Note on the continuous-time quantum Monte Carlo (CT-QMC)
simulation method
– Application to the Anderson model –
Annamária Kiss, December 23, 2009
1
The CT-QMC method
The Anderson model is given by the Hamiltonian
H=
X
ε k c+
kσ ckσ + εf
fσ+ fσ + U nf,↑ nf,↓ +
X
σ
kσ
Xh
i
∗ +
Vk c+
k,σ fσ + Vk fσ ck,σ .
(1)
kσ
The physical properties of the Anderson model are discussed in Appendix A.
We separate the Hamiltonian into an unperturbed part H0 and a perturbation H1 as H = H0 +H1 .
Thus, the partition function is given by
Z = Tr e
−βH0
Tτ e
−
Rβ
0
dτ H1 (τ )
"
= Tr e
−βH0
∞
X
Z β
(−1)n
−
dτ H1 (τ )
Tτ
n!
0
n=0
!n #
,
(2)
where H1 (τ ) is given in the interaction picture as H1 (τ ) = eτ H0 H1 e−τ H0 . The CT-QMC simulation
samples the infinite sum of multiple integrals in eq. (2) stochastically.
Two basically different methods exist in CT-QMC. Namely,
∗ weak-coupling method: we make expansion in eq. (2) in the Coulomb interaction [1];
∗ hybridization method: we make expansion in eq. (2) in the hybridization term
1
[2], [3].
In the following we discuss the details of the hybridization method.
2
Hybridization method
2.1
Partition function
Following the example of the Anderson model given by the Hamiltonian (1), we take
H1 =
Xh
i
∗
∗ +
V k c+
k,σ fσ + Vk fσ ck,σ ≡ h1 + h1
(3)
kσ
in the partition function
"
Z = Trc Trf e
1
−βH0
!n #
∞
X
Z β
(−1)n
Tτ
−
dτ H1 (τ )
n!
0
n=0
.
(4)
We note that this method is further developed for the case of Kondo and Coqblin-Schrieffer models, where the
expansion is in the Kondo coupling J [6].
1
Only terms with the same power give contribution to the trace:
Z β
(−1)n
dτ H1 (τ )
−
n!
0
!n
!q
Z β
Z β
(−1)2q
=
dτ h1 (τ )
−
dτ h∗1 (τ )
c(2q, q) −
q!
0
0
!q
!q
Z β
Z β
1
∗
=
dτ h1 (τ )
−
dτ h1 (τ ) ≡ Zq ,
−
(q!)2
0
0
!q
(5)
where c(k, q) = k!/(q!(k − q)!). Thus, the partition function has the form

Z = Trc Trf e−βH0 Tτ
∞
X

Zq  .
(6)
q=0
The detailed form of Zq is given as
Zq
Z
1 Z
=
D[q̃] D[q̃]0 Vk1 σ1 ...Vkq σq Vk∗1 σ1 ...Vk∗q σq ×
(q!)2
+
+
+
0
0
0
0
× c+
k1 ,σ1 (τ1 )ck01 ,σ10 (τ1 )...ckq ,σq (τq )ck0q ,σq0 (τq )fσ1 (τ1 )fσ 0 (τ1 )...fσq (τq )fσq0 (τq )
1
Z β
Z β
Z β
X
X
1 Zβ
0
0
dτ
dτ
Vk1 σ1 ...Vkqα σqα Vk∗1 σ1 ...Vk∗qα σqα ×
dτ
...
dτ
...
=
q
1
α
1
qα
(q!)2 0
0
0
0
0
k1 ...kqα k ...k0
qα
1
(
Y
×
α
)(
Y
+
0
0
c+
k1 ,α (τ1 )ck01 ,α (τ1 )...ckqα ,α (τqα )ck0qα ,α (τqα )
)
fα (τ1 )fα+ (τ10 )...fα (τqα )fα+ (τq0α )
,
(7)
α
where we introduced the notation
Z
D[q̃] ≡
Z β
0
dτ1 ...
Z β
0
X
dτq
X
.
(8)
k1 ...kq σ1 ...σq
For the simplicity we take Vk1 σ1 = Vk2 σ2 = ... = Vkq σq = Vk∗1 σ1 = Vk∗2 σ2 = ... = Vk∗q σq = V in the
following.
Z
Zc Zf
=
∞
h
i
1 X
Trc Trf e−βHc e−βHf Tτ Zq
Zc Zf q=0
∞
X
Z
1 Z
+
0
0
=
D[q̃] D[q̃]0 (V 2 )q hTτ c+
k1 (τ1 )ck01 (τ1 )...ckq (τq )ck0q (τq )ic ×
2
q=0 (q!)
× hTτ fσ1 (τ1 )fσ+0 (τ10 )...fσq (τq )fσ+q0 (τq0 )if
1
1
=
(q!)2
{qα }
q!
Q
α (qα !)
X
*
×
Tτ
(
Y
α
!2 Z
β
0
dτ1 ...
Z β
0
Z β
dτqα
0
Z β
dτ10 ...
0
dτq0α
X
X
(V 2 )qα ×
k1 ...kqα k01 ...k0qα
)+ *
+
0
0
c+
k1 ,α (τ1 )ck01 ,α (τ1 )...ckqα ,α (τqα )ck0qα ,α (τqα )
Tτ
c
(
Y
)+
fα (τ1 )fα+ (τ10 )...fα (τqα )fα+ (τq0α )
α
,
f
(9)
where h...ic(f ) = Trc(f ) e−βHc(f ) ...
’qα ’ by the change
P
q
→
P
{qα }
q!/
1
.
Zc(f )
Q
In the last part of eq. (9) we summed for the state number
α (qα !),
where
P
2
α qα
= q.
We can perform the trace calculation over the conduction electron operators by using Wick’s
theorem:
*
X
2 qα
X
Y
Tτ
(V )
)+
(
α
k1 ...kqα k01 ...k0qα
+
0
0
c+
k1 ,α (τ1 )ck01 ,α (τ1 )...ckqα ,α (τqα )ck0qα ,α (τqα )
c
*
X
=
2 qα
X
(V )
k1 ...kqα k01 ...k0qα
X
sign(c) Tτ
)+
(
Y
+
0
0
c+
k1 (τ1 )ck01 (τ1 )ic ...hTτ ckqα (τqα )ck0qα (τqα )
α
{c}
=
c
Y
ˆ (qα ) ,
det ∆
α
α
(10)
ˆ (qα ) has the form
where {c} means the all possible connections, and matrix ∆
ˆ (qα ) =
∆
α
gα (τq0α − τ1 )

.
.
.
.
.
.
. gα (τq0α − τqα )



,




gα (τ10 − τ1 )



