Coulomb interactions, Dirac sea polarization, and $SU\left(4\right)$ symmetry breaking of the integer quantum Hall states of graphene

We investigate effects of the filled Dirac sea on the $SU\left(4\right)$ symmetry breaking for the integer quantum Hall states of graphene with long-ranged electrostatic Coulomb interactions. Our model also includes Hubbard and nearest-neighbor repulsive interactions with strengths $U$ and $V$, respectively. We find the symmetry breaking of $n=0$ Landau level is accompanied by an $SU\left(4\right)$ polarization of the filled Dirac sea. We compute the phase diagram for filling factor $\nu =0,\pm 1$ in the $U\text{−}V$ space at magnetic fields relevant to experiments. The Dirac sea polarization significantly influences the phase diagram. It suppresses the Kekulé ordering at $\nu =0$ and induces it at $\nu =\pm 1$. We also calculate the excitation gaps at small tilted magnetic fields. This enables us to conclude that the ground state is a charge density wave at $\nu =0$ and valley-spin polarized at $\nu =\pm 1$. We delineate a region in the $U\text{−}V$ space consistent with the experiments.

Figure 1

This figure shows the ground-state energy density as a function of the variational parameter $m$ for $B=20$ T and dielectric constants ${\varepsilon }_r=1,2,4$. Here, we are considering the case where graphene is placed on a substrate with dielectric constant ${\epsilon }_r$. The minima of the energy density function decrease with increasing ${\varepsilon }_r$. The curves have been adjusted to align the minima along the $m$ axis.

Figure 2

This figure shows the variation of activation gap with magnetic field. The magnitude of gaps at $\nu =0$ and $-1$ are same. The solid points are values for the gap obtained from our calculations. The solid line shows the best fit for the gaps as function of magnetic field of the form $a\sqrt{B}+bB$. $a,b$ are best-fit parameters shown in the inset for three dielectric constants. We have shown the variation of gaps with change in the dielectric constant of the substrate. The dielectric constant ${\varepsilon }_r=1$ corresponds to the suspended graphene. The best fit showed the dominant $\sqrt{B}$ contribution and linear $B$ decreases as the dielectric constant of substrate increased. The dashed lines show the gaps when the effects of the filled Dirac sea are ignored.

Figure 3

The figure shows the phase diagram for possible ground states at $\nu =0$ in $U\text{−}V$ parameter space. This phase diagram is for the magnetic field $B=30$ T and ${\varepsilon }_r=1.0$. The region marked FM is the ferromagnetic ordered state, CDW is the charge ordered state, and Canted is the canted spin ordered state. The three regions marked with dots represent solutions obtained with numerical minimization corresponding to three phases. The bold line separating the three phases was obtained analytically after equating each pair of analytical expressions for the variational energies. Equations for these lines are given in Eqs. (86), (89), and (88). Phase transition across the FM and Canted regions is continuous, whereas others are first-order transition.

Figure 4

One of the arrangements of $SU\left(4\right)$ components of the $n=0$ Landau levels for the doubly degenerate charge ordered ground state shown in (a), antiparallel spins located on same sublattice. The site order parameter for charge ordering shown in (b). Two colors represent the staggered charge densities on two sublattices.

Figure 5

The $SU\left(4\right)$ components of the $n=0$ Landau level for the ferromagnetic ground states is shown in (a). The spin pointing along the direction of the magnetic field on each sub-lattice shown with different colors. The site order parameter for the ferromagnetic state, the total spin shown in (b). The small circle size indicates that this site order parameter magnitude comes from the half-filled $n=0$ Landau level quartet.

Figure 6

The $SU\left(4\right)$ components of the $n=0$ Landau level for the canted ground state shown in (a). The spins on sub-lattice shown in different colors with canted angle $2{\theta }_0$ between them. The site order parameter in (b), the total spin along the direction of the magnetic field on each sublattice which has contributions on from the occupied quartets of the $n=0$ Landau levels shown with small circles. (c) Shows the order parameter corresponding to Néel ordering in the direction perpendicular to the magnetic field. Larger circle size compared to order parameter in (b) is to indicate contributions from filled Dirac sea to this site order parameter.

Figure 7

(a) Shows the phase diagram for $\nu =0$ at $B=10$ T from the variational state analysis of our interacting model. (b) Shows phase diagram when the effects of filled Dirac sea are ignored. The lines separating the phase transition does not change with magnetic field in (b).

