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4.3. MODEL AND METHODS 87
comb lattice with N =72andN = 144 sites. The N = 72 cluster is defined
by the translation vector T 1 = nb 1 + nb 2 (b i are defined in Fig. 5.2) and the
vector T 2 = nb 1 + mb 2 orthogonal to T 1 . We have used anti-periodic boundary
conditions along T 1 and periodic ones along T 2 . This allows to reduce the finite
size effect. The N = 144 cluster uses 2T 1 and 2T 2 .
4.3.2 Variational wavefunctions
Next we introduce the variational subspace that we use to study the ground state
of the t−J model. Our variational wavefunctions are built from the ground-states
|ψ MF 〉 of the mean-field like Hamiltonian 1.23. The set of variational parameters
is given by χ ij allowing anisotropic nearest neighbor hopping, when χ ij = |χ ij |e iθ ij
is complex, it is breaking time-reversal symmetry leading to a so-called flux phase,
∆ ij the singlet BCS pairing, and h i responsible for SDW instabilities. In general
|ψ MF 〉 is not a state of fixed number of particles due to the presence of the BCS
pairing. In VMC simulations it is however necessary to work with states of fixed
particle number and we therefore apply the projector P N which projects on the
subspace of N electrons. Further, since the t − J model allows at most one
fermion per site, we discard also all configurations with doubly occupied sites by
applying the complete Gutzwiller projector P G = ∏ i (1 − n i↑n i↓ ). To summarize,
our variational wavefunction is written as
|ψ var 〉 = P G P N |ψ MF 〉 . (4.1)
Now we give a list of all the variational states which we found to be relevant in
the honeycomb lattice.
• FS: The Gutzwiller projected Fermi sea is our reference state, i.e. t ij ≡ 1,
all other parameters equal zero, and thus no minimization is necessary.
• AF: A staggered antiferromagnetic order, i.e. ∆ ij ≡ 0andh i ≡ (−1) i h.
• RVB: A singlet superconducting phase 1 , i.e. h i ≡ 0, and
∆ ij ≡ ∆e iφα (4.2)
where the singlet order parameter has a uniform amplitude but each nearest
neighbor bond α =1, 2, 3isallowedtohaveitsownphase.
• RVB/AF: A state mixing superconductivity and antiferromagnetism.
• F: A ferromagnetic state with partial or full polarization, i.e. ∆ ij ≡ 0and
h i ≡ h.
1 We have also checked the possibility of triplet pairing, however the minimum of the energy
was always found for singlet pairing trial functions.