2
V 


gα (τ10 − τ2 )
.
.
gα (τ10 − τqα )
gα (τ20 − τ1 ) .
(11)
where
gα (τ 0 − τ ) =
X
gα,k (τ 0 − τ ) = −hTτ ck,α (τ 0 )ck,α (τ )+ ic
(12)
k
is the bare conduction electron Green’s function.
The sequence of τ -integrals in the last part of eq. (9) is arbitrary. By time-ordering the arguments
we obtain
Z β
0
dτ1 ...
Z β
0
dτqα = (qα !)
Z ordered
dτ1 ...
Z ordered
dτqα ≡ (qα !)
Z ordered
D[τ ],
(13)
where β > τ1 > τ2 > ... > τqα ≥ 0. Thus, we obtain the partition function as
Z
Z c Zf
=
X Z ordered
D[τ ]
Z ordered
0
D[τ ]
≡
α)
ˆ (q
det ∆
α
*
Tτ
(
Y
)+
fα (τ1 )fα+ (τ10 )...fα (τqα )fα+ (τq0α )
α
α
{qα }
Z
Y
f
D[q]W [q],
(14)
where
W [q] =
Y
ˆ (qα )
det ∆
α
*
Tτ
α
(
Y
)+
fα (τ1 )fα+ (τ10 )...fα (τqα )fα+ (τq0α )
α
Z
D[q] =
X Z ordered
,
(15)
f
D[τ ]
{qα }
3
Z ordered
D[τ 0 ].
(16)
2.2
Metropolis algorithm
The average of quantity O is given by
1 Z
hOi =
D[q]W [q]O(q),
Z
where the partition function has the form Z =
R
(17)
D[q]W [q]. In Monte Carlo simulation we evaluate
the average given by eq. (17) stochastically. It is not possible to consider all the configurations q.
f [q]. In this
Instead, we choose a set {q} where the configurations q are generated by the weight W
case the average is reweighted as
R
hOi =
f [q]O(q)
D[q]W [q]/W
.
R
f [q]
D[q]W [q]/W
(18)
The simplest case when we generate the configurations randomly. However, the sampling is more efficient when we generate the configurations with the weight they contribute to the partition function,
f [q] = W [q]. This method is called importance sampling. The importance sampling
namely when W
method can be achieved when we assure the
∗ ergodicity: each configuration can be reached during the generates of configurations;
∗ detailed balance condition: Wxy /Wyx = W [y]/W [x], where Wxy is the transition probability
from configuration x to configuration y, and W [x] is the weight of the configuration x.
The detailed balance condition can be realized by the Metropolis algorithm. In this algorithm we
divide Wxy into a proposal part and an acceptance part as
Wxy = Wprop (x → y)Wacc (x → y).
(19)
We can find that
!
W [y]Wprop (y → x)
Wacc (x → y) = min 1,
.
W [x]Wprop (x → y)
(20)
When we generate a new configuration y from the configuration x, we accept this new configuration
y by the acceptance rate Wacc (x → y). Namely, we choose a random number r from the interval
[0, 1], and accept the new configuration if r < Wacc . The average in the Monte Carlo simulation is
obtained as a simple arithmetic average
hOiMC =
1 X
O(qk ),
N q1 ,...qN
f [q] = W [q] in eq. (18).
since W
4
(21)
2.3
Segment picture
In the case of the Anderson model the configuration is a set of the imaginary times: {τ1 , τ10 , ...τk , τk0 }.
The f -operators f and f + should occur in alternating order, otherwise the expectation value in
eq. (14) is zero. This statement is held also for models where the Hamiltonian has only densitydensity interaction terms (no exchange terms), i.e., it is diagonal in component indices. In this
case the configuration can be represented by segment picture. The two possible configurations are
shown in Fig. 1. There are two components for the f -electron operators, namely σ =↑, ↓. These two
Figure 1: Segment representation of a kth-order configuration. The lines correspond to occupied
f -electron state (particle number 1), while the empty spaces to particle number 0.
components lead to two independent channels in the segment picture as it is shown in Fig. 2.
Figure 2: The two spin indices leads to two independent channels in the segment picture. The right
part of the figure shows all possible connections of operators c+ (f ) and c(f + ), where the weighted
contributions of the diagrams are summed up into the determinant of matrix ∆(k)
α with k =↑, ↓.
Open and filled diamonds correspond to operators c+ (f ) and c(f + ), respectively, while the dashed
lines mean the hybridization lines V 2 gα (τj0 − τi ).
5
There are two basic operations when we generate a new configuration: addition or removal a
segment. When we add a new segment to a kth-order configuration, first we add the operator
f + (τ 0 ) by choosing τ 0 from the interval (β, 0]. If f + (τ 0 ) is on an existing segment, we reject the new
configuration. If the operator f + (τ 0 ) is located between the operators f + (τl0 ) and f (τl+1 ), we put the
operator f (τ ) somewhere in the interval [0, lmax ], where lmax = mod(τl0 + β − τ 0 , β). The transition
probability of this process is given by
Wprop (k → k + 1) =
1
.
βlmax
(22)
When we remove a segment from a k + 1th-order configuration, we obtain the transition probability
as
Wprop (k + 1 → k) =
1
k+1
(23)
since we have k + 1 choice to remove one segment. The acceptance rate given by eq. (20) becomes
for the Anderson model as
!
W [k + 1] βlmax
Wacc (k → k + 1) = min 1,
.
W [k] k + 1
(24)
First, let us consider the case of U = 0, when it is enough to consider only one channel. In this case
we can write
E
D
0
)
ˆ (k+1) Tτ f (τ1 )f + (τ10 )...f (τk+1 )f + (τk+1
det ∆
W [k + 1]
=
ˆ (k)
W [k]
hTτ f (τ1 )f + (τ10 )...f (τk )f + (τk0 )if
det ∆
f
,
(25)
expectation value in eq. (25) can be evaluated as
hO1 (τ1 )...O2k (τ2k )i = Tr e−Hf (β−τ1 ) O1 (τ1 )e−Hf (τ1 −τ2 ) ...O2k (τ2k )e−Hf (τ2k ) ,
(26)
where
Hf = εf
X
fσ+ fσ ,
(27)
σ
and operators Ok denote the f -operators f or f + . The trace is evaluated as Trf =
D
E
Tτ f (τ1 )f + (τ10 )...f (τk )f + (τk0 )
=
n=0,1 hn|...|ni.
P
X
hn|e
−Hf (β−τ1 )
f
O1 (τ1 )|nihn|e−Hf (τ1 −τ2 ) O2 (τ2 )|nihn|...e−Hf (τ2k−1 −τ2k ) O2k (τ2k )e−Hf (τ2k ) |ni
n=0,1
= e−E0 (β−τ1 ) hn(0) |O1 (τ1 )|n(1) ie−E1 (τ1 −τ2 ) hn(1) |O2 (τ2 )|n(2) i...hn(2k−1) |O2k (τ2k )|n(2k) ie−E2k (τ2k ) .(28)
The non-zero matrix elements in eq. (28) are when hn(l−1) |Ol (τl )|n(l) i is h0|f |1i or h1|f + |0i with the
energy eigenvalue El−1 = 0 and El−1 = εf , respectively. The first and last operators are O1 = f
6
and O2k = f + which gives that E0 = E2k = 0, and the operators have the sequence f f + f f + ...f f + .
Thus, expression (28) becomes
D
Tτ f (τ1 )f
+
E
(τ10 )...f (τk )f + (τk0 )
f
−εf (τ1 −τ10 ) −εf (τ2 −τ20 )
=e
e
...e
−εf (τk −τk0 )
−εf
=e
P
l
q q
,
(29)
where lq = τq − τq0 is the length of the qth segment.
When we consider also the Coulomb interaction term (U 6= 0), then the trace over the f states
becomes
*
Tτ

