Semimetal–Mott insulator quantum phase transition of the Hubbard model on the honeycomb lattice

We take advantage of recent improvements in the grand canonical hybrid Monte Carlo algorithm, to perform a precision study of the single-particle gap in the hexagonal Hubbard model, with on-site electron-electron interactions. After carefully controlled analyses of the Trotter error, the thermodynamic limit, and finite-size scaling with inverse temperature, we find a critical coupling of $U_c/\kappa =3.834\left(14\right)$ and the critical exponent $z\nu =1.185\left(43\right)$. Under the assumption that this corresponds to the expected antiferromagnetic Mott transition, we are also able to provide a preliminary estimate $\beta =1.095\left(37\right)$ for the critical exponent of the order parameter. We consider our findings in view of the $\text{SU}\left(2\right)$ Gross-Neveu, or chiral Heisenberg, universality class. We also discuss the computational scaling of the hybrid Monte Carlo algorithm, and possible extensions of our work to carbon nanotubes, fullerenes, and topological insulators.

Figure 1

Examples of effective mass determinations from the single-particle correlators, extracted from ensembles of auxiliary field configurations. The blue line with error band gives $m_{\text{eff}}$ with statistical error, obtained from a hyperbolic cosine effective mass (15). The length of the blue band indicates the fitting region. For comparison, a constant fit to the effective mass (11) is shown by the dashed orange line. The dot-dashed red line shows the estimation of the systematic error, as explained in Appendix pp2. Note that the red and orange lines have been extended outside of the fitting region, for clearer visibility. (Left) $\kappa \beta =8$, $L=15$, $N_t=64$, and $U/\kappa =3.5$. (Right) $\kappa \beta =12$, $L=6$, $N_t=72$, and $U/\kappa =3.85$. The time slice index $\tau$ is integer-valued. The effective mass $m_{\text{eff}}$ is given in units of $\kappa$.

Figure 2

Simultaneous two-dimensional fit of $\mathrm{\Delta }(N_t,L)$ (in units of $\kappa$) using Eq. (21), for $\kappa \beta =8$ and $U/\kappa =3.5$. Note that Eq. (21) only incorporates effects of $\mathcal{O}\left(L^{-3}\right)$ and $\mathcal{O}\left(N_t^{-2}\right)$. Data points for $L<9$ have been omitted from the fit, but not from the plot. Very small lattices lead to large values of $\mathrm{\Delta }$, which are not visible on the scale of the plot. This fit has ${\chi }^2/\text{d.o.f.}\simeq 1.1$, corresponding to a p-value of $\sim 0.34$.

Figure 3

The single-particle gap ${\mathrm{\Delta }}_0(U,\beta )$, with all quantities in units of $\kappa$, after the thermodynamic and continuum limit extrapolations. We also show ${\mathrm{\Delta }}_0(U,\beta \to \infty )$ as a solid black line with error band (see Sec. 4a). For $U<U_c\simeq 3.834\left(14\right)$ the zero-temperature gap vanishes.

Figure 4

$f=\beta {\mathrm{\Delta }}_0$ (left) and $g={\beta }^{\zeta }{\mathrm{\Delta }}_0/U$ (right) as a function of $u={\beta }^{\mu }(U-U_c)$ according to Eqs. (31) and (33), respectively. All quantities are plotted with the parameters for optimal data collapse, and expressed in units of $\kappa$.

Figure 5

Single-particle gap ${\mathrm{\Delta }}_0$ scaled by $\beta$ (left panel) and by ${\beta }^{\zeta }/U$ (right panel), as a function of $U$. Note that ${\mathrm{\Delta }}_0$ has been extrapolated to infinite volume and $\delta \to 0$ (continuum limit). Each line represents a given inverse temperature $\beta$. Here, $\zeta =0.924\left(17\right)$ as obtained from the data collapse fit. The averages of all the crossing points $U_c^f$ and $U_c^g$ have been marked by vertical lines. For $U_c^g$, data with $\beta \le 4$ have been omitted due to the thermal gap discussed in Appendix pp3. All quantities are given in units of $\kappa$.

Figure 6

Illustration of the thermal gap in the weakly coupled regime, as given by Eq. (C5). Our MC data for the single-particle gap $\mathrm{\Delta }$ from Fig. 3 is shown multiplied by ${\beta }^2$. All quantities are expressed in appropriate units of $\kappa$.

Figure 7

Simultaneous two-dimensional fit of $\mathrm{\Delta }(N_t,L)$ (in units of $\kappa$) using Eq. (21), for $\kappa \beta =10$ and $U/\kappa =3.9$ (left) and $\kappa \beta =4$ and $U/\kappa =5.0$ (right). Only the extrapolations in $L$ are shown. Data points for $L<9$ have been omitted from the fits, but not from the plots. These fits have ${\chi }^2/\text{d.o.f.}\simeq 0.83$ and $p$ value of $\simeq 0.62$ (left), and ${\chi }^2/\text{d.o.f.}\simeq 1.1$ and $p$ value of $\simeq 0.36$ (right).