f.f. assaad
DESCRIPTION
Numerical approaches to the correlated electron problem: Quantum Monte Carlo. F.F. Assaad. The Monte Carlo method. Basic. Spin Systems. World-lines, loops and stochastic series expansions. The auxiliary field method I The auxiliary filed method II Ground state - PowerPoint PPT PresentationTRANSCRIPT
F.F. Assaad.
MPI-Stuttgart. Universität-Stuttgart.
21.10.2002
Numerical approaches to the correlated electron problem:
Quantum Monte Carlo.
The Monte Carlo method. Basic.
Spin Systems. World-lines, loops and stochastic series expansions.
The auxiliary field method I
The auxiliary filed method II
Ground state
Finite temperature
Hirsch-Fye.
Special topics (Kondo / Metal-Insulator transition) and outlooks.
Ground state method:CPU V3
TT
TT
eeOeH
HH
O||
||lim
22
0
00 T
t
)( ,S Hubbard 6X6
e
OeO H
H
Tr
Tr
Finite temperature: CPU V3
Ground state.
Finite temperature.
Hubbard.
0 , , ,i, j, i, i,j i
2
i, ,
,( 1/ 2)( 1/ 2)
j
i i i
id
it U n n nH c c ce c
A l
),0,0( B AB
The choice of the trial wave function for the Projector method.
Magnetic fields and size effects.
cctH jiji
,
t010.
LFK vT 2
ccetH jid
j
i
i
ji
lA0
2
,
),0,0( B AB
Scaling:
LB 2
0
Electronic system:X-Y plane.
L=16: More than an order of magnitude gain in temperature before results get dominated by size effects.
Cv/T
T T
0BL = 4,6,...16
02LB
L = 4,6,...16
Thermodynamic quantities.
T T
s
0B 02LB
L = 4,6,. ..16
L = 4,6,...16
FFA PRB 02
I. Basic formalism for the case of the Hubbard model.
),0,0( B AB
H t HU
Magnetic field in z-direction:
cccce nnnUtH iiidlA
ij
i
,i,i,i
,,,j,i,ji,
2
,)2/1)(2/1(0
)(ΨΨΨΨ τΔ)( 2HtτΔHτΔ U OZ T
L
TTH
T eee
)(TrTr τΔ)( 2HtτΔHτΔ U OZLH eee
τΔL
Trotter.
Ground state:
Finite temperature:
Hubbard.
1
)1('
1
)(τ~~s
si
s
sH eeennnnU
e
eU
U
2/
2/
)'cos(
)cosh(
0,)2/1)(2/1( UnnUHU
Breaks SU(2) spin symmetry.Symmetry is restored after summation over HS. Fields.
Complex but conservs SU(2) spin symmetry.
The choice of Hubbard Stratonovich transformation. (Decouples many body propagator into sum of single particle propagator interacting with extermal field.)
e φdφπ
A Aφe 22/
2 2
2
1 )τ()η()(γ 4τ
2,1
2τ
Oe AllA
le
Generic.
s
HH see tU )(B
ee cTccAcss
)(
)(Βwith:
γ γ( 1) 1 6 / 3 ( 2) 1 6 / 3
( 1) = 2(3- 6) ( 2) = 2(3+ 6)
01
,,
NAcAc
p
yyxxT
yyxx Pece
(1) Propagation of a Slater determinant with single body operator remains a Slater determinant.
Properties of Slater Determinants.
)'*det(Ψ'Ψ PPTT(2) Overlap:
01 1
,
N p
y
N
xyxxT Pc
, ,Tr det 1A Bc c c cA Bx y x yx y x ye e e e
(3) Trace over the Fock space:
01 1
,
N p
y
N
xyxxT Pc
Trial wave functionis slater determinant:
P is N x Np matrix.
ee TAsΒs)(
)(
Ground state.
ss
PsBsBPssss LL
LTLTZ...
1*
1...
11
)(....)(det)(....)( ΨΨ BB
Finite temperature.
ss
sBsBL
LZ...
1
1
)(....)(1det
ee cTccAcss
)(
)(Β
1
*1
...
1 ....det ( ) ( ) ( )L
LB B P O sP s sZ s s
O
1 1
1
.. ..( ) ( ) ( ) ( )( , )
...( ) ( )
LT T
LT T
Os s s sO s
s s
B B B B
B B
1
( ) 1G U U U U
For a given HS configuration Wick‘s theorem holds. Thus is suffices to compute Green functions.
,( ) ( )
i j i jG c c
1 1.... ...., LB B P B Bs s s sPU U
Observables ground state.
Observables finite temperature.
1 1
1
.. ..Tr ( ) ( ) ( ) ( )( , )
..Tr ( ) ( )L
L
Os s s sO s
s s
B B B B
B B
1( ) 1 ( ,0) ( , )G B B
1 1.... ....( ,0) , ( , ) LB B B B B Bs s s s
Wick´s Theorm
Upgrading, single spin flip.
)(1)()( ' sss lll BBB so that G
UU
UU
W
W
Old
New
11det
det
1det
Thus the Green function is the central quantity. It allows calculation of observables and determines the Monte Carlo dynamics. Same is valid in the finite temperature approach.
This form holds for both the finite temperature and ground state algorithm!
If the spin flip is accepted, we will have to upgrade the equal time Green function
1 11 1
1,
where1 i ji j
u vA A
v uAA u v u v u vA
Upgrading of the Green function is based on the Sherman Morrison formula.
