Universal Features of Entanglement Entropy in the Honeycomb Hubbard Model

The entanglement entropy is a unique probe to reveal universal features of strongly interacting many-body systems. In two or more dimensions these features are subtle, and detecting them numerically requires extreme precision, a notoriously difficult task. This is especially challenging in models of interacting fermions, where many such universal features have yet to be observed. In this Letter we tackle this challenge by introducing a new method to compute the Rényi entanglement entropy in auxiliary-field quantum Monte Carlo simulations, where we treat the entangling region itself as a stochastic variable. We demonstrate the efficiency of this method by extracting, for the first time, universal subleading logarithmic terms in a two-dimensional model of interacting fermions, focusing on the half-filled honeycomb Hubbard model at $T=0$. We detect the universal corner contribution due to gapless fermions throughout the Dirac semi-metal phase and at the Gross-Neveu-Yukawa critical point, where the latter shows a pronounced enhancement depending on the type of entangling cut. Finally, we observe the universal Goldstone mode contribution in the antiferromagnetic Mott insulating phase.

Figure 1

(a) A triangular region on a $6\times 6$ lattice with a zigzag edge. (b) A triangular region with a bearded edge.

Figure 2

The new DQMC computation of $S_2$ as compared to DMRG [36] for the system pictured. Here we use open boundaries in the $x$ direction. The inset shows a closer look at the precise agreement at $U=8$ using only a single value of $\lambda =0.5$ and $N_{\mathrm{seed}}$ independent simulations.

Figure 3

Left panel: $S_2$ computed for both kinds of triangular regions depicted in Figs. 1 and 1 at $U=1$ and twisted boundary conditions with $\varphi =0.1$. The inset shows the result of a three parameter fit to a linear plus log scaling and subtracting away the area law piece (a shift of $\tilde{c}=1$ is given to the bearded triangle for clarity of comparison). The blue dashed line shows the field theory prediction for the semimetal phase. Right panel: the same analysis but for with $U=3.8$ at the GNY point with $\varphi =0$. The bearded triangle shows an enhanced logarithmic contribution in this case. This is compared to the field theory value for free fermions plus a three-component boson shown by the gray dotted-dashed line in the inset.

Figure 4

Left panel: $S_2$ for bearded triangles depicted in Fig. 1 using ${\mathrm{\Delta }}_{\tau }=0.5$ and $\varphi =0.15$ as a function of $U$. Middle panel: The fit as a function of $L$ performed for each value of $U$ with the area law piece subtracted away, with a constant shift of $U$ added for clarity. Right panel: The extracted corner coefficient as a function of $U$, showing an enhanced value at the GNY point.

Figure 5

Half-system entropy $S_2$ on rectangular systems as depicted, with $U=8$. This plot is similar to Fig. 3, but now the subleading log term has the opposite sign, as can be seen by the slope in the inset. The thin line in the main panel helps to visualize the bend in the data coming from the logarithmic term. The coefficient of the log term counts the Goldstone modes with the coefficient $N_g/2$, here giving 0.95(5).