Continuous-time QMC Solvers for Electronic Systems in
Fermionic and Bosonic Baths
F.F. Assaad
(DMFT@25, Juelich, September 2014)
Goal: Detailed overview of CT-INT and CT-AUX (CT-HYB à See notes)
Outline
Ø  Weak coupling CT-QMC – Basics.
– Add ons: Retarded interactions:
phonon degrees of freedom.
Ø  Selected applications. Topological insulators, electron-phonon interaction.
Ø  Conclusions.
I. Weak coupling CT-QMC (CT-INT).
H = H 0 + U d↑†d↑ d↓†d↓
!!
Consider:
H0 is a single body Hamiltonian
n↑
n↑
G σ (τ 2 ,τ 1 ) = T dσ† (τ 2 )dσ (τ 1 )
We would like to compute for example:
Prior knowledge:
Examples:
σ
0
G (τ 2 ,τ 1 ) = T d (τ 2 )dσ (τ 1 )
†
σ
0
We will assume time reversal
symmetry such that G0 is diagonal
in spin degrees of freedom.
Single impurity Anderson model ( à DMFT)
Magnetic impurity on the edge of a topological insulator (Important for understanding
spin transport)
H0
2
Ud d d d
r
n
†
†
↑ ↑ ↓ ↓
V
t
a2
a1
F. Goth et al. Phys. Rev. B (2013)
I. Weak coupling CT-QMC (CT-INT).
Dyson. Expansion around U=0.
Tr ⎡⎣e − βH ⎤⎦
Tr ⎡e
⎣
Wick
n=1
− βH 0
⎤
⎦
=
τ n −1
β
∑ ∫ dτ
n
0

1
∫
(Notes à Path integral formulation
Here à More pedestrian second quantized formulation)
d τ n (−U )n
n↑ (τ 1 )n↓ (τ 1 )n↑ (τ n )n↓ (τ n )
0
0
G0↑
⎛ G0↑ (τ 1 ,τ 1 )
⎞
0
U = −U det ⎜
⎟ ≡ −U det ⎡⎣ M 1 (τ 1 )⎤⎦
↓
⎜
⎟
0
G
(
τ
,
τ
)
↓
0
1 1 ⎠
⎝
G
σ
0
G0 (τ 2 ,τ 1 ) = T d̂ σ+ (τ 2 ) d̂ σ (τ 1 )
0
I. Weak coupling CT-QMC (CT-INT).
Dyson. Expansion around U=0.
Tr ⎡⎣e − βH ⎤⎦
Tr ⎡e
⎣
− βH 0
⎤
⎦
=
τ n −1
β
∑ ∫ dτ
n
0

1
∫
d τ n (−U )n
n↑ (τ 1 )n↓ (τ 1 )n↑ (τ n )n↓ (τ n )
0
0
Wick
n=2
+
+
= U 2 det ⎡⎣ M 2 (τ 1 ,τ 2 ) ⎤⎦
+
⎛ Gσ (τ ,τ ) Gσ (τ ,τ )
0
1 1
0
1 2
det ⎡⎣ M 2 (τ 1 ,τ 2 ) ⎤⎦ = ∏ det ⎜ σ
⎜⎝ G0 (τ 2 ,τ 1 ) G0σ (τ 2 ,τ 2 )
σ
⎞
⎟
⎟⎠
Sum of connected &
disconnected diagrams
Tr ⎡⎣e − βH ⎤⎦
Tr ⎡e
⎣
− βH 0
⎤
⎦
=
∑
n
β
τ n −1
0
0
∫ dτ 1 
∫
(
)
d τ n (−U )n det ⎡⎣M n τ 1,, τ n ⎤⎦
Sum with Monte Carlo
Weight
with
⎛ Gσ (τ ,τ ) ! Gσ (τ ,τ )
0
1
n
⎜ 0 1 1
"
#
"
det ⎡⎣ M n (τ 1 ,τ 2 ,!,τ n ) ⎤⎦ = ∏ det ⎜
σ
⎜ Gσ (τ ,τ ) … Gσ (τ ,τ )
0
n
n
⎝ 0 n 1
⎞
⎟
⎟
⎟
⎠
= Sum over all connected and disconnected diagrams at order n.
We can now see how to define the configuration space
let C=:{𝜏1, 𝜏2,…, 𝜏n}, and we will carry out the sum over C with Monte Carlo
importance sampling.
Tr ⎡⎣e − βH ⎤⎦
Tr ⎡e
⎣
− βH 0
⎤
⎦
=
∑
n
β
τ n −1
0
0
∫ dτ 1 
∫
(
)
d τ n (−U )n det ⎡⎣M n τ 1,, τ n ⎤⎦
Sum with Monte Carlo
Weight
Weight / Sign.
Ø 
HU
U
K
d
d
→
−
1/
2
−
δ
−
1/
2
+
δ
=
−
n
s
n
s
[
]
[
]
)( ↓
)
∑ ( ↑
2 s =±1
2β
K =U β (δ 2 − 1/ 4), cosh(α ) − 1 =
1
,
2(δ 2 − 1/ 4)
e
∑
s
δ > 1/2
à  New dynamical variable s. Exact mapping onto CT-AUX (K. Mikelsons et al. PRE 09)
(Rombouts et al. PRL 99, Gull et. al EPL 08)
à Sign problem behaves as in Hirsch-Fye, and auxiliary field methods
(Absent for one-dimensional chains, particle-hole symmetry, impurity models)
(
sα n↑d −n↓d
)
Tr ⎡⎣e − βH ⎤⎦
Tr ⎡e
⎣
− βH 0
⎤
⎦
=
∑
n
β
τ n −1
0
0
∫ dτ 1 
∫
(
)
d τ n (−U )n det ⎡⎣M n τ 1,, τ n ⎤⎦
Sum with Monte Carlo
Weight
Weight / Sign.
Ø 
HU
U
→
∑
2 s=±1
(n
d
↑
)(
− ⎡⎣1 / 2 − sδ ⎤⎦ n
K =U β (δ 2 − 1/ 4), cosh(α ) − 1 =
Ø 
d
↓
)
K
− ⎡⎣1 / 2 + sδ ⎤⎦ = −
2β
1
,
2(δ 2 − 1/ 4)
∑e (
sα n↑d −n↓d
s
δ > 1/2
HU =U (n↑d − [1/2 − δ ]) (n↓d − [1 / 2 + δ ]) + U δ (n↑d − n↓d )
Absorb in H0
Ø 
Particle-Hole symmetry
δ =0
and only even powers of n occur in expansion.
)
Tr ⎡⎣e − βH ⎤⎦
Tr ⎡e
⎣
− βH 0
⎤
⎦
τ n −1
β
=
∑ ∫ dτ ∑
n
1
0
s1

∫ dτ ∑
n
0
sn
Sum with Monte Carlo
(
n
⎛ U⎞
⎡
⎤
⎜⎝ − 2 ⎟⎠ det ⎣M n τ 1,s1 , τ n ,sn ⎦
(
)
Weight
)
det ⎡⎣M n τ 1,s1,!, τ n ,sn ⎤⎦ =
T ⎡⎣n↑ (τ 1 ) − α ↑ (s1 )⎤⎦ ⎡⎣n↓ (τ 1 ) − α ↓ (s1 )⎤⎦ ! ⎡⎣n↑ (τ n ) − α ↑ (sn )⎤⎦ ⎡⎣n↓ (τ n ) − α ↓ (sn )⎤⎦
σ
⎛ G σ (τ , τ ) − α (s ) !
