Dynamics in the quantum Hall effect and the phase diagram of graphene

The dynamics responsible for lifting the degeneracy of the Landau levels in the quantum Hall (QH) effect in graphene is studied by utilizing a low-energy effective model with a contact interaction. A detailed analysis of the solutions of the gap equation for Dirac quasiparticles is performed at both zero and nonzero temperatures. The characteristic feature of the solutions is that the order parameters connected with the QH ferromagnetism and magnetic catalysis scenarios necessarily coexist. The solutions reproduce correctly the experimentally observed QH plateaus in graphene in strong magnetic fields. The phase diagram of this system in the plane of temperature and electron chemical potential is analyzed. The phase transitions corresponding to the transitions between different QH plateaus in graphene are described.

Figure 1

The diagrammatic form of the gap equation in the Hartree-Fock (mean-field) approximation. The upper (lower) diagram corresponds to the form of the gap equation with the long-range Coulomb (contact) interaction. The indices denote quasiparticle spins.

Figure 2

(Color online) Free-energy density versus the electron chemical potential ${\mu }_0$ for several different solutions found analytically (top panel) and numerically (bottom panel) in a range of ${\mu }_0$ relevant to the dynamics in the lowest Landau level. The numerical results are shown for a nonzero but small temperature, $T=1\text{K}$. The values of the electron chemical potential are given in units of the Landau energy scale ${ϵ}_B$, and the free-energy densities are given in units of ${ϵ}_B/l^2$, where $l=\sqrt{\hslash c/\left|e{B}_{\perp }\right|}$ is the magnetic length.

Figure 3

(Color online) Temperature dependence of the nontrivial order parameters in the $\nu =0$ QH state described by the $S1$ solution. The results in a model with a vanishing Zeeman energy $\left(Z=0\right)$ are shown in the top panel, and the results in a realistic model with a nonzero Zeeman energy $\left(Z\ne 0\right)$ are shown in the bottom panel. Note that ${\tilde{\mu }}_{\pm }={\tilde{\Delta }}_{\pm }=0$ in both cases. The values of the temperature and the order parameters are given in units of the dynamical scale $M$.

Figure 4

(Color online) Order parameters for the singlet solution $S1/S2$ as functions of the electron chemical potential ${\mu }_0$ for several different values of temperature.

Figure 5

(Color online) Order parameters for solution $H1$ as functions of the electron chemical potential ${\mu }_0$ for several different values of temperature.

Figure 6

(Color online) Nontrivial order parameters of metastable solutions $H2$ (top panel) and $S3$ (bottom panel) as functions of the electron chemical potential ${\mu }_0$. In calculation, the temperature is taken nonzero but small, $T=1\text{K}$. The values of the electron chemical potential are given in units of the Landau energy scale ${ϵ}_B$, while the order parameters are given in units of the dynamical scale $M$.

Figure 7

(Color online) Nontrivial order parameters of the $S$-type numerical solution that contains the analytical solutions (f-i), (f-iii), and (f-v) as parts, connected by two intermediate solutions.

Figure 8

(Color online) Nontrivial order parameters of the extended hybrid solution (f-ii) which determines the ground state for the $\nu =3$ QH plateau in graphene.

Figure 9

(Color online) Nontrivial order parameters of the extended hybrid solution (f-iv) which determines the ground state for the $\nu =5$ QH plateau in graphene.

Figure 10

(Color online) The difference between the free-energy density of three hybrid-type solutions and the free-energy density of the $S$-type solution in the range of ${\mu }_0$, associated with the dynamics of the $n=1$ LL. In calculations, the temperature is taken nonzero but small, $T=1\text{K}$. The values of the free-energy density and the electron chemical potential are given in the same units as in Fig 2.

Figure 11

(Color online) Schematic phase diagram of graphene in the plane of temperature and electron chemical potential. The values of chemical potential are given in units of the Landau energy scale ${ϵ}_B$, and the values of temperature are given in units of the dynamical scale $M$.