Outer product.
II. Comments.
:....1 PsBsBU l:P
Gram Schmidt. > >d vU U :U
UUUU 11
UUUUGreen functions remains invariant. .
(A) Numerical stabilzation T=0.
Since the algorithm depends only on the equal time Green function everything remains
invariant!
< <d vU USimilarly:
The Gram Schmidt orthogonalization.
Numerical stabilization finite T.
Use:
To calculate the B matrices without mixing scales use.
Diagonal elements are equal time Green functions.
As we will see later, the off-diagonal elements correspond to time displaced Green functions.
You cannot throw away scales.
The inversion.
Measuring time displaced Green functions.
(a) Finite temperature.
1 2 Note:
,
1 2,s x yG
Thus we have:
But we have already calculated the time displace Green functions. See Eq 81.
Time displaced Green functions for ground state (projector) algorithm.
Consider the free electron case: so that 0 00( ) and assume that 0,1k k k k
k
k c c c cH
L=8, t-U-W model: W/t =0.35, U/t=2, <n>=1, T=0
Ψ)0,(),(Ψ3
1ln 00 QSQS
Ψ)0,(),(Ψln 00 QQ SS zz
Ψ)0,(),(Ψ3
1ln 00 QSQS
t
SU(2) invariant code.
SU(2) non-invariant code.
SU(2) non-invariant code.
(C) Imaginary time displaced correlation functions.
1
)1('τΔ~
s
siH eennU
Note: Same CPU time for both simulations.
1
)(τΔ~s
sH eennU
. Gaps. Dynamics (MaxEnt).
(B) Sign problem.
5 βt
6 βt
10 βt
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0<n>
<si
gn>
U/t = 4, 6 X 6
a) Repulsive Hubbard.
1
)(τΔ~s
sH eennU
1
)1('τΔ~
s
siH eennU
)(....)(1detWeight 12
sBsB L
Particle-hole symmetry:
is real even in the presence of a magnetic field.
)(....)(1det 1sBsB L
b) Attractive Hubbard, U<0.
1
)1(''τΔ~
s
sH eennU
)(....)(1detWeight 12
sBsB L )(....)(1det 1sBsB L is real for all band fillings(no magntic field.)
General: Models with attractive interactions which couples independently to an internal symmetry with an even number of states leads to no sign problem.
Away from half-filling.
cc i
i
i 1
)(....)(1det)(....)(1det 11 sBsBsBsB LL
Half-filling.
Impurity models such as Anderson or Kondo model (Hirsch-Fye).
( , ),, ,
i j
f
I
c
i jt c c JH
S S No charge fluctuations on f-sites.
)(||1
)( ||2
,EEmn
ZS mn
f
I
En
mnf Se
)(S f
t/
T <TK
T >TK
Dynamical f-spin structure factor J/t=2
00
SSf
I
f
IId
Numerical (Hirsch-Fye impurity algorithm) CPU:V0(Nimp3
T/TK
J/t = 1.2J/t = 1.6J/t = 2.0
T
TK/t 0.21
TK/t 0.06TK/t 0.12
is the only low energy scaleeTJt
K
/
1 120, 0, 2
( ),
)( )(, ,f f
ff f nc c Vt c cH nU
i, j
i j
The Hirsch Fye Impurity Algorithm.
H
H U
fff nnUcffcVcctH ))((,, 2
121
0
,0,0),(
ji,
ji
Finite temperature:
ss
ssZL
LH
1
)()(1deteTr 1- BB We only need Green functions on f-sites
(1) Upgrading (equal time).(2) Observables.
)(1)(/)'(1)(11
)'( sGsDsDsGsG
G(s): f-Green function for HS s. (L x L matrix)
Start with D(s) = 1. G(s) is the f-Green function of H0. Exact solution in thermodynamic limit.
From G(s) compute G(s´) at the expense of LxL matrix inversion
CPU time ~ L3 (i.e. 3).
1
)(τΔ~s
sH eennU
With
e,ediag)( ,1 sssD L in say up spin sector.
T/TK
J/t = 1.2J/t = 1.6J/t = 2.0
T
Single impurity (Hirsch-Fye algorithm)
TK/t 0.21
TK/t 0.06TK/t 0.12
SSjiji,
f
IIcJcctH
,,),(
00
SSf
I
f
IId
(IV) Related algorithm: Hirsch Fye Impurity Algorithm.
(III) Approximate strategies to circumvent sign problem
sT
L
EL
L
TL
EL
LT
se
e
Ψlim
ΨlimΨΨΨ
)(1
Δ
Δ
00
0
0
BBB
BB
0ΨΨ,)( 0ΔΔ
Ts
HH see tU BB
Recall:
Assume that we know and for s´
then we can omit all paths evolving from this point since:
0,0
~)'(
10)(Δ
GStoOrthogonal1
Δ
10
0
lim
Ψlim
EE
s
e
e
EEL
L
TL
EL
L
BBB
But: We do not know ! Approximate it to impose constraint and method becomes approximative.
CPQMC (Zhang, Gubernatis). Approximate by a single Slater determinant to impose constraint.
Ψ0
0Ψ)'(Ψ0 TsB
Ψ0
Ψ0
4x4 Hubbard: U/t = 8, <n>=0.625
Ene
rgy/
t
First orderTrotter.
Second orderTrotter.
Exact: -17.510t, Extrapolated : -17.520(2)t