Stability of Dirac Liquids with Strong Coulomb Interaction

We develop and apply the diagrammatic Monte Carlo technique to address the problem of the stability of the Dirac liquid state (in a graphene-type system) against the strong long-range part of the Coulomb interaction. So far, all attempts to deal with this problem in the field-theoretical framework were limited either to perturbative or random phase approximation and functional renormalization group treatments, with diametrically opposite conclusions. Our calculations aim at the approximation-free solution with controlled accuracy by computing vertex corrections from higher-order skeleton diagrams and establishing the renormalization group flow of the effective Coulomb coupling constant. We unambiguously show that with increasing the system size $L$ (up to $\ln (L)\sim 40$), the coupling constant always flows towards zero; i.e., the two-dimensional Dirac liquid is an asymptotically free $T=0$ state with divergent Fermi velocity.

Figure 1

(a) Honeycomb lattice of graphene with the nearest-neighbor hopping amplitude $t$ ($|{\mathbf{a}}_1|=|{\mathbf{a}}_2|=a$). (b) The basic building block for the $G^{2}W$ skeleton expansion is based on three-point vertexes characterized by the unit cell index $\mathbf{R}$, imaginary time $\tau$, and sublattice index $\xi =\{\mathbf{A},\mathbf{B}\}$. Green’s functions $G$ and screened interactions $W$ are connecting three-point vertexes pairwise. (c) Typical skeleton free-energy diagram.

Figure 2

The order parameter $\mathrm{\Delta }m$ as a function of inverse skeleton order $1/N$. Calculations were performed for system sizes $L=8$ (green triangles) and 16 (blue circles) and for different values of the symmetry breaking field $h=0.1$, 0.2, 0.3, 0.4. By filled red stripes we show the corresponding results (with error bounds) from Ref. [17] for $L=18$.

Figure 3

RG flow of $\alpha$ with the system size for various bare coupling constants ${\alpha }_0$ within the GW approximation.

Figure 4

RG flow of $\alpha$ with the system size for $N=1$ (red), $N=2$ (green), and $N=3$ (blue). GW curves were obtained for $L=16$, 32, 64, 128, 256; for $L=16$ there are visible deviations from the master curve because this size is not in the scaling limit yet. Curves for $N=2$ and 3 were obtained for $L=32$, 64, and 128 (see text). Black dashed line is the RG flow based on the $\text{second}\text{−}\text{order}+\mathrm{RPA}$ approximation [1, 7, 8]. Purple solid line (courtesy of A. Sharma and P. Kopietz) is the flow obtained within the functional renormalization group approach [10].