blume-emery-griffiths(beg) model
TRANSCRIPT
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
BlumeBlume--EmeryEmery--GriffithsGriffiths (BEG) Model(BEG) ModelModel on Model on CellularCellular AutomataAutomata andand algorithmalgorithmSimulationsSimulations andand resultsresults
OutlineOutline
BlumeBlume--EmeryEmery--GriffithsGriffiths (BEG) Model(BEG) ModelModel on Model on CellularCellular AutomataAutomata andand algorithmalgorithmSimulationsSimulations andand resultsresults
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 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 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 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
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
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
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
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
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
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
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
• 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:
• 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:
• 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:
• 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:
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
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
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
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
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
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.
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
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
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
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
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
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
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!!