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Classical and Quantum Monte Carlo Methods. Or: Why we know as little as we do about interacting fermions. Erez Berg Student/Postdoc Journal Club, Oct. 2007. Outline. Introduction to MC Quantum and classical statistical mechanics Classical Monte Carlo algorithm for the Ising model - PowerPoint PPT Presentation
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Classical and Quantum Monte Carlo Methods
Or: Why we know as little as we do about interacting fermions
Erez BergStudent/Postdoc Journal Club, Oct. 2007
Outline
• Introduction to MC• Quantum and classical statistical
mechanics• Classical Monte Carlo algorithm for
the Ising model• Quantum Monte Carlo algorithm
for the Hubbard model• “Sign problems”
Introduction: Monte Carlo
www.wikipedia.org
Monte Carlo, Monaco
“Monte Carlo” solution:
i =
Introduction: Monte Carlo
Suppose we are given the problem of calculating
10000000
1i
i
S a
And have nothing but a pen and paper.
… And we may need to sum much fewer numbers.
Statistical mechanics
ˆTr HZ e
Thermodynamic quantities
1
2
2 2
ln
, ,...V
F Z
F FC T MT H
Correlation functions
ˆ1ˆ ˆTr HO OeZ
Statistical mechanicsExample: Classical Ising model
1ˆ { ,..., }N i j
ij
H J
Problem: calculate 1
1
ˆ { ... }1 1
...
1L
L
Hk keZ
Number of terms = 2100=1030
On a supercomputer that does 1015 summations/sec, this takes 107 years…
2D lattice with 10x10 sites:
Stochastic summation
{ }
{ }S f
Trick: write{ }
{ } { }{ } P
f fS PP P
Pick N configurations randomly with probability { }P
1{ } ...{ }N
Is an arbitrary probability distribution { }P
Calculate1...
{ }1{ }
i
i N i
fSN P
Stochastic summation (cont.)Mean and standard deviation:
(Central limit theorem)
2 1std{ } stdN
fS S SPN
1...
{ }1{ }
iN
i N Pi
f fSN P P
So… for any choice of P. NS S
How to choose P?
Importance Sampling
We should choose P such that is minimized. std fP
For example, if , then !std 0fP
P f
… This is a cheat, because to normalize P we need to sum over f.
But it shows the correct trend: choose P which is large where f is large.
Sampling Technique
Back to the Ising model: 1
1
ˆ { ... }1 1
...
1L
L
Hk keZ
A natural choice of P:ˆ { }
{ }HePZ
How to choose random configurations with probability ?{ }P
Solution: Generate a Markov process that converges to { }P
The Metropolis Algorithm
1. Start from a random configuration2. Pick a spin j. Propose a new configuration
that differs by one spin flip3. If , accept the new
configuration:4. If , accept the new configuration
with probability 5. And back to step 2…
1{ }
{ }'
1{ }' { }P P 2{ } : { }'
1{ }' { }P P
1{ }'/ { }P P
1 2 3{ } { } { } ...
“Random walk” in configuration space:
Outline
• Introduction to MC• Quantum and classical statistical
mechanics• Classical Monte Carlo algorithm for
the Ising model• Quantum Monte Carlo algorithm
for the Hubbard model• “Sign problems”
Quantum statistical mechanicsˆTr HZ e
…But now, H is an operator.
In general, we don’t even know how to calculate exp(-H).
Example: Single particle Schrodinger equation
2
2ˆ2 RH V Rm
Quantum statistical mechanics
2ˆ ˆ ˆ ˆ
0
Tr Tr .... ... exp2
H H H H mZ e e e e DR d R V R
Path integral formulation:
2/1
11
( )... exp ( )2
Pi i
Pi
m R RZ dR dR V R
Discrete time version:
P12
P
1,ir 2,ir
The Hubbard Model †
, ,
1 1ˆ . .2 2i j i i i
ij i i
H t c c H c n U n n
•“Prototype” model for correlated electrons•Relation to real materials: HTC, organic SC,…•No exact (or even approximate) solution for D>1
How to formulate QMC algorithm?
