Spin and valley skyrmions in Landau levels $\left\vert N\right\vert \geq 1$ of graphene and bilayer graphene

In a two-dimensional electron gas (2DEG), skyrmions are the lowest-energy charged excitations at filling factor $\nu =1$ while, for the chiral 2DEG in graphene, valley and spin skyrmions can exist up to Landau level $\left\vert N\right\vert =3$ $[1]$. In this talk, we discuss the excitation energy of spin and valley skyrmions in Landau level $\left\vert N\right\vert \geq 1$ in both graphene and bilayer graphene. In graphene, we consider a finite Zeeman term in order to compute the range of Zeeman coupling for which skyrmions are the lowest-energy charged excitations. We also show how the excitation energy is modified when screening is considered $[2]$. In bilayer graphene, we first derive the phase diagram of the chiral 2DEG at integer filling of the quartet of states in Landau levels $\left\vert N\right\vert \geq 1$ and show how a finite potential difference applied between the two layers can control the spin and pseudospin polarizations. We then compute the excitation energy of valley and spin skyrmions by using an anisotropic $\sigma $ model derived from the Hartree-Fock Hamiltonian and adding screening corrections.\\[4pt] [1] Kun Yang, S. Das Sarma, and A. H. MacDonald, Phys. Rev. B 74,075423 (2006).\\[0pt] [2] Wenchen Luo and R. Cote, Phys. Rev. B 88, 115417(2013).