ising model monte carlo methods

8/10/2019 Ising Model Monte Carlo Methods

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Ising Model

Dieter W. Heermann

Monte Carlo Methods

2009

Dieter W. Heermann (Monte Carlo Methods) Ising Model 2009 1 / 17
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Outline

1 Introduction

2 The Model

3 Local Monte Carlo AlgorithmsFixed Energy Monte CarloMetropolis-Hastings Monte Carlo

4 Global Algorithms

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The Model

The Ising model [1] is defined as follows:

Let G=Ld be a d-dimensional lattice.

Associated with each lattice site i is a spin siwhich can take on the

values +1 or 1.The spins interact via an exchange coupling J. In addition, we allowfor an external field H.

The Hamiltonian reads

H=Ji,j

sisj+Hi

si (1)

The first sum on the right-hand side of the equation runs over nearestneighbours only.

The symbol denotes the magnetic moment of a spin. If theexchange constant Jis positive, the Hamiltonian is a model forferromagnetism, i.e., the spins tend to align parallel.

For Jnegative the exchange is anti ferromagnetic and the spins tendto align antiparallel. In what follows we assume a ferromagnetic

interaction J>0.Dieter W. Heermann (Monte Carlo Methods) Ising Model 2009 3 / 17
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The Model

i



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L l M t C l Al ith Fi d E M t C l


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Local Monte Carlo Algorithms Fixed Energy Monte Carlo

Let Ebe the fixed energy and suppose that a spin configurations= (s1,..., sN) was constructed with the required energy.

We set the demon energy to zero and let it travel through the lattice.At each site the demon attempts to flip the spin at that site.

If the spin flip lowers the system energy, then the demon takes up theenergy and flips the spin.

On the other hand, if a flip does not lower the system energy the spinis only flipped if the demon carries enough energy.

A spin is flipped if

ED H>0 (2)

and the new demon energy is

ED=ED H (3)


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After having visited all sites one time unit has elapsed and a newconfiguration is generated.In Monte-Carlo method language the time unit is called the MC step

per spin.After the system has relaxed to thermal equilibrium, i.e., after n0Monte-Carlo Steps (MCS), the averaging is started. For example, wemight be interested in the magnetization.Let n be the total number of MCS, then the approximation for the

magnetization is

= 1

n n0

nin0

m(si) (4)

where siis the ith generated spin configuration. Since the demonenergies ultimately become Boltzmann distributed, it is easy to showthat

J

kBT =

1

4ln1 + 4 J

ED (5)


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Local Monte Carlo Algorithms Metropolis-Hastings Monte Carlo
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Metropolis-Hastings Monte Carlo

The simplest and most convenient choice for the actual simulation is

a transition probability involving only a single spin; all other spinsremain fixed.

It should depend only on the momentary state of the nearestneighbours.

After all spins have been given the possibility of a flip a new state iscreated. Symbolically, the single-spin-fliptransition probability iswritten as

Wi(si) : (s1,..., si,..., sN)(s1,..., si, ..., sN)

where Wiis the probability per unit time that the ith spin changesfrom si to si.

With such a choice the model is called the single-spin-flip Ising model(Glauber).


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Local Monte Carlo Algorithms Metropolis Hastings Monte Carlo

Let P(s) be the probability of the state s. In thermal equilibrium atthe fixed temperature T and field K, the probability that the i-th spin

takes on the value si is proportional to the Boltzmann factor

Peq(si) = 1

Zexp

H(si)

kBT

The fixed spin variables are suppressed.

We require that the detailed balance condition be fulfilled:

Wi(si)Peq(si) =Wi(si)Peq(si)

or

Wi(si)

Wi(si)=

Peq(si)

Peq(si)


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g p g

It follows that

Wi(si)

Wi(s

i)

=exp(si/Ei)

exp(si/E

i)

where

Ei =Ji,j

sj

The derived conditions do not uniquely specify the transitionprobability W.The Metropolis function

Wi(si) =min 1, 1exp(H/kBT)and the Glauber function

Wi(si) =(1 sitanh Ei/kBT)

2

where is an arbitrary factor determining thetimescale.Dieter W. Heermann (Monte Carlo Methods) Ising Model 2009 11 / 17

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g p g

Algorithmically the Metropolis MC method looks as follows:

1: Specify an initial configuration.2: Choose a lattice site i.3: Compute Wi.4: Generate a random number R[0, 1].5: if Wi(si)>R then6: si si7: else

8: Otherwise, proceed with Step 2 untilMCSMAXattempts have been

made.9: end if




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g g

Java program can be found

here


http://wwwcp.tphys.uni-heidelberg.de/comp-phys/Examples/ising/index.htmhttp://wwwcp.tphys.uni-heidelberg.de/comp-phys/Examples/ising/index.htmhttp://wwwcp.tphys.uni-heidelberg.de/comp-phys/Examples/ising/index.htmhttp://wwwcp.tphys.uni-heidelberg.de/comp-phys/Examples/ising/index.htmhttp://find/


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#include

#include

# defineL 10

int main(intargc, char*argv[])

{

intmcs,i,j,k,ip,jp,kp,in,jn,kn;

intold_spin,new_spin,spin_sum; intold_energy,new_energy;

intmcsmax;

intspin[L][L][L];

intseed;

doubler;

doublebeta;

doubleenergy_diff;

doublemag;

mcsmax = 100;

beta = 0.12; // beta = J/kT KC = 0.2216544 Talapov and Blte (1996)

seed = 4711;

srand(seed);

for(i=0;i


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Global Algorithms
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Global Algorithms

See lecture on cluster and multi-grid algorithms


Global Algorithms
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E. Ising: Z. Phys. 31,253 (1925)

M. Creutz: Phys. Rev. Lett. 50, 1411 (1983)

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ising model monte carlo methods

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