Scaling Theory of the Mott Transition and Breakdown of the Grüneisen Scaling Near a Finite-Temperature Critical End Point

We discuss a scaling theory of the lattice response in the vicinity of a finite-temperature critical end point. The thermal expansivity is shown to be more singular than the specific heat such that the Grüneisen ratio diverges as the critical point is approached, except for its immediate vicinity. More generally, we express the thermal expansivity in terms of a scaling function which we explicitly evaluate for the two-dimensional Ising universality class. Recent thermal expansivity measurements on the layered organic conductor $\kappa \mathrm{\text{−}}(\mathrm{BEDT}\mathrm{\text{−}}\mathrm{TTF}{)}_{2}X$ close to the Mott transition are well described by our theory.

Figure 1

Simplified phase diagram of the $\kappa \mathrm{\text{−}}(\mathrm{ET}{)}_{2}X$ family as mapped out in Refs. 6, 7, 8. While the salt with $X=\mathrm{Cu}[\mathrm{N}(\mathrm{CN}{)}_2]\mathrm{Cl}$ (located at $p=0$) is a Mott insulator, the hydrogenated variant of the $X=\mathrm{Cu}[\mathrm{N}(\mathrm{CN}{)}_2]\mathrm{Br}$ salt ($h8\mathrm{\text{−}}\mathrm{Br}$) is a superconductor. Its deuterated variant ($d8\mathrm{\text{−}}\mathrm{Br}$), investigated in Ref. 10, is located at ambient pressure (dotted line) very close to the critical region. For simplicity, regions of inhomogeneous phase coexistence are left out. The thick, solid line corresponds to a first-order transition. It terminates in a critical end point (red dot), above which there is only a crossover between the Mott insulator and the metal (dashed line). The crossover line differs from the dashed-dotted line $t\approx 1.74|h{|}^{8/15}$ where signatures in the lattice response are expected to be the strongest.

Figure 2

Plot of the scaling function ${\Psi }_{\alpha }(x)$ for the 2D Ising universality class.

Figure 3

Plot of the thermal expansivity ${\alpha }_{\mathrm{sing}}$ as a function of $t$ and $h$, which diverges at the critical end point $t=h=0$. The line of first-order phase transitions along the negative $t$ axis [where the volume of the system changes according to $\Delta V\propto (-t{)}^{1/8}$] is clearly accompanied by a branch cut in the thermal expansivity. Note that due to the scale invariance near the critical point, this plot looks the same on all length scales; i.e., rescaling $t\to b^{y_t}t$ and $h\to b^{y_h}h$ with arbitrary $b$, we obtain exactly the same graph.

Figure 4

Fit to the expansivity data for $d8\mathrm{\text{−}}\mathrm{Br}$ crystal #1 (diamonds ◇) and #3 (triangles ▽) from Ref. 10 by Eq. (11). The fits turn out to be only weakly sensitive to the parameter $\lambda$. A least-squares fit gives, for the fitting parameters of crystal #1, $T_c+\zeta (p-p_c)=27.5\mathrm{K}$, $(p-p_c)/\lambda =26.7\mathrm{K}$, $A/{\lambda }^{7/15}=874\times {10}^{-6}{\mathrm{K}}^{-8/15}$, $B/{\lambda }^{8/15}=3.88{\mathrm{K}}^{-7/15}$, $C=13.2\times {10}^{-6}{\mathrm{K}}^{-1}$, and $D=0.85\times {10}^{-6}{\mathrm{K}}^{-2}$. The straight dashed lines denote the linear background contribution.