Electron-Electron Interactions in Graphene: Current Status and Perspectives

The problem of electron-electron interactions in graphene is reviewed. Starting from the screening of long-range interactions in these systems, the existence of an emerging Dirac liquid of Lorentz invariant quasiparticles in the weak-coupling regime is discussed, as well as the formation of strongly correlated electronic states in the strong-coupling regime. The analogy and connections between the many-body problem and the Coulomb impurity problem are also analyzed. The problem of the magnetic instability and Kondo effect of impurities and/or adatoms in graphene is also discussed in analogy with classical models of many-body effects in ordinary metals. Lorentz invariance is shown to play a fundamental role and leads to effects that span the whole spectrum, from the ultraviolet to the infrared. The effect of an emerging Lorentz invariance is also discussed in the context of finite size and edge effects as well as mesoscopic physics. The effects of strong magnetic fields in single layers and some of the main aspects of the many-body problem in graphene bilayers are briefly reviewed. In addition to reviewing the fully understood aspects of the many-body problem in graphene, a plethora of interesting issues are shown to remain open, both theoretically and experimentally, and the field of graphene research is still exciting and vibrant.

Figure 1

(a) Honeycomb lattice with the two sublattices in graphene. The arrows are nearest neighbor vectors. (b) Tight-binding spectrum for the $\pi -{\pi }^{*}$ bands. The horizontal line intersecting the $K$ point corresponds to the Fermi level at half-filling. (c) Brillouin zone centered around the $\Gamma$ point. (d) Dirac cone resulting from the linearization of the tight-binding spectrum around the $K$ points (circles).

Figure 2

Diagram for the polarization bubble corresponding to Eq. (2.12).

Figure 3

Polarization bubble ${\Pi }^{(1)}(q,\omega )$ for graphene, within the Dirac approximation. (a), (b) A density plot of the real and imaginary parts of the polarization bubble ${\Pi }^{(1)}(q,\omega )$ defined in Eq. (2.12), and normalized to the DOS at the Fermi level $\rho (\mu )$. (c), (d) Present constant frequency cuts at $\omega /\mu =0.5$, 1.0, 1.5, 2.0, 2.5, 3.0. (e) The static limit ${\Pi }^{(1)}(q,0)$, whose closed form expression is written in Eq. (2.16). Notice the transition from a constant value ($q<2k_F$) to the linear in $q$ dependence at large momenta. The derivative of the polarization is shown in the same panel, and can be seen to vary continuously. (f) The real (solid) and imaginary (dashed) parts of the uniform limit (2.17).

Figure 4

Polarization bubble ${\Pi }^{(1)}(q,\omega )$ for the conventional 2DEG. (a), (b) A density plot of the real and imaginary parts of the polarization bubble ${\Pi }^{(1)}(q,\omega )$ normalized to the DOS at the Fermi level. (c), (d) Constant frequency cuts at $\omega /\mu =0.5$, 1.0, 1.5, 2.0, 2.5, 3.0. (e) The static limit ${\Pi }^{(1)}(q,0)$ and (f) the uniform limit ${\Pi }^{(1)}(0,\omega )$.

Figure 5

Colored regions represent the particle-hole continuum of graphene due to interband and intraband transitions. Left: half-filled case; right: finite $\mu$ case, away from half-filling. Dashed line: acoustic plasmon for the single layer (${\omega }_p\propto \sqrt{\mu q}$).

Figure 6

Plasmon band (hatched region) for an infinite stack of graphene layers. Solid line: optical mode $k_z=0$. Dashed line: acoustic mode $k_z=\pi /d$, ${\omega }_p\propto \sqrt{\mu }q$, with linear dispersion, at the edge of the band. All the other modes in between are acoustic. Adapted from 383.

Figure 7

Self-energy diagrams: (a) first order Hartree-Fock, (b) second order loop diagram (first diagram in the RPA series), (c) second order exchange (vertex correction) diagram, and (d) rainbow diagram.

