Phase diagram for the $\nu =0$ quantum Hall state in monolayer graphene

The $\nu =0$ quantum Hall state in a defect-free graphene sample is studied within the framework of quantum Hall ferromagnetism. We perform a systematic analysis of the “isospin” anisotropies, which arise from the valley and sublattice asymmetric short-range electron-electron (e-e) and electron-phonon (e-ph) interactions. The phase diagram, obtained in the presence of generic isospin anisotropy and the Zeeman effect, consists of four phases characterized by the following orders: spin-polarized ferromagnetic, canted antiferromagnetic, charge density wave, and Kekulé distortion. We take into account the Landau level mixing effects and show that they result in the key renormalizations of parameters. First, the absolute values of the anisotropy energies become greatly enhanced and can significantly exceed the Zeeman energy. Second, the signs of the anisotropy energies due to e-e interactions can change upon renormalization. A crucial consequence of the latter is that the short-range e-e interactions alone could favor any state on the phase diagram, depending on the details of interactions at the lattice scale. On the other hand, the leading e-ph interactions always favor the Kekulé distortion order. The possibility of inducing phase transitions by tilting the magnetic field is discussed.

Figure 1

Landau levels (LLs) [Eq. (1)] in monolayer graphene. Neglecting the Zeeman effect, the orbitals (i.e., eigenstates of the orbital part of the single-particle Hamiltonian) of each LL are fourfold degenerate due to two projections of spin and two valleys (splitting shown for illustration purpose and not implied). At $\nu =0$ filling factor, each orbital of the $n=0$ LL is occupied on average by two electrons, while LLs with $n<0$ ($n>0$) are filled (empty). The many-body states with identical occupation of the $K{K}^{\prime }\otimes s$ isospin-spin subspaces of all orbitals (only one possible occupation is shown) exactly minimize the energy of the Coulomb interactions, SU(4) symmetric in the $K{K}^{\prime }\otimes s$ space. Such states form the family of degenerate ground states of the $\nu =0$ quantum Hall ferromagnet.

Figure 2

Graphene honeycomb lattice and the structure of the wave functions of the $n=0$ LL at the lattice scale. In each valley, $K$ or $K^{\prime }$, the wave functions reside on only one sublattice, $A$ (left) or $B$ (right).

Figure 3

In-plane optical phonon modes with the strongest e-ph coupling. (left) Linear combination of the two degenerate $E_2$ modes ${\widehat{u}}_{x,y}\left(\mathbf{r}\right)$ with the phonon wave vector at $\Gamma$ point. (right) Linear combination of degenerate $A_1$, $B_1$ modes ${\widehat{u}}_{a,b}\left(\mathbf{r}\right)$ with the wave vector at $K,K^{\prime }$ points.

Figure 4

Diagrammatic representation of the terms in the energy functional (44) of the $\nu =0$ QHFM, defining the bare (lowest order in interactions) values of parameters. (a) Diagram for the gradient term ${\mathcal{E}}_{\circ }\left(P\right)$ [Eq. (45)], determining the bare stiffness ${\rho }_s^{\left(0\right)}$ [Eq. (49)]; the wavy line stands for the Coulomb interaction [Eq. (31)]. (b) and (c) Diagrams for the isospin anisotropy term ${\mathcal{E}}_{\diamond }\left(P\right)$ [Eqs. (47) and (48)], determining the bare anisotropy energies $u_{\perp ,z}^{\left(0\right)}$ [Eqs. (50, 51, 52)]. The dashed line represents either the short-range e-e [Eq. (32)] or e-ph [Eq. (33)] interactions. The Hartree (b) and Fock (c) contributions produce the first and second terms in Eq. (48), respectively.

Figure 5

Diagrams for the energy functional (44), second order in the Coulomb interactions (wavy lines). Diagram (a0) represents the first-order large-$N$ correction to the Coulomb propagator. Diagrams (a1)–(a4) diverge logarithmically, but cancel each other within the logarithmic accuracy.