Determinantal MC
Blankenbecler, Scalapino, Sugar (1981)
ˆ ˆ ˆ ˆTr Tr ....H H H HZ e e e e
ˆ ˆ ˆH K V
†
, ,
ˆ . .
1 1ˆ2 2
i j iij i
i ii
K t c c H c n
V U n n
ˆ ˆ ˆ 2H K Ve e e O
Trotter-Suzuki decomposition:
Determinantal MC (2)
The term is quadratic, and can be handled exactly. K
What to do with the term?V
ˆ ˆ ˆ ˆ ˆ ˆTr ....K V K V K VZ e e e e e e
Hubbard-Stratonovich transformation:
1 1 12 2 4
1
12
U n n U s n n
s
e e e
/ 2cosh Ue
Note that this works only for U>0
Determinantal MC (3)Hubbard-Stratonovich transformation for any U:
1 1 1
12 2 4
1
12
U n n U s n n
s
e e e
/ 2cosh Ue
U<0
U>0
1 1 12 2 4
1
12
U n n U s n n
s
e e e
Determinantal MC (4)For the U>0 case, the partition function becomes:
† † † †1 14 Tr ....
d i ij j i ij j i ij j i ij jij ij ij ij
ik
c K c c V c c K c c V N cL U
s
Z e e e e e
Here † † †. .i ij j i j i iij ij i
c K c t c c H c c c
†, ,i ij j ik i i
ij i
c V k c s n n
i
sikk
k+1
k-1
i+1
Determinantal MC (5)Now, since the action is quadratic, the fermions can be traced out.
†
Tr det 1i ij j ijc A c Ae e
†
Tr det 1i ij j ijc A k c A k
k k
e e
14 det det
d
ik
L U
ik iks
Z e M s M s
,
, 1 ikV sKik
k
M s e e
Monte Carlo Evaluation
ˆ
ˆ
det detˆTrˆ
det detTrik
ik
H ik ik iks
Hik ik
s
O s M s M sOeO
M s M se
14 det det
d
ik
L U
ik iks
Z e M s M s
And, by a variation of Wick’s theorem,
How to calculate this sum?
Monte Carlo: interpret as a probability P{s} det detM M
Z
Sign Problem
sgnˆ
sgnP
P
O sO
Problem: is not necessarily positive. det detM MZ
Solution: det det sgn det detM M M M
det detM M
P sZ
Probability distribution:
And evaluate the numerator and denominator by MC!
At low temperatures and large U, the denominator becomes extremely small, causing large errors in .
sgnP
O
Sign Problem (2)But…
4x4 Hubbard model (Loh et al., 1990)
Sign Problem (3)Note that for U<0,
det detM M
Therefore
And there is no sign problem!
det det 0M M
Summary
No sign problem Sign problemHubbard model (U<0) Hubbard model (U>0): generic
fillingHubbard model (U>0): half
fillingHeisenberg model, triangular
lattice“Bose-Hubbard” model (any U) Most “frustrated” spin models
Heisenberg model, square lattice
• “Sign problem free” models can be considered as essentially solved!
• In models with sign problems, in many cases, the low temperature physics is still unclear.
• Unfortunately, many interesting models belong to the second type.
SummaryQuoting M. Troyer:“If you want you can try your luck: the person who finds a general solution to the sign problem will surely get a Nobel prize!”
References•M. Troyer, “Quantum and classical monte carlo algorithms, www.itp.phys.ethz.ch/staff/troyer/publications/troyerP27.pdf
• N. Prokofiev, lecture notes on “Worm algorithms for classical and quantum statistical models”, Les Houches summer school on quantum magnetism (2006).
•R. R. Dos Santos, Braz. J. Phys. 33, 36 (2003).
•R. T. Scalettar, “How to write a determinant QMC code”, http://leopard.physics.ucdavis.edu/rts/p210/howto1.pdf
•E. Dagotto, Rev. Mod. Phys. 66, 763 (1994).
•J. W. Negele and H. Orland, “Quantum many particle systems”, Addison-Wesley (1988).