G
(τ 1, τ n )
0
1
1
σ
1
0
⎜
det ⎜
"
#
"
∏
σ
σ
σ
⎜
G
(
τ
,
τ
)
…
G
(τ n , τ n ) − α σ (sn )
0
n
1
0
⎝
Note: α σ (s) = ⎡⎣1 / 2 − σ sδ ⎤⎦
⎞
⎟
⎟
⎟
⎠
0
=
Tr ⎡⎣e − βH ⎤⎦
Tr ⎡e
⎣
− βH 0
⎤
⎦
τ n −1
β
=
∑ ∫ dτ ∑
n
1
0
s1

∫ dτ ∑
n
0
sn
n
⎛ U⎞
⎡
⎤
⎜⎝ − 2 ⎟⎠ det ⎣M n τ 1,s1 , τ n ,sn ⎦
Sum with Monte Carlo
(
)
Weight
Importance sampling (See notes Appendix A)
Configuration C: set of n-vertices at imaginary times
⎡⎣τ 1 ,s1 ⎤⎦ ⎡⎣τ 2 ,s2 ⎤⎦ ⎡⎣τ 3 ,s3 ⎤⎦
⎡⎣τ 1 ,s1 ⎤⎦ ⎡⎣τ 2 ,s2 ⎤⎦, ⎡⎣τ n ,sn ⎤⎦
0
of models considered, we have not noticed substantial
differences in performance between static and dynamical
For the Hubbard m
H
of Ising
fields. τ nWe
however favorn the dynamical
β
Eq. (2) we obtain:
−1
⎤
Tr ⎡⎣e − βchoices
⎛ U⎞
⎦
⎡M invariversion
sincedit
keep the
= ∑
τ 1 allows
 one
d τton ∑
− SU(2)
detspin
τ 1,s1 , τ n ,sn ⎤⎦
∑
∫
∫
(*
n
⎜
⎟
− βH 0
⎣
2
⎡
⎤
⎝
⎠
s1
n of0 the non-interacting
sn Hamiltonian Ĥ .
Tr e ant form
0
0
⟨n⟩ = −βU
⟨
⎣
⎦
(
Sum with Monte Carlo
)
i
Weight
Using the same te
methods [9] the C
Importance sampling (See notes Appendix A)
acceptance
probab
In principle
moves,at the
addition
and ⎡removal
of
⎤
⎡
⎤
⎡
⎤
τ
,s
τ
,s
τ
,s
,
Configuration
C: set of two
n-vertices
imaginary
times
n⎦
⎣ 1 1 ⎦ ⎣ 2 2 ⎦ or
⎣ nremoval
of a ve
Hubbard vertices, are sufficient [16]. In the Metropolis
Eq. (19) a sweep
scheme,
the acceptance
for Carlo)
a given
move
reads
Importance
sampling
produces aratio
(Monte
time
series
of configurations.
results in an effort
'
& 0
TC ′ →C W (C ′ )
Mσ−1 is far better
Transition Prob. PC→C ′ = min
, CN-1, ……, C2, C1
, 1 . à CN(16)
0
minantal methods
TC→C
W
(C)
′
from scratch after
0
W (C)
where
T
toCthe
probability
of proposP(C) = fort of the order n
C ′ →C
In this time series,
thecorresponds
configuration
occurs
with probability:
W (C)
∑limiting
ing a move from configuration C ′ to configuration C and
factors th
B.
Monte Carlo Sampling
C
O
∑ W (C)O(C)
=
∑ W (C)
C
C
=
N
1
∑ O(Cn ) ±
N n=1
(O −
O
N
)
2
Assumptions
1)  The moves are ergodic
2)  The configurations Ci are statistically independent
3)  The variance exits
à Irrespective on the dimension of the configuration
space the precision scales as 1/ CPU
See notes, Appendix A
Tr ⎡⎣e − βH ⎤⎦
Tr ⎡e
⎣
− βH 0
⎤
⎦
τ n −1
β
=
∑ ∫ dτ ∑
n
1
0
s1

∫ dτ ∑
n
0
sn
n
⎛ U⎞
⎡
⎤
⎜⎝ − 2 ⎟⎠ det ⎣M n τ 1,s1 , τ n ,sn ⎦
Sum with Monte Carlo
(
)
Weight
Importance sampling (See notes Appendix A)
à Importance of high precision tests. L=2, βt=10, U/t=2
Double occupancy:
0.378765 +/- 0.000050,
Double occupancy:
0.378564 +/- 0.000027
Exact: 0.378732
(Bad random number generator!)
O
∑ W (C)O(C)
=
∑ W (C)
C
C
=
N
1
∑ O(Cn ) ±
N n=1
(O −
O
N
)
2
Assumptions
1)  The moves are ergodic
2)  The configurations Ci are statistically independent
3)  The variance exits
See notes, Appendix A
ant form of the non-interacting Hamiltonian Ĥ0 .
Tr ⎡⎣e − βH ⎤⎦
Tr ⎡e
⎣
− βH 0
⎤
⎦
τ n −1
β
=
∑ ∫ dτ ∑
n
1
0
s1

∫ dτ ∑
n
0
sn
n
⎛ U⎞
⎡
⎤
B. Monte
Carlo Sampling
⎜⎝ − 2 ⎟⎠ det ⎣M n τ 1,s1 , τ n ,sn ⎦
(
)
In principle two moves, the addition and removal
Weight
Hubbard vertices,
are sufficient [16]. In the Metrop
scheme, the acceptance ratio for a given move reads
Importance sampling (See notes Appendix A)
'
& 0
′
TC ′ →C W (C )
PC→C ′ = min
,1 .
(
0
TC→C ′ W (C)
Sum with Monte Carlo
Adding
1 1
T0 (Cn → Cn+1 )where
=
β 2
TC0 ′ →C corresponds to the probability of prop
ing a move from configuration C ′ to configuration C a
[τ1, s1 ] [τ 2 , s2 ] [τ 3 , s3 ]
⎡⎣τ 1 ,s1 ⎤⎦ ⎡⎣τ 2 ,s2 ⎤⎦ ⎡⎣τ 3 ,s3 ⎤⎦ ,
Removing
T0 (Cn → Cn−1 ) =
[τ1, s1 ] [τ 2 , s2 ] [τ 3 , s3 ]
1
n
⎡⎣τ 1 ,s1 ⎤⎦ ⎡⎣τ 2 ,s2 ⎤⎦
⎡⎣τ 4 ,s4 ⎤⎦
MC
ant form of the non-interacting Hamiltonian Ĥ0 .
