Finite-temperature crossover and the quantum Widom line near the Mott transition

The experimentally established phase diagram of the half-filled Hubbard model features the existence of three distinct finite-temperature regimes, separated by extended crossover regions. A number of crossover lines can be defined to span those regions, which we explore in quantitative detail within the framework of dynamical mean-field theory. Most significantly, the high-temperature crossover between the bad metal and Mott-insulator regimes displays a number of phenomena marking the gradual development of the Mott insulating state. We discuss the quantum critical scaling behavior found in this regime, and propose methods to facilitate its possible experimental observation. We also introduce the concept of quantum Widom lines and present a detailed discussion that highlights its physical meaning when used in the context of quantum-phase transitions.

Figure 1

Phase diagram of the half-filled maximally frustrated Hubbard model. The background is an actual color map of the resistivity obtained using the IPT impurity solver (see the text): Blue, small resistivity; red, large resistivity.

Figure 2

Resistivity (in units of ${\rho }_{{}_{\text{Mott}}}$) calculated in the entire $U-T$ plane. The white stripes follow the lines of equal resistivity and separate the orders of magnitude in the resistivity. Spinodals are denoted with thick black lines, and the first-order phase transition line is dashed.

Figure 3

The effective resistivity exponent ($\beta =d\log \rho /d\log T$) calculated in the entire $U-T$ plane illustrates the different transport regimes (see the text).

Figure 4

Various definitions for the crossover lines between the Fermi liquid and the bad metal. The meaning of each definition is illustrated on a smaller panel to the right. The results are obtained with the QMC.

Figure 5

Various definitions for the crossover lines between the bad metal and the Mott insulator. The meaning of each definition is illustrated on a smaller panel to the right. The results are obtained with the QMC.

Figure 6

(a) Experimental results: Conductivity scaling in high-mobility Si MOSFETs presents a textbook example of quantum critical scaling (taken from Ref. 47). (b) DMFT QMC results: Resistivity scaling strongly reminiscent of what is seen in MOSFETs. After dividing $\rho (U,T)$ with the value of resistivity on the instability line ${\rho }_c\left(T\right)$ (see the text) and then rescaling the temperature with an appropriately chosen parameter $T_0\left(\delta U\right)$, the resistivity curves collapse onto two branches.

Figure 7

Density of states (QMC results) along the instability line $U^{*}\left(T\right)$ (middle column), and along the parallel trajectory for smaller (left column) and larger $U$ (right column).

Figure 8

Free-energy landscape (IPT results): (a) Along the “zero field” line ($\delta U=0$). At $T>T_c$, the curvature of the free energy increases with temperature, and it is zero at $T=T_c$. Below $T_c$, at the first-order transition line, metallic and insulating solutions have the same free energy. (b) Along the “finite field” line ($\delta U=-0.05$). At $T>T_c$, the curvature of the free energy is greater than in the “zero field” case. In the coexistence region one of the minima is energetically favored. Note that the spacing between $\Delta \mathcal{F}$ curves for different temperatures is arbitrary.

Figure 9

Possible phase diagram of a generalized Hubbard model. The observed scaling (valid in the green region) may be due to a quantum critical point that is unreachable by the simple two-parameter half-filled Hubbard model. An additional, third parameter (here marked with $X$) could drive $T_c$ to zero at some critical value, and extend the region of validity of the scaling formula in the $U-T$ plane.

Figure 10

The symmetric and asymmetric part of the scaling function, $h_s$ and $h_a$, at various temperatures. The small value of $h_s\left(y\right)$ shows that the mirror symmetry of resistivity curves is present. The $h_a\left(y\right)$ curves collapse around the inflection-point line, which shows that the exponent, $z\nu =0.953$, is well evaluated. Fitting a third-order polynomial to $h_a\left(y\right)$ in the range where these curves collapse can reveal the exact form of the scaling formula. In our calculations only the linear term is significant.

Figure 11

The instability line lies among the other crossover lines. $\log \rho \left(U\right)$ is linear in this crossover region, which allows for the scaling formula to be valid.

Figure 12

The derivative of resistivity with respect to $U$ [${\partial \rho (U,T)/\partial U|}_{U_{\mathrm{infl}}}$] along the inflection-point line. Above roughly $2T_c$, it fits well to a power-law curve of exponent $-0.95$. This can be used to evaluate the value of the scaling formula exponent. At lower temperatures the decrease in resistivity is faster, and the behavior deviates from the power law, and the scaling formula fails at temperatures below $2T_c$.

Figure 13

Relative error of the scaling formula color coded in the $U-T$ plane. The dotted lines are the boundary of the scaling region. The two green filaments below $2T_c$ are where the scaling formula intersects with the actual DMFT result.

Figure 14

Resistivity (in units of ${\rho }_{{}_{\text{Mott}}}$) along the crossover lines is weakly dependent on temperature and much larger than the Mott limit.