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Quantum Monte Quantum Monte Carlo MethodsCarlo MethodsJian-Sheng WangJian-Sheng Wang

Dept of Computational Dept of Computational Science, National University Science, National University

of Singaporeof Singapore

Quantum Monte Quantum Monte Carlo MethodsCarlo MethodsJian-Sheng WangJian-Sheng Wang

Dept of Computational Dept of Computational Science, National University Science, National University

of Singaporeof Singapore

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Outline• Introduction to Monte Carlo

method• Diffusion Quantum Monte Carlo• Application to Quantum Dots• Quantum to Classical --Trotter-

Suzuki formula

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Stanislaw Ulam (1909-1984)

S. Ulam is credited as the inventor of Monte Carlo method in 1940s, which solves mathematical problems using statistical sampling.

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Nicholas Metropolis (1915-1999)

The algorithm by Metropolis (and A Rosenbluth, M Rosenbluth, A Teller and E Teller, 1953) has been cited as among the top 10 algorithms having the "greatest influence on the development and practice of science and engineering in the 20th century."

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Markov Chain Monte Carlo

• Generate a sequence of states X0, X1, …, Xn, such that the limiting distribution is given by P(X)

• Move X by the transition probability W(X -> X’)

• Starting from arbitrary P0(X), we have

Pn+1(X) = ∑X’ Pn(X’) W(X’ -> X)• Pn(X) approaches P(X) as n go to ∞

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• Ergodicity[Wn](X - > X’) > 0For all n > nmax, all X and X’

• Detailed BalanceP(X) W(X -> X’) = P(X’) W(X’ -> X)

Necessary and sufficient conditions for convergence

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Taking Statistics• After equilibration, we estimate:

• It is necessary that we take data for each sample or at uniform interval. It is an error to omit samples (condition on things).

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1( ) ( )P( )d ( )

N

ii

Q X Q X X X Q XN

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Metropolis Algorithm (1953)

• Metropolis algorithm takes

W(X->X’) = T(X->X’) min(1,

P(X’)/P(X))where X ≠ X’, and T is a symmetric stochastic matrixT(X -> X’) = T(X’ -> X)

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The Statistical Mechanics of Classical Gas/(complex) Fluids/Solids

Compute multi-dimensional integral

where potential energy

( 1, 1,...)

1 1 2 2 1 1

( 1, 1,...)

1 1

( , , , ,...)e ...

e ...

B

B

E x yk T

N N

E x yk T

N N

Q x y x y dx dy dx dyQ

dx dy dx dy

1( ,...) ( )N

iji j

E x V d

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Advanced MC Techniques

• Cluster algorithms• Histogram reweighting• Transition matrix MC• Extended ensemble methods

(multi-canonical, replica MC, Wang-Landau method, etc)

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2. Quantum Monte 2. Quantum Monte Carlo MethodCarlo Method

2. Quantum Monte 2. Quantum Monte Carlo MethodCarlo Method

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Variational Principle• For any trial wave-function Ψ, the

expectation value of the Hamiltonian operator Ĥ provides an upper bound to the ground state energy E0:

0

ˆ| H|

|E

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Quantum Expectation by Monte Carlo

*

*

ˆ ˆ| H | ( )H ( )

| ( ) ( )

P( )E ( )L

dX X X

dX X X

dX X X

where

2

1 ˆE ( ) H ( )( )

P( ) | ( ) |

L X XX

X X

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Zero-Variance Principle• The variance of EL(X) approaches zero

as Ψ approaches the ground state wave-function Ψ0.

σE2 = <EL

2>-<EL>2 ≈ <E02>-<E0>2 = 0

Such property can be used to construct better algorithm (see Assaraf & Caffarel, PRL 83 (1999) 4682).

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Schrödinger Equation in Imaginary Time

H

H , ( ) (0)t

ii t e

t

Let = it, the evolution becomesH

( ) (0)t e

As -> , only the ground state survive.

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Diffusion Equation with Drift

• The Schrödinger equation in imaginary time becomes a diffusion equation:

21( )

2 TV X E

We have let ħ=1, mass m =1 for N identical particles, X is set of all coordinates (may including spins). We also introduce a energy shift ET.

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Fixed Node/Fixed Phase Approximation

• We introduce a non-negative function f, such thatf = Ψ ΦT* ≥ 0

Ψ ΦT

ff is interpreted as walker density.

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Equation for f

21( )

2where

1 1 ˆ and H

L T

LT TT T

fff E X E f

E

v

v

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Monte Carlo Simulation of the Diffusion

Equation• If we have only the first term -

½2f, it is a pure random walk.• If we have first and second term, it

describes a diffusion with drift velocity v.

