Two-site entanglement in the two-dimensional Hubbard model

Abstract

The study of entanglement in strongly correlated electron systems typically requires knowledge of the reduced density matrix. Here, we apply the parquet dynamical vertex approximation to study the two-site reduced density matrix at varying distance, in the Hubbard model at weak coupling. This allows us to investigate the spatial structure of entanglement in dependence of interaction strength, electron filling, and temperature. We compare results from different entanglement measures, and benchmark against quantum Monte Carlo.

Figures (12)

• Figure 1: Tracing out all but two lattice sites $i,j$ provides the two-site reduced density matrix $\rho_{i,j}$, computed here from two- and four-point Green's functions with the $p$D A. From $\rho_{i,j}$, entanglement measures such as the mutual information $I$ and fermionic negativity $N^F$ between any two lattice sites can be evaluated. This entanglement includes spin and charge degrees of freedom.
• Figure 2: Rényi mutual information $I_{\textrm{R}}$ as a function of distance $d$ comparing $p$D A (color) and QMC (grey) for the 16$\times$16-site Hubbard model at half-filling and, from left to right, $U=2,\ 3,\ 4$. Solid lines and crosses are measurements along the diagonal of the lattice $\mathbf{\Delta}=(\Delta,\Delta)$, boxes and dotted lines are measured along the NN path $\mathbf{\Delta}=(\Delta,0)$. We denote the real distance $d=\sqrt{\Delta_x^2+\Delta_y^2}$ from the reference site on the $x$-axis. Insets are the same plot with an enlarged y-axis. For $U=2,3$ the QMC error bars are smaller than the symbols. Errors only become significant in the inset for $U=4$, here both the statistic and Trotter error are relevant.
• Figure 3: Rényi mutual information $I_{\textrm{R}}$ (false colors) between a reference point (star) and a second lattice site at distance $(\Delta_x,\Delta_y)$, comparing $p$D A and QMC for $U=2$, $\beta=5$ and two fillings, i.e., $n=1$ (half-filling, two leftmost panels) and $n=0.8$ (two rightmost panels) footnoteRS. Note that NN $I^R$ is outside the color bar and marked in yellow; from left to right we have for NN: $I^{R}_{01} = 0.5484$, $I^{R}_{01}=0.5540$; $I^{R}_{R,01} = 0.4982$, $I^{R}_{R,01}=0.5031$.
• Figure 4: a)-d) Mutual information $I$ and e)-h) fermionic negativity $N^F$ plotted as real space entanglement between lattice sites separated by $\mathbf{\Delta}=(\Delta_x,\Delta_y)$ from a reference site marked by $\star$. From left to right we display (a,e) $U=3$, $\beta=5$, $n=1$ where the yellow NN values are $I_{01} = 0.3014$, $N^F_{01}=0.1646$; (b,f) $U=2$$\beta=5$, $n=1$ with $I_{01} = 0.3114$, $N^F_{01}=0.1663$; (c,g) $U=2$, $\beta=14$, $n=1.0$ with $I_{01} = 0.3196$, $N^F_{01}=0.1732$; (d,h) and $U=2$, $\beta=5$, $n=0.8$ with $I_{01} = 0.3181$, $N^F_{01}=0.1417$footnoteRS.
• Figure 5: Three different paths (dashed colored lines) connect sites $i$ and $j$ with three hoppings each, whereas there is just one path that connects $i$ and $l$ (solid line) with three hoppings. The Manhattan distance between $i$ and $j$ is the same as that between $i$ and $l$, the Euclidean distance is even larger for the former pair.