Figure 8

(a) Self-energy and (b) vertex corrections to the polarization bubble.

Figure 9

An attractive line of fixed pints for interactions and gauge field disorder.

Figure 10

RPA self-energy, which includes an infinite resummation of polarization bubbles.

Figure 11

RG flow of the coupling $\lambda$ and the quasiparticle residue $Z$ as a function of the RG scale $l$; the infrared limit is at $l\to \infty$. From 149.

Figure 12

Exact evaluation of the RPA equations for (a) the quasiparticle residue and (b) the Fermi velocity. On the horizontal axis $f$ is defined as $f\equiv N\alpha$. $\Lambda$ is in units of $k_F$. Values of $\Lambda$ from $\sim {10}^2$ to ${10}^1$ correspond to density $n$ from $n\sim {10}^{11}$ to $\sim {10}^{13}{\mathrm{cm}}^{-2}$ (while $\Lambda \sim {10}^3$ is ultralow density $n\sim {10}^9{\mathrm{cm}}^{-2}$). The values of $\Lambda$ (in units of $k_F$) can be converted into density $n$ via $\Lambda /k_F\approx 220/\sqrt{\tilde{n}}$, $\tilde{n}=n/({10}^{10}{\mathrm{cm}}^{-2})$. The curves labeled 2DES refer to the case of 2DEG with parabolic bands, where $f=\sqrt{2}{r}_s$ and $r_s\sim 1/\sqrt{n}$. From 335.

Figure 13

(a) Density dependence of the velocity for suspended graphene. The solid line is the result of RG treatment within RPA [Eq. (3.27)]. (b) Reshaping of the Dirac cone due to the interaction-driven renormalization (increase) of the Fermi velocity at low momenta. The outer cone represents the linear Dirac spectrum without many-body effects. From 108.

Figure 14

Schematic phase diagram in the $\alpha -N$ plane.

Figure 15

ARPES data showing strong features at the Dirac point, which is below the Fermi energy (at 0). The splitting shown in (H) is attributed to the presence of “plasmarons” (quasiparticles strongly bound to plasmons) and depends on the value of $\alpha$ ($\alpha \approx 0.5$ fits the data). From 46.

Figure 16

Inverse compressibility, measured by 268. The solid line is the compressibility of noninteracting Dirac fermions.

Figure 17

Upper panel: Inverse compressibility calculated at different levels of approximation as a function of density. The inset enlarges the low-density region. Lower panel : Compressibility calculated within RPA, relative to the free level for different couplings and densities. Here $N=4$ is the Dirac fermion degeneracy. The numbers refer to the values of $\Lambda /k_F$, which can be converted into density $n$ via $\Lambda /k_F\approx 220/\sqrt{\tilde{n}}$, $\tilde{n}=n/({10}^{10}{\mathrm{cm}}^{-2})$. This implies $\Lambda /k_F\sim {10}^2$ for $n\sim {10}^{11}{\mathrm{cm}}^{-2}$, and $\Lambda /k_F\sim 10$ for $n\sim {10}^{13}{\mathrm{cm}}^{-2}$. Adapted from 23.

Figure 18

Schematic drawing of the level diving process in the supercritical regime and of the resulting quasispectrum of levels for massive and massless fermions.

Figure 19

(a) LDOS, $N(\epsilon ,r)$ at $r=a$ for several couplings $g<g_c$. The inset shows $N(\epsilon ,r)$ for $g=0.27$ and different $r$. For comparison, the exact LDOS calculated in the full tight-binding lattice for the same parameters is shown as dashed lines. In the horizontal axis the energy is in units of the hopping $t$. (b) The weak-coupling transport cross section as a function of $g$. The inset shows the phase shifts for different $j$. (c) LDOS, $N(\epsilon ,r)$ at several distances $r$, for $g=1>g_c$. The inset shows the oscillating LDOS correction for $\epsilon >0$. (d) Energy dependence of the phase shifts (top) and the supercritical contribution ${\overline{n}}_j(\epsilon ,r)$ to the LDOS (bottom) for $g=1.0$.