 Y

α=σ,σ
+

fα (τ1 )fα+ (τ10 )...fα (τqα )fα+ (τq0α )
= e−εf
P
l
q q
e−U lov ,
(30)
f
where lov is the overlap between the two channels σ and σ since the Coulomb interaction acts when
we have | ↑i and | ↓i f -electron states at the same imaginary time. Thus, eq. (24) becomes
ˆ (k+1)
βlmax det ∆
e
Wacc (k → k + 1) = min 1, sign(τ − τ )
e−εf l e−U ∆lov ,
(k)
ˆ
k + 1 det ∆
!
0
(31)
where le is the length of the newly inserted segment, and ∆lov means the change in the overlap between
the two channels. The term sign(τ − τ 0 ) is inserted because a minus sign comes in case (b) of Fig. 1
when we rearrange the f -operators into the same sequence as it is in case (a).
The acceptance rate for the process k → k − 1 can be obtained by similar consideration as
ˆ (k−1) e
k det ∆
Wacc (k → k − 1) = min 1, sign(τ − τ )
eεf l eU ∆lov .
(k)
ˆ
βlmax det ∆
!
0
(32)
There are a further operation besides the segment addition and removal that we use in the simulation.
This is the shift operation when we shift one of the τ s or τ 0 s, which makes the sampling more efficient.
The acceptance rate of the shift operation is given as
(k)

Wacc (k → k)(shift)

