Bosonic field theory of tunable edge magnetism in graphene

A bosonic field theory is derived for the tunable edge magnetism at graphene zigzag edges. The derivation starts from an effective fermionic theory for the interacting graphene edge states, derived previously from a two-dimensional Hubbard model for graphene. The essential feature of this effective model, which gives rise to the weak edge magnetism, is the momentum-dependent nonlocal electron-electron interaction. It is shown that this momentum dependence may be treated by an extension of the bosonization technique and leads to interactions of the bosonic fields. These interactions are reminiscent of a ${\phi }^4$ field theory. Focusing on the regime close to the quantum phase transition between the ferromagnetic and the paramagnetic Luttinger liquid, a semiclassical interpretation of the interacting bosonic theory is given. Furthermore, it is argued that the universal critical behavior at the quantum phase transition between the paramagnetic and the ferromagnetic Luttinger liquid is governed by a small number of terms in this theory, which are accessible by quantum Monte Carlo methods.

Figure 1

Left: The linearized dispersion, described by $H_0$. The dashed line indicates the original cosine dispersion of direct model derived in Ref. 8. For ${\Gamma }_1>0$, the electrons at high energy are weakly interacting while the low-energy sector is strongly interacting. The momenta of the left- and right-movers are centered around their respective Fermi momenta. Right: The allowed interaction processes $g_1$ (backscattering), $g_2,g_4$ (forward scattering). Umklapp processes are forbidden for the edge states (see text).

Figure 2

Spin polarization $m$ from the fermionic mean-field theory. The different $m$ curves correspond to different momentum dependencies ${\Gamma }_1=0,...,\frac{6}{\pi }$.