Ising ModelIsing Model

Presentation in coursePresentation in course

Advanced Solid State PhysicsAdvanced Solid State PhysicsBy Michael HeBy Michael Heßß

History of critical point History of critical point behaviorbehavior

history: three diff periods in the development of the history: three diff periods in the development of the theory of critical point behaviortheory of critical point behavior

1. v.d.Waals mean field theory to liquid-gas phase 1. v.d.Waals mean field theory to liquid-gas phase transition but after 1965 numerical calculations and transition but after 1965 numerical calculations and experiments proved that this mean field theory was experiments proved that this mean field theory was quantitatively incorrect around the critical pointquantitatively incorrect around the critical point

2. after 1965 more phenomenological theories based 2. after 1965 more phenomenological theories based on scaling invarianceon scaling invariance

3. same period: another key in critical phenomena 3. same period: another key in critical phenomena theory based on universality (universality: dissimilar theory based on universality (universality: dissimilar systems show similarities near their critical points)systems show similarities near their critical points)

4. 1971/72: K.Wilson took semi-phenomenological 4. 1971/72: K.Wilson took semi-phenomenological concepts of scaling and universality and converted concepts of scaling and universality and converted these ideas into real calculations of critical point these ideas into real calculations of critical point behaviorbehavior

Ernst IsingErnst Ising

Student of Wilhelm Lenz in Hamburg. PhD 1924, Thesis work on linear chains of coupled

magnetic moments. This is known as the Ising model.

The name ‘Ising model’ was coined by Rudolf Peierls in his 1936 publication ‘On Ising’s model of ferromagnetism’.

He survived World War II but it removed him from research. He learned in 1949 (25 years after the publication of his model) that his model had become famous.

10.05.1900 in Köln – 11.05.1998 in Peoria (IL)

S. G. Brush, History of the Lenz-Ising Model, Rev. Mod. Phys 39, 883-893 (1962)

Solving of Ising modelSolving of Ising model invented by W. Lenz and his student E. Ising (1920)invented by W. Lenz and his student E. Ising (1920) 1D: solved analytically by Ising (1925): no phase transition 1D: solved analytically by Ising (1925): no phase transition

in 1D and he concluded incorrectly that in higher D also no in 1D and he concluded incorrectly that in higher D also no phase transitionphase transition

2D square lattice: solved by L. Onsager (1944), exhibits 2D square lattice: solved by L. Onsager (1944), exhibits phase transitionphase transition

Also in higher dimensions phase transition can be modeledAlso in higher dimensions phase transition can be modeled Istrail showed that computation of the free energy of an Istrail showed that computation of the free energy of an

arbitrary subgraph based on Ising model will not be arbitrary subgraph based on Ising model will not be approximated computationally intractable (not solvable) by approximated computationally intractable (not solvable) by any method for the case 3D and higherany method for the case 3D and higher- impossible to efficiently compute all possible - impossible to efficiently compute all possible thermodynamic quantities with arbitrary external fieldsthermodynamic quantities with arbitrary external fields- it does not mean that the critical exponents or spin-spin - it does not mean that the critical exponents or spin-spin correlations cannot be computed near criticality.correlations cannot be computed near criticality.

1. Critical point and phase 1. Critical point and phase transitiontransition

Critical point of fluid (p=const)Critical point of fluid (p=const)

1. temp dependence1. temp dependence- Phases for SFPhases for SF66 the curve is the curve is

- Magnetization for DyAlO3Magnetization for DyAlO3

rho = densityTc = critical temp

Fig 1: phase diagram of fluid at p=const

Fig 2: Onset of magnetization in ferromagnet

2. Ising model2. Ising modelIsing model: Ising model: - for ferromagnetics inventedfor ferromagnetics invented- Simple Hamiltonian of spins Simple Hamiltonian of spins s(r) at lattice point at lattice point rr - Assumptions: - only two statesAssumptions: - only two states

- only nearest neighbor interactions- only nearest neighbor interactions

- every pair counts only one time- every pair counts only one time

- Hamiltonian:- Hamiltonian:...

,,,

kji

kjiijkji

jiiji

ii sssJssJsH

s = spinsJ = energy of interaction (<0 if σi = σj , >0 if σi = -σj )H = external magnetic field (decreasing H if spins

lined up, increasing H if not)

1s

Coupling to field pair interaction3-body interaction

Ising model in 2D

2. Ising model2. Ising model

Magnetic interactions:Magnetic interactions:

- seek to align spins relative to one another. - seek to align spins relative to one another.

- spins become effectively "randomized" when thermal energy is greater - spins become effectively "randomized" when thermal energy is greater than the strength of the interaction.than the strength of the interaction.