ˆe
k det ∆
e

0
= min 1, (τ − τ )
e−εf (l−l) e−U ∆lov 
,
(k)
ˆ
βlmax det ∆
(33)
where l and le are the lengths of the segment before and after the shift operation, respectively.
ˆ −1 which
The determinant ratios in eqs. (31), (32), and (33), and also the matrix inverse M̂ = (∆)
will be important in the calculation of the f -electron Green’s function can be calculated by using
fast-update formulas. The derivation is given in Appendix B.
The probability distribution of the random walk in the configuration space according to the
acceptance rates Wacc (k → k + 1) and Wacc (k → k − 1) is shown in Fig. 3.
We note that special attention is necessary when all the segments are absent in one of the channels.
In this case the trace calculation over the f states is modified as follows. First we discuss the case
7
U = 0, εf = 0
β = 10
β = 50
β = 100
β = 500
β = 1000
0.4
0.35
120
kaverage (matrix sixe)
0.45
ρ(k)
0.3
0.25
0.2
0.15
0.1
0.05
100
80
60
40
20
0
0
0
20
40
60
80
100
120
0
140
200
400
600
800
1000
β
k
Figure 3: Left: Probability distribution ρ of the random walk in the configuration space as a function
of the expansion order (k) at temperatures β = 10, 50, 100, 500, 1000. Right: The position of the
maximum probability as a function of the inverse temperature. The parameter values are U = 0,
V 2 = 0.1, εf = 0.
of segment addition. We note the number of segments in channel A by kA , and write the acceptance
rate as
ˆ (k+1)
βlmax det ∆
Wacc (k → k + 1) = min 1, sign(τ − τ )
Q ,
ˆ (k)
k + 1 det ∆
!
0
(34)
where
Q=
0
)
Trf e−βHf f (τ1 )f + (τ10 )...f (τk+1 )f + (τk+1
Trf (e−βHf f (τ1 )f + (τ10 )...f (τk )f + (τk0 ))
.
(35)
We have to distinguish the following cases:
• kσ = 0, kσ 6= 0
The empty channel σ gives the contributions
h0nσ |...|0nσ i → 1
h1nσ |...|1nσ i → e−βεf ,
where in the last term the empty configuration is taken as the full segment (= filled from β to 0)
since the empty and full configurations are degenerate. The notation ... means e−βHf f f + ...f f + ,
which should be rewritten as in eq. (26). Combining these results with the other channel σ, we
obtain before the segment addition
h01|...|01i → 0
8
h11|...|11i → 0
h00|...|00i → e−lσ εf
h10|...|10i → e−lσ εf e−βεf e−lσ U .
Adding a new segment with length lσ to the channel σ, we obtain
h01|...|01i → 0
h11|...|11i → 0
h00|...|00i → e−lσ εf e−lσ εf e−lov U
h10|...|10i → 0.
Thus,
Q=
e−lσ εf e−lσ εf e−lov U
e−lσ εf e−lov U
=
.
e−lσ εf + e−lσ εf e−βεf e−lσ U
1 + e−βεf e−lσ U
(36)
• kσ = 0, kσ = 0
Before the segment addition we obtain
h01|...|01i → e−βεf
h11|...|11i → e−βεf e−βεf e−βU
h00|...|00i → 1
h10|...|10i → e−βεf ,
and after adding a new segment with length lσ to the channel σ:
h01|...|01i → e−lσ εf e−βεf e−lσ U
h11|...|11i → 0
h00|...|00i → e−lσ εf
h10|...|10i → 0.
Thus, we obtain
Q=
e−lσ εf (1 + e−βεf e−lσ U )
.
1 + 2e−βεf + e−β(2εf +U )
• kσ 6= 0, kσ = 0
9
(37)
Before the segment addition we obtain
h01|...|01i → e−lσ εf e−βεf e−lσ U
h11|...|11i → 0
h00|...|00i → e−lσ εf
h10|...|10i → 0,
and after adding a new segment with length lσ to the channel σ:
h01|...|01i → e−(lσ +lσ )εf e−βεf e−(lσ +lσ )U
h11|...|11i → 0
h00|...|00i → e−(lσ +lσ )εf
h10|...|10i → 0.
Thus, we obtain
Q=
e−elσ εf (1 + e−βεf e−(lσ +elσ )U )
.
1 + e−βεf e−lσ U
(38)
• kσ 6= 0, kσ 6= 0
We already discussed this case and the result is given in eq. (31).
In the case of segment removal, the acceptance rate is given by
ˆ (k−1)
k det ∆
Q0 ,
Wacc (k → k − 1) = min 1, sign(τ − τ 0 )
ˆ (k)
βlmax det ∆
!
where Q0 = Q−1 .
In the case of the shift operation we have to distinguish the case
• kσ 6= 0, kσ = 0
Before the segment addition we obtain
h01|...|01i → e−lσ εf e−βεf e−lσ U
h11|...|11i → 0
h00|...|00i → e−lσ εf
h10|...|10i → 0,
10
(39)
and after adding a new segment with length lσ to the channel σ:
h01|...|01i → e−lσ εf e−βεf e−lσ U
h11|...|11i → 0
h00|...|00i → e−lσ εf
h10|...|10i → 0.
Thus, we obtain
Q=
2.4
e−(elσ −lσ )εf (1 + e−βεf e−elσ U )
.
1 + e−βεf e−lσ U
(40)
Measuring the f -electron Green’s function
The f -electron Green’s function is given by
Gf (τ 0 − τ ) = −hTτ f (τ 0 )f (τ )+ i.
(41)
In order to measure this Green’s function, we need a configuration where operators f (τ 0 ) and f (τ )+
are unconnected. In the hybridization method it is achieved by removing one of the hybridization
lines.
We consider a kth-order partition function configuration, which weight is given by
D
E
pZ = Tτ f (τ1 )f + (τ10 )...f (τk )f + (τk0 )
ˆ (k) .
det ∆
f
(42)
We remove the hybridization line V 2 gα (τi0 − τj ), and then we can write the weight of the ”Green’s
function configuration” as
pG =
D
0
Tτ f (τj )f (τi0 )+ f (τ1 )f + (τ10 )...f (τk−1 )f + (τk−1
)
D
E
= (−1)i+j Tτ f (τ1 )f + (τ10 )...f (τk )f + (τk0 )
f
E
f
ˆ (k−1)
det ∆
ˆ (k−1) .
det ∆
(43)
The contribution to the Green’s function at τ = τi0 − τj is given by
ˆ (k−1)
(−1)i+j det ∆
pG
6=ij
=
,
(k)
ˆ
pZ
det ∆
(44)
ˆ (k−1) means that line i and column j are removed from the matrix ∆
ˆ (k) . Using the
where notation ∆
6=ij
matrix identity
(k−1)
ˆ
(−1)i+j det∆
6=ij
ˆ (k) ∆
ˆ (k)
= det∆
11
−1
ji
,
(45)
we obtain the f -electron Green’s function as
*
Gf (τ ) =
2.5
2
k 1X
ˆ (k) −1 δ(τ, τ 0 − τj )
∆
i
ji
β ij
+
*
=
MC
k
1X
(k)
M̂ δ(τ, τi0 − τj )
β ij ji
+
.
(46)
MC
Correlation functions
Before discussing the correlation functions, we consider the density operator n = f + f . In CT-QMC
