in Eq. (7). Moreover, the resulting scaling function is in good agreement with that obtained using the high-temperature series.

Figure 2

Resistivity vs temperature for the infinite-connectivity Bethe lattice at $U=\infty$, for several densities as calculated within NRG-DMFT (symbols). The results are compared to the ones from the high-temperature series, both to leading order, $\rho \propto T/D$ (straight dotted lines), and including the first two subleading corrections, $D/T$ and ${(D/T)}^3$ (curved dashed lines).

Figure 3

Leading-order series result for the temperature dependence of the dc resistivity for the finite-$U$ Hubbard model on an infinite-dimensional hypercubic lattice. (Left) Half-filled case for various values of $U$. The resistivity first decays exponentially with the temperature followed by a linear regime with a slope which asymptotically approaches 8, independent of the value of $U$. The inset is a close-up to the range $\frac{T}{D}\le 1$. (Right) Case of $U=4$ for various values of the doping $\delta$. The resistivity curve consists of two linear regimes connected by a regime of exponential decay, which is both wider and more pronounced for smaller $\delta$.

Figure 4

The function $h^{\left(0\right)}\left(\frac{x}{D}\right)$ associated with the self-energy spectral density: ${\rho }_{\mathrm{\Sigma }}(-\mu +x)=D{h}^{\left(0\right)}\left(\frac{x}{D}\right)+\cdots$, for the infinite-$U$ Hubbard model on a hypercubic lattice for $d=\infty$. This function is even and displayed here for $x>0$ only. The solid curves are obtained using the MEM with the zeroth through eighth moments, while the dashed curves are the Gaussian approximation obtained using the MEM with just the zeroth and second moments. The latter agrees well with the former at all but the highest density.

Figure 5

The function $h^{\left(0\right)}\left(\frac{x}{D}\right)$ associated for the infinite-$U$ Hubbard model on the infinite-connectivity Bethe lattice.

Figure 6

The function $f^{\left(1\right)}\left(\frac{\omega }{D}\right)$ associated with the optical conductivity $\frac{\sigma \left(\omega \right)}{\pi {\sigma }_0}=\frac{D}{T}{f}^{\left(1\right)}\left(\frac{\omega }{D}\right)+\cdots$ for the infinite-$U$ Hubbard model on a $d=\infty$ hypercubic lattice. The dashed curves are obtained using the MEM with the zeroth through eighth moments, while the solid curves are obtained using Eq. (87). As the density is lowered, the MEM becomes less reliable, and is therefore not displayed for $n\le 0.3$.

Figure 7

Slope of the $T$-linear dc resistivity for the infinite-$U$ Hubbard model on a hypercubic lattice for $d=\infty$, in units of $\frac{{\rho }_0}{\pi D}$ plotted as a function of $n$. The blue curve is plotted using Eq. (88), while the red curve is plotted using the approximation ${\rho }_{\mathrm{dc}}\approx {\rho }_{\mathrm{dc},0}/(1-n)$, where ${\rho }_{\mathrm{dc},0}/{\rho }_0=4.76064\frac{T}{\pi D}\simeq 1.51\frac{T}{D}$ is the $n=0$ value of the dc resistivity.

Figure 8

High-temperature optical conductivity $\frac{\sigma \left(\omega \right)}{\pi {\sigma }_0}=\frac{D}{T}{f}^{\left(1\right)}\left(\frac{\omega }{D}\right)+\cdots$ for the infinite-$U$ Hubbard model on a $d=\infty$ hypercubic lattice. The dashed curves are obtained using the Mori reconstruction Eq. (87), while the solid curves are obtained from the single-particle self-energy using the “bubble formula” (93).

Figure 9

The slope of the dc resistivity for the infinite-$U$ Hubbard model on a hypercubic lattice for $d=\infty$ in units of $\frac{{\rho }_0}{\pi D}$ plotted as a function of $n$. The blue and red curves are plotted using Eq. (93), the former with $h^{\left(0\right)}\left(\frac{x}{D}\right)$ reconstructed using the MEM, the latter by assuming $h^{\left(0\right)}\left(\frac{x}{D}\right)$ to be Gaussian, which is an excellent approximation. The green curve is plotted using the “Mori reconstruction method” [Eq. (88)]. We find reasonable agreement between these two complementary methods.

Figure 10

$f^{\left(1\right)}\left(\frac{\omega }{D}\right)$ for the infinite-$U$ Hubbard model on a Bethe lattice for $d=\infty$, where $\frac{\sigma \left(\omega \right)}{\pi {\sigma }_0}=\frac{D}{T}{f}^{\left(1\right)}\left(\frac{\omega }{D}\right)+\cdots$. The curves are obtained using Eq. (93).

Figure 11

The slope of the dc resistivity for the infinite-$U$ Hubbard model on a Bethe lattice for $d=\infty$ in units of $\frac{{\rho }_0}{\pi D}$ plotted as a function of $n$. The blue and red curves are plotted using Eq. (93), the former with $h^{\left(0\right)}\left(\frac{x}{D}\right)$ reconstructed using the MEM, the latter by assuming $h^{\left(0\right)}\left(\frac{x}{D}\right)$ to be Gaussian. The Gaussian form is an excellent approximation.