- w/o magnetic field the Ising model is symmetric for interchange of - w/o magnetic field the Ising model is symmetric for interchange of ± but ± but magnetic field breaks this symmetrymagnetic field breaks this symmetry

Energy of interaction J:Energy of interaction J:

- J- Jijij > 0 the interaction is called > 0 the interaction is called ferromagneticferromagnetic (aligned spins)(aligned spins)

- J- Jijij < 0 the interaction is called < 0 the interaction is called antiferromagneticantiferromagnetic(antialigned spins)(antialigned spins)

- J- Jijij = 0 the spins are = 0 the spins are noninteractingnoninteracting

2. Ising model2. Ising modelThermodynamics of Ising model can be obtained:Thermodynamics of Ising model can be obtained:- for this system, the operation for this system, the operation TrTr means: means:

- So the free energy is given bySo the free energy is given by

- Thermodynamic properties derived by differentiation e.g. average Thermodynamic properties derived by differentiation e.g. average magnetization at site i is derived by magnetization at site i is derived by δδF/ F/ δδHiHi

(method of sources)(method of sources)

1 }1{111 )(2S SSS

TriN

eTrTkJHTF Biji ln},...){},{,(

e

Tk

sTr

eTrTk

H

Fs

B

iB

ii

1

Ising ModelIsing Model- Free energy functional:Free energy functional:

- Goal: - model phase changes of real latticesGoal: - model phase changes of real lattices - the 2D square lattice Ising model is simplest model to show - the 2D square lattice Ising model is simplest model to show

phase changesphase changes

]/exp[}]{exp[ kTsF

21}{ KShSsF

Extensive operators Extensive operators (entire lattice)(entire lattice)

Local operatorsLocal operators Interaction scalarsInteraction scalars

r

rsS )(1

NN

rsrsS )'()(2

)(rs

NN

rsrsr )()()( 211

kTHh /

kTJK /

Ising model in 1DIsing model in 1D

We defined h=H/kT and K=J/kT. The partition function is given by

can be calculated exacty.

Ising model in 1D:

In the following, we will take a look at boundary conditions, thermodynamics and correlations.

N

i i iii SSKSheTrZ 1 1

Ising model in 1D: Periodic boundaries

Periodic boundary conditions are defined by

Ising model in 1D with PBC:

We assume that there is no external field (h=0). Then, we have

We can solve this. Define where i=1…N-1. Then we have

Substitution to the partition function gives

1 iiss

11 SSN

NN KKZ )sinh2()cosh2(

1

1

1 11

2

...S S

sKsssK

S N

N

i NiieZ

Ising model in 1D: Free boundaries

Ising model in 1D with free b.c.:

Again, we assume that there is no external field (h=0). Then, we have

Using the same transformation as before, i.e. where i=1…N-1.

We get

We have the partition function now. Next, we take a look at free energy and thermodynamics.

1 iiss

1)cosh2(2 NKZ

1

1

1 1

2

...S S

ssK

S N

N

i iieZ

Ising model in 1D: Free energy

Since we have the partition function, we also have the free energy

a) For PBC

b) For free b.c.

The difference between boundary conditions becomes negligible at the thermodynamic limit.

The more general way is do this with transfer matrix. Works also for nonzero field.

)cosh2{ln(]})(tanh1ln[)cosh2{ln( KTNkKKTNkF BN

B

)cosh2{ln()}ln(cosh1

)2{ln( KTNkKN

NTNkF BB

Thermodyn limit

Ising ModelIsing Model

- Visualization of Ising model during phase transitionVisualization of Ising model during phase transitionhttp://http://www.pha.jhu.edu/~javalab/ising/ising.htmlwww.pha.jhu.edu/~javalab/ising/ising.html

- Ising model with external field hIsing model with external field h

http://www.physics.uci.edu/~etolleru/IsingApplet/IsingApplet.htmlhttp://www.physics.uci.edu/~etolleru/IsingApplet/IsingApplet.html

Universality of Ising ModelUniversality of Ising Model

Universality:Universality:

- the fluctuations close to the phase transition are described by - the fluctuations close to the phase transition are described by a continuum field with a free energy or Lagrangian as a a continuum field with a free energy or Lagrangian as a function of the field values.function of the field values.

- Ising model decribes exactly the fluctuations around the - Ising model decribes exactly the fluctuations around the critical pointcritical point

In contrast: the mean field doesn’t describe fluctuation at TIn contrast: the mean field doesn’t describe fluctuation at Tcc

ApplicationsApplicationsMagnetism:Magnetism:- In 19In 19thth century: two theories: Ampere postulated that permanent century: two theories: Ampere postulated that permanent

magnets are due to permanent internal atomic currents vs. theroy of magnets are due to permanent internal atomic currents vs. theroy of permanent magnetic momentpermanent magnetic moment

- Electron spin discovered to describe magnetismElectron spin discovered to describe magnetism- Ising model: investigate if electrons could made toIsing model: investigate if electrons could made to

spin in same direction by simple local forcesspin in same direction by simple local forces

Lattice gas:Lattice gas:- Interpret Ising model as a statistical modelInterpret Ising model as a statistical model- B = [0,1],[unoccupied,occupied] B = [0,1],[unoccupied,occupied] B = ( B = (σσ + 1)/2 + 1)/2 σσ = [-1,1] = [-1,1]

- The density of atoms can be controlled by chem pot The density of atoms can be controlled by chem pot

ji

jiBBJ,

i

iji

ji BBBJ ,

ApplicationsApplicationsBiologie – neurons in brain:Biologie – neurons in brain:- states: firing, not firingstates: firing, not firing- To reproduce average firing rate for each neuron To reproduce average firing rate for each neuron

includes activity of each neuron (statistically includes activity of each neuron (statistically independent)independent)

- To allow for pair interactions when a neuron To allow for pair interactions when a neuron tends to fire along with anothertends to fire along with another

This energy function only introduces probability biases for a spin having a value and for a pair of spins having the same value.

i

iiji

jiij ShSSJ,2

1

J – NN interaction of firing rateh – self-firing rate

i

iiSh