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• 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 M

T 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 P

P P

Pick N configurations randomly with probability { }P

1{ } ...{ }N

Is an arbitrary probability distribution { }P

Calculate1...

{ }1

{ }i

i N i

fS

N P

Stochastic summation (cont.)Mean and standard deviation:

(Central limit theorem)

2 1std{ } std

N

fS S S

PN

1...

{ }1

{ }i

Ni N Pi

f fS

N P P

So… for any choice of P. NS S

How to choose P?

Importance Sampling

We should choose P such that is minimized. stdf

P

For example, if , then !std 0f

P

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 { }

{ }He

PZ

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 configuration

2. Pick a spin j. Propose a new configuration that differs by one spin flip

3. 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 mechanicsˆ

Tr 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 R

H V Rm

Quantum statistical mechanics

2

ˆ 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:

P

1

2

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

1

2

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

1

2

U n n U s n n

s

e e e

/ 2cosh Ue

U<0

U>0

1 1 12 2 4

1

1

2

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

1

4 det detd

ik

L U

ik iks

Z e M s M s

,

,1 ikV sK

ikk

M s e e

Monte Carlo Evaluation

ˆ

ˆ

det detˆTrˆ

det detTr

ik

ik

H ik ik iks

Hik ik

s

O s M s M sOe

OM s M se

1

4 det detd

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 M

Z

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

filling

Hubbard model (U>0): half filling

Heisenberg 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.

Summary

Quoting 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).