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