[PDF] [PDF] The Ising Model

[PDF] Ising Modelpptx - MIT

-Magnetism and the Ising Model (today's lecture) -Thermodynamic results in magnetism, such as the critical And the magnetization per spin is simply

[PDF] The Ising Model

of spins, and m is the dimensionless magnetization per spin Given N+ and N_ ( that is, Nand m), we can write the total entropy,

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The quantity χ in (5 18) is the zero-field susceptibility Because a system of noninteracting spins is paramagnetic, such a model is not applicable to materials such as iron that can have a nonzero magnetization even when the magnetic field is zero Ferromagnetism is due to the interactions between the spins

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Figure 14 4: Two–dimensional square lattice Ising model magnetization per spin Helmholtz free energy F = E − TS where S is now the entropy

[PDF] LECTURE 18 The Ising Model (References: Kerson Huang

Consider a lattice of N sites with a spin S on each site Each spin there is no spontaneous magnetization and the one-dimensional Ising model never exhibits

[PDF] The Ising model in the canonical ensemble

4 mai 2016 · 1: The temperature dependence of m(T), the mean magnetization per spin Ising model: phase transition Tc = 2 269 J/kB Low T: spin 

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of Λ We then specify some fixed choice for each spin just outside of Λ For example we Λ Two that are of particular interest in physics are the magnetization

[PDF] Magnetization process in the exactly solved spin-1/2 Ising

through the Heisenberg spin pairs placed in-between each couple of the Ising spins The total Hamiltonian of the spin-1 2 Ising-Heisenberg model on 

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and, above a certain temperature Tc, the magnetization vanishes entirely Each spin is a magnetic dipole and therefore produces a magnetic field which influ-

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LECTURE18

TheIsingModel

de¯nedtobe E

IfSig=¡X

hi;jiJ ijSiSj¡N X i=1B iSi(1) J B E

IfSig=¡JX

hi;jiS iSj¡BN X i=1S i(2)

Thepartitionfunctionisgivenby

Z=+1 X s

1=¡1+1

X s

2=¡1:::+1

X s

N=¡1e¡¯EIfSig(3)

wecanwrite E

IfSig=¡JN

X i=1S iSi+1¡BN X i=1S i(4)

Theperiodicboundaryconditionmeansthat

S

N+1=S1(5)

Thepartitionfunctionis

Z=+1 X s

1=¡1+1

X s

2=¡1:::+1

X s

N=¡1exp"

¯N X i=1(JSiSi+1+BSi)# (6) canbeexpressedintermsofmatrices: Z=+1 X s

1=¡1+1

X s

2=¡1:::+1

X s

N=¡1exp"

¯N X i=1µ

JSiSi+1+1

(7) itsmatrixelementsaregivenby hSjPjS0i=exp½

JSS0+1

2B(S+S0)¸¾

(8) elements: h+1jPj+1i=exp[¯(J+B)] h¡1jPj¡1i=exp[¯(J¡B)]

ThusanexplicitrepresentationforPis

P=Ãe¯(J+B)e¡¯J

e

¡¯Je¯(J¡B)!

(10) 2 Z=+1 X s

1=¡1+1

X s

2=¡1:::+1

X s

N=¡1hS1jPjS2ihS2jPjS3i:::hSNjPjS1i

+1 X s

1=¡1hS1jPNjS1i

=TrPN =¸N ++¸N

¡(11)

Eq.(5).Theeigenvalueequationis

det ¯e

¯(J+B)¡¸e¡¯J

e

¡¯Je¯(J¡B)¡¸¯

Solvingthisquadraticequationfor¸gives

§=e¯J·

cosh(¯B)§q (13)

WhenB=0,

+=2cosh(¯J)(14)

¡=2sinh(¯J)(15)

spin: F

NkBT=limN!11NlnZ

=limN!11 Nln8 :¸N +2

41+ø¡

N3 59
=ln¸++limN!11Nln2

41+ø¡

N3 5 =ln¸+(16)

SotheHelmholtzfreeenergyperspinis

F

N=¡kBTNlnZ=¡kBTln¸+

=¡J¡kBTln· cosh(¯B)+q (17) 3

Themagnetizationperspinis

m=M N 1

¯N@lnZ@B

=¡1 N@F@B sinh(¯B) q

ApplicationsoftheIsingModel

LatticeGas

V(r)=8

:1(r=0)

¡"o(r=a)

0otherwise(19)

n i=(1ifsiteiisoccupied

0ifsiteiisunoccupied(20)

4

Theinteractionenergyis

E

Gfng=¡"oX

hi;jin inj(21) S i=2ni¡1(22) functions,itturnsoutthat²o=4J.

BinaryAlloy

sothatNA+NB=N.Theoccupationofeachsiteis n i=(1ifsiteiisoccupiedbyatomA

0ifsiteiisoccupiedbyatomB(23)

GeneralizationstoOtherSpinModels

5 onecouldhavesomethinglike J ij=A j~ri¡~rjjn(24)

Yosida.TheRKKYinteractionhastheform

J(r)»sin(2kFr)¡2kFrcos(2kFr)

(kFr)4 cos(2kFr) (kFr)3(25) actionoscillatesanddecaysasapowerlaw.

FrustrationandSpinGlasses

H=¡X

i>jJ ijSi¢Sj(26) J con¯gurationispossible. 6 E configuration coordinate memory.

MonteCarloSimulations

(PA+PB)Theresultisknownas`detailedbalance': P n(t)Wn!m=Pm(t)Wm!n(28) P n(t)=e¡En=kBT Z(29) probabilityisgivenbytheratio P n(t)

Pm(t)=e¡En=kBTe¡Em=kBT=Wm!nWn!m

=e¡(En¡Em)=kBT =e¡¢E=kBT(30) where¢E=(En¡Em). W n!m=(exp(¡¢E=kBT)¢E>0

1¢E<0(31)

1.Chooseaninitialstate.

2.Chooseasitei.

8

4.If¢E<0,°ipthespin.If¢E>0,then

(a)Generatearandomnumberrsuchthat05.Gotothenextsiteandgoto(3). hAi=P inthesimulation.

H=¡X

i>jJ ijSi¢Sj =¡1 2X i;jJ ijSi¢Sj =¡X iS i¢0 @X j1

2JijSj1

A =¡X iS i¢hi(32) h i=1 2X jJ ijSj(33) local¯eldsoftheneighboringspins. 9quotesdbs_dbs19.pdfusesText_25