Magnetic Field Induced Semimetal-to-Canted-Antiferromagnet ...

3 Functional Integral Formulation
Now we interpret the Ising field on time slice n as a N-dimensional vector sn the elements
of which take the values ±1. Finally the grand canonical partition function reads
�
Z = T r e −β
� ��
H−µN
��e−∆τHI −∆τHT = T r e �m �
⎡⎛
= T r ⎣⎝C
�
s1,s2,··· ,sN =±1
+ O(∆τ 2 )
= C m �
m� �
T r e c† V(sn)c −∆τc
e † �
Tc
n=1
�
m
= C
s1,s2,··· ,sn
�
m
= C
s1,s2,··· ,sm
sn
P
α
e i si(ni,↑−ni,↓) −∆τ(−t
e P
〈i,j〉,σ c†
i,σcj,σ) �
m�
T r e c† V(sn)c −∆τc
e † �
Tc
n=1
� �� �
Us(β,0)
⎞m⎤
T r [Us(β, 0)] (3.17)
In line two, the chemical potential can be absorbed in a redefinition of HT . The partition
function now is the trace over a the sum of propagators Us(β, 0) in imaginary time. Using
the following relation 2 for the bilinear operators c † A1c, · · · , c † Anc,
T r[e c† A1c e c † A2c · · · e c † Anc ] = det[1 + e A1 e A2 · · · e An ] , (3.18)
the trace (3.17) can be evaluated explicitly by writing it as determinant of matrices. This
technique is known as integrating out the fermionic degrees of freedom. With the matrix
representation of the propagator,
Bs(β, 0) =
m�
n=1
e V(sn) e −∆τT , (3.19)
the final version of (3.17) is (with m∆τ = β)
Z = C m
�
det[1 + Bs(β, 0)] . (3.20)
s1,s2,··· ,sm
This is the general finite temperature result which is the basis of the finite temperature
QMC (FTQMC) algorithm. This method relies on the grand canonical ensemble. However
if one is soley interested in ground state results it is more efficient to use a canonical
approach which is subject of Chapter 4.
2 A detailed proof may be found in [5]
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