Tr ⎡⎣e − βH ⎤⎦
Tr ⎡e
⎣
− βH 0
⎤
⎦
τ n −1
β
=
∑ ∫ dτ ∑
n
1
0
s1

∫ dτ ∑
n
0
sn
n
⎛ U⎞
⎡
⎤
B. Monte
Carlo Sampling
⎜⎝ − 2 ⎟⎠ det ⎣M n τ 1,s1 , τ n ,sn ⎦
(
)
7.11
corresponds to the probability of choosing vertex
i undertwo
the moves,
assumption
that eachand
vertex
In principle
the addition
removal
Sum with Monte Carlo
Weight
Hubbard
vertices,
sufficientacceptance
[16]. In the
Metrop
ally probable. As shown in Appendix A (see
Eq. (132)
), the are
Metropolis
reads
the acceptance
ratio for a given move reads
✓ scheme,
◆
0
0
Importance sampling (See notes Appendix
TC 0 !C WA)
(C )
'
& 0
′
,
1
.
(44)
PC!C 0 = min
TC ′ →C W (C )
0
TC!C 0 W (C)
PC→C ′ = min
,1 .
(
0
TC→C ′ W (C)
PCn !Cn+1 = min
PCn+1 !Cn = min
✓
✓
where
Q TC0 ′→C corresponds◆to the probability of prop
′
detfrom
M (C
U ing a move
n+1 )
configuration
C
to configuration C a
Q
(n + 1)
det M (Cn )
,1
Q
◆
det M (Cn )
(n + 1)
Q
,1 .
U
det M (Cn+1 )
hat in our Thus,
formulation
theorordering
the vertices
since
to accept
remove aofmove
we have is
toimportant
compute the
ratiowe
of have
two defined the
R
R
determinants.
ation without
time-ordering 0 d⌧1 · · · 0 d⌧n as opposed to a time-ordered formulation
R ⌧n 1
R ⌧1
d⌧n . The reader is encouraged to show that both formulations lead to
1 0 d⌧2 · · · 0
me acceptance/rejection ratios. In practical implementations one will also include a move
(17)
)fdeterminant
(⌧ )ii = identity
hhT
f (⌧
C
he ratio of the two determinants, we
the
nuse
det M (CnC)n det M (Cn )
1
u
(19)
det(A
+
u
⌦
v)
=
det(A)
1
+
v
·
A
where
1
0
1
0
Useful identities:
0
g0 (⌧1 , ⌧ 0 )
g
(⌧
,
⌧
)
0
1
Sherman-Morrison formula
C
B
C
B
..
.
C
B
.. 1 C
BM (Cn )
.
M 1(Cn )
C.
B
B
(18)
C.
B
u ⌦ vA
1 (Cn ) =
1 BA
B (C
(18)
)
=
C
n
n×n
n
n
0
C
B
(20), v ∈!
=A
g0 (⌧nA, ∈!
⌧ )A , u ∈!
(1) (A + u ⌦ v)
@ 1 + v ·gA
1u
, ⌧ 0 )A
@
0 (⌧n
u ⌧⊗0 )v∈! n×n , (u ⊗ v)i, j = ui v j
(⌧,
⌧
)
.
.
.
g
g
0(⌧, ⌧n ) g0 (⌧,
0
1
0
g (⌧, ⌧ ) . . . g (⌧, ⌧ ) g (⌧, ⌧ )
0
1
0
n
0
u i v = identity
ui vi
To compute the ratio of the two determinants, we use the determinant
∑
he1 ratio
two
we use the determinant identity
i
⌦ v1 of
+ the
u2 ⌦
v2determinants,
)=
u
(21)
1
⇤
⇥
det(A
1
+
v
·
A
1
1 + u ⌦ v) = det(A)
11
1 u
1+v ·A u
v ·A u
v ·A u
,
A) 1 + v · A u
(2)1
det(A + u ⌦ v) = det(A) 1 + v · A u
1
2
2
2
1
1
2
(19)
(19)
P
as
well
as
the
Sherman-Morrison
formula
er product is given by (u ⌦ v) = ui vj and the scalar product by u·v = i ui vi .
e Sherman-Morrison formula i,j
⌦ vA
(19)1can equally
be formally derived by Taylor-expanding (1 +1A 1 u 1⌦ v)A 11. uEq.
(A + u ⌦Av)1 u =
(20)
A
1
1u
⌦
vA
1
1
1
+
v
·
A
emonstrated
by
using
fact
that
(A
+and
u the
⌦
v)
(20)
= Adet(A) = exp Tr 1log (A).
From
(1)
(2)
follows
1
+
v
·
A
u
the B to(Cobtain
n ) matrix as
1
0
det (A + u1 ⌦ v1 + u2 ⌦ v02 ) =
C
B ⇥
(21)
⇤
.
v
)
=
+ u1 ⌦ v1 + u2 ⌦
1
1
1
1
.
C
B
2
.C
(C1n+) v1 · A u
M
1u
+ v⌦
v1 ·(22)
A u2 ,
det(A)
1 +
2 ·vA +u
2 ⌦ vv2 · A u1
B
B
(C
)
=
u
(21)
n
1
1
2
2
⇤
⇥
B1
10C
1
1
P
A
@
1 + vis2 given
v2 · =
,
· Aouter
u1product
· A by
u2(u ⌦ v)
Au v
u1 andvthe
u
(A) 1where
+ v1 the
1·A
2
scalar product by u·v = i ui vi .
i j
i,j
0
...0 1
P
1
1
u
⌦
v)
can equally
Eq.
(20)
can
be
formally
derived
by
Taylor-expanding
(1
+
A
ter product is given by (u ⌦ v)i,j = ui vj and the scalar product by u·v = . iEq.
ui v(19)
i.
be formally
demonstrated
by using the
that1 u
det(A)
1 exp Tr log (A).
⌦ v) =
. Eq. (19) can equally
be formally
derived
by Taylor-expanding
(1fact
+A
Decomposing the B (Cn ) matrix as
demonstrated by using the fact that det(A) = exp Tr log (A).
The Monte
dynamics
relies
on the calculation
ofEq.
ratios
of For
determinants.
Suchvertex
ratiosaddition
can
beCarlo
computed
using the
determinant
identities of
(21).
instance, under
we will
have
compute for identities
each spin sector
be computed
using
thetodeterminant
of Eq. (21). For instance, under vertex addition
1
0
we will
have toacompute
Adding
Vertex for each spin sector
g0 (⌧1 , ⌧ 0 )
1 C
0
B
.
C
B M (Cn )
0 ..
g0 (⌧1 , ⌧ )
C
B
det
B
C
B
0
..g0 (⌧n , ⌧ ) C C
A
@ )
B M (C
.0 0
n
C0
B
0
0
det
(s )
g0 (⌧ , ⌧1 ) . . . g0 (⌧ , ⌧n ) g0 (⌧ , ⌧ 0) ↵ C
det M (Cn+1 )B
g0 (⌧n , ⌧ )
A
@=
det M (Cn )
det M (C )
g0 (⌧ 0 , ⌧1 ) . . . g0 (⌧ 0 , ⌧n ) g0 (⌧n0 , ⌧ 0 ) ↵ (s0 )
det M (Cn+1 )
1
00
1
=
det M (Cn )
det M (Cn )0
BB
C
.. C
C
BBM (Cn )
C
1
1
.