we obtain the density operator as
1Zβ 0
1X
hni =
dτ n(τ 0 ) =
(τi − τi0 ),
β 0
β i
(47)
which is nothing else than the total length of the segments.
Now we discuss the density-density correlation function:
χ(τ ) = hn(τ )n(0)i = hTτ n(τ )n(0)i = hTτ f + (τ )f (τ )f + (0)f (0)i = −hTτ f + (τ )f + (0)f (τ )f (0)i
= −hTτ f + (τ )f (0)ihTτ f + (0)f (τ )i + hTτ f + (τ )f (τ )ihTτ f + (0)f (0)i
= −hTτ f (0)f + (τ )ihTτ f (τ )f + (0)i + hni2 = (−Gf (−τ ))(−Gf (τ )) + hni2 = Gf (τ )2 + hni2
(48)
e ) ≡ χ(τ ) − hni2 .
We introduce χ(τ
In CT-QMC we can evaluate hn(τ )n(0)i as
* Z
+
1Zβ 0
1 β 0 hTτ n(τ + τ 0 )n(τ 0 )f f + ...f f + if
0
0
dτ n(τ + τ )n(τ ) =
dτ
hn(τ )n(0)i =
.
β 0
β 0
hTτ f f + ...f f + if
MC
(49)
This ratio is evaluated in CT-QMC as the overlap between a configuration and its shift by τ as it is
shown in Fig. 4.
In the limits of τ = 0 and τ = β we obtain
e
e
χ(0)
= χ(β)
= hn2 i − hni2 = hni − hni2 = 1/2 − (1/2)2 = (1/2)2 .
(50)
The result for the f -electron Green’s function Gf (τ ) and the density-density correlation function
e ) obtained by CT-QMC is shown in Fig. 5.
χ(τ
R
ˆ (k−1) Tτ f (τj )f (τ 0 )+ f (τ1 )f + (τ 0 )...f (τk−1 )f + (τ 0 ) =
Another derivation of this formula is: Gf (τi0 −τj ) = − D det ∆
1
i
k−1 f
6=ij
−1 R
0
ˆ (k) ∆
ˆ (k)
− D(−1)i+j det ∆
Tτ f (τj )f (τi0 )+ f (τ1 )f + (τ10 )...f (τk−1 )f + (τk−1
) f
ji
−1
−1 D
E
R
(k)
(k) ˆ (k)
+ 0
+ 0
(k)
ˆ
ˆ
= − D det ∆
∆
hTτ f (τ1 )f (τ1 )...f (τk )f (τk )if = −
∆
= − M̂ji
2
ji
ji
12
Figure 4: Calculation of the susceptibility hn(τ )n(0)i in CT-QMC as the calculation of overlap
between a configuration and its shift by τ .
U = 0, εf = 0, β = 50
Gf, exact
Gf
χf, exact
χf
0.3
Gf (τ), χf (τ)
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
0
10
20
30
40
50
τ
Figure 5: Imaginary-time dependence of the f -electron Green’s function Gf (τ ) and the densitye ). The parameter values are U = 0, β = 50, V 2 = 0.1, εf = 0. The
density correlation function χ(τ
exact results are also shown by orange color.
2.6
Spectral function
The spectral function (density of states) as a function of the real frequency is given by
1
ρ(ω) = − Im[G(ω + iδ)].
π
(51)
The spectral function in the case of the symmetric Anderson model for different temperatures and
Coulomb interactions is shown in Fig. 6.
13
kBT/∆ = 0.02 (β = 3000)
U/(π ∆) = 3
β =1000, U=0
β =1000
β =500
β =100
β =50
0.8
U=0
U/(π ∆) = 1
1
π ∆ ρimp(ω)
π ∆ ρimp(ω)
1
0.6
0.4
0.2
0.8
0.6
0.4
0.2
εf
0
-10
-5
εf + U
0
0
5
10
-4
-2
0
ω/∆
2
4
ω/∆
Figure 6: Impurity density of states as a function of frequency in the case of the symmetric Anderson
model for U/(π∆) = 3 with temperatures β = 50, 100, 500, 1000 (upper panel) and for β = 3000 with
Coulomb interactions U/(π∆) = 1 and U = 0 (bottom panel). The parameter values are V 2 = 0.01,
D = 1, (∆ = 0.0156).
3
Dynamical mean field theory (DMFT)
In general, it is very difficult to solve lattice fermion problems. Thus, we need to use some approximation. A commonly used method is the dynamical mean field theory (DMFT), which approximate
the original lattice problem by an effective impurity problem plus a self-consistency condition (see
Fig. 7), where the effective impurity problem can be solved efficiently by the CT-QMC impurity
solver. In infinite dimension the diagonal elements of the Green’s function are much larger than
the off-diagonal elements (Gii (τ ) Gij (τ )). Thus, the self-energy becomes local (Σ(k, τ ) → Σ(τ )),
which leads to that DMFT is exact in d = ∞. First, we discuss the formulation for the f -electrons.
Formulation for the f -electrons:
In the effective impurity picture the site is embedded into an effective medium described by the
cavity Green’s function G, which incorporates the effect of interaction U at the surrounding sites.
Thus, the Green’s function is given by
Gf (z)−1 = Gf (z)−1 − Σf (z).
(52)
The self-consistency condition means that the Green’s function Gf (z) is the same as the site-diagonal
part of the Green’s function of the original lattice problem, which is the case in infinite dimension.
Namely,
Gf (z) = Gf (k, z) =
Z
V2
dερ(ε) z − εf − Σf (z) −
z−ε
14
!−1
.
(53)
Figure 7: Left side shows the original lattice problem, which is approximated by an effective impurity
problem in DMFT shown in the right side.
We use iteration to perform the DMFT procedure. First, we start with a trial cavity Green’s
function Gf (z) and use the CT-QMC solver to obtain the Green’s function Gf (z). Using eq. (52)
we obtain the self-energy Σf (z), what we use in eq. (53) to obtain a further approximate for the
Green’s function Gf (z). Then we calculate the cavity Green’s function from equation Gf (z)−1 =
Gf (z)−1 + Σf (z), what we use in the next step of the iteration. We continue this procedure until a
convergent result is obtained for Gf (z).
Right part of Fig. 8 shows the self-energy
Σf (z) = Gf (z)−1 − Gf (z)−1 ,
(54)
which is very noisy for large frequencies because the inversion of Gf (z) in eq. (54) amplifies the
statistical errors. Thus, we may have problem in DMFT that we can cross over if we reformulate the
DMFT procedure for the conduction electrons. We can do it as far as the f -level has no dispersion,
i.e. it is sharp.
Formulation for the conduction electrons:
First we derive a relation between the f -electron Green’s function and the conduction electron Green’s
function:
1
Gf (z) =
z − εf − Σf (z) −

