Graphene via large $N$: A renormalization group study

We analyze the competing effects of moderate to strong Coulomb electron-electron interactions and weak quenched disorder in graphene. By using a one-loop renormalization group calculation controlled within the large-$N$ approximation, we demonstrate that at successively lower energy (temperature or chemical potential) scales, a type of non-Abelian vector potential disorder always asserts itself as the dominant elastic scattering mechanism for generic short-ranged microscopic defect distributions. Vector potential disorder is tied to both elastic lattice deformations (“ripples”) and topological lattice defects. We identify several well-defined scaling regimes, for which we provide scaling predictions for the electrical conductivity and thermopower, valid when the inelastic lifetime due to interactions exceeds the elastic lifetime due to disorder. Coulomb interaction effects should strongly figure into the physics of suspended graphene films, where $r_s>1$; we expect the vector potential disorder to play an important role in the description of transport in such films.

Figure 1

Diagrammatic elements of the bare Feynman rules. The associated amplitudes are summarized in Table .

Figure 2

The fermion propagator at $N=\infty$.

Figure 3

The gauge (Coulomb) propagator at $N=\infty$.

Figure 4

Screening of the scalar potential disorder $g_u$ due to Coulomb interactions in the $N=\infty$ limit.

Figure 5

Corrections to the electronic self-energy at $\mathit{O}(\frac{1}{N},\{\widetilde{g_u},{\mathcal{G}}_{\mu }^{\nu }\})$.

Figure 6

Coulomb vertex corrections at $\mathit{O}(\frac{1}{N},\{\widetilde{g_u},{\mathcal{G}}_{\mu }^{\nu }\})$.

Figure 7

(a) Cancellation between counterpropagating three-electron rings involving only identity matrix (1) insertions into the trace. (b) Three-electron rings involving two identity matrix insertions and one disorder matrix of the form ${\widehat{M}}_{\mu \nu }={\widehat{\sigma }}^{\mu }{\widehat{\kappa }}_{\nu }$ vanish identically.

Figure 8

Corrections to the gauge (Coulomb) field self-energy to $\mathit{O}(\frac{1}{N},\{\widetilde{g_u},{\mathcal{G}}_{\mu }^{\nu }\})$.

Figure 9

Corrections to the gauge (Coulomb) field self-energy to $\mathit{O}(\frac{1}{N},\{\widetilde{g_u},{\mathcal{G}}_{\mu }^{\nu }\})$.

Figure 10

Evaluation of two-loop diagrams ${\mathsf{D}}_{8.5}$ and ${\mathsf{D}}_{8.6}$ in Fig. 8. These graphs can be expressed in terms of products of an ultraviolet-convergent polarization bubble and an ultraviolet-divergent vertex correction (see Fig. 6).

Figure 11

Renormalization of the disorder strength at $\mathit{O}(\frac{g}{N},g^2)$, where $g∊\{\widetilde{g_u},{\mathcal{G}}_{\mu }^{\nu }\}$.

Figure 12

Renormalization of the disorder strength at $\mathit{O}(\frac{g}{N},g^2)$, where $g∊\{\widetilde{g_u},{\mathcal{G}}_{\mu }^{\nu }\}$.

Figure 13

Renormalization of the disorder strength at $\mathit{O}(\frac{g}{N},g^2)$, where $g∊\{\widetilde{g_u},{\mathcal{G}}_{\mu }^{\nu }\}$.

Figure 14

Plot of $\overline{w}{f}_3\left(\overline{w}\right)$.

Figure 15

Set of ten RG flow trajectories obtained via numerical integration for different initial Coulomb interaction strengths $\overline{w}$. Here, we study the case with SU(2) valley-space symmetry restored on average, using Eqs. (4.11a)–(4.11e). The initial conditions for the disorder coupling strengths are $\widetilde{g_u}\left(0\right)=g_A\left(0\right)=g_m\left(0\right)=0.01$ for all trajectories shown in the figure. We have set the expansion parameter $\eta =0.3$ [see Eq. (4.3)]; the shape of the trajectories is only weakly dependent on $\eta$ (within our one-loop approximation). Lowercase Latin letters label trajectories with different initial Coulomb interaction strengths given by $\overline{w}=\left(\mathrm{a}\right)$ 0.010, (b) 0.079, (c) 0.13, (d) 0.16, (e) 0.20, (f) 0.25, (g) 0.32, (h) 0.40, (i) 0.63, and (j) 1.0. We do not exhibit the RG evolution of the random mass parameter $g_m$ [see Eqs. (2.9, 2.11d, 2.11e, 4.11e)] because we find that its behavior is always subleading compared to the scalar $\left(\widetilde{g_u}\right)$ or vector $\left(g_A\right)$ potential parameters.

Figure 16

Same RG trajectories shown in Fig. 15 but projected into the $\widetilde{g_u}\text{−}{g}_A$ disorder plane.