C + u1 ⌦ v1 + u2 ⌦ v2 C
B
det B
0
C
BB
C
0A
A
C
BB @@
C
.. C
BBM (Cn ) 0
C
. .. C
.0 1
B
C
det B
C + u1 ⌦ v1 + u2 ⌦ v2 C
B
=B
0det
A M (Cn )
@@
A
00
7.12 =
7.12
with
1 + v1 · M
= 1and
+ v1 · M
with
and
1
1
=n )u1
(C
0
1 + v2 · M
1
...0 1
(Cn )u2
v ·M
det M (Cn2 )
1
(Cn )u1
v1 · M
1
1 0
0
0
1 1+
v1 )·i M
(C
(C
= 1(g0 (⌧
, ⌧v
),2 .·.M
. , g0 (⌧
),2g0 (⌧ 0 , ⌧v02) · M
↵ (s(C
) n )u
1)1, (v
=
un1 )u
n ,n⌧)u
(u2 )i =
i,n+1 ,
1
Fakher
F.
(C
n )u2 .
Assaad
Fakher F. Assaad
1
(C )u2 .
i,n+1n
v2 = (g0 (⌧ 0 , ⌧1 ), . . . , g0 (⌧ 0 , ⌧n ), 0).
0 0
0
0
⌧1)), . . 1)
. , g, 0 (⌧
, g20)(⌧
), g,0 (⌧v02, ⌧=0 )(g0 (⌧
↵ ,(s
(v,1⌧)ni ),
=0).i,n+1
u1 = (g0 (⌧1 , ⌧ 0 ), . . .(u
i =
n , ⌧i,n+1
Carrying out the calculation yields
Carrying out the calculation yields
detdet
MM(C(C
n+1 )
0
0
n+1 )= g0 (⌧ 0, ⌧ 0)
= g0 (⌧ , ⌧ )
detdet
MM(C(C
n )n )
n
X
n
⇥
⇤
X
0 ⇥
1 ⇤
0
↵
(s
g
(⌧
(C
)
g
(⌧
0 )
0 , ⌧i ) M
1
0 j, ⌧ ) .
0
n
0
↵ (s )
g0 (⌧ , ⌧i ) M (Cn ) i,j g0i,j
(⌧j , ⌧ ) .
(45) (45)
0
i,j=1
i,j=1
(Cnn))isisknown,
known,computing
computing
ratio
involves
n2 operations.
Hence,
provided
that
thethe
ratio
involves
n2 operations.
Hence,
provided
thatthe
thematrix
matrix M 11(C
-1
2
If M (Cn) is known then computing the ratio involves n operations.
th
vertex
of theof the
vertex
removaltakes
takes aa very
very simple
simple form.
that
wewe
remove
the n
TheThe
vertex
removal
form. Assume
Assume
that
remove
the
nth vertex
. Thenfor
foraa given
given spin
have
to compute
configuration
spin sector,
sector,we
wewill
will
have
to compute
configuration
CnC.nThen
Hence, provided that the matrix M (Cnn) is known, computing the ratio involves n operations.
vertex of
of the
the
The vertex
vertex removal
removal takes
takes aa very
very simple
simple form.
form. Assume
Assume that
that we
we remove
remove the
the nnthth vertex
The
Thenfor
foraagiven
givenspin
spinsector,
sector,we
wewill
willhave
havetotocompute
compute
configuration
Removing
a Vertex
configuration
CCnn. .Then
0
1
0
1
(⌧1, ,⌧⌧1)) ↵↵ (s
(s11)) . .. .. .
(⌧11, ,⌧⌧nn 11))
gg0(⌧
gg00(⌧
00
B 0 1 1 ..
C
B
.. ..C
.. ..
.
B
C
.
B
C
.
.
.
.
.
.
B
C
C
detB
det
B
C
B
C
(⌧nn 11, ,⌧⌧11))
(⌧nn 11, ,⌧⌧nn 11)) ↵↵ (s
(snn 11)) 00A
@ gg00(⌧
A
. .. .. . gg00(⌧
@
00
. .. .. .
00
11
detM
M (C
(Cnn 11))
det
==
detM
M (C
(Cnn))
detM
M (C
(Cnn))
det
det
==
with
with
and
and
det[M
[M (C
(Cnn))++uu11⌦⌦vv11++uu22⌦⌦vv22] ]
det
detM
M (C
(Cnn))
det
⇣⇣
⌘⌘
[M (C
(Cnn)])]1,n
[M (C
(Cnn)])]n,n
(v11))i i==
uu11==
[M
1,n, ,. .. .. ., ,[M
n,n 11 , , (v
(46)
(46)
i,n
i,n
⇣⇣
⌘⌘
=
[M (C
(Cnn)])]n,1
[M (C
(Cnn)])]n,n
(u22))i i== i,n
[M
(u
i,n, , vv22=
n,1, ,. .. .. ., ,[M
n,n 11, ,00 . .
Evaluatingthe
theabove
abovegives
gives
Evaluating
(Cnn 11)) ⇥⇥ 11
detM
M (C
⇤⇤
det
M (C
(Cnn)) n,n. .
== M
n,n
detM
M (C
(Cnn))
det
(47)
(47)
(Cnn))atathand,
hand,the
thecomputational
computationalcost
costfor
forcomputcomputAgain,provided
providedthat
thatwe
wehave
havethe
thematrix
matrixM
M 11(C
Again,
-1
If M (Cn) is known then computing the ratio is negligible.
ingthe
theratio
ratiofor
forvertex
vertexremoval
removalisisnegligible.
negligible.
ing
Havingcomputed
computedthe
theratio
ratioof
ofdeterminants,
determinants,we
wecan
cancompute
computethe
theacceptance
acceptanceprobability,
probability,draw
draw
Having
pseudo-randomnumber,
number,and
andaccept
acceptor
orreject
rejectthe
themove.
move. IfIfaccepted,
accepted,we
wewill
willhave
havetotoupgrade
upgrade
aapseudo-random
11
det M (Cn )
0
−1
Upgrading M σ (Cn±1 )
0
1
g0 (⌧1 , ⌧ )
C
B
..
C
BM (Cn )
.
C.
B
B (Cn ) = B
0 C−1
g
(⌧
,
⌧
)M
If the move is accepted
then you have to upgrade
Aσ (Cn±1 )
@
0 n
g0 (⌧, ⌧1 ) . . . g0 (⌧, ⌧n ) g0 (⌧, ⌧ 0 )
(18)
±
mpute the ratio
use
Butof the
M two
(C determinants,
) = M (Cwe
)+
u ±the
⊗determinant
v ± + u ± ⊗ videntity
and
σ
n±1
σ
n
1
−1
det(A
+isu present
⌦ v) =indet(A)
1
memory.
σ
n
M (C )
1
2
2
u
(19)
A 1 u ⌦ vA 1
1 + v · A 1u
recursively to carry out
(20)
+v·A
1
as the Sherman-Morrison formula
in
à  Use
(A + u ⌦ v)
1
=A
1
the update.
t (A + u1 ⌦ v1 + u2 ⌦ v2 ) =
⇥à Required number
2
v2 · A 1 un2
det(A) 1 + v1 · A 1 u1 1of+operations
v2 · A
1
u1
v1 · A
1
u2
⇤
,
P
(21)
the outer product is given by (u ⌦ v)i,j = ui vj and the scalar product by u·v = i ui vi .