V2
z−εk
2
2
z − εk − z−εfV−Σf (z) + z−εfV−Σf (z)
1


=
2
z − εf − Σf (z)
z − εk − z−ε V−Σ (z)
f

2
f

=
1
V
1


+
2
2
z − εf − Σf (z) (z − εf − Σf (z)) z − εk − z−ε V−Σ
f
f (z)
=
1
1
V2
2
+
G
(z)
≡
Σ
(z)
+
Σ
(z)
G
(z)
,
c
c
c
c
z − εf − Σf (z) (z − εf − Σf (z))2
V2
15

(55)
β = 500
log[-Im[Gf]]
log[-Im[Gf,0]]
1/(iεn)
5
4
0.4
Im[Σf (iεn)]
log[-Im[Gf (iεn)]]
6
β = 500
3
2
1
0
-1
-2
0.2
0
-0.2
-0.4
-6
-5
-4
-3
-2
-1
0
1
2
3
0
log[iεn]
500
1000
1500
2000
n
Figure 8: Left: Log-log plot of the imaginary part of the interacting and non-interacting (U=0)
f -electron Green’s functions as a function of the Matsubara frequencies iεn = i(2n − 1)πβ. Right:
Imaginary part of the f -electron Matsubara self-energy Σf (iεn ) as a function of n. The parameter
values are U/(π∆) = 3, f = −0.075, β = 500, V 2 = 0.01, D = 1, (∆ = 0.0156).
where we defined Σc as
V2
Σc (z) ≡
.
z − εf − Σf (z)
(56)
V 2 Gf (z) = t = Σc + Σ2c Gc ,
(57)
Thus, we have the relation
where t is the t-matrix. We define the conduction electron cavity Green’s function as
Gf (z) =
1
.
z − εf − V 2 Gc (z)
(58)
Thus, eq. (56) can be rewritten as
Σc ≡
V 2 Gf
V2
V2
t
= −1
=
=
−1
−1
2
2
z − εf − Σ f
1 + V Gf Gc
1 + tGc
Gf + V Gc − Gf + Gf
(59)
From eqs. (57) and (59) we have
Σc =
Σc + Σ2c Gc
t
−→ 1 + tGc = 1 + Σc Gc −→ tGc =
Gc −→ Gc = Gc + Gc tGc .
1 + tGc
1 + tGc
(60)
From eq. (59) we write
t=
1
,
Σ−1
c − Gc
16
(61)
and using this relation in eq. (60), we have
Gc
1
−→ Σc = Gc−1 − G−1
(62)
Gc = Gc + −1 −1
Gc = Gc + Gc −1
c
Σc − Gc
Gc Σc − 1
as we expect. In the formulation for the conduction electrons the self-consistency condition is the
following
Gc (z) = Gc (k, z) =
Z
dερ(ε) (z − ε − Σc (z))−1 .
(63)
We perform the iteration as follows. We start with a trial conduction electron cavity Green’s function
Gc , and obtain the f -electron Green’s function Gf by applying the CT-QMC impurity solver. Then,
we use the t-matrix t = V 2 Gf in eq. (59), and obtain Σc . With this self-energy we calculate the
site-diagonal conduction electron Green’s function Gc using eq. (63), which gives the new trial cavity
Green’s function through the relation Gc−1 = G−1
c + Σc . Then, we start the iteration again until a
convergent result is obtained for Gf (z).
In order to use the CT-QMC for solving the DMFT problem of the periodic Anderson model, we
assume semi-circle density of states for the conduction electrons, which is the exact result for Bethe
lattice in d = ∞:
2 √ 2
D − ω2.
(64)
πD2
The DMFT results for the Anderson lattice with U = 0 using the CT-QMC solver together with
ρ0 (ω) =
the analytic results (see Appendix A) are shown in Fig. 9. We can see that the hybridization gap is
reproduced.
References
[1] A. N. Rubtsov, V. V. Savkin, and A. I. Lichtenstein, Phys. Rev. B 72 (2005) 035122.
[2] P. Werner, A. Comanac, L. de Medici, M. Troyer, and A. J. Millis, Phys. Rev. Lett. 97 (2006)
076405.
[3] P. Werner and A. J. Millis, Phys. Rev. B 74 (2006) 155107.
[4] E. Gull, P. Werner, A. J. Millis, and M. Troyer: cond-mat/0609438.
[5] H. Kusunose: private communication.
[6] J. Otsuki, H. Kusunose, P. Werner, and Y. Kuramoto, J. Phys. Soc. Jpn 76 (2007) 114707.
[7] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68 (1996) 13.
[8] H. J. Vidberg and J. W. Serene: J. Low Temp. Phys. 29 (1977) 179.
17
ρ(ω)D
U = 0, εf = 0, β = 500
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
f, exact
c, exact
c(free)
f
c
-1
-0.5
0
0.5
1
ω/D
Figure 9: Conduction electron and f -electron density of states as a function of frequency in the
case of the Anderson lattice with parameters U = 0, f = 0, V 2 = 0.1, D = 1, (β = 500). The
black dashed, gray dashed, and pink solid lines show the analytic results for ρc (ω), ρf (ω), and ρ0 (ω),
respectively, given by eqs. (A.83), (A.84), and (64).
A
The Anderson model
A.1
Single-impurity Anderson model
One of the simplest models to deal with an impurity embedded into the sea of conduction electrons
is the Anderson model, which is described by the Hamiltonian
H=
X
ε k c+
kσ ckσ + εf
X
fσ+ fσ + U nf,↑ nf,↓ +
σ
kσ
Xh
i
∗ +
Vk c+
k,σ fσ + Vk fσ ck,σ .
(A.65)
kσ
In the atomic limit (V = 0) the possible configurations for the f -electrons are the emty, singly or
doubly occupied states:
|f 2 i : E(f 2 ) = 2εf + U
|f 0 i : E(f 0 ) = 0
|f 1 ↑i, |f 1 ↓i : E(f 1 ) = εf .
For example, when the conditions
E(f 2 ) − E(f 1 ) = εf + U > 0
E(f 0 ) − E(f 1 ) = −εf > 0
are satisfied, the ground state is the singly occupied state which possesses magnetic moment. In the
following we consider this case.
18
Figure 10: Diagrammatic representation of the f and conduction electron Green’s functions for the
single-impurity Anderson model.
When we allow coupling between the f and conduction electrons, the sharp f -state broadens due
to the hybridization. Using the resolvent form of the Green’s function (z − H)Ĝ(z) = 1̂, we can write
the Green’s function matrix of the single-impurity Anderson model as

Ĝ(z) =







z − εk .
.
−Vk∗
.
.
.
.
.
. z − εk 0
−Vk
.
−Vk0
−Vk∗0
z − εf − Σf (z)
−1








=
Ĝc (z)
Ĝcf (z)∗
Ĝcf (z)
Ĝf (z)

,
(A.66)
where we find that
Gf (z) = hf |Ĝ(z)|f i = z − εf − Σf (z) −
X
k
|Vk |2
z − εk
!−1
;
Gc (z, k, k0 ) = hk|Ĝ(z)|k0 i = g(z, k)δk,k0 + g(z, k)Vk∗ Gf (z)Vk0 g(z, k0 ),
(A.67)
(A.68)
where g(z, k) is the bare conduction electron Green’s function with the form g(z, k) = (z − εk )−1 ,
and the effect of the Coulomb interaction is inserted into Σf . The corresponding diagrams are shown
in Fig. 10.
In the further discussion we distinguish two cases, i.e., the cases of small and large Coulomb
interaction.
Small Coulomb interaction (U < π∆)
In this case we regard U as perturbation. We write the Green’s function as
Gf (z) = (z − εf − Σf (z) − ΣV )−1 ,
where ΣV =
P
k
(A.69)
|Vk |2 /(z −εk ) is the contribution from the hybridization. The Hartree approximation
for the f -electron self-energy shown in Fig. 11 gives that εf is shifted as εf → Ef = εf + U nf,−σ . We
used the constant density of states for the conduction electrons:
1
for − D < ω < D;
2D
ρ0 (ω) = 0 otherwise,
ρ0 (ω) =
19
(A.70)
Figure 11: Hartree approximation for the self-energy Σf in the case of the single-impurity Anderson
model.
and evaluate ΣV for real frequencies as
ΣV (ω + iδ) =
X
k
X
X V2
D + ω V2
− i∆, (A.71)
δ(ω − εk ) = V 2 ρ0 ln =P
− iπV 2
ω + iδ − εk
ω
−
ε
D
−
ω
k
k
k
where we introduced ∆ ≡ πV 2 ρ0 . When the bandwidth 2D is large the real part of ΣV is much
smaller than the imaginary part. Thus, we can approximate
ΣV (ω + iδ) ≈ −i∆.
(A.72)
The impurity density of states is obtained as
1
1
∆
ρf (ω) = − ImGf (ω + iδ) = − Im(ω − Ef + i∆)−1 =
,
π
π
(ω − Ef )2 + ∆2
(A.73)
which has Lorentzian form centered around Ef .
Large Coulomb interaction (U π∆)
In the atomic limit the impurity Green’s function has the form
Gf (ω) =
1 − hnf,−σ i
hnf,−σ i
.
+
ω − εf
ω − εf − U
(A.74)
Thus, when the Coulomb interaction is large, the single peak in the impurity density of states splits
into two peaks centered around εf and εf + U which correspond to the excitation energies on adding
an electron to the system in the atomic limit.
We consider the symmetric case: εf = −U/2, when hnf,σ i = 1/2 for all temperatures. There is no
any singularities when we increase the Coulomb interaction from small values towards large values.
This means that there is one-to-one correspondence with the excitations of the non-interacting case
U = 0 and that we have a local Fermi-liquid. We expand Σf in powers of (ω − εf ) in eq. (A.69), and
obtain
#−1
"
∂Re[Σ(εf )]
Gf (ω) = (ω − εf + i∆ − Σf (ω)) ≈ ω − εf − Re[Σ(εf )] + i∆ − (ω − εf )
− iO(ω 2 )
∂ω
z
z
=
=
,
(A.75)
e
ω − εf − z(εf − εF + Re[Σ(εf )]) + iz∆ − iO(ω 2 )
ω − εf − εef + i∆
−1
20
where we used that Im[Σf (ω)] = 0 at ω = εf , and z = 1/(1 − ∂Re[Σ(εf )]/∂ω) is called wavefunction
renormalization. The Green’s function Gf (ω) looks like as the U = 0 Green’s function. In the
symmetric case we obtain at low energies:
Gf (ω) =
z
,
ω + iz∆ − iO(ω 2 )
(A.76)
which gives that the impurity density of states possesses a peak around ω ∼ 0 which is called Kondo
resonance, whose height does not depend on the interaction U :
1
1
ρf (ω ∼ 0) = − Im [Gf (ω + iδ)]ω=0 =
.
(A.77)
π
π∆
This peak exists for low temperatures, and disappears when T > 4TK , where TK is the Kondo
temperature. The weight of the Kondo resonance sharpens as U increases due to the renormalization
∆ → z∆ in eq. (A.76). These behaviors can be seen in Fig. 6.
A.2
Periodic Anderson model
The periodic Anderson lattice is described by the Hamiltonian
X
H=
ε k c+
kσ ckσ + εf
X
+
fiσ
fiσ +
X
iσ
kσ
U nif,↑ nif,↓ + V
Xh
i
i
+
c+
iσ fiσ + fiσ ciσ .
(A.78)
iσ
We can diagonalizing the Hamiltonian (A.78) in the non-interacting case (U = 0):

