blume-emery-griffiths(beg) model

PhasePhase transitionstransitions of of BlumeBlume--EmeryEmery--GriffithsGriffiths Model on a Model on a CellularCellular

AutomataAutomata

Nurgül SEFEROĞLUNurgül SEFEROĞLU

Gazi Gazi UniversityUniversityInstituteInstitute of of ScienceScience andand TechnologyTechnology

DepartmentDepartment of of AdvancedAdvanced TechnologiesTechnologiesTurkeyTurkey

JuneJune 20082008

OutlineOutline

BlumeBlume--EmeryEmery--GriffithsGriffiths (BEG) Model(BEG) ModelModel on Model on CellularCellular AutomataAutomata andand algorithmalgorithmSimulationsSimulations andand resultsresults

OutlineOutline


OutlineOutline


BLUMEBLUME--EMERYEMERY--GRIFFITHSGRIFFITHS(BEG)(BEG) MODELMODEL

Hamiltonian of the model

∑∑ ∑ ++=>< >< i

2i

2j

ij ij

2ijiI SDSSKSSJH

J bilinear and K biquadratic interaction constantsD single-ion anisotropy constant

Si= -1, 0, 1

BEG model(1971): He3-He4 mixtures and other physicalsystems

The model has been studied by differenttechniques. Most of these analysis predict that, the model on three dimension shows a variety of interesting features:

single and double re-entrancy regionferrimagnetic phasesincluding a tricitical point, critical end point orbicritical end point for certain model parameters







The model is simulated on a cellular automata by usingimproved algorithm(it is improved from Creutzalgorithm)

The calculations are done on a simple cubic lattice of the linear dimensions L=12,16,18 and 24 with periodicboundary conditions.

Blume-Emery-Griffiths Model on a cellularautomata

EachEach site of site of thethe latticelattice has has threethree variablesvariables: : AllAll variablesvariables areare an an integerinteger

TheThe firstfirst oneone is is IsingIsing spinspin BiBi, , BiBi = Si + 1, Si= = Si + 1, Si= --1, 0 1, 0 andand 1 1 BiBi= 0, 1 = 0, 1 andand 22TheThe secondsecond oneone is is HHkk ((kinetickinetic energyenergyassociatedassociated withwith thethe demondemon) ) ItIt is is equalequaltoto thethe changingchanging in in thethe IsingIsing energyenergy forforanyany spinspin flipflip. . ItIt takestakes integerinteger valuesvalues in in thethe intervalinterval (0,m)(0,m)TheThe thirdthird variablevariable, w, , w, is is parityparityitsits valuevalue maymay be 0 be 0 oror 1.1.

UpdatingUpdating rulerule is is appliedappliedonlyonly blackblack sitessitesandand thenthen theirtheir colorcolor is is changedchanged intointo whitewhite;;WhiteWhite sitessites arearechangedchanged intointo blackblackwithoutwithout updatingupdating

it it providesprovides a a checkboardcheckboard stylestyle updatingupdating

w=1w=1w=0w=0






w=1w=1w=0w=0






w=1w=1w=0w=0






w=1w=1w=0w=0






w=1w=1w=0w=0






w=1w=1w=0w=0






w=1w=1w=0w=0






w=1w=1w=0w=0

• For a site to be updated, its spin is changed to one of the other two states with ½ probability.

• The change in the Ising energy, dHI is calculated.

• If this energy is transferable to or from themomentum variable(Hk) associated with this site, thischange is done and the momentum is appropriatelychanged. Otherwise, the spin and momentum are not changed.

• During the updating process, total energy of thesystem H=HI+Hk is conserved.

updating rule:





updating rule:





updating rule:





updating rule:

Heating algorithm

All spins take the F/SQ ordered structure according to theselected (J,K,D) parameter set.

Kinetic energy per site is given to the certain percent of thelattice via second variable.

This configuration is run during 10000 CA time steps.

At the end of this step, the configuration at low temperature is obtained.

This configuration has been chosen as a starting configuration forthe heating run.

During the heating cycle, energy is added to the system throughthe second variable of each site(Hk) after “t” cellular automatonsteps. This process is realized by increasing of certain values in thekinetic energy of each site.

Heating algorithm







Heating algorithm







Heating algorithm







Heating algorithm







Heating algorithm







Detail of simulationsFor estimating the kind of PTs, the temperaturevariations of the some quantites are calculated:

order parameters,

∑=

=N

1iiSN

1m ∑=

=N

1i

2iSN

1q

N=L3

susceptibility,kT/)mm(N

hm 22 ><−><=

∂∂

=χ

internal energy,

0I H/HU =

specific heat

∑∑ ∑ ++=>< >< i

2i

2j

ij ij

2ijiI SDSSKSSJH

222I )kT/()UU(NT/HC ><−><=∂∂=

In addition, finite-size scaling theory is used forestimating the static critical exponents.