0) can be formally derived by Taylor-expanding (1 + A 1 u ⌦ v) 1 . Eq. (19) can equally
mally demonstrated by using the fact that det(A) = exp Tr log (A).
posing the B (Cn ) matrix as
1
0
2.6
Average expansion parameter
What is the average expansion parameter?
A crucial issue concerns the average expansion parameter hni since it will determine the average
size of the matrix M (Cn ). The computational effort to visit each vertex – a sweep – will then
Consider H =3 H0 + H1
scale as hni . For a general interaction term Ĥ1 , the average expansion parameter is [6]
hni =
(m=n-1)
=
=
Z
Z
1 X ( 1)n n
d⌧1
···
d⌧n hT Ĥ1 (⌧1 ) · · · Ĥ1 (⌧n )i0
Z n
n!
0
0
Z
Z
Z
1 X ( 1)m
d⌧1 · · ·
d⌧m
d⌧ hT Ĥ1 (⌧1 ) · · · Ĥ1 (⌧m )Ĥ1 (⌧ )i0
Z m m!
0
0
0
Z
d⌧ hĤ1 (⌧ )i .
(49)
0
For the Hubbard model, replacing Ĥ1 by the form of Eq. (7), we obtain
hni =
⇥
U h(n̂"
1/2)(n̂#
1/2)i
2
⇤
,
(50)
where we have set ⇢ = 1/2. Thus, the computational time for a sweep scales as in the HirschFye approach, namely as ( U )3 . The algorithm can be used for lattice models with N correlated
sites. In this case, the computational time for a sweep scales as (N U )3 , which is more expensive than the auxiliary-field approach, which scales as U N 3 . As mentioned in the introduction,
the advantage of the CT-INT method lies in the fact that it is action-based such that fermionic
and bosonic baths can be easily implemented.
2.6
Average expansion parameter
crucial issueWhat
concerns
average
expansion
parameter hni since it will determine the average
is the the
average
expansion
parameter?
A crucial issue concerns the average expansion parameter hni since it will determine the average
e of the matrix
M (Cn ). The computational effort to visit each vertex – a sweep – will then
size of the matrix M (Cn ). The computational effort to visit each vertex – a sweep – will then
H =3 H0 +interaction
H1
. scale
For aasgeneral
term Ĥ
the
average expansion parameter is [6]
le as hni3Consider
1 , Ĥ
interaction
term
hni . For a general
1 , the average expansion parameter is [6]
Z( 1)n n
Z
1 nX
1X
(
1)
n
d⌧1
···
d⌧n hT Ĥ1 (⌧1 ) · · · Ĥ1 (⌧n )i0
hni =
d⌧
· · · Ĥ1 (⌧n )i0
hni =
Z n
n!d⌧1 0 · · ·
0 n hT Ĥ1 (⌧1 )
Z n
n! X 0 m Z
Z
0 Z
1
( 1)
(m=n-1)
Zm d⌧ hT Ĥ1 (⌧1 ) · · · Ĥ1 (⌧m )Ĥ1 (⌧ )i0
Z
Z1 · · · d⌧
=
d⌧
m
X
Z m m!
1
( 1)
0
0
0
=
d⌧
·
·
·
d⌧
d⌧ hT Ĥ1 (⌧1 ) · · · Ĥ1 (⌧m )Ĥ1 (⌧ )i0
Z
1
m
Z m = m! d⌧0hĤ (⌧ )i . 0
0
(49)
1
Z
0
= For the Hubbard
d⌧ hĤ1 (⌧
)i . replacing Ĥ1 by the form of Eq. (7), we obtain
model,
Z
For
0
(
Z
)(
)
⇤
⇥
U
d
d
2
⎡
⎤
⎡
⎤
,
1/2)(n̂
1/2)i
hni
=
U
h(n̂
n
−
1
/
2
−
s
δ
n
−
1
/
2
+
s
δ
"⎦
#⎣
∑
↑
↓
⎣
⎦
r the Hubbard model,2replacing
Ĥ1 by the form of Eq. (7), we obtain
s=±1
H 1 = HU =
(49)
(50)
where we have set ⇢ = 1/2. Thus, the computational time for a sweep scales as in the Hirsch⇤
⇥
Fye approach, namely as ( U )3 . The algorithm can be used for lattice2 models with N correlated
,
(50)
hni =
U h(n̂" 1/2)(n̂# 1/2)i
sites. In this case, the computational time for a sweep scales as (N U )3 , which is more expensive than the auxiliary-field approach, which scales as U N 3 . As mentioned in the introduction,
ere we havetheset
⇢ = 1/2.
themethod
computational
a sweep3 scales
asfermionic
in the Hirschadvantage
of theThus,
CT-INT
lies in the facttime
that itfor
is 3action-based
such that
à Time to update
each
vertex
(a
sweep)
~
n
~
(𝛽U)
bosonic
easily
implemented.
algorithm
can be used for lattice models with N correlated
e approach,and
namely
asbaths
( Ucan
)3 .beThe
es. In this case, the computational time for a sweep scales as (N U )3 , which is more expen-
crucial issueWhat
concerns
average
expansion
parameter hni since it will determine the average
is the the
average
expansion
parameter?
e of the matrix M (Cn ). The computational effort to visit each vertex – a sweep – will then
le as hni3 . For a general interaction term Ĥ1 , the average expansion parameter is [6]
hni =
=
=
For
Z
Z
1 X ( 1)n n
d⌧1
···
d⌧n hT Ĥ1 (⌧1 ) · · · Ĥ1 (⌧n )i0
Z n
n!
0
0
Z
Z
m Z
X
1
( 1)
d⌧1 · · ·
d⌧m
d⌧ hT Ĥ1 (⌧1 ) · · · Ĥ1 (⌧m )Ĥ1 (⌧ )i0
parameter.
Z m m!
0Histogram of expansion
0
0
Z
d⌧ hĤ1 (⌧ )i .
0
(
)(
(49)
)
U
n↑d − ⎡⎣1 / 2 − sδ ⎤⎦ n↓d − ⎡⎣1 / 2 + sδ ⎤⎦
∑
r the Hubbard model,2replacing
Ĥ1 by the form of Eq. (7), we obtain
s=±1
H 1 = HU =
hni =
⇥
U h(n̂"
1/2)(n̂#
1/2)i
2
⇤
,
(50)
ere we have set ⇢ = 1/2. Thus, the computational time for3a sweep3 scales as in the Hirschà Time to update
each vertex (a sweep) ~ n ~ (𝛽U)
3
e approach, namely as ( U ) . The algorithm can be used for lattice models with N correlated
es. In this case, the computational time for a sweep scales as (N U )3 , which is more expen-
Tr ⎡⎣e − βH ⎤⎦
− βH
Tr ⎡e 0 ⎤
⎣
⎦
=
∑
n
β
∫ dτ 1 ∑ !
