Fermi-liquid ground state of interacting Dirac fermions in two dimensions

An unbiased zero-temperature auxiliary-field quantum Monte Carlo method is employed to analyze the nature of the semimetallic phase of the two-dimensional Hubbard model on the honeycomb lattice at half filling. It is shown that the quasiparticle weight $Z$ of the massless Dirac fermions at the Fermi level, which characterizes the coherence of zero-energy single-particle excitations, can be evaluated in terms of the long-distance equal-time single-particle Green's function. If this quantity remains finite in the thermodynamic limit, the low-energy single-particle excitations of the correlated semimetallic phase are described by a Fermi-liquid-type single-particle Green's function. Based on the unprecedentedly large-scale numerical simulations on finite-size clusters containing more than 10 000 sites, we show that the quasiparticle weight remains finite in the semimetallic phase below a critical interaction strength. This is also supported by the long-distance algebraic behavior ($\sim r^{-2}$, where $r$ is distance) of the equal-time single-particle Green's function that is expected for the Fermi liquid. Our result thus provides a numerical confirmation of Fermi-liquid theory in two-dimensional correlated metals.

Figure 1

A finite-size cluster and a unit cell of the honeycomb lattice. ${\mathit{a}}_1=a(\frac{3}{2},\frac{\sqrt{3}}{2})$ and ${\mathit{a}}_2=a(\frac{3}{2},-\frac{\sqrt{3}}{2})$ are the primitive translational vectors with $a$ being the lattice constant. The $x$ and $y$ axes are indicated in the lower-left part of the figure. The small parallelogram defined by ${\mathit{a}}_1$ and ${\mathit{a}}_2$ is the unit cell. The large parallelogram defined by $L{\mathit{a}}_1$ and $L{\mathit{a}}_2$ is a finite-size cluster of $L=4$. The filled (empty) circles represent lattice sites belonging to sublattice $A$ ($B$).

Figure 2

(a) Computational time of one space-time Monte Carlo sweep with the two multiplication schemes (right axis) and speedup of the sparse-matrix case in the polynomial expansion scheme relative to the dense-matrix case in the conventional scheme (left axis) for $N_{\mathrm{site}}=2L^2$ with $L=20,26,32,38,44,50,56,62,68$, and 74. The dashed line at $\mathrm{Speedup}=1$ is a guide to the eye. A single thread is used for the calculations. (b) Speedup with multithreading relative to the single-thread case. These benchmark calculations are performed at the HOKUSAI GreatWave facility with SPARC64 XIfx processors in RIKEN.

Figure 3

$lnL$ dependence of $ln\left|D_{AB,\sigma }\left(r_{\max }\right)\right|$ for (a) $U/t=3.5$, (b) $U/t=3.7$, and (c) $U/t=4$. For comparison, the result for the noninteracting case is also shown in (d). Lines are linear fit to the data of the form $-\alpha \left(L_{\min }\right)lnL+b\left(L_{\min }\right)$, where $\alpha \left(L_{\min }\right)$ and $b\left(L_{\min }\right)$ are fitting parameters with $L_{\min }$ being the minimum $L$ used for the fit. The maximum $L$ used for the fit is 74 for all cases, including (d).

Figure 4

$L_{\min }$ dependence of $\alpha \left(L_{\min }\right)$ for different values of $U$ indicated in the figure. For comparison, $\alpha \left(L_{\min }\right)$ for $U=0$ is also shown by grey dots. The dashed lines indicate $\alpha =2$ and $\alpha =2+{\eta }_{\psi }$ with ${\eta }_{\psi }=0.2$.

Figure 5

The quasiparticle weight $Z\left(L\right)$ given in Eq. (18) as a function of $1/L$. Lines are polynomial fits to the data for $U/t=3.5$ and 3.6. The extrapolated values in the thermodynamic limit are also shown at $1/L=0$. The quasiparticle weights estimated previously from the jump of the momentum distribution function [24] are also shown by stars next to the present results.

Figure 6

Log-log plot of $\left|D_{AB,\sigma }^{\left(0\right)}\left(\mathit{r}\right)\right|$ with $\mathit{r}=n({\mathit{a}}_1+{\mathit{a}}_2)=(3na,0)$ calculated directly using Eq. (11) on an $L=1000$ cluster up to $\left|\mathit{r}\right|/a⩽249$ (red circles). The asymptotic algebraic decay of Eq. (A11) is also shown by dashed line.