• The last term represents birth-death of the walkers.

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Walker Space

X The population of the walkers is proportional to the solution f(X).

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Diffusion Quantum Monte Carlo Algorithm

1. Initialize a population of walkers {Xi}

2. X’ = X + η ½ + v(X) 3. Duplicate X’ to M copies: M = int( ξ +

exp[-((EL(X)+EL(X’))/2-ET) ] )

4. Compute statistics5. Adjust ET to make average

population constant.

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Statistics• The diffusion Quantum Monte

Carlo provides estimator for

0

0

1

ˆ| Q |( ) ( )

|( )

1 ( )

T

T

N

ii

dX Q X f XQ

dX f X

Q XN

where 1 ˆ( ) Q TT

Q X

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Trial Wave-Function• The common choice for interacting

fermions (electrons) is the Slater-Jastrow form:

1 1 1 2 1

2 1( )

1

( ) ( ) ( )

( )( )

( ) ( )

N

J X

N N N

X e

r r r

r

r r

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Example: Quantum Dots

• 2D electron gas with Coulomb interaction in magnetic field

1

22 2 2

0

1ˆH h| |

where ( , ) and

ˆ1 1ˆ ˆh ( ) L2 2 4 2 2

Nk

Nk i j i j

i i i

zz s

x y

B Br V g

r r

r

r

We have used atomic units: ħ=c=m=e=1.

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Trial Wave-Function• A Slater determinant of Fock-Darwin solution

(J(X)=0):

where

• L is Laguerre polynomial• Energy level En,m,s=(n+2|m|+1)h+ g B(m+s)B

21| | | | 2 2

, ,2

( , , ) L ( ) ( )im rm m

n ms nm n s

er c r r e

22 2

0 4B

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Six-Electrons Ground-state Energy

Using parameters for GaAs.

The (L,S) values are the total orbital angular momentum L and total Pauli spin S.

From J S Wang, A D Güçlü and H Guo, unpublished

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Addition Spectrum EN+1-EN

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Comparison of Electron Density

Electron charge density from trial wavefunction (Slater determinant of Fock-Darwin solution), exact diagonalisation calculation, and QMC.

N=5 L=6 S=3

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QD - Disordered Potential

Random gaussian peak perturbed quantum dot. From A D Güçlü, J-S Wang, H Guo, PRB 68 (2003) 035304.

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Quantum System at Finite Temperature

• Partition function

• Expectation value

ˆ( ) H

H

| |

Tr

E X

X

Z e e

e

HTr

H

Q

Tr

e

eQ

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D Dimensional Quantum System to D+1

Dimensional Classical system

HH

ˆ ˆ ˆH H H

, , ,

| | | ( ) |

| | | | | |

MM

M M Mi i j k

i j k

e e

e e e

Φi is a complete set of wave-functions

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Zassenhaus formula

• If the operators  and Bˆ are order 1/M, the error of the approximation is of order O(1/M2).

11 ˆ ˆˆ ˆ ˆˆ [A 2B,[A,B]][A,B]ˆ ˆˆ ˆA B A B 62

ˆA B

...

e e e e e

e e

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Trotter-Suzuki Formula

where  and Bˆ are non-commuting operators

ˆ ˆˆ ˆA B A/ B/limM

M M

Me e e

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Quantum Ising Chain in Transverse Field

• Hamiltonian

• where

1 0ˆ ˆ ˆˆ ˆ ˆH V Hz z x

i i ii i

J

0 1 0 1 0ˆ ˆ ˆ, ,

1 0 0 0 1yx zi

i

Pauli matrices at different sites commute.

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Complete Set of States• We choose the eigenstates of

operator σz:

• Insert the complete set in the products:

1 2 1 2ˆ | |zi N i N

0 0ˆ ˆˆ ˆH HV VM M M Me e e e

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A Typical Term1

, , 1ˆ 2, , 1

logtanh( )1| | sinh(2 )

2

xi i k i ka

i k i k

ae a e

(i,k)

Space direction

Trotter or β direction

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Classical Partition Function

,

1 2, 1, , , 1H , ,

0{ }

1 2

Tr

where

, logcoth

i k

i k i k i k i ki k i k

K K

Z e Z e

JK K

M M

Note that K1 1/M, K2 log M for large M.

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Summary• Briefly introduced (classical) MC

method• Quantum MC (variational,

diffusional, and Trotter-Suzuki)• Application to quantum dot

models