Figure 8

The figure shows the phase diagram for the ground state at $\nu =-1$ in $U\text{−}V$ parameter space. The phase diagram shown here is for magnetic field $B=30$ T and ${\varepsilon }_r=1.0$. The red dots marking the region VSP correspond to the area in parameter space where the numerical minimization yielded angle parameters for a valley-spin polarized state. The region marked KO corresponds where the angle parameters result in Kekulé ordered state. The bold line separating the two phases is obtained by equating the energies of these two phases and equation for this line is given in Eq. (99). Phase transition across this line is continuous.

Figure 9

(a) Shows the $SU\left(4\right)$ component of $n=0$ Landau level for one of the two degenerate ground states for valley-spin polarized ground state. (b) The lattice manifestation of the site order parameter for the total spin along the direction of the magnetic field. (c) Represents the site order parameter which is the staggered charge density on the sublattice points. (d) Shows the Néel ordering of the spins along the direction of the magnetic field for the valley-spin polarized ground state.

Figure 10

(a) Shows the arrangement of $SU\left(4\right)$ components of the occupied quartet of the $n=0$ Landau level for the valley-spin canted ground state. The colors of arrows indicate spins localized on the two sublattice points and different lengths represent unequal weights on the sublattice points. The site order parameter on the lattice for the valley-spin canted ground state is same as for the valley-spin polarized ground state, shown in Figs. 9–9. (b) Shows the bond order parameters which result in Kekulé ordering that breaks the lattice translation symmetry.

Figure 11

(a) Shows the region for all excitations from the $\nu =0$ ground state at $B_{\perp }=10$ T and ${\epsilon }_r=1.0$. (b) Shows the variation of particle-hole gaps with respect to $B_T$ (tilt angle $<{45}^{\circ }$) for all the three phases for the Hall state at $\nu =0$. The gap for ferromagnetic spin ordered state, labeled $E_1$, increases linearly with $B_T$, whereas gap for the charge ordered state, labeled $E_2$, decreases linearly. The gap for canted spin phase, labeled $E_3$, shows an increase with $B_T$ with a quadratic dependence. In the canted phase, the gap is inversely proportional to the Hubbard interaction strength $U$. In the figure we have chosen $U=2$ eV for the shown curve.

Figure 12

(a) Shows the excitations described in the main text for the ground states at $\nu =-1$ at $B_{\perp }=10$ T and ${\epsilon }_r=1.0$ in the $U\text{−}V$ parameter space. (b) Shows the variation of gaps with $B_T$ (tilt angle $<{45}^{\circ }$). The excitations ${\tilde{E}}_1,{\tilde{E}}_2$, and ${\tilde{E}}_3$ are for the valley-spin polarized state and ${\tilde{E}}_4$ and ${\tilde{E}}_5$ for the valley-spin canted state. The excitations ${\tilde{E}}_2$ and ${\tilde{E}}_3$ of the valley-spin polarized state have a linear variation with $B_T$ whereas ${\tilde{E}}_1$ has none. The excitation ${\tilde{E}}_5$ of the valley-spin canted state has a quadratic dependence on $B_T$ whereas ${\tilde{E}}_4$ has competing increasing quadratic and decreasing linear $B_T$ dependence. For small tilt angles, the decreasing linear $B_T$ dependence dominates as shown in (b).

Figure 13

The figure shows the delineated region in $U\text{−}V$ parameter space that is consistent with behavior of gaps at $\nu =0$ and $-1$ discussed in the main text. This is a representative plot shown for $B_{\perp }=10$ T and ${\varepsilon }_r=1.0$. For the values of $U$ and $V$ marked by the shaded region results in a charge ordered ground state at $\nu =0$ and valley-spin polarized ground state at $\nu =\pm 1$.

Figure 14

This figure illustrates the effects of filled Dirac sea on the ground phases due to electron-phonon interactions for the numerical experiment described in Eq. (114). (a) Shows the result when the effects of filled Dirac sea is ignored, i.e., calculations restricted to $n=0$ Landau level. This reproduces the results shown in Ref. [22], when the $u_z=0$ [22]. (b) Shows the results when the Dirac sea polarization is taken into account. In the absence of Zeeman term, CDW state has lower energy when compared to Kekulé order (KO) state. In presence of Zeeman term, the canted spin state has lower energy even for large values of ${\kappa }_P$.