SU(4) spin waves in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>?</mml:mi><mml:mo>=</mml:mo><mml:mo>±</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> quantum Hall ferromagnet in graphene

The lowest energy excitations of a quantum Hall ferromagnet are generalized SU(4) spin waves that can be real spin, valley pseudospin waves, or mixed spin-valley (see image). In this paper, the dispersion relations these different spin-wave types investigated based on phase diagram at filling factor $\ensuremath{\nu}=\ifmmode\pm\else\textpm\fi{}1$. three four phases, there one more linearly dispersing modes low energy, due to remaining symmetry in residual anisotropies.