0
τ n −1
∫
s1
0
d τ n ∑ (−1)n T HU ⎡⎣τ 1,s1 ⎤⎦ !HU ⎡⎣τ n ,sn ⎤⎦
sn
≡ W(C )
≡∑
C
U
HU ⎡⎣τ 1,s1 ⎤⎦ = ⎡⎣n↑(τ 1 ) − α ↑(s1 )⎤⎦ ⎡⎣n↓(τ 1 ) − α ↓(s1 )⎤⎦
2
Measurements.
Gσ (τ ,τ ') ≡ T d̂ σ+ (τ ) d̂ σ (τ ') =
0
∑ (−1)
C
n
T HU ⎡⎣τ 1 ,s1 ⎤⎦ !HU ⎡⎣τ n ,sn ⎤⎦ d̂ σ+ (τ ) d̂ σ (τ ')
∑ W (C)
C
0
∑ W (C) G (τ ,τ ')
=
∑ W (C)
σ
C
C
C
n
T HU ⎡⎣τ 1 ,s1 ⎤⎦ !HU ⎡⎣τ n ,sn ⎤⎦ d̂ σ+ (τ ) d̂ σ (τ ')
σ
σ
0
= G 0 (τ ,τ ') − ∑ Gσ 0 (τ ,τ α ) M σ n −1
G C (τ ,τ ') ≡
T HU ⎡⎣τ 1 ,s1 ⎤⎦ !HU ⎡⎣τ n ,sn ⎤⎦
α ,β =1
(
0
)
σ
G
(τ β ,τ ')
αβ
0
rem, which holds when expanding around a Gaussian theory.
Using the determinant identity given by Eq. (10), one will readily see that the single-particle
Green function
by ⎡the
n
⎡τ given
⎤ d̂ +of
,s ⎤ !H
τ ,sratio
(τ )two
d̂ (τ determinants:
')
T H is
Gσ C (τ ,τ ') ≡
U
⎣
σ
⎦ σ
0
= Gσ 0 (τ ,τ ') − ∑ Gσ 0 (τ ,τ α ) M σ n −1
det Bα ,(C
T HU ⎡⎣τ 1 ,s1 ⎤⎦ !HU ⎡⎣τˆn†,sn ⎤⎦ ˆ 0
β =1 n )
hhT f (⌧ )f0 (⌧ )iiCn =
1
1
⎦
U
⎣
n
(
n
)
αβ
Gσ 0 (τ β ,τ ')
(17)
det M (Cn )
where from:
Follows
σ
⎛ M σ (C ) !
G1
(τ ,τ ')
n
0
0
⎜
g0 (⌧1 , ⌧ ) 1
"
C#
B
⎜
..
det )⎜
BM (C
σ
. GC
n
(τ ,τ ')
B
0C .n
B (Cn ) = B
⎜
C
0 σ
σ
σg0 (⌧n , ⌧ )A
+
@
⎜
G
(
τ
,
τ
)
!
G
(
τ
,
τ
)
G0 (τ ,τ ')
T HU ⎡⎣τ 1 ,s1 ⎤⎦ !HU ⎡⎣τ n ,sn ⎤⎦ d̂ σ (τ ) d̂ σ (τ ')
1
0
n
⎝ 0
0
= ⌧1 ) . . . g0 (⌧, ⌧n ) σ g0 (⌧, ⌧ 0 )
g0 (⌧,
det M (Cn )
T HU ⎡⎣τ 1 ,s1 ⎤⎦ !HU ⎡⎣τ n ,sn ⎤⎦
0
0
(
)
⎞
⎟
⎟
⎟
⎟
⎟⎠
(18)
To compute the ratio of the two determinants, we use the determinant identity
⎤
⎥
⎟
"
# ⎟ +u ⊗v +u ⊗v ⎥
1
1
2
2⎥
0 ⎟
⎥
⎟ 1
1
A! u0 ⌦1vA
⎠
⎥⎦
1det
+ vM ·σ (C
An )1 u
1
⎡⎛ det(A)
det(A + u ⌦ v) =
!+ v ·0A⎞ u
M σ (Cn ) 1
⎢⎜
⎢⎜
det
as well as the Sherman-Morrison formula ⎢⎜
⎢⎜
0
1 ⎢⎣ ⎝
(A + u ⌦ v)
=A 1
=
toUse:
obtain
det (A + u1 ⌦ v1 + u2 ⌦ v2 ) =
⇥
det(A) 1 + v1 · A 1 u1 1 + v2 · A 1 u2
(
(19)
(20)
)
1
v 2 · A u1
1
v 1 · A u2
⇤
,
P
(21)
Tr ⎡⎣e − βH ⎤⎦
Tr ⎡e
⎣
− βH 0
τ n −1
β
=
⎤
⎦
∑ ∫ dτ ∑
n
1
0
s1

∫ dτ ∑
n
0
sn
Sum with Monte Carlo
n
⎛ U⎞
⎡
⎤
⎜⎝ − 2 ⎟⎠ det ⎣M n τ 1,s1 , τ n ,sn ⎦
(
)
Weight
Measurements.
n
T HU ⎡⎣τ 1 ,s1 ⎤⎦ !HU ⎡⎣τ n ,sn ⎤⎦ d̂ σ+ (τ ) d̂ σ (τ ')
σ
σ
0
= G 0 (τ ,τ ') − ∑ Gσ 0 (τ ,τ α ) M σ n −1
G C (τ ,τ ') ≡
T HU ⎡⎣τ 1 ,s1 ⎤⎦ !HU ⎡⎣τ n ,sn ⎤⎦
α ,β =1
(
0
Wick theorem applies for each configuration C of vertices (See notes)
Direct calculation of Matsubara Green functions is possible
n
G C (iωm ) = G 0 (iωm ) − G 0 (iωm ) ∑ e−iωmτα ( M σ n −1 ) α β Gσ 0 (τ β ,0)
σ
σ
σ
α , β =1
)
σ
G
(τ β ,τ ')
αβ
0
Examples.
a) Particle-hole symmetric Anderson Model, U/t=4.
G(τ )
β t = 400
n = 270
Hirsch-Fye:
LTrot = 400 / 0.2 (Δτ t = 0.2)
Speedup:
( 2000 / 270)
3
≈ 400
b) Off particle-hole Symmetry, U/t=4 βt=40.
G(t)
-Im(G(iwm))
n = 65
Speedup
( 200 / 65)
3
≈ 30
G (iωm ) =
∫ dω
A(ω )
iωm − ω
Direct calculation of G(iwm) is possible.
Sign problem – a simple example.
Consider:
†
†
H = −t1 ∑ ci†ci+1 + ci+1
ci − t2 ∑ ci†ci+2 + ci+2
ci ,
t1,t2 > 0,
i
i
{c ,c } = δ
†
i
j
i,j
World line configuration:
O
Z =
W (c)O(c) 1Z ' ∑ W (c) sign(c)O(c)
∑
=
≡
c
c
Z
∑W (c)
Z'=
c
∑ W (c)
Z
Z'
and sign(c) = W (c)
c
W (c)
has negative weight.
