Hierarchy of spin and valley symmetry breaking in quantum Hall single-layer graphene

Several microscopic mechanisms for breaking the fourfold degeneracy of the central Landau level in a single-layer graphene is examined. We consider valley-scattering random potential, Zeeman interaction, and electron-phonon coupling as potential candidates to break the SU(4) symmetry of the Coulomb exchange interaction. A phase diagram is worked out, giving either the “spin-first” or “valley-first” splitting of the central Landau levels depending on the relative strengths of the electron-phonon coupling and the Zeeman interaction. Existence of a midgap band at the domain wall separating Landau levels of opposite valley polarity is demonstrated.

Figure 1

(Color online) Plot of the energy levels at half filling obtained from self-consistent HF calculation. The $x$ axis represents the guiding center coordinate $m$ which takes values from 1 to $N_{\varphi }$. The Hamiltonian under consideration is (from left to right) $H^{\text{EX}}$ (black solid lines), $H^{\text{EX}}+H^{\text{B}}$ (red dash lines), $H^{\text{EX}}+H^{\text{imp}}$ (magenta crosses), and $H^{\text{EX}}+H^{\text{B}}+H^{\text{imp}}$ (blue filled squares). The impurity broadening is $W=0.1$ and the Zeeman field is $B_{\sigma }=0.08$. The system size used is $N_{\varphi }=50$.

Figure 2

(Color online) (a) Phase diagram of the central LL at half filling with varying Zeeman field $\left(2B_{\sigma }\right)$ and electron-phonon coupling strength $\left(U\right)$. In region I $\left(2B_{\sigma }\gtrsim U\right)$, only the spin degeneracy is broken while the valley symmetry remains preserved. In region II $\left(U\gtrsim 2B_{\sigma }\right)$, both spin and valley symmetries are lost but the main level splitting takes place in the valley direction. A first-order phase boundary separates the two regions. (b) Schematic LL energy levels in each region, where the index $\pm {\tau }_x$ refers to the two valley eigenstates polarized along the $x$ direction. The energy-level separations and spin-valley quantum numbers for each LL are specified.

Figure 3

(Color online) Phase diagram of the central LL at quarter filling. Meaning of the symbols are the same as in Fig. 2. Only one LL lies below the Fermi level here.

Figure 4

(Color online) (a) The energy ${\epsilon }_k$ obtained from Eq. (14) with $m_{\text{v}}=1.0$ and (b) the energy spectra in the cases of armchair and zigzag boundaries with $m_{\text{v}}=1.0$.