Fig. 1: Brillouin zone showing fermi surface nesting. 
The model considered here consists three parts as follows:
Here H_{it} is an itinerant band model with a Fermi surface consisting of one electron pocket at (0,0) and hole pockets at (0,π) and (π,0) (see Fig. 1). [1] A nesting vector Q = (π,0) exists between two kinds of Fermi pocket. This Fermi pocket characteristic is thought to be an important feature of the band structure of ironbased superconductors. H_{J2} is a next nearest neighbor antiferromagnetic Quantum Heisenberg Model, and H_{J0} is a Hund'slike ferromagnetic coupling between the local and itinerant parts.
This model is proposed by Kou et al. as a trial model for ironbased superconductors. [2] The issue is whether the electrons in these materials are itinerant or localized in nature.
The local and itinerant dynamics are coupled to one another by H_{J0}. At mean field level, we can think of either part as being in a selfconsistent magnetic field established by the other. In such sense, we can separate H into two parts describing either parts respectively:
Here p_{i} = exp(i Q·r_{i}) is a stripelike staggering factor induced by the nesting vector. The selfconsistent fields (order parameters) are taken as stripe antiferromagneticlike, which is a magnetic order observed in quite a few ironbased superconductors.
The two order parameters are taken as parallel to each other on every site, since  J_{0} < 0 prefers a ferromagnetic configuration.
The subscript A on the local part indicates that J_{2} couples to nextnearestneighbor sites, so that in the mean field approximation all sites separate into two identical and independent sublattices A and B (two checkboard components of the original lattice). The dynamics of the two parts are exactly the same, so we only take care of the part.
An exact diagnalization of the itinerant part can be done via Bogoliubov transformation. As for the local part, further approximation is needed. Ref. [1] employed the NonLinear Sigma Model, while here we do a calculation using a different technique, the Schwinger Boson mean field method [3].
After diagnalizing both parts, we get selfconsistent equations of the order parameters, which, in turn, determine the phase transition point. We find that the magnetic orders of the two parts enhance each other through the coupling term.
Here BZA is the first Brillouin zone of sublattice A. This can be diagonalized by a canonical Bogoliubov transformation, giving the result
Since the staggered field ⟨S_{i}⟩ will induce antiferromagnetic local spin order, we should do an antiferromagnetic Schwinger Boson transformation:
Here SA and SB are further sublattices ("subsublattice") of A as a bipartite itself (see below), and a_{i} and b_{i} are Boson operators.


Fig. 2: Illustration of symmetry breaking patterns showing interpenetrating order parameters A and B. 
Under such representation the Hamiltonian is written as (neglecting additive constants)
Here
is the socalled bond operator. The λ term is Lagrange multiplier for the constraint
Notice that the staggering factor is eliminated by p_{i}^{2} = 1.
The Schwinger Boson mean field approximation takes an order parameter
giving the mean field Hamiltonian
This Hamiltonian can again be diagnalized by a Bosonic Bogoliubov transformation. The result is
Here
BZA' is the Brillouin zone of sublattice reduced by (π,0).
© 2010 Xiao Ge. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] S. Raghu and S. C. Zhang, Phys. Rev. B 77, 220503(R) (2008).
[2] S. P. Kou, T. Li and Z.Y.Weng, ArXiv: 0811.4111.
[3] A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer, 1994).