Mott quantum criticality in the one-band Hubbard model: Dynamical mean-field theory, power-law spectra, and scaling

Recent studies of electrical transport, both theoretical and experimental, near the bandwidth-tuned Mott metal-insulator transition have uncovered apparent quantum critical scaling of the electrical resistivity at elevated temperatures, despite the fact that the actual low-temperature phase transition is of first order. This raises the question whether there is a hidden Mott quantum critical point. Here we argue that the dynamical mean-field theory of the Hubbard model admits, in the low-temperature limit, asymptotically scale-invariant (i.e., power-law) solutions, corresponding to the metastable insulator at the boundary of the metal-insulator coexistence region; these solutions can be linked to the physics of the pseudogap Anderson model. While our state-of-the-art numerical renormalization-group calculations reveal that this asymptotic regime is restricted to very small energies and temperatures and hence is difficult to access numerically, we uncover the existence of a wide crossover regime where the single-particle spectrum displays a different power law. We show that it is this power-law regime, corresponding to approximate local quantum criticality, which is continuously connected to and responsible for the apparent quantum critical scaling above the classical critical end point. We connect our findings to experiments on tunable Mott materials.

Figure 1

DMFT phase diagram of the half-filled Hubbard model as function of interaction strength $U$ and temperature $T$, both measured in units of the half-bandwidth $W$, with the first-order transition line (blue), the critical endpoint (dot), and the boundary of the coexistence region (black dashed). The color map shows the exponent of the spectral function, $r_T=dlnA\left(\omega \right)/dln\omega$, at $\omega =2T$. A wide regime with $r_T=0.7...0.8$ (red) is visible which extends down to very low $T$, corresponding to the crossover power law which controls the approximate quantum critical behavior. In the coexistence region below $T_c$, data for the insulating solution is shown, which is metastable in the green hatched region. Data have been cut off deep in the insulator where $A(\omega =2T)<{10}^{-8}$ (white). For details, see Sec. 5a.

Figure 2

Hybridization function $A\left(\omega \right)$ at $T=0$ for different $U>U_{c1}$. As the boundary of the coexistence region is approached, $A\left(\omega \right)$ develops a power law with exponent $r_{\mathrm{hyb}}=0.79\left(3\right)$ at intermediate frequencies (dashed line; fitting range indicated by vertical blue lines). The inset shows the same data on linear scale.

Figure 3

Effective exponent of the hybridization, $r=dlnA\left(\omega \right)/dln\omega$, as function of $U$ and $\omega$ at $T=0$. Values larger than $r_{max}=2$ and smaller than $r_{min}=-1$ were set to $r_{max}$ (light yellow) and $r_{min}$ (light blue), respectively. The extent of the power-law region (red) grows as $U$ is decreased toward $U_{c1}$. Data have been cut off deep in the insulator where $A(\omega =2T)<{10}^{-8}$ (white).

Figure 4

$T=0$ Mott gap ${\mathrm{\Delta }}_U$ extracted from the spectra in Fig. 2 and shown as function of $(U-U_{c1})$. A nonlinear fit (green) to ${\mathrm{\Delta }}_U=c{(U-U_{c1})}^{\nu z}$ results in $c=1.3\left(2\right)$, $U_{c1}=2.379\left(5\right)$ and $\nu z=0.79\left(5\right)$. Only data points in the scaling regime (red) were included in the fit.

Figure 5

Hybridization function as shown in Fig. 2, but now plotted as $A\left(\omega \right)/{\omega }^{r_{\mathrm{hyb}}}$ vs. $\omega /{\mathrm{\Delta }}_U$, with $r_{\mathrm{hyb}}=0.784$ and ${\mathrm{\Delta }}_U$ from Fig. 4. While there is reasonable data collapse for $\omega /{\mathrm{\Delta }}_U\lesssim 1$, there are systematic deviations from scaling both at large $U$ and for $U$ close to $U_{c1}$. The green shading is a guide to the eye highlighting the universal part of the data.

Figure 6

Imaginary part of the self-energy as function of frequency at different $U>U_{c1}$. Close to $U_{c1}$ a power law develops, but with an exponent $r_{\mathrm{\Sigma }}=-0.37\left(4\right)$ deviating significantly from the expected value $-r_{\mathrm{hyb}}$ of an asymptotic power law (dashed line; fitting range indicated by vertical blue lines).

Figure 7

Hybridization function for (a) $T=0.0192\approx 0.6T_c$, (b) $T=0.04\approx 1.3T_c$, and (c) $T=0.08\approx 2.7T_c$. In (a), a power law is visible at intermediate frequencies for larger $U$, i.e., for insulating-like solutions, and a first-order transition to the metal occurs as $U$ is lowered. In (b) and (c), a power law is visible at intermediate frequencies in the crossover between metallic and insulating phase; this crossover gets washed out as temperature is increased. The vertical lines mark the temperatures.

Figure 8

Logarithmic derivative $dlnA\left(\omega \right)/dln\omega$ as function of $U$ and $\omega$ at (a) $T=0.0192\approx 0.6T_c$ and (b) $T=0.08\approx 2.7T_c$. The horizontal dashed line marks the temperature, the vertical dashed line in (a) the boundary of the coexistence region $U_{c1}\left(T\right)$. The power-law region (red) reaches its maximum extent close to $U_{c1}\left(T\right)$. Note that in (b) the exponent in the power law region is slightly smaller than in (a).

Figure 9

Resistivity on lines parallel to the Widom line with different $\delta U=U-U^{*}\left(T\right)$ as a function of temperature. The resistivity is normalized to ${\rho }_c\left(T\right)$, the resistivity on the Widom line $U^{*}\left(T\right)=2.386$. The black curves mark the reference curve onto which all data points are collapsed; it is obtained by a spline interpolation of the data for $\delta U=\pm 0.04$. The different colors correspond to different values of $\delta U$, as shown in the color bar. The inset shows the unscaled data for $\rho (\delta U,T)$.

Figure 10

Value of the resistivity ${\rho }_c\left(T\right)$ on the Widom line $U^{*}\left(T\right)=2.386$, used to normalize the resistivity in Fig. 9. It shows a power-law dependence on temperature both above and below the critical point, with exponent $-1.3\left(1\right)$.

Figure 11

Collapse of data points if plotted as $\rho /{\rho }_c$ vs $T/T_0$. The color code and markers are the same as in Fig. 9. The scaling collapse is convincing, with deviations visible at low $T$ and large $\delta U$ where the resistivity data are numerically less reliable. Inset: Power-law fits to the rescaling factors $T_0$ give $\nu z_{+}=0.66\left(2\right)$ for $\delta U>0$ and $\nu z_{-}=0.65\left(4\right)$ for $\delta U<0$.

Figure 12

Different densities of states used to test the universality of the spectral exponents; for details, see text.

Figure 13

Hybridization at $T=0$ as function of frequency at different $U>U_{c1}$ for the DOS ${\rho }_{\mathrm{narrow}}\left(\epsilon \right)$ (see text). The behavior is qualitatively similar to the semicircular case, but the exponent of the crossover power law is slightly different. The dashed line shows a power-law fit with $r_{\mathrm{hyb}}=-0.77\left(3\right)$ (fitting range indicated by vertical blue lines).

Figure 14

Imaginary part of the self-energy as function of frequency at different $U>U_{c1}$ for the DOS ${\rho }_{\mathrm{narrow}}\left(\epsilon \right)$. The dashed line shows a power-law fit with $r_{\mathrm{\Sigma }}=-0.40\left(2\right)$ (fitting range indicated by vertical blue lines).