Figure 20

(a) Induced electron density $\delta n(\mathit{r})$ plotted as a function of distance to the Coulomb center, for different impurity strengths, $g<g_c$. Data obtained from full diagonalization of the tight-binding Hamiltonian in a lattice with ${124}^2$ atoms. The black lines are $\propto 1/r^3$ and guides for the eye. The inset shows the saturation of the integrated charge accumulated inside $r<R_{\max }$, as a function of $R_{\max }$. (b) Same as (a), but for the supercritical case, $g>g_c$, and the dashed line is now $\propto 1/r^2$.

Figure 21

(a) Total integrated charge in the vicinity of the impurity $Q$ obtained from exact diagonalization in the lattice (dots), from RPA (4.16), and from the exact Green’s function in the Coulomb field (4.19). (b) The self-consistent $Z_{\mathrm{eff}}$, obtained from Eq. (4.21) (403). Numerical data (dots) are plotted after accounting for finite-size renormalization of $g_c$ (326).

Figure 22

Semimetal (SM) insulator transition predicted by the renormalization group analysis of the extended Hubbard model, in large $N$ expansion. $U$ is the on-site Hubbard coupling and $V$ is the nearest neighbor site repulsion. $U_c$ stands for the critical coupling. AF: antiferromagnetic phase; CDW: charge density wave state. From 184.

Figure 23

(a) Honeycomb lattice with an impurity atom. Black: sublattice A; white: sublattice B. Intersection of the Dirac cone spectrum $E(\mathbf{k})=\pm v|\mathbf{k}|$ with the localized level $E_f={\epsilon }_0$: (b) ${\epsilon }_0>0$, (c) ${\epsilon }_0<0$.

Figure 24

Boundary between magnetic and nonmagnetic impurity states in the scaling variables $x=\Delta D/U$ and $y=(\mu -{\epsilon }_0)/U$ for (a) ${\epsilon }_0>0$ and (b) ${\epsilon }_0<0$. $|{\epsilon }_0|/D=0.029$, 0.043, 0.029 and $V/D=0.14$, 0.14, 0.04 for circles, squares, and triangles, respectively. The upturn close to $y=1$ and $x\to 0$ in (b) signals a crossover to the Fermi-liquid regime $\mu$, $U\gg |{\epsilon }_0|>0$, where the Dirac points are physically irrelevant. This feature is not visible in this scale when $V$ is very small (triangles). From 417.

Figure 25

Two adatom configurations in graphene: (a) the adatom (circle) sits on top of a carbon atom, and (b) the adatom (circle) sits at the center of the honeycomb hexagon, hybridizing equally with the two sublattices. Arrows: nearest neighbor vectors.

Figure 26

Schematic phase diagram around the Kondo QCP at half-filling: temperature vs Kondo coupling. LM: local moment phase, where the Kondo cloud is suppressed. In the critical LM phase, quantum critical fluctuations dominate. From 200.

Figure 27

Kondo coupling vs chemical potential in graphene for $U=1\mathrm{eV}$ and $V=1\mathrm{eV}$. The Kondo coupling can be controlled by gate voltage across the weak ($J\ll J_c$) and strong coupling ($J\gtrsim J_c$) Kondo regimes, where $J_c$ is the critical coupling at half-filling.

Figure 28

Superconducting order parameter ${\Delta }_{1,j}={\Delta }_1{e}^{i{\theta }_j}$ ($j=1$, 2, 3), with phases along the three different bond directions in the lattice.