Figure 17

As in Fig. 15 but with the initial conditions for the disorder coupling strengths $\widetilde{g_u}\left(0\right)=g_m\left(0\right)=0.1$, $g_A=0.01$. Lowercase Latin letters label trajectories with different initial Coulomb interaction strengths given by $\overline{w}=\left(\mathrm{a}\right)0.010$, (b) 0.25, (c) 0.40, (d) 0.50, (e) 0.63, (f) 0.79, (g) 1.0, (h) 1.3, (i) 1.6, (j) 2.5, and (k) 6.3.

Figure 18

Same RG trajectories shown in Fig. 17 but projected into the $\widetilde{g_u}\text{−}{g}_A$ disorder plane.

Figure 19

Schematic phase diagram for disordered interacting graphene projected into the disorder $(\widetilde{g_u},g_A)$ interaction $\left(\overline{w}\right)$ coupling constant space, based in part on the perturbative RG calculations performed at weak coupling (Refs. 19, 33) and within the large-$N$ approximation (as detailed in this paper). The labeled small dots represent “known” phases, while the large black globes represent “nonlinear sigma model $\left(\mathrm{NL}\sigma \mathrm{M}\right)$ arenas,” wherein the system is described by a well-known (if poorly understood) model, which may possess multiple phases in $d=2$ spatial dimensions (Refs. 45, 52). Point (a) represents noninteracting clean (not disordered) graphene, as located by Eq. (4.16). This theory is a sink for the clean Coulomb-interacting line (the vertical axis), as well as a portion of the $\widetilde{g_u}\text{−}\overline{w}$ plane. It is unstable to generic disorder perturbations. The dashed line in the $\widetilde{g_u}\text{−}\overline{w}$ is the unstable fixed line located by Eqs. (4.12a, 4.12b). Point (b) is the noninteracting non-Abelian SU(2) vector potential disorder CFT (Refs. 18, 26). Both points (a) and (b) are embedded in the particle-hole symmetric plane $\widetilde{g_u}=g_m=0$ represented by the shaded plane in the figure. Point (c) represents the noninteracting Anderson insulator for the orthogonal metal class (AI) in two dimensions. Globe (d) should be described by the symplectic (spin-orbit) class (AII) noninteracting $\mathrm{NL}\sigma \mathrm{M}$, possibly augmented by a topological term (Refs. 19, 20, 21, 22, 23, 24). In principle, both diffusive metallic and Anderson insulating phases are possible (Ref. 45). Globe (e) represents the symplectic class Finkel’stein nonlinear sigma model $\left(\mathrm{FNL}\sigma \mathrm{M}\right)$ for interacting electrons (Refs. 52, 54, 55); globe (f) represents the orthogonal class (AI) $\mathrm{FNL}\sigma \mathrm{M}$ (Refs. 52, 56, 57). The numerical RG flows exhibited in Figs. 15, 17 should be “superimposed” on this figure.

Figure 20

Sketch of possible logarithmic scaling regimes of the dc conductivity ${\sigma }_{\mathrm{dc}}$ vs temperature in undoped graphene. Labels 1–3 identify the qualitative scaling behaviors pictured here with the three regimes, respectively, discussed in Secs. 5B1, 5B2, 5B3 in the text. Nonmonotonic behavior, i.e., the peak separating the curve segments labeled 1 and 3 in the figure, can occur if a sample microscopically dominated by strong Coulomb interactions and weak scalar potential disorder (“QED” regime), distinguished by a conductivity that increases with decreasing temperature [Eq. (5.5)], crosses over at lower temperatures to the strong-coupling (“QCD”) regime dominated by non-Abelian vector potential disorder, which is characterized by a decreasing conductivity [Eq. (5.10)]. An intermediate scaling regime (2) is also possible, which is characterized by a monotonically decreasing conductivity [Eq. (5.8)]. These predictions require that ${\tau }_{\mathit{el}}<{\tau }_{\mathit{in}}$ (see text).

Figure 21

Schematic RG flow diagram projected into the 3D coupling constant space formed from the screened scalar potential disorder strength $\left(\widetilde{g_u}\right)$, the non-Abelian vector potential disorder strength $\left(g_A\right)$, and Coulomb interaction strength $(\overline{w}=\pi r_s∕2)$ [Eq. (2.24)]. This diagram should be compared to the flow plots, obtained via numerical integration of Eqs. (4.11a)–(4.11e), depicted in Figs. 15, 17 of Sec. 4E. In this figure, we have sketched characteristic flows designated by labels 1–3, corresponding to the “QED”, intermediate, and “QCD” scaling regimes discussed in Sec. 5. The dashed curve represents the unstable fixed line discussed below Eqs. (4.12a, 4.12b) in Sec. 4D1.