Antiferromagnetism in the Hubbard model on the honeycomb lattice: A two-particle self-consistent study

The semimetal to antiferromagnet quantum phase transition of the Hubbard model on the honeycomb lattice has come to the forefront in the context of the proposal that a semimetal to spin liquid transition can occur before the transition to the antiferromagnetic phase. To study the semimetal to antiferromagnet transition, we generalize the two-particle self-consistent (TPSC) approach to the honeycomb lattice (a structure that can be realized in graphene for example). We show that the critical interaction strength where the transition occurs is $U_c/t=3.79\pm 0.01$, quite close to the value $U_c/t=3.869\pm 0.013$ reported using large-scale quantum Monte Carlo simulations. This reinforces the conclusion that the semimetal to spin-liquid transition is preempted by the transition to the antiferromagnet. Since TPSC satisfies the Mermin-Wagner theorem, we find temperature-dependent results for the antiferromagnetic and ferromagnetic correlation lengths as well as the dependence of double occupancy and of the renormalized spin and charge interactions on the bare interaction strength. We also estimate the value of the crossover temperature to the renormalized classical regime as a function of interaction strength.

Figure 1

Plot of ${\chi }_{afm}^0(\mathbf{q},i\nu =0)$ for $T=0.005$.

Figure 2

Plot of $\langle n_{\uparrow }{n}_{\downarrow }\rangle$ as a function of $U$ for given temperatures $\left(N=100\right)$. The temperature dependence is very small, as can be checked from the inset.

Figure 3

Plot of the irreducible vertex for the spin ${\mathcal{U}}_s$ as a function of $U$ for given temperatures $\left(N=100\right)$. The temperature dependence is extremely small as can be seen from the inset.

Figure 4

Semilogarithmic plot of the spin correlation length in the antiferromagnetic channel $\xi$ as a function of $U$ for various temperatures $\left(N=100\right)$.

Figure 5

Plot of the ferromagnetic correlation length ${\xi }_{fm}^s$ as a function of $U$ for given values of temperature $\left(N=100\right)$.

Figure 6

Plot of ${\xi }_0$ (dashed blue line with circles) and of ${\mathrm{\Gamma }}_0$ (dot-dashed green line with squares) as a function of $T$ for the range $T=0.001$ to $0.1$. Each quantity has its own vertical axis: left for ${\xi }_0$ and right for ${\mathrm{\Gamma }}_0$.

Figure 7

Plots of the crossover temperature as a function of interaction strength for given values of $N$. The crossover temperature was determined from $\xi =v_F/T$ scanning $U$ at fixed $T$, and scanning $T$ at fixed $U$ to obtain an estimate of the error in finding the intersection $\xi (U,T)=v_F/T$.

Figure 8

Plots of $ln\xi$ as a function of $lnT$ for various values of $U \left(N=100\text{and}200\right)$. The straight dashed magenta line corresponds to a pure power law, $z=1$, hence to the value $U_c$ for the quantum critical point.