Binder cumulant

22

4

L MM1g

><><

−=

Detail of simulations

The results of simulations

The phase diagram in the ground state of model on a simplecubic lattice

Ferromagneticorder(F), all spinsare “+1” or “-1”

Perfect zeroorder(S), “0”

Stagger quadrupolar(SQ), twosublattice(A and B), randomly, in A Si=±1 and in B Si=0













At T≠0, the variaous phases of the model are definedaccording to the order parameters at a selected model parameter set:

For F region,

Ferromagnetic phase (F) m≠0, q≠2/3

Quadrupolar phase (Q) m=0, q≠2/3;

For SQ region,

Ferromagnetic phase (F) mA = mB ≠ 0 and qA = qB

Quadrupolar phase (Q) mA = mB = 0 and qA = qB

Staggered quadrupolar phase(SQ)mA = mB = 0 and qA ≠ qB


For F region,

Ferromagnetic phase (F)m≠0, q≠2/3

Quadrupolar phase (Q)m=0, q≠2/3;

For SQ region,

Ferromagnetic phase (F) mA = mB ≠ 0 and qA = qB

Quadrupolar phase (Q) mA = mB = 0 and qA = qB



For F region,

Ferromagnetic phase (F)m≠0, q≠2/3

Quadrupolar phase (Q)m=0, q≠2/3;

For SQ region,

Ferromagnetic phase (F)mA = mB ≠ 0 and qA = qB

Quadrupolar phase (Q)mA = mB = 0 and qA = qB


Phase transition in BEG model

The obtained results in the F region andnear the F and PZ phase boundary

• Phase diagrams are obtained for certain model parameters

TCP: at which the PT changes from second order to first order

second-order Q→F(at certain parameters) Q: m=0, q≠2/3;

F: m≠0, q≠2/3

re-entrant Q→F→Q(at certain parameters)

double re-entrant Q→F→Q→F(at certain parameters)

double re-entrantQ→F→Q→F

(at certain parameters)

The obtained results in the SQ region and near theF and SQ phase boundary

phase diagrams are obtained for certain model parameters

BCP: at which two second order lines meet on the first order line

successive Q → F → SQ PT

Q→F second orderF→SQ first order

re-entrant Q → F → Q → SQ PT

re-entrant Q → F → Q → SQ PT

Static critical exponents

The infinite lattice critical point(Tc) are obtained from theintersection of the Binder cumulant curves for different latticesizes.

for the F→Q PTfor the Q→F PT

Exponent ν:

)L(gg /1L

νε=νν can be can be obtainedobtained usingusing thethe finitefinite size size scalingscaling relationrelation forforBinderBinder cumulantcumulant, , whichwhich is is defineddefined byby

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

-40 -20 0 20 40ε 1L1/ν

g L

L=12L=16L=18L=24

ν=0.64

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

-40 -20 0 20 40ε 2L1 / ν

g LL=12L=16L=18L=24

ν=0.64

TheThe scalingscaling data data forfor thethe finitefinite--size size latticeslattices lieslieson a on a singlesingle curvecurvenearnear thethe criticalcriticaltemperaturestemperatureswhenwhen νν=0.64 =0.64

ε=(T-Tc)/Tc

for the F→Q PTfor the Q→F PT

exponent β: )L(XLm /1/ ννβ− ε=

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.0 0.5 1.0 1.5 2.0Log(ε1L1 /ν)

Log(

mLβ

/ν)

L=12L=16L=18L=24Linear(slope=0.55 T>Tc1)Linear(slope=0.31 T<Tc1)

d = 0k = -1.01

(c)

-1.5

-1.3

-1.1

-0.9

-0.7

-0.5

-0.3

-0.1

0.1

0.3

0.5

0.5 1.0 1.5 2.0Log(ε 2L1 / ν)

Log

(mLβ

/ν)

L=12L=16L=18L=24Linear (slope=0.31 T>Tc2)Linear (slope=0.55 T<Tc2)

d=0k=-1.01

(a)

for the Q→F PT for the F→Q PT

exponent γ : )L(YLkT /1/ ννγ ε=χ

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0 1.0 2.0

Log(ε1L1 /ν)

Log(

k BT χ

L−γ/

ν )

L=12L=16L=18L=24Linear(slope=1.25)Linear (slope=1.25)

T>Tc1

T<Tc1

(d)

for the Q→F PT for the F→Q PT

-2.6

-2.4

-2.2

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

0.0 1.0 2.0Log(ε 2L1/ν)

Log

(kBT

χL

−γ/

ν )L=12L=16L=18L=24Linear (slope=1.25) Linear (slope=1.25)

T>Tc2

T<Tc2

(b)

For the all continous Q →F PT, the estimatedvalues of critical exponents are equal touniversal values (β=0.31, γ=1.25, α=0.12,ν = 0.64)

ThankThank youyou forfor youryour attentionattention!!

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blume-emery-griffiths(beg) model