H=
X
+ 
c+
kσ , fkσ
kσ
εk V

V

εf
ckσ

fkσ


=
X
+

a+
kσ , bkσ
kσ
Ek+
0

akσ
0
Ek−

bkσ

,
(A.79)
where
s
εk − εf 2
εk + εf
Ek± =
±
+ V 2.
(A.80)
2
2
Thus, the hybridization of the f -electron and the conduction electron bands leads to the appearance
of two bands, which are separated by a hybridization gap δ (δ ∼ 2V 2 /D if D V, εf ).
The f -electron and conduction electron Green’s functions and density of states are obtained as
Gc (k, z) =
V2
z − εk −
z − εf
!−1
;
(A.81)
!−1
V2
1
V2
Gf (k, z) = z − εf −
=
+
Gc (k, z);
z − εk
z − εf
(z − εf )2
"
#
!
X
1 1
V2
ρc (ω) = −
Im
Gc (k, ω + iδ) = ρ0 ω −
;
πN
ω − εf
k
X
1 1
V2
V2
ρf (ω) = −
Im
Gf (k, ω + iδ) =
ρ
ω
−
0
πN
(ω − εf )2
ω − εf
k
"
#
for U = 0.
21
(A.82)
(A.83)
!
(A.84)
B
Fast-update formulas
The fast-update formulas are the generalization of the Shermann-Morrison formula, which gives a
recipe for the case when we know the inverse of a matrix Â, and we want to calculate the inverse of
another matrix which only slightly differs from the matrix Â. Namely,
Â
−→
Â−1
 + u ⊗ v
−→
(Â + u ⊗ v)−1 = Â−1 −
(Â−1 · u) ⊗ (v · Â−1 )
,
1+λ
(B.85)
where
λ = v · Â−1 · u.
(B.86)
With this method we can calculate the inverse of a matrix with N 2 operations instead of N 3 operations, where N is the matrix size.
ˆ (k) )−1 , where (∆
ˆ (k) )ij =
In each Monte Carlo step we calculate and store the matrix M̂ (k) = (∆
V 2 g(τj0 − τi ). We discuss separately the following cases:
• shift operation I: f + (τn0 ) → f + (τen0 )
ˆ (k) as ∆(k)
We change the jth column of ∆
nj → ∆nj where n = 1, 2, ..., k. Thus,
u : un = ∆nj − ∆nj ,
(k)
(B.87)
v : vn = 1 if n = j, vn = 0 otherwise
(B.88)
in the formula (B.85). Furthermore,
λ=
(k)
(∆−1 )jl ∆lj − ∆lj
X
l
=
X
(∆−1 )jl ∆lj − 1 =
(k)
X
l
Mjl ∆lj − 1 ≡ λI − 1
(B.89)
l
in the formula (B.86). Using the formula (B.85) we obtain
!
Mnm
X
1
(k)
= (∆ )nm −
(∆−1 )nl ∆lj − ∆lj
(∆−1 )jm
(1 + λ) l
−1
=
(k)
Mnm
1
−
λI
X
(k)
Mnl ∆lj
!
(k)
− 0 Mjm
when n 6= j,
(B.90)
l
!
Mjm
X
1
(k)
= (∆ )jm −
(∆−1 )jl ∆lj − ∆lj
(∆−1 )jm
(1 + λ) l
−1
=
(k)
Mjm
1
−
λI
X
(k)
Mjl ∆lj
!
(k)
(k)
− 1 Mjm = Mjm −
l
22
1
(k)
(k) 1
(λI − 1) Mjm = Mjm (. B.91)
λI
λI
• shift operation II: f (τn ) → f (τen )
ˆ (k) as ∆(k)
We change the ith row of ∆
in → ∆in where n = 1, 2, ..., k. Thus,
u : un = 1 if n = i, un = 0 otherwise,
(k)
v : vn = ∆in − ∆in
(B.92)
(B.93)
in the formula (B.85). Furthermore,
λ=
X
(k)
∆il − ∆il
(∆−1 )li =
l
X
∆il (∆−1 )li − 1 =
X
l
(k)
∆il Mli − 1 ≡ λII − 1
(B.94)
l
in the formula (B.86). Using the formula (B.85) we obtain
Mnm
X
1
(k)
(∆−1 )ni
= (∆ )nm −
∆il − ∆il (∆−1 )lm
(1 + λ)
l
!
−1
!
1
(k)
(k) X
(k)
∆il Mlm − 0
= Mnm
−
Mni
λII
l
Mni
when m 6= i,
X
1
(k)
= (∆ )ni −
(∆−1 )ni
∆il − ∆il (∆−1 )li
(1 + λ)
l
(B.95)
!
−1
=
(k)
Mni
!
1
1
(k) X
(k)
(k) 1
(k)
(k)
Mni
Mni (λII − 1) = Mni (B.96)
.
∆il Mli − 1 = Mni −
−
λII
λII
λII
l
• segment addition (k → k + 1)
ˆ (k) as
We add ith row and jth column to the matrix ∆
ˆ (k) =
∆

.







.
. 0 .



