Critical opalescence across the doping-driven Mott transition in optical lattices of ultracold atoms

Phase transitions and their associated crossovers are imprinted in the behavior of fluctuations. Motivated by recent experiments on ultracold atoms in optical lattices, we compute the thermodynamic density fluctuations $\delta N^2$ of the two-dimensional fermionic Hubbard model with plaquette cellular dynamical mean-field theory. To understand the length scale of these fluctuations, we separate the local from the nonlocal contributions to $\delta N^2$. We determine the effects of particle statistics, interaction strength $U$, temperature $T$, and density $n$. At high temperature, our theoretical framework reproduces the experimental observations in the doping-driven crossover regime between metal and Mott insulator. At low temperature, there is an increase of thermodynamic density fluctuations, analogous to critical opalescence, accompanied by a surprising reduction of the absolute value of their nonlocal contributions. This is a precursory sign of an underlying phase transition between a pseudogap phase and a metallic phase in doped Mott insulators, which should play an important role in the cuprate high-temperature superconductors. Predictions for ultracold atom experiments are made.

Figure 1

(a) Thermodynamic density fluctuations $\delta N^2$ versus occupation $n$ at temperature $T=1/1.5$ for different values of the interaction strength $U$. Thermodynamic fluctuations can be decomposed into a local part, $\delta n^2$, plus a nonlocal part, $\delta n_{nl}^2$. (b) Local density fluctuations $\delta n^2$ versus $n$ for the same model parameters as in panel (a). Dotted red line and dashed gray line are exact results for $U=0$ and $U=\infty$, respectively. (c) Nonlocal density fluctuations $\delta n_{nl}^2$ versus $n$ for the same model parameters as in panel (a). Full symbols are experimental data with ultracold atoms in Ref. [28]. We work with units where $A=k_B=1$.

Figure 2

Nonlocal density fluctuations $\delta n_{nl}^2$ versus temperature $T$ for different values of $U$, at half filling $n=1$. Full symbols are experimental data with ultracold atoms of Ref. [28].

Figure 3

(a) Sketch of the low-temperature $U\text{−}n$ phase diagram of the two-dimensional Hubbard model in the normal state solved with plaquette CDMFT [39, 40]. The defining feature of the model is the blue line, marking the first-order transition between a metal (orange region) and Mott insulator at $n=1$ (green vertical line at $n=1$), and between a metal and a pseudogap (blue region) for $n\ne 1$. The former refers to the $U$-driven Mott transition (see vertical double arrow), the latter denotes the doping (or filling $n$) driven Mott transition (see horizontal double arrow). (b) Color map of the charge compressibility ${\kappa }_T$ in the $T\text{−}n$ phase diagram for $U=6.2>U_{\mathrm{MIT}}$. The first-order transition is bounded by the spinodal lines $n_{c1}$ and $n_{c2}$ (lines with solid triangles). The region between $n_{c1}$ and $n_{c2}$ is an instability region. The first-order transition ends in a critical endpoint at $(n_c,T_c)$, from which a line of a rapid crossover emerges (the Widom line, $T_W$, marked by open circles). (c) Semilogarithmic plot of isothermal charge compressibility ${\kappa }_T$ versus $n$ at $U=6.2$ for several temperatures. The maxima of the compressibility define the Widom line appearing in panel (b).

Figure 4

(a) Thermodynamic density fluctuations $\delta N^2$ versus $n$ for $U=6.2>U_{\mathrm{MIT}}$ at temperature $T=1/60$, which is close to the critical endpoint temperature $T_c$. (b) Local density fluctuations $\delta n^2$ versus $n$ for $U=6.2$ and $T=1/60$. (c) Nonlocal fluctuations $\delta n_{nl}^2$ versus $n$ for $U=6.2$ and $T=1/60$. Panels (d), (e), (f) are a zoom into the shaded orange regions of panels (a), (b), (c), for several temperatures, $T=1/30,1/40,1/50,1/60$ which lie above $T_c$, and for $T=1/100$ which lies below $T_c$ (and thus it is discontinuous as a function of $n$).