But: Z´ is the partition function of H with fermions replaced by hard core bosons. Thus:
F
B
Z / Z ' ≈ e−β V δ with δ = (E0 − E0 ) > 0
V
For practical purposes we will need:
Δ (Z / Z ' )
Z /Z '
<< 1 but Δ (Z / Z ' ) ≈ 1
CPU
so that CPU >> e2 β V δ
Note: Ø  Had we formulated everything in Fourier space........
Ø  Hamiltonian H with t1>0 and t2<0 and hard core bosons yields a sign problem. This
corresponds essentially to a frustrated spin chain. Thus the sign problem is not
limited to fermionic systems.
Ø  δ is formulation dependent.
Continuous-time QMC Solvers for Electronic Systems in
Fermionic and Bosonic Baths
F.F. Assaad
(DMFT@25, Juelich, September 2014)
Goal: Detailed overview of CT-INT.
Outline
Ø  Weak coupling CT-QMC – Basics.
– Add ons: Retarded interactions:
phonon degrees of freedom.
Ø  Selected applications. Topological insulators, electron-phonon interaction.
Ø  Conclusions.
Phonons (bosonic baths) Integrate out phonons in favor of a retarded interaction.
Hubbard-Holstein:
Hˆ =
∑σ t
i, j
i, j,
+ U ∑ nˆi ,↑ nˆi ,↓ + g ∑ Qˆ i (nˆi − 1) +
+
i ,σ
cˆ cˆ j ,σ
i
i
∑
i
Pˆ i2 k ˆ 2
+ Qi
2M 2
Integrate out the phonons
β
⎡
Z = ∫ ⎣⎡dc+ dc ⎦⎤ exp ⎢−S0 −U ∫ dτ ni↑ (τ ) ni↓ (τ ) +
0
⎣
D (i − j ,τ − τ ') = δ i , j
0
P (τ ) =
ω0
β
β
⎤
[ni (τ ) − 1] D (i − j,τ − τ ')[n j (τ ') − 1] ⎥
∫0 dτ ∫0 dτ ' ∑
i, j
⎦
0
g2
P (τ − τ ')
2k
−|τ |ω0
− ( β −|τ |)ω0
⎡
⎤⎦ , ω0 = k / M
e
e
+
− βω0 ⎣
2(1 − e
)
Attractive, retarded interaction (time scale 1/w0).
Antiadiabatic limit: lim
ω0 →∞
P(τ ) = δ (τ )
à Attractive Hubbard.
Phonons (bosonic baths) Integrate out phonons in favor of a retarded interaction.
β
⎡
Z = ∫ ⎣⎡dc dc ⎦⎤ exp ⎢−S0 −U ∫ dτ ni↑ (τ ) ni↓ (τ ) +
0
⎣
+
QMC:
β
β
⎤
[ni (τ ) − 1] D (i − j,τ − τ ')[n j (τ ') − 1] ⎥
∫0 dτ ∫0 dτ ' ∑
i, j
⎦
0
Expand both in Hubbard and retarded phonon interaction.
Vertices:
Phonon.
Hubbard.
D0 (i − j ,τ − τ ')
i,τ
j ,τ '
i,τ
U
i,τ
i,τ
i,τ
j ,τ '
j ,τ '
Continuous-time QMC Solvers for Electronic Systems in
Fermionic and Bosonic Baths
F.F. Assaad
(DMFT@25, Juelich, September 2014)
Goal: Detailed overview of CT-INT.
Outline
Ø  Weak coupling CT-QMC – Basics.
– Add ons: Retarded interactions:
phonon degrees of freedom.
Ø  Selected applications. Electron-phonon interaction, Topological insulators.
Ø  Conclusions.
Bosonic Baths
à Electron-phonon problems
M. Hohenadler, FFA. J. Phys.: Condens. Matter 25, 014005 (2013)
Peierls to superfluid crossover in the one-dimensional quarter filled Holstein model @ g2/Wk=0.35
P̂ 2i k 2
+ Q̂i ,
Ĥ = ∑ ti, j ĉ ĉ j,σ + g ∑ Q̂i ( n̂i − 1) + ∑
2M 2
i, j,σ
i
eierls to superfluid crossover
in the 1D
Holstein imodel
+
i,σ
1
Charge correlation
L=28, 𝛽t=40
(a)
10
0.6
ω0 / t = 0.1
ω0 / t = 0.5
ω0 / t = 1.0
ω0 / t = 4.0
0.4
0.2
0
0.5
ω0 << t
P(r)
N(q)
0.8
0
0.5
1
q/π
10
10
10
1.5
-1
k
M
s-wave pairing correlations
(b)
ω0 / t = 0.1
ω0 / t = 0.25
ω0 / t = 0.5
ω0 / t = 1.0
ω0 / t = 4.0
ω0 / t = ∞
-2
-3
-4
2
ω0 =
0
20
10
r
(c)
(d)
Pairs of electrons form a commensurate
CDW (diagonal LRO) à Peierls instability
2
0.4
0.3
0.2
Pairs condense to form an s-wave superconductor
(off diagonal LRO).
1.5
ω0 / t = 0.1
ω0 / t = 0.5
n(k)
S(q)
ω0 >> t
1
0.5
ω0 / t = 0.1
ω0 / t = 0.5
ω / t = 1.0
5
Charge dynamical structure factor @ 𝜆/t =0.35 N ( q, ω ) =
1
Z
∑e β
− Em
n nˆ (q) m
2
δ ( En − Em − ω )
β t = 40, ρ = 0.5
n ,m
Superconducting
Peierls (CDW)
ω0 = 0.1
λ =0
ω0 = 0.5
Charge
Velocity.
π /2
q
π
q
Piling up of spectral weight
at 2kf. Slow dynamics of
the bipolaronic CDW.
q
Growth of charge velocity.
Mobility of bipolarons.
Optical Conductivity.
Continuity equation:
σ '(q, ω ) =
ω
q2
(1 − e ) N (q, ω)
− βω
Long wavelength limit: N (q, ω ) ≈ N (q)δ ( vcq − ω )
with
N (q) ≈ α q
à σ '(ω ) = lim q →0 σ '(q, ω ) ≈ α vcδ (ω ) at T=0.
Peierls to superfluid crossover in the 1D Holstein mod
(a)
0.8
10
0.6
10
ω0 / t = 0.1
ω0 / t = 0.5
ω0 / t = 1.0
ω0 / t = 4.0
0.4
0.2
0
0.5
0.4
P(r)
N(q)
1
0
0.5
(c)
1
q/π
10
10
1.5
-1
(b)
-2
-3
-4
2
0
2
(d)
Spin dynamical structure factor @ 𝜆/t =0.35 S ( q, ω ) =
1
Z
∑ e− β Em n Sˆz (q) m
2
β t = 40, ρ = 0.5
δ ( En − Em − ω )
n,m
Superconductivity
Peierls
π /2
q
ω0 = 0.5
ω0 = 0.1
λ =0
π
q
Suppression of low energy spectral weight.