Figure 12

${\tilde{G}}_{\mathrm{loc},L}\left(\overline{\tau }\right)\equiv G_{\mathrm{loc},L}\left(\overline{\tau }\right)e^{-{\overline{\mu }}^{\left(0\right)}\overline{\tau }}=G_{\mathrm{loc}}\left(\overline{\tau }\right)e^{-{\overline{\mu }}^{\left(0\right)}\overline{\tau }}+e^{-\beta U\overline{\tau }}\frac{n}{2}$ plotted versus $(1-\overline{\tau })$, where $G_{\mathrm{loc}}\left(\overline{\tau }\right)$ is obtained from DMFT calculations (using CT-INT quantum Monte Carlo) for the Hubbard model on a Bethe lattice with $n=0.8,U=20$, and $\frac{T}{D}=1.6$. The DMFT data, represented by the blue dots, is fit to the functional form $\left\{{\gamma }_0+{\left(\beta D\right)}^2\left[{\gamma }_{22}{(1-\overline{\tau })}^2+{\gamma }_{21}(1-\overline{\tau })\right]\right\}$, with fit parameters ${\gamma }_0=-0.199997,{\gamma }_{21}=0.02242$, and ${\gamma }_{22}=-0.0255256$.

Figure 13

$\frac{1}{D}Im\mathrm{\Sigma }\left(\omega \right)$ vs $\frac{\omega +\mu }{D}$ for the infinite-$U$ Hubbard model on the Bethe lattice for $n=.9$. The different colored curves are the NRG results at different temperatures, while the black dashed line is $\frac{-\pi }{1-\frac{n}{2}}{h}^{\left(0\right)}\left(x\right)$, which is the asymptotic high-$T$ result. The inset shows the corresponding local density of states at each temperature. At $\frac{T}{D}=0.4$, the high-$T$ result for $\mathrm{\Sigma }$ is almost in perfect agreement with the actual self-energy and remains a good approximation down to $\frac{T}{D}\approx 0.2$. At the two lowest temperatures displayed, the self-energy acquires a quasipole followed by a sharp minimum. The latter corresponds to the quasiparticle peak in the density of states, while the former corresponds to the dip between the quasiparticle peak and the lower Hubbard band [66].

Figure 14

The chemical potential $\mu$ vs $\frac{T}{D}$ plotted for $n=0.8$ and 0.9 for the infinite-$U$ Hubbard model on the Bethe lattice. For each density, the NRG result is compared to the high-$T$ series, $\mu \left(T\right)/D={\overline{\mu }}^{\left(0\right)}\frac{T}{D}+{\overline{\mu }}^{\left(2\right)}\frac{D}{T}+{\overline{\mu }}^{\left(4\right)}{\left(\frac{D}{T}\right)}^3$ (dashed curved lines), and the asymptotic high-$T$ result $\mu =Tln\left[\frac{n}{2(1-n)}\right]$ (thin dashed straight lines). The series extends to a lower temperature for $n=0.9$ since the effective Fermi temperature shrinks like $(1-n)$ with increasing density.

Figure 15

(a) and (b) Imaginary part of the self-energy for the Bethe lattice for $n=0.8\text{and}\frac{T}{D}=2$. The analytically continued QMC-DMFT data are compared to the $U=\infty$ high-$T$ result $\frac{-\pi h^{\left(0\right)}\left(\frac{\omega +\mu }{D}\right)}{1-\frac{n}{2}}$. (c) ${\int }_{-\infty }^{\omega }\mathrm{Im}\mathrm{\Sigma }\left(\omega \right)d\omega$.

Figure 16

Resistivity vs temperature for the infinite-$U$ Hubbard model on the infinite-connectivity Bethe lattice for $n=0.9$. The blue dots are the NRG result. The red line is the leading order high-$T$ result, while the black line contains the first two subleading corrections as in Eq. (99). The high-temperature series is in very good agreement with the NRG solution for $\frac{T}{D}\gtrsim 0.2$, throughout most of the “bad-metal” regime.

Figure 17

Resistivity of the infinite-connectivity Bethe lattice at $U=\infty$ and $n=0.7$ as obtained from a NRG solution of the DMFT equations, plotted together with the absolute value of the kinetic energy $|E_K|$ and compressibility $\kappa$ (multiplied by $T$). The resulting diffusion constant and transport scattering time obtained as $\mathcal{D}=\sigma /\kappa$ and ${\tau }_{tr}=(\sigma /{\sigma }_0)/\left|E_K\right|$ are also displayed. For the plotted quantities, we measure energy in units of the half-bandwidth $D$.

Figure 18

Resistivity of the infinite-connectivity Bethe lattice at $U=\infty$ and $n=0.7$ as obtained from a NRG solution of the DMFT equations, plotted together with the inverse diffusion constant and the compressibility. The quadratic fit displays the $T^2$ behavior of the inverse diffusion constant and the resistivity in the Fermi liquid regime. In the range $T/D\sim 0.01\text{–}0.05$, both the resistivity and the inverse diffusion constant are quasi-linear in the temperature, while the compressibility is essentially constant.