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116 CHAPTER 5. THE FLUX PHASE FOR THE CUPRATES
be considered simultaneously leading to the following MF hamiltonian,
H MF = −t ∑ 〈ij〉σ
g t ij(c † i,σ c j,σ + h.c.)+ ∑ iσ
ɛ i n i,σ
− 3 4 J ∑ 〈ij〉σ
g J i,j (χ jic † i,σ c j,σ + h.c. −|χ ij | 2 ) (5.4)
− 3 4 J ∑ 〈ij〉σ
g J i,j (∆ jic † i,σ c† j,−σ + h.c. −|∆ ij| 2 ),
where the previous Gutzwiller weights have been expressed in terms of local
fugacities z i =2x i /(1 + x i )(x i is the local hole density 1 −〈n i 〉), g t i,j = √ z i z j and
g J i,j =(2− z i)(2 − z j ), to allow for small non-uniform charge modulations [130].
The Bogoliubov-de Gennes self-consistency conditions are implemented as χ ji =
〈c † j,σ c i,σ〉 and ∆ ji = 〈c j,−σ c i,σ 〉 = 〈c i,−σ c j,σ 〉.
In principle, this MF treatment allows for a description of modulated phases
with coexisting superconducting order, namely supersolid phases. Previous investigations
[126] failed to stabilize such phases in the case of the pure 2D square
lattice (i.e. defect-free). Moreover, in this Section, we will restrict ourselves to
∆ ij = 0. The case where both ∆ ij and χ ij are non-zero is left for a future work,
where the effect of a defect, such as for instance a vortex, will be studied.
In the case of finite V 0 ,theon-sitetermsɛ i may vary spatially as −µ+e i ,where
µ is the chemical potential and e i are on-site energies which are self-consistently
given by,
e i = ∑ 〈 〉
V i,j nj . (5.5)
j≠i
In that case, a constant ∑ i≠j V i,j(〈n i 〉〈n j 〉 + n 2 ) has to be added to the MF
energy. Note that we assume here a fixed chemical potential µ. In a recent work
[131], additional degrees of freedom where assumed (for V 0 = 0) implementing an
unconstrained minimization with respect to the on-site fugacities. However, we
believe that the energy gain is too small to be really conclusive (certainly below
the accuracy one can expect from such a simple MF approach). We argue that we
can safely neglect the spatial variation of µ in first approximation, and this will
be confirmed by the more accurate VMC calculations in Section 5.4. Incidently,
Ref. [131] emphasizes a deep connection between the stability of checkerboard
structures [126] and the instability of the SFP due to nesting properties 4 of some
parts of its Fermi surface [132].
4 Note that the Fermi surface of the SFP is made of four small elliptic-like pockets centered
around (±π/2, ±π/2)