. 0 . 
,

0 0 1 0 

.
(B.97)
. 0 .
ˆ (k) . Then, we add a new row i and column
which operation does not change the determinant of ∆
ˆ (k) and obtain ∆
ˆ (k+1)
j to ∆
by adding the new elements ∆i1 , ∆i2 , ...∆ik+1 and ∆1j , ∆2j , ...∆k+1j .
+ij
We consider this operation in two separate steps.
(k)
ˆ (k) as ∆nj
step 1: We change the jth column of ∆
→ ∆nj with n = 1, 2, ..., k + 1 similarly to
(k+1)
shift operation I, and obtain the matrix δ̂+ij
Thus,
u : un = ∆nj − ∆nj ,
(k)
(B.98)
v : vn = 1 if n = j, vn = 0 otherwise
(B.99)
in the formula (B.85). Parameter λ in formula (B.86) becomes
λ=
(k)
(∆−1 )jl ∆lj − ∆lj
X
(k)
= (∆−1 )ji ∆ij − ∆ij
l
23
= ∆ij − 1 ≡ λI − 1.
(B.100)
For later convenience we introduce the row vector R and column vector L as
Rt =
Ls =
k
X
l=1
k
X
(k)
∆il Mlt
(B.101)
(k)
(B.102)
Msl ∆lj .
l=1
Using expressions (B.90) and (B.91) we obtain
(k)
when n 6= j and m 6= i;
Mnm = Mnm
X
1
1
(k)
Mni = −
Mnl ∆lj = − Ln
when n 6= j;
λI l
λI
1
Mji =
.
λI
(k+1)
step 2: Now we change the ith row of δ̂+ij
ˆ (k+1)
the matrix ∆
+ij . Thus,
(B.103)
(B.104)
(B.105)
(k)
as ∆in → ∆in where n = 1, 2, ..., k, and obtain
u : un = 1 if n = i, un = 0 otherwise,
(k)
v : vn = ∆in − ∆in
(B.106)
(B.107)
in the formula (B.85). Parameter λ in formula (B.86) becomes
λ =
X
(k)
∆il − ∆il
(k+1)
((δ̂+ij )−1 )li = −
l
1 X
1 X
(k)
∆il − ∆il Mllk0 ∆l0 j = −
∆il Mllk0 ∆l0 j
λI ll0
λI ll0
≡ λII − 1,
(B.108)
where we used eq. (B.104). Using expressions (B.95) and (B.96) together with the expressions
(B.103), (B.104), and (B.105), we obtain
!
Mnm =
Mni =
Mjm =
Mji =
=
!
X
1
1 X (k)
1 1
(k)
(k)
−
∆il Mlm = Mnm
Ln Rm
Mnl ∆lj
+
−
λII
λI l
λII λI
l
when n 6= j and m 6= i;
(B.109)
1
1
1
1 1
− Ln (λII − 1) = −
Ln
when n 6= j;
(B.110)
− Ln −
λI
λII
λI
λI λII
!
X
1
1 1
(k+1)
(k)
(k+1)
0−
((δ̂+ij )−1 )ji
∆il − ∆il ((δ̂+ij )−1 )lm = −
Rm
λII
λII λI
l
when m 6= i;
(B.111)
!
X
1
1
1
1 1
(k+1)
(k)
(k+1)
−
((δ̂+ij )−1 )ji
∆il − ∆il ((δ̂+ij )−1 )li =
−
(λII − 1)
λI
λII
λI
λII λI
l
1 1
.
(B.112)
λI λII
(k)
Mnm
24
Thus, we obtained the inverse of the new matrix as

−L(i) /λ+
M̂ 0


M̂ k+1 =  −R(j) /λ+

(k)


1/λ+
−L(i) /λ+
M̂ 0

M̂ 0
,
−R(j) /λ+ 

(B.113)
M̂ 0
(i)
where Mts0 = Mts + Lt Rs(j) /λ+ , and we introduced the notation
λI λII = ∆ij −
X
(k)
∆sj Mts ∆it ≡ λ+ .
(B.114)
st
We can also obtain the determinant ratio as
(k+1)
ˆ +ij
det ∆
1
= (−1)i+j (k+1) = (−1)i+j λ+ ,
(k)
ˆ
det ∆
M
(B.115)
ji
where we used the relation
(k)
(k−1)
(k+1)
ˆ (k) Mji = (−1)i+j det ∆
ˆ
det ∆
6=ij
(k+1)
ˆ +ij Mji
−→ det ∆
ˆ (k) .
= (−1)i+j det ∆
(B.116)
• segment removal (k → k − 1)
ˆ (k) and obtain ∆
ˆ (k−1) . The inverse of matrix
We erase the ith row and jth column from ∆
6=ij
M̂ (k−1) is given by
(k)
(k−1)
Mst
=
(k)
Mst
(k)
M M
− si jt ,
λ−
(B.117)
(k)
where λ− = Mji .
The determinant ratio is obtained in this case as
(k−1)
ˆ
det ∆
1
1
(k)
6=ij
=
Mji =
λ− = (−1)i+j λ− .
i+j
i+j
(k)
ˆ
(−1)
(−1)
det ∆
C
(B.118)
Padé approximation
In CT-QMC we obtain the Green’s function as a function of imaginary time τ . Thus, we have to
perform Fourier transform (FT) to obtain the Green’s function in the Matsubara representation
G(iεn ) =
Z β
dτ eiεn τ G(τ ),
(C.119)
0
and then make analytic continuation to obtain the Green’s function along the real frequency axis
G(iεn ) → G(ω + iδ).
25
(C.120)
To perform analytic continuation, we use the Padé approximation. The problem is the following. We
know the values of a complex function C at N complex points zi : C(z1 ) = u1 , C(z2 ), = u2 ...C(zN ) =
uN , and we want to obtain C at real points. The Padé approximant is expressed as a rational function
CN (z) =
AN (z)
,
BN (z)
(C.121)
where AN (z) and BN (z) are polynomials of z of order (N − 1)/2 and (N − 1)/2 when N is odd, and
(N − 2)/2 and N/2 when N is even [8]. We express CN (z) as continued fraction:
CN (z) =
a1 a2 (z − z1 ) aN (z − zN −1 )
a1
,
...
=
a
(z−z1 )
1+
1+
1+
1 + 21+...
(C.122)
where the coefficients ai are determined that CN (zn ) = un by the recursion formulas
ai = gi (zi ), g1 (zi ) = ui , gk (z) =
gk−1 (zk−1 ) − gk−1 (z)
.
(z − zk−1 )gk−1 (z)
(C.123)
Then, CN (z) at arbitrary z is evaluated as
CN (z) =
AN (z)
,
BN (z)
(C.124)
where
An+1 (z) = An (z) + (z − zn )an+1 An−1 (z)
(C.125)
Bn+1 (z) = Bn (z) + (z − zn )an+1 Bn−1 (z)
(C.126)
with A0 = 0, A1 = a1 , B0 = B1 = 0.
26