Figure 1

NLCE (lines) and DQMC (symbols) results for the entropy of the Hubbard model in the honeycomb (HC) lattice as a function of $T$ for $U/w=1/2$, 1, $3/2$, and 2 at half filling. The NLCE results were obtained by applying Euler's resummation to the last six terms in each order of the expansion.[27, 30] Here we report results for the last order (thick lines) and the next to last order (thin black lines). We also show NLCE results for the entropy of the antiferromagnetic Heisenberg model (AFHM) for $J/w=1/18$. The site expansion for this model was carried out up to the seventeenth order. The results in this case were obtained using Wynn's algorithm and we report those for the highest term after eight (thick line) and seven (thin black line) orders of improvement.[27, 30] Unless otherwise specified, all NLCE results reported in the following for the honeycomb lattice are obtained as explained above and are presented using the same convention. Inset: NLCE results for the entropy vs density ($n$) at $T/w=0.126$ for the same values of $U$ and legends used in the main panel.Reuse & Permissions

Figure 2

Specific heat vs $T$ for (a) the honeycomb lattice and (b) the square lattice.[17, 38, 39] Results are presented for the Hubbard model with $U/w=1/2$, 1, $3/2$, and 2 at half filling, and for the Heisenberg model with $J/w=1/18$ (honeycomb) and $J/w=1/32$ (square). The NLCE site expansion for the Heisenberg model in the square lattice was carried out up to the fifteenth order.[30] Lines (symbols) depict results from NLCE (DQMC).Reuse & Permissions

Figure 3

(a) Temperature and (b) $C_v$ at the high-temperature crossing point between the specific heat curves for consecutive values of $U$ (see text), as obtained from NLCE for the honeycomb lattice.Reuse & Permissions

Figure 4

(a) Uniform spin susceptibility, and (b) NN spin correlations in the honeycomb lattice at half filling vs $T$ for $U/w=1/2$, 1, $3/2$, and 2. In (a), we also include results for the Heisenberg model with $J/w=1/18$. The inset in (b) shows $T^{*}$, which is the temperature at the maximum of the uniform susceptibility, and $T^{\prime }$, which is the temperature at which $\partial S_{nn}^{zz}/\partial T$ is maximal, as a function of $U$ (for $U>U_c$). Lines (symbols) depict results from NLCE (DQMC).Reuse & Permissions

Figure 5

NLCE results for the staggered susceptibility as a function of temperature in the honeycomb lattice at half filling for $U/w$ from $1/2$ up to 2. The inset shows ${\chi }^{\mathrm{stg}}$ as a function of $U/w$ for $T/w=0.05$.Reuse & Permissions

Figure 6

Antiferromagnetic structure factor as a function of temperature in the honeycomb lattice at half filling with $U/w=1/2$, 1, $3/2$, and 2. Lines (symbols) correspond to NLCE (extrapolated DQMC) results.Reuse & Permissions

Figure 7

NLCE (lines) and DQMC (symbols) results for normalized double-occupancy $4D/n^2$ vs $T$ at different densities for (a) $U/w=1$ and (b) $U/w=3/2$ on the honeycomb lattice. Inset of (a) shows $T/w$ at the minimum of $4D/n^2$ as a function of $n$.Reuse & Permissions

Figure 8

Isentropic curves for $T/w$ vs $U/w$ in the honeycomb lattice at density $n=1.00$ (thin dashed lines) and $n=0.90$ (thick solid lines) for constant entropies $S=0.50$, $0.60$, $0.65$, $ln\left(2\right)$, $0.75$, $0.85$, and $0.90$. Those results were obtained using NLCE. For $n=0.90$, we also present results from DQMC (symbols).Reuse & Permissions

Figure 9

NLCE results for the isentropic curves of $T/w$ vs $U/w$ in trapped systems in the honeycomb lattice for constant average entropies per particle $s=0.5$, $0.6$, $0.7$, $0.8$, and $0.9$; and for (a) $\tilde{\rho }=9.3$ and (b) $\tilde{\rho }=4.3$. The shaded region emphasizes temperatures below $T^{*}$, for ($U>U_c$), in the half-filled homogeneous system [see the inset in Fig. 4b].Reuse & Permissions

Figure 10

Local density for trapped fermions in the honeycomb lattice with (a) $\tilde{\rho }=9.3$ and $s=0.6$, and (b) $\tilde{\rho }=4.3$ and $s=0.8$. Results are presented for $U/w$ from 0 to $7/6$. NLCE results are depicted as lines and DQMC results as symbols.Reuse & Permissions

Figure 11

(a) NLCE results for the isentropic curves in the square lattice, similar to Fig. 9 but with $\tilde{\rho }=3.6$; (b) local density in the square lattice with $\tilde{\rho }=3.6$ and $s=0.8$. Lines and symbols are the same as in Fig. 10.Reuse & Permissions