Edge structure of graphene monolayers in the $\nu =0$ quantum Hall state

Monolayer graphene at neutrality in the quantum Hall regime has many competing ground states with various types of ordering. The outcome of this competition is modified by the presence of the sample boundaries. In this paper we use a Hartree-Fock treatment of the electronic correlations allowing for space-dependent ordering. The armchair edge influence is modeled by a simple perturbative effective magnetic field in valley space. We find that all phases found in the bulk of the sample, ferromagnetic, canted antiferromagnetic, charge-density wave, and Kekulé distortion, are smoothly connected to a Kekulé-distorted edge. The single-particle excitations are computed taking into account the spatial variation of the order parameters. An eventual metal-insulator transition as a function of the Zeeman energy is not simply related to the type of bulk order.

Figure 1

Shape of the kinetic energy edge potential $E_{\mathrm{kin}}$ in ${\mathcal{H}}_{\mathrm{kin}}$ of Eq. (8). The kinetic energy rises from $E_{\mathrm{kin}}=0$ in the bulk (equivalent to the case of infinitely extended, translationally invariant graphene in the $n=0$ LL) to the energy of the $n=1$ LL exactly at the edge. The curve was obtained from solving the problem of free electrons on a graphene lattice in the presence of a magnetic field in the presence of boundary, i.e., by applying appropriate boundary conditions for an armchair edge (see inset). Zigzag edges have similar nonzero energy states but also additional zero-energy modes that are beyond the scope of our work.

Figure 2

Bulk GS phase diagram as a function of the coupling energies $u_{\perp }$ and $u_z$ of the $\nu =0$ QH state for a Hamiltonian as in Eq. (7). Different colors distinguish between the different possible GS phases. The white, dotted lines indicate the parameters we choose throughout this paper to perform cuts through the phase diagram in order to explore the behavior of all possible GS phases when starting from the bulk and moving towards an edge of the graphene lattice. The phase diagram for the different GS phases in the bulk of graphene was first presented in Ref. [7].

Figure 3

Evolution of the spin and isospin as well as the concurrence $C$ as functions of $R=\sqrt{2}r/{\ell }_B$, with $r$ the distance from the edge. We fix $u_z=5E_Z$ and vary $u_{\perp }$. (Colored lines: $S_z$, solid; $T_z$, dashed; $T_x$, dotted. Black, thin, solid line: Concurrence $C$.) There is a transition between the bulk phase on the right-hand side [CAF for $u_{\perp }=-E_Z$ in the upper left panel (green colors) and F for all other choices of $u_{\perp }$ shown (blue colors)] to a KD phase at the edge. In an intermediate regime spin and isospin are canted with $0<S_z<1$ and $0<T_x<0$ at the same time and nonzero values of the concurrence. With growing $u_{\perp }$, this domain wall grows narrower in space and moves closer to the boundary.

Figure 4

Same as Fig. 3 for transverse coupling energy $u_z=-2E_Z$ and different perpendicular coupling energies $u_{\perp }$, such that the phases of the lower half plane of the GS phase diagram are established in the bulk. Left panel: $u_{\perp }=0.6$, leading to a CDW in the bulk (red colors). Right panel: $u_{\perp }=1.2E_Z$, at which the bulk is in a F phase (blue colors).

Figure 5

Maximum concurrence $C_{\max }$ in the different regimes from the bulk to the edge. For three different values of the coupling energy $u_z$, we vary $u_{\perp }$. Different symbols represent different phases in the bulk at the respective value of $u_{\perp }$: F (blue circles), CAF (green squares), CDW (red diamonds), and KD (gray triangles). Empty, shaded, or filled symbols distinguish between the cuts at different $u_z$. The dashed lines connect the data points as a guide to the eye. Vertical, gray lines mark the values of $\{u_z,u_{\perp }\}$, at which, in the bulk, transitions between the respective phases occur. We observe that nonzero values of $C$ happen only if the system in the bulk is in a CAF or in a F phase. In this case, the CAF/F transition is smooth. If the bulk is CDW or KD, the concurrence remains equal to zero all the way from the bulk to the edge. This leads to a discontinuous jump in the curve of $C_{\max }$ at the point of the CDW/F transition.

Figure 6

SP energy levels ${\varepsilon }_{\pm \pm }$ for the different phase regimes as they are obtained from the analytical formulas in Table 1. Upper left panel: CAF (solid, green lines), F (blue, dashed lines). Upper right panel: CDW for $\left|\mathrm{\Delta }\right|>E_Z$ (solid, red lines) and $\left|\mathrm{\Delta }\right|<E_Z$ (dashed, black lines). Lower left panel: KD for $\left|\mathrm{\Delta }\right|<E_Z$ with $\mathrm{\Delta }>0$ (solid, orange lines) or $\mathrm{\Delta }<0$ (dashed, black lines). Lower right panel: KD for $\left|\mathrm{\Delta }\right|>E_Z$ with $\mathrm{\Delta }>0$ (solid, orange lines) or $\mathrm{\Delta }<0$ (dashed, black lines). Depending on the sign and the magnitude of $\mathrm{\Delta }$, the different cases for the SP spectra differ in the number of level crossings, even within one and the same phase.