Interpretation: Δsp >0 , Δc=0
q
Continuous-time QMC Solvers for Electronic Systems in
Fermionic and Bosonic Baths
F.F. Assaad
(DMFT@25, Juelich, September 2014)
Goal: Detailed overview of CT-INT.
Outline
Ø  Weak coupling CT-QMC – Basics.
– Add ons: Retarded interactions:
phonon degrees of freedom.
Ø  Selected applications. Electron-phonon interaction, Topological insulators.
Ø  Conclusions.
à Correlation effects on edge state of quantum Spin Hall states
Phases of the Kane Mele Hubbard model
S. Rachel & K. Le Hur, PRB 82, 075106, (2010)
M. Hohenadler et al. PRB 85 (2012)
8
Magnetic Insulator
6
U/t
Electronic Correlations
Fermionic Baths
xyz AFM
xy AFM
4
2
Semimetal
Quantum Spin Hall Insulator
0
0.0
0.1
λ/t
Spin-orbit coupling
0.2
Fermionic Baths
à Correlation effects on edge state of quantum Spin Hall states
Phases of the Kane Mele Hubbard model
S. Rachel & K. Le Hur, PRB 82, 075106, (2010)
M. Hohenadler et al. PRB 85 (2012)
Antiferromagnetic
Mott insulator
8
T. Paiva et al.
Phys. Rev. B, 72, 085123
( 2005)
Z. Y. Meng et al.
Nature 464, 847 (2010)
S. Sorella et al.
Scientific Reports 2, 992 (2012)
F. Assaad & I. Herbut
PRX 3, 031010 (2013)
6
U/t
S. Sorella and E. Tosatti.
EPL, 19, 699, (1992)
Electronic Correlations
Magnetic Insulator
xyz AFM
xy AFM
4
2
Semimetal
Quantum Spin Hall Insulator
0
0.0
0.1
λ/t
B. Clark arXiv:1305.0278
Dirac fermions
Tight binding model on
Honeycomb lattice at
half-filling
Symmetries
SU(2) spin ✓
Sublattice ✓
Time reversal ✓
Spin-orbit coupling
0.2
Fermionic Baths
à Correlation effects on edge state of quantum Spin Hall states
Phases of the Kane Mele Hubbard model
Antiferromagnetic
Mott insulator
S. Rachel & K. Le Hur, PRB 82, 075106, (2010)
M. Hohenadler et al. PRB 85 (2012)
i
j
(
8
Z. Y. Meng et al.
Nature 464, 847 (2010)
S. Sorella et al.
Scientific Reports 2, 992 (2012)
F. Assaad & I. Herbut
PRX 3, 031010 (2013)
U/t
Electronic Correlations
T. Paiva et al.
Phys. Rev. B, 72, 085123
( 2005)
6
xyz AFM
U
t
Magnetic Insulator
S. Sorella and E. Tosatti.
EPL, 19, 699, (1992)
)
= i λ ĉi†σ z ĉ j − ĉ †jσ z ĉi
xy AFM
4
2
Semimetal
Haldane model in each
spin sector
Quantum Spin Hall Insulator
0
0.0
0.1
0.2
λ/t
B. Clark arXiv:1305.0278
Dirac fermions
Tight binding model on
Honeycomb lattice at
half-filling
Symmetries
SU(2) spin ✓
Sublattice ✓
Time reversal ✓
Spin-orbit coupling
A 1,↑(q,ω )
SU(2) spin à U(1) spin
Sublattice ✗
Time reversal ✓
Opens gap à Quantum spin Hall
with robust helical edge states.
A 1,↓(q, ω)
0
π
q
2π
Fermionic Baths
à Correlation effects on edge state of quantum Spin Hall states
Phases of the Kane Mele Hubbard model
M. Hohenadler et al. PRB 85 (2012)
8
Magnetic Insulator
6
U/t
Question: Elastic
backward scattering
is forbidden due to
time reversal
symmetry. What
about inelastic
scattering?
S. Rachel & K. Le Hur, PRB 82, 075106, (2010)
xyz AFM
xy AFM
4
2
Semimetal
Quantum Spin Hall Insulator
0
0.0
0.1
0.2
λ/t
A 1,↑(q,ω )
(vF ~ 𝜆)
Bulk: U/W << 1
Edge: U/vF >> 1
Ground state of bulk is well
described by mean field
Edge states are exponentially localized
Strongly correlated problem
A 1,↓(q, ω)
Fermionic Baths
à Correlation effects on edge state of quantum Spin Hall states
M. Hohenadler, T. C. Lang and F. F. Assaad
Phys. Rev. Lett. 106, 100403 (2011)
M. Hohenadler and F. F. Assaad
Phys. Rev. B 85, 081106(R) (2012)
à Paramagnetic mean field for bulk. Retain all the fluctuations on the edge.
Open,n
Periodic, r
β
β
S = ∫ dτ ∫ dτ '
0
0
∑
r ,r ',σ
β
cr†,σ (τ )G0,−1σ (r − r ',τ − τ ')cr ',σ (τ ') + U ∫ d τ
0
⎛
1⎞ ⎛
1⎞
n
(
τ
)
−
(
τ
)
−
n
∑ ⎜⎝ r ,↑
2 ⎟⎠ ⎜⎝ r ,↓
2 ⎟⎠
r
Green function of the model on the ribbon (e.g. paramagnetic mean-field)
Fermionic Baths
à Correlation effects on edge state of quantum Spin Hall states
Single par>cle spectral func>on 8
Magnetic Insulator
xyz AFM
xy AFM
4
Strong coupling: λ / t = 0.05, U / t = 2
2
0.01
Semimetal
6
10
(b) A ↑(q,ω)
ω/t
0.1
1
0.5
4
0
0.0
0.1
(a) N(q,ω)
Quantum Spin Hall Insulator
0.2
λ/t
2
-
0
4
Dynamic spin-­‐structure factor at λ / t = 0.1, U / t = 2
-
(c) Sx(q,ω)
(d) Sz(q,ω)
3
ω/t
U/t
6
-0.5
2
1
0
0
π
q
2π 0
à  Inelastic scattering between left (spin do)
and right (spin
up) movers reduces
substantially the spectral weight of the helical
edge state.
2k f
(
)
Sσx q,ω =
1
−β E
e n m S x (q) n
∑
Z n,m
2
δ ( Em − En − ω )
π
q
2π
Single particle spectral function U/t=2
HU = −
U
Sq+ Sq− + Sq− Sq+
∑
2 q
Second order perturbation theory.
G0↑
G0↓
∑↑
=
G0↓
Continuous-time QMC Solvers for Electronic Systems in
Fermionic and Bosonic Baths
F.F. Assaad
(DMFT@25, Juelich, September 2014)
Goal: Detailed overview of CT-INT and CT-AUX (CT-HYB à See notes)
Outline
Ø  Weak coupling CT-QMC – Basics.
– Add ons: Retarded interactions:
phonon degrees of freedom.
Ø  Selected applications. Topological insulators, electron-phonon interaction.
Ø  Conclusions.
CT-INT is a very flexible action based QMC method which allows to tackle
correlated electron problems embedded in fermionic and bosonic baths. In the
absence of negative sign problem, it scales polynomially in the number of
interacting degrees of freedom.