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46 CHAPTER 2. NUMERICAL METHODS
known quantum sign problem. Moreover, this method is only valid for Hubbard
theories, and cannot be applied to the t−J models. This method will only be
considered in Chapter 6, where the three band Hubbard model is considered.
Nevertheless, AFQMC is certainly a powerful technique. The main advantage
of the techniques is first that it can deal with both real and complex wavefunctions,
and second it can deal with non-collinear magnetism and twisted spin
boundary conditions [59]. Finally, there is no special difficulty in AFQMC to
measure the physical observables. We discuss here shortly some of the outcomes
of this technique. In AFQMC, the following variational wavefunctions are considered:
ψ (1)
S
= e −λK e −αV ψ MF (2.71)
ψ (2)
S
= e −λ′K e −α′V e −λK e −αV ψ MF (2.72)
ψ (3)
S
= e −λ′′K e −α′′V e −λ′K e −α′V e −λK e −αV ψ MF (2.73)
ψ (m)
S
= e −λ1K e −α1V ...e −λmK e −αmV ψ MF (2.74)
|ψ MF 〉 is the non-interacting wavefunction, and it is given by the slater determinant
of the matrix ψ MF , its columns being the single particle states. The matrix
ψ MF is a square matrix with size N e × N e when |ψ MF 〉 is a tight-binding slater
determinant, and the matrix is rectangular with size 2N × N e when |ψ MF 〉 is
a BCS state, where N is the number of sites of the lattice and N e the number
of particles. Moreover, K denotes the kinetic part of the Hamiltonian, and
V denotes the on-site Hubbard interaction part. The parameters {λ i } i=1,m and
{α i } i=1,m are variational parameters. For large m, we recover the Suzuki-Trotter
decomposition of the full Hamiltonian, and thus we expect the wavefunction ψS
m
to converge to the true ground-state when m is sufficiently large. Unfortunately,
the quantum sign problem becomes more severe when the number of iterations
is increased.
The AFQMC method is based on the decoupling of the interacting V part,
that maps the system to a free electronic problem coupled to fluctuating Ising
fields. More formally, we apply the Hubbard-Stratanovitch transformation, and
the Gutzwiller factor can be written in the possible following forms:
e −∆τUn i↑n i↓
∑
= e −∆τU(n i↑+n i↓ )/2 1
2 eγx i(n i↑ −n i↓)
(2.75)
xi=±1
e −∆τUn i↑n i↓
∑
= e −∆τU(n i↑+n i↓ −1)/2 1
2 eγx i(n i↑ +n i↓ −1)
(2.76)
xi=±1
The former decoupling is real when U > 0 and the latter is real when U < 0.
Therefore, it is possible to work with real matrices for both the repulsive (U >0)
or attractive (U