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Computational Solid State Physics
計算物性学特論 第８回
8. Many-body effect II:Quantum Monte Carlo method
Quantum diffusion Monte Carlo method
Diffusion Monte Carlo method to calculate the ground state
Importance sampling method How to treat the Pauli principle:
fixed node approximation
Schrödinger equation in atomic unit
EH
eN
iiJ
1
2
2
1
e ee N
i
N
ij ij
N
ii rrvV
11
1)(
VJH
How to solve the Schrödinger equation for many electrons?
),()(),(
tEti
tT x
x
0
)()(),(k
tEEikk
TkeCt xx )0( tC kk
),()(),(
xx
TE
0
)()(),(k
EEkk
TkeC xx
)(
100
0)(),( EE
kkk
keCC
xx
Imaginary Time
00),(lim
Cx
The ground state wave function can be obtained in the limit of infinite time.
Time-dependent Schrödinger equation
),())((),(),( 2
xxxx
VED T
.0),( x
diffusion
TExVm
H )(2
22
mD
2
2
branching
Diffusion equation holds for
Diffusion equation with branching process for the ground state wave function
Diffusion equation for particles
),(),( tntnDd
xvxF
Fx
div),(
t
tn
),(),(),( 2 tntnD
t
tnd
xvxx
Flux:
Conservation of number of particles:
D : diffusion constant, vd: drift velocity
Diffusion equation
diffusion flux drift flux
Rate equation
)exp(0 Rtnn
Rndt
dn
R>0 : growth rate
R<0: decay rate
Time-dependent Green’s function
),;,()(),;,(
122
12
xy
xyGE
GT
xxxyy dG ),(),;,(),( 1122
)(),;,( 11 yxxy G : Boundary Condition
xxxyy de TE ),(),( 1))((
212
xyxy ))((12
12),;,( TEeG
xxxyy dG ),();,(),(
)()( 1))((
212 TEe
Time evolution of wave function
Short time approximation
xyxy ))((12
12),;,( TEeG
BdiffEVJEVJ GGeee TT )()(
)(],[2
1 32 OJVGGGBdiff
BdiffGGG
);,();,( 2
xyxy
diffdiff GD
G
DNdiff eDG 4/)(2/3 2
)4();,( xyxy
The transition probability from x to y can be simulated by random walk in 3N dimensions for N electron system.
Green’s function of the classical diffusion equation
))]()([2
1(
);,(TEVV
B eG
yx
xy
BTB GVE
G)(
The branching process can be simulated by the creation or destruction of walkers with probability GB
Green’s function of the rate equation
Importance sampling
)(),(),( xxxf
/HEL
/2QF
),())(())(),((),(),( 2
xxxFxxx
fEEfDfDf
LTQ
: Local energy
: Quantum force
: analytical trial fn.
Diffusion Drift Branching
Diffusion equation with branching process
FF DDDJ )(~ 2
Jdiff eG
~);,(
~ xy
DDNdiff
QeDG 4/))((2/32
)4();,(~ xFxyxy
Kinetic energy operator
Drift term
The transition probability from x to y can be simulated by biased random walk with quantum force F in 3N dimensions for N electron system.
Biased diffusion Green’s function
Detailed balance condition
To guarantee equilibrium
);,(~
);,(~
)(
)();,( 2
2
xy
yx
x
yxy
G
Gq
));,(,1min();,( xyyx qA
Acceptance ratio of move of the walker from x to y
);,(~
);,(~ yxxy diffdiff GG
DMC Importance-sampled DMC
))]()([( 21
);,(~
TLL EEEB eG yxxy
))]()([2
1(
);,(TEVV
B eG
yx
xy
suppression of branching process
DMC and Importance-sampled DMC for the hydrogen atom
Branching process:
Walker 1
Walker 2
Walker 3
Walker 4 Branching
Biased diffusion
Schematic of the Green’s function Monte Carlo calculation with importance sampling for 3 electrons
0
00
00
)()()()(
)()()()(
)()(
)()()(
E
dd
dd
dfdEfE LfL
xxxxxx
xxxxxx
xx
xxxxx
M
kkL
MfL EM
E1
)(1
lim X
Evaluation of the ground state energy
How to remove the condition ?
Fixed node approximation
to treat wave functions with nodes Fixed phase approximation
to treat complex wave functions
0),( x
Fixed node approximation
0)(),(),( xxxfD
Wave function φ is assumed to have the same nodes with ΨＤ .
Importance sampling on condition
Pauli principle for n like-spin electrons
)()()(
)()()(
)()()(
det)(
21
22212
12111
nnnn
n
n
D
xxx
xxx
xxx
x
)()( xx ji
)()( ikDkiD xxxx
Slater determinant
ji xx 0)( xD
Slater determinant has nodes.
Fixed phase approximation
0)(*),(),( xxxf
Wave function φ is assumed to have the same phase with ).(x
Importance sampling on condition
Ground states of free electrons
n
rs 1
3
4 3
D.M.Ceperley, B.J.Alder: PRL 45(1980)566
Transition of the ground state of free electrons
Unpolarized Fermi fluidPolarized Fermi fluidWigner crystal
70sr
9070 sr
sr90
n
rs 1
3
4 3
n: concentration of free electrons
Problems 8
Calculate the ground state wave function of a hydrogen atom, using the diffusion Monte Carlo method.
Consider how to calculate the ground state energy. Calculate the ground state of a hydrogen atom,
using the diffusion Monte Carlo method with importance sampling method. Assume the trial function as follows.
Derive the diffusion equation for in importance sampling method.
)exp( 2rtr
),( xf
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