Figure 29

Order parameter (OP) amplitude $|{\Delta }_{1,\mathbf{k}}|$ in the BZ: (left panel) ${\Delta }_{1,j}={\Delta }_1$ with $j=1$, 2, 3 indexing the three different bond directions of the crystal [see Eq. (5.28)] and (right panel) ${\Delta }_{1,j}=e^{i2\pi j/3}$, which describes a flux phase. Light colors represent higher amplitude. Dirac points are located at the $K$ points, at the edges of the BZ. In all dark spots, the OP has $p+ip$ symmetry around the respective high symmetry points. In the three light spots on the right panel, the OP has $s$-wave symmetry around the $K^{\prime }$ points.

Figure 30

Phase diagram between the $s$ wave and effective $p+ip$ phases in the spin singlet channel. Left: $\mu =0$ case, which is quantum critical. Right: $\mu \ne 0$ case. Continuous lines represent second order transitions, and dashed lines represent first order transitions. From 415.

Figure 31

Dependence of the gap, normalized by the band cutoff $D$, on $g$ in the weak- $(g<g_c)$ and strong-coupling $(g>g_c$) sectors for $\mu =0$, $\mu /D=0.1$ and 0.3. The value of $\mu$ decreases from the top to bottom curve.The model has a QCP at half-filling. From 414.

Figure 32

Sketch of the magnetization at the zigzag edges of a graphene ribbon.

Figure 33

Sketch of the magnetization induced near a vacancy.

Figure 34

Sketch of the magnetization induced at the edges of a quantum dot.

Figure 35

Single energy peaks and Coulomb diamonds in a graphene quantum dot. From 340.

Figure 36

Sketch of the extension of edge states in a graphene quantum dot.

Figure 37

Sketch of a graphene ribbon with disordered edges as a series of quantum dots.

Figure 38

Graphene point contact coupled to a quantum dot. From 395.

Figure 39

Sketch of the successive splittings of the Landau levels as the magnetic field is increased. (a) Spin states are split first, and then the valley degeneracy is broken. (b) Valley degeneracy is lifted first, followed by the breaking of spin degeneracy.

Figure 40

Splittings of the Landau levels in graphene as function of magnetic field. From 460.

Figure 41

Resistance of a suspended graphene sample as a function of carrier density for two different magnetic fields. From R. V. Gorbachev, D. C. Elias, A. S. Mayorov, A. A. Zhukov, K. S. Novoselov, and A. K. Geim, unpublished.

Figure 42

(a) Top view of a graphene bilayer. White and solid black circles: top layer carbon atoms; gray and thin circles around the black ones: bottom layer. (b) Four-band spectrum of the bilayer $\pm E_{\gamma }(\mathbf{p})$, with $\gamma =\pm$ as shown in Eq. (9.4), near the corner of the Brillouin zone. (c) Brillouin zone with high symmetry points. (d) Illustration of the four-band spectrum around the $K$ point.

Figure 43

The polarization ${\Pi }^{(1)}(q,\omega )$ of bilayer graphene, obtained within the two-band approximation, for finite chemical potential, and zero temperature. All panels are normalized to the DOS at the Fermi energy, $\mu$. (a) Density plot of the imaginary part and, in (b) we have cuts of the same at constant frequency, for $\omega /\mu =0.5$, 1.0, 1.5, 2.0, 2.5, 3.0. (c) The static limit ${\Pi }^{(1)}(q,0)$ in Eq. (9.12), and includes the intraband contribution (dashed), the interband contribution (dotted), and the full polarization (solid). (d) The real and imaginary parts of the polarization in the undoped case (9.14) as a function of $\nu =2m\omega /q^2$.

Figure 44

Mass renormalization for $\alpha =0.5$ in the bilayer, calculated with a static Thomas-Fermi screened Coulomb interaction $V(q)=e^2/{ϵ}_0(q+\lambda q_{\mathrm{TF}})$, as a function of the electronic density. Circles: $\lambda =1$; squares: $\lambda =0.01$; triangles: $\lambda ={10}^{-4}$. The inset shows ${log}_{10}(m^{*}/m)$ as a function of ${log}_{10}(n)$ for two of the $\lambda$ values; the mass saturates at a finite value for $n\to 0$. From 42.