Local Entanglement Entropy and Mutual Information across the Mott Transition in the Two-Dimensional Hubbard Model

Entanglement and information are powerful lenses to probe phases transitions in many-body systems. Motivated by recent cold atom experiments, which are now able to measure the corresponding information-theoretic quantities, we study the Mott transition in the half-filled two-dimensional Hubbard model using cellular dynamical mean-field theory, and focus on two key measures of quantum correlations: entanglement entropy and a measure of total mutual information. We show that they detect the first-order nature of the transition, the universality class of the end point, and the crossover emanating from the end point.

Figure 1

(a) Temperature $T$ versus interaction strength $U$ phase diagram of the 2D Hubbard model at half filling ($n=1$) within plaquette CDMFT. Coexistence associated with the first-order transition between a metal and a Mott insulator terminates at the second-order critical end point $(U_c,T_c)$. From the end point emerges a crossover line (Widom line) defined by the maxima in the correlation length. Here we estimate the Widom line by the locus of the inflection point in the double occupancy $D(U{)}_T$ at different temperatures (open red circles, $T_W$). This line overlaps with the loci of the inflection points in the local entropy $s_1(U{)}_T$ (blue crosses, $T_{s_1}$) and in the total mutual information ${\overline{I}}_1(U{)}_T=s_1-s$ (green squares, $T_{{\overline{I}}_1}$), where $s$ is thermal entropy. (b) Double occupancy $D$ versus $U$ at $n=1$ and different temperatures. For $T<T_c$, hysteresis appears when sweeping $U$ in the forward and reverse directions. Arrows show the sweep direction.

Figure 2

(a) Local entropy $s_1$ versus $U$ at $n=1$ and $T=1/10$ (circles). Horizontal lines mark $\ln 4$ (for $U=0$) and $\ln 2$ (for $U=\infty$). (b) Enlargement of $s_1$ versus $U$ at $n=1$ for $T=1/10$ and $T=1/100$ (triangles) close to the Mott transition. Inflection point for $T>T_c$ and jumps for $T<T_c$ are visible. (c) Increase in the slope $\partial s_1/\partial U$ as one approaches $T_c$ from above. The positions of the minima are shown as blue crosses in Fig. 1.

Figure 3

(a) Total mutual information ${\overline{I}}_1$ versus $U$ at $n=1$ for different temperatures. Filled squares are experimental data of ultracold atoms in Ref. [17] at $T\approx 1/1.4$ (see also Supplemental Material [56]). (b) Enlargement of panel (a) showing $T=1/10$, $1/12$, and $T=1/50$. (c) Thermodynamic entropy $s$ versus $U$ at $n=1$ for the same temperatures of panel (b). In all panels, a dashed vertical line marks the inflection point while the shaded area marks the coexistence between metal and insulator. The loci of the inflections in ${\overline{I}}_1(U{)}_T$ are shown as green squares in Fig. 1.