Figure 6

Diagrams for the energy functional (44), first order in the Coulomb interactions (wavy line) and in either the short-range e-e or e-ph interactions (dashed line). Diagrams (b1)–(b4) diverge logarithmically and represent the lowest-order correction to the Hartree contribution [Fig. 4b] to the anisotropy energy.

Figure 7

Same as in Fig. 6, but with respect to the Fock contribution [Fig. 4c].

Figure 8

The function $F\left(w\right)$ [Eq. (68)] of the Coulomb coupling constant $w$ [Eq. (69)] describes the renormalizations of the Dirac velocity $v\left(l\right)$, $w\left(l\right)$ itself, and the short-range e-e and e-ph couplings.

Figure 9

Renormalization of the Coulomb coupling constant $w\left(l\right)$ [Eq. (69)]. The bare value $w\left(a\right)=3.4$ was used.

Figure 10

RG flows [Eqs. (71, 72, 73)] of the coupling constants ${\overline{g}}_{\alpha \perp }\left(l\right)$ and ${\overline{g}}_{\alpha z}\left(l\right)$ of the short-range electron-electron interactions (11).

Figure 11

The functions ${\mathcal{F}}_{\mathrm{e}-\mathrm{e}}(l_B/a,w)$ and ${\mathcal{F}}_{\mathrm{e}-\mathrm{ph}}(l_B/a,w)$ [Eqs. (73) and (79)], characterizing the strength of renormalizations of the isospin anisotropy energies $u_{\alpha }^{(\mathrm{e}-\mathrm{e})}$ and $u_{\alpha }^{(\mathrm{e}-\mathrm{ph})}$ [Eqs. (80) and (81)] due to short-range e-e and e-ph interactions. The value $w=3.4$ for the bare Coulomb coupling [Eq. (69)] was used. In a typical experimental range $l_B/a\sim 10\text{--}100$, the anisotropy energies are enhanced by about one order compared to their bare values $u_{\alpha }^{(\mathrm{e}-\mathrm{e},0)}$ and $u_{\alpha }^{(\mathrm{e}-\mathrm{ph},0)}$ [Eqs. (50, 51, 52, 53), and (82)].

Figure 12

Charge-density-wave (CDW) phase of the $\nu =0$ QHFM.

Figure 13

Kekulé-distortion (KD) phase of the $\nu =0$ QHFM.

Figure 14

Spin-polarized ferromagnetic (F) phase of the $\nu =0$ QHFM.

Figure 15

Antiferromagnetic (AF) phase of the $\nu =0$ QHFM.

Figure 16

Phase diagram of the $\nu =0$ QHFM states minimizing the isospin anisotropy energy ${\mathcal{E}}_{\diamond }\left(P\right)$ [Eqs. (47) and (48)], in the space of the anisotropy energies $(u_{\perp },u_z)$. Physical orders of the phases are shown in Figs. 12, 13, 14, 15.

Figure 17

Canted antiferromagnetic (CAF) phase of the $\nu =0$ QHFM.

Figure 18

Phase diagram of the $\nu =0$ quantum Hall ferromagnet in monolayer graphene in the presence of the isospin anisotropy and Zeeman effect, in the space of the anisotropy energies $(u_{\perp },u_z)$ and with the Zeeman energy ${\epsilon }_Z$ as parameter. Physical orders of the phases are shown in Figs. 12, 13, 14, and 17. Top graph shows the optimal value $s_z^{*}$ [Eq. (104)] of the spin projection on the direction of the magnetic field in the CAF and F phases.

Figure 19

Phase transitions induced by tilting the magnetic field, which increases the Zeeman energy ${\epsilon }_Z$ relative to the anisotropy energies $u_{\perp ,z}$. The transitions from the CAF (point $a$) or CDW (point $c$) to the F phase occur directly, while the transition from the KD (point $b$) to the F phase can occur only through the CAF phase.