Download Diffusion quantum Monte Carlo

Quantum Monte Carlo
Methods
Jian-Sheng Wang
Dept of Computational Science,
National University of Singapore
1
Outline
•
•
•
•
Introduction to Monte Carlo method
Diffusion Quantum Monte Carlo
Application to Quantum Dots
Quantum to Classical --TrotterSuzuki formula
2
Stanislaw Ulam (19091984)
S. Ulam is credited
as the inventor of
Monte Carlo method
in 1940s, which
solves mathematical
problems using
statistical sampling.
3
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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Necessary and sufficient
conditions for convergence
• Ergodicity
[Wn](X - > X’) > 0
For all n > nmax, all X and X’
• Detailed Balance
P(X) W(X -> X’) = P(X’) W(X’ -> X)
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Taking Statistics
• After equilibration, we estimate:
1
Q (X )  Q (X ) P(X ) d X 
N
N
Q (X )
i 1
i
• 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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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 matrix
T(X -> X’) = T(X’ -> X)
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The Statistical Mechanics of
Classical Gas/(complex) Fluids/Solids
Compute multi-dimensional integral
Q
Q (x , y , x , y ,...) e


1
1
2
e

2
E ( x 1,y 1,...)
kBT

E ( x 1, y 1,...)
kBT
dx1dy1 ...dxN dyN
dx1dy1 ...dxN dyN
where potential energy
N
E (x1 ,...)  V (dij )
i j
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Advanced MC Techniques
•
•
•
•
Cluster algorithms
Histogram reweighting
Transition matrix MC
Extended ensemble methods (multicanonical, replica MC, Wang-Landau
method, etc)
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2. Quantum Monte
Carlo 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:
E0 
ˆ|
 |H
|
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Quantum Expectation by
Monte Carlo
ˆ| 
 |H
|

*
ˆ ( X )
dX

(
X
)H

*
dX

( X ) ( X )

  dX P(X )EL (X )
where
1 ˆ
EL (X ) 
H  (X )
 (X )
P(X ) | (X ) |2
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Zero-Variance Principle
• The variance of EL(X) approaches zero as
Ψ approaches the ground state wavefunction Ψ0.
σE2 = <EL2>-<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

i
 Ĥ ,  (t )  e
t
i
Ĥt
(0)
Let  = it, the evolution becomes
(t )  e

Ĥ
(0)
As  -> , only the ground state survive.
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Diffusion Equation with
Drift
• The Schrödinger equation in
imaginary time  becomes a diffusion
equation:
  1 2


   V (X )  ET  
  2

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 that
f = Ψ ΦT* ≥ 0
f is interpreted
as walker
density.
f
Ψ
ΦT
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Equation for f
f
1 2

   f    vf   E L (X )  ET f

2
where
1
1 ˆ
v
T and E L 
HT
T
T
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Monte Carlo Simulation
of the Diffusion Equation
• If we have only the first term -½2f,
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 birthdeath 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 statistics
5. Adjust ET to make average population
constant.
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Statistics
• The diffusion Quantum Monte Carlo
provides estimator for
Q 
 dX Q (X )f (X )
 dX f (X )
1

N

ˆ |
0 | Q
T
 0 |T
N
Q ( X )
i 1
where
i
1 ˆ
Q (X ) 
Q T
T
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Trial Wave-Function
• The common choice for interacting
fermions (electrons) is the SlaterJastrow form:
1 (r1 ) 1 (r2 )
1 (rN )
2 (r1 )
J (X )
 (X )  e
N (r1 )
N (rN )
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Example: Quantum Dots
• 2D electron gas with Coulomb
interaction in magnetic field
N
1
i  j | ri  rj |
ĤN   hˆk  
k 1
where ri  (xi , yi ) and
2

 2
ˆz 
1
1
B
B ˆ
2
2
ˆ
h      0 
 r  V (r)   Lz  gs

2
2
4 
2
2 
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):
1
 r 2
e im |m| |m|
2
 n ,m ,s (r ,  ,  )  cnm
r Ln (r ) e 2 s ( )
2
where
2
B
 2  02 
4
• L is Laguerre polynomial
• Energy level En,m,s=(n+2|m|+1)h + g
B(m+s)B
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Six-Electrons Groundstate 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
N=5
L=6
S=3
Electron charge
density from trial
wavefunction
(Slater
determinant of
Fock-Darwin
solution), exact
diagonalisation
calculation, and
QMC.
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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
Z  e
 E ( X )
X
   |e
ˆ
 H
|

 Tr e
• Expectation value
  Ĥ
Q 




Ĥ

Tr Q̂ e




Tre   Ĥ
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D Dimensional Quantum
System to D+1 Dimensional
Classical system
 | e   Ĥ |   | (e


i , j , ,k
 |e

ˆ
H
M

 Ĥ
M
)M |
| i i | e

ˆ
H
M
| j
k | e

ˆ
H
M
|
Φi is a complete set of
wave-functions
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Zassenhaus formula
e
ˆ
ˆ B
A
ˆ
A
ˆ
B
e e e
e e
Â
1 ˆ ˆ
 [A,B]
2
e
1 ˆ ˆ ˆ ˆ
 [A
2B,[A,B]]
6
...
ˆ
B
• If the operators  and Bˆ are order
1/M, the error of the approximation
is of order O(1/M2).
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Trotter-Suzuki Formula
e
ˆ
ˆ B
A

 lim e
M 
ˆ M
A/
e
ˆ/ M
B

M
where  and Bˆ are non-commuting
operators
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Quantum Ising Chain in
Transverse Field
• Hamiltonian
z z
x
ˆ
ˆ H
ˆ
H  J ˆi ˆi 1  ˆi  V
0
i
i
• where
0 1 
 0 i 
1 0 
y
z
ˆ  
, ˆ  
, ˆ  



1
0
i
0
0

1






x
Pauli matrices at different sites
commute.
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Complete Set of States
• We choose the eigenstates of
operator σz:
ˆ | 1 2
z
i
 N  i | 1 2
N
• Insert the complete set in the
products:
e
ˆ
H
 0
M
e
Vˆ

M
e
ˆ
H
 0
M
e
Vˆ

M
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A Typical Term
 i ,k | e
a ˆix
|  i ,k 1
Trotter or β
direction
1

sinh(2a ) e
2

1
 
logtanh(a )
2 i ,k i ,k 1
(i,k)
Space direction
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Classical Partition
Function
Z  Tr e   Ĥ  Z 0
{


i ,k
e
K1 i ,k i 1,k K2 i ,k i ,k 1
i ,k
i ,k
}
where
K1 
J
M
, K2  logcoth

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
38