Figure 7

Spatial behavior of the energetic SP spectra in the presence of a boundary for positive transverse coupling $u_z=5E_Z$ and different values of the perpendicular coupling $u_{\perp }$: CAF bulk phase at $u_{\perp }=-E_Z$ in the upper left panel (green lines), F bulk phase at $u_{\perp }=-0.2E_Z, u_{\perp }=0.5E_Z$, and $u_{\perp }=1.5E_Z$ in the remaining panels, respectively (blue lines). Thick, colorful lines show our numerical results. Here, different line shapes distinguish between the different single-particle energy levels ${\varepsilon }_1\le {\varepsilon }_2\le {\varepsilon }_3\le {\varepsilon }_4$. Thin, black lines compare to the analytical formulas for ${\varepsilon }_{\pm \pm }\left(R\right)$ listed in Table 1 for the different phases, respectively, in which no modulation of the underlying spin/isospin texture towards the edge is taken into account. We see two bulk levels, being twofold degenerate and separated by a gap of width $\mathrm{\Delta }{\varepsilon }_{bulk}$ in the bulk, split into four branches when approaching the edge. The branches exhibit kinks and regimes of different behavior corresponding to the transitions between different spin and isospin phases during the evolution from the bulk to the edge (see Fig. 3). The edge gap between the two intermediate levels ${\varepsilon }_2$ and ${\varepsilon }_3$ (dotted and dashed lines, respectively) $\mathrm{\Delta }{\varepsilon }_{\mathrm{edge}}=\text{min}\left(\right|{\varepsilon }_3-{\varepsilon }_2\left|\right]$ remains finite over a certain range of $u_{\perp }$, reducing gradually as $u_{\perp }$ increases until it finally closes completely. The lower right panel shows a configuration were the levels cross and form gapless edge states. Special attention should be paid to the upper right panel and the lower left panel in which the numerical results show configurations with a F phase in the bulk were finite edge gaps $\mathrm{\Delta }{\varepsilon }_{\mathrm{edge}}\ne 0$ remain, whereas the analytical curves cross as soon as the bulk passes into an F phase. The behavior of the $\mathrm{\Delta }{\varepsilon }_{\mathrm{edge}}$ as a function of $u_{\perp }$ is studied further in Fig. 9.

Figure 8

Same as Fig. 7 for $u_z=-2E_Z$ and different $u_{\perp }$, hence displaying the SP spectra for the bulk phase regimes of the lower half plane of the GS phase diagram. Left panel: CDW bulk phase at $u_{\perp }=-0.6E_Z$ (red lines). Right panel: F bulk phase at $u_{\perp }=-1.3E_Z$ (blue lines). Thin, black lines compare to the analytical formulas given in Table 1 for the different phases, respectively, in which the evolution of the bulk's phase towards the edge is neglected. For the CDW wave phase in the left panel we observe four nondegenerate SP levels in the bulk; furthermore, as these levels do not bend towards each other but disperse when approaching the edge, the minimum energy gap to SP excitations is given by the bulk gap $\mathrm{\Delta }{\varepsilon }_{\mathrm{edge}}=\mathrm{\Delta }{\varepsilon }_{bulk}$.

Figure 9

Behavior of the gap $\mathrm{\Delta }{\varepsilon }_{\mathrm{edge}}$ in the SP spectra close to the edge. We use three couplings: $u_z=-2E_Z, u_z=2E_Z$, and $u_z=5E_Z$ (filled, shaded, and empty symbols, respectively). Different colors and symbols stand for different bulk phases: KD (gray triangles), CDW (red diamonds), CAF (green squares), and F phase (blue circles). Dashed, colored lines connect the data points as a guide to the eye. The dotted, black lines represent the behavior of the data in the linear regimes (they are shifted by a constant offset with respect to the curves for better visibility). Gray vertical lines indicate the critical values for bulk phase transitions. For all phases the gap $\mathrm{\Delta }{\varepsilon }_{\mathrm{edge}}$ monotonically decreases as $u_{\perp }$ grows until it finally closes in the regime where the bulk is in an F phase. The inset is a close-up on how the blue lines smoothly approach $\mathrm{\Delta }\varepsilon =0$. The transitions from $\mathrm{\Delta }{\varepsilon }_{\mathrm{edge}}\ne 0$ to $\mathrm{\Delta }{\varepsilon }_{\mathrm{edge}}=0$ take place at $u_{\perp }\approx 0.3E_Z, u_{\perp }\approx E_Z, u_{\perp }\approx 1.625E_Z$.

Figure 10

Close-up on the SP spectra in the regime with multiple crossings for $u_z=2E_Z$. Left side: $u_{\perp }=1.2E_Z$, right side: $u_{\perp }=3E_Z$. The blue lines show the SP energy levels ${\varepsilon }_i$. The different background colors mark the spatial regimes in which the GS establishes different spin/isospin textures: there is a F phase (blue region). In the intermediate (white) region, the system undergoes a transition before it finally ends up in a KD phase (yellow region). We observe the crossings of the SP levels to occur in regions of different spin/isospin phases—their origins lie in the different symmetries of the F and KD phases.

Figure 11

Evolution of the single-electron spin and isospin components $s_z\left(i\right)$ and $t_x\left(i\right)$ of the SP eigenstates from the bulk towards the edge for the two anisotropy energies $u_{\perp }=-E_Z$ (CAF bulk GS, left panels with green lines) and $u_{\perp }=-0.2E_Z$ (F bulk GS, right panels with blue lines). Different line shapes distinguish between the four SP energy levels ${\varepsilon }_1\le {\varepsilon }_2\le {\varepsilon }_3\le {\varepsilon }_4$. Green/blue lines correspond to the observables of the two lowest-lying states which are occupied orbitals in the HF GS. Gray lines indicate the behavior of the higher-lying SP states. Arrows show the behavior of the spin and isospin polarizations. The second and third levels $|2\rangle$ and $|3\rangle$ are oppositely polarized in spin and isospin at the edge. In all plots we set $u_z=5E_Z$.