pdf, 9 MiB - Infoscience - EPFL

2.6. AUXILIARY-FIELD QUANTUM MONTE CARLO 47
as one-body part in the AFQMC simulations. For clarity we describe below
the method when the former decoupling is used. Moreover, γ is defined by :
cosh(γ) =exp(∆τ |U |/2) ; s i is a classical auxiliary Ising field configuration that
takes the value s i = ±1. These identities come from the more general identity
that transforms any quadratic operator to a superposition of linear operators:
e − ∆τ
2 λˆυ2 =
∫ ∞
−∞
dx e−x2 /2
√
2π
e x√ −∆τλˆυ
(2.77)
We emphasize that the mapping (2.75) is not unique, as seen in equation (2.75),
and the sign problem that can occur in AFQMC depends strongly on the choice
of this mapping. The weight of the wavefunction ψ m is therefore given by:
〈ψ m |ψ m 〉 = ∑
〈ψ MF |e V (si1) e λ1K ...e V (sim) e λmK e λmK e V (s i ) m+1 ...e V (s i ) 2m |ψ MF 〉
(s i1 )..(s i2m )
(2.78)
where s i1 is an Ising spin configuration on the lattice and the potential V (u) is
given by :
(
V 1 (u) = exp 2γσ ∑ )
u i n iσ (2.79)
i
(
V 2 (u) = exp −γ ∑ ) (
u i exp 2γ ∑ )
u i n iσ (2.80)
i
i
depending on whether the attractive (V 1 (u)) or repulsive (V 2 (u)) Gutzwiller projection
is used. In the former case, the Hubbard-Stratanovitch transformation
maps the correlated problem on uncorrelated fermions with the spin degrees of
freedom coupled to a fluctuating field, and in the latter case the charge degrees of
the fermions are coupled to the field. The V 1 (u) andV 2 (u) operators are diagonal
matrices. For instance V 1 (u) isgivenby:
V 1 (u, γ) =diag (2γu 1 , ··· , 2γu N , −2γu N+1 , ··· , −2γu 2N ) (2.81)
Several properties of the Slater determinants are worth mentioning. For any pair
of real non-orthogonal Slater determinants, |φ〉 and |φ ′ 〉,itcanbeshownthat
their overlap integral is:
〈φ|φ ′ 〉 = det ( Φ T Φ ′) (2.82)
The operation on any Slater determinants by any operator ˆB of the form:
( )
∑
ˆB =exp c † i U ijc j (2.83)
ij
simply leads to another Slater determinant [68, 69, 70, 71, 64] :
ˆB|φ〉 = φ † 1 ′ φ † 2 ′ ...φ † ′ M
|0〉 = |φ ′ 〉 (2.84)