Ising Landau 1

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The Ising model



Ernst IsingMay 10, 1900 in Köln-May 11 1998 in Peoria (IL)

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

• Student of Wilhelm Lenz in Hamburg. PhD 1924.Thesis work on linear chains of coupled magneticmoments. 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 himfrom research. He learned in 1949 - 25 years afterthe publication of his model - that his model hadbecome famous.

• Lars Onsager solved the Ising model (zero eld)in two dimensions in 1944.



Ising modelA general Ising model is dened as

• No phase transition at d=1 for T>0

• For J1ijk =0 , phase transition(s) for

• For d>4, mean eld results are exact

∑i= j

| J i j | < ∞

H = − ∑i

H isi − ∑i, j

J i. j sis j − ∑i, j , k

J 1i , j , k sis j sk + ....

It has the following general properties

pair interactions 3-body interactionscoupling to a eld

Lower critical dimension is d L =1 and the upper critical dimension is d U=4.

Thermodynamics of the Ising model can be obtained from

F = − k B T ln [Tr e − βH ] s i = −∂ F

∂ H i T

for example

Ising model in 2D



Ising model in 1DDene

can be calculated exactly.

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

Ising model in 1D

h ≡ βH and K ≡ β J . The partition function is given by

Z (h,

K , N ) = ∑

{s }

e h ∑ N i= 1 s i+ K ∑ i s i s i+ 1



Ising model in 1D: Periodic boundariesPeriodic boundary conditions are dened by

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

s N + 1 = s 1

Ising model in 1D with PBC

1 2 3 N-1 N N+1

......

Z = ∑s 1

∑s 2

.... ∑s N

e K ∑ N − 1i = 1 s i s i+ 1 + Ks N s 1

PBCnote

We can solve this. Dene η i= s

is

i + 1where i = 1

, ..., N − 1

η i +1 when s i=s i+1-1 when s i=-s i+1

= }. Then, we have

Substitution to the partition function gives Z = ( 2cosh K ) N + ( 2sinh K ) N



Ising model in 1D: Free boundaries

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

Ising model in 1D with free boundary conditions

1 2 3 N-1 N......

Using the same transformation as before, i.e., η i = s i s i + 1 where i = 1, ...,

N − 1

η i +1 when s i=s i+1-1 when s i=-s i+1

= }that is,

Z = ∑s 1 ∑s 2.... ∑s N

eK ∑ N − 1

i = 1s i s i + 1

we have Z = 2 (2cosh K ) N − 1

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



Ising model in 1D: Free energy Since we have the partition function, we also have the free energy

For PBC:

F = − k B T N ln (2cosh K ) + ln [1 + ( tanh K ) N ] → − Nk B T ln (2cosh K )

thermodynamic limit

For free (or open) boundary conditions:

F = − k B T N ln 2 + N − 1 N

ln (cosh K ) → − Nk B T ln (2cosh K )

thermodynamic limit

The difference between boundary conditions becomes negligible

at the thermodynamic limit.

The more general way is do this with transfer matrix. Works alsofor nonzero eld.



Ising model in 1D: Pair correlation functionThe two-point spin-spin correlation function is dened as

G(i , j ) ≡ (si − si )( s j − s j ) = si s j − si s j

If the system is spatially homogeneous (has translational invariance), then

si = s j ≡ s

Above T c we haves

=0 G

(i , j

) =s

is

j

What does G(i,j) measure?

The probability for spins i and j to have the same value is

Pi j = δ si s j

= 12

(1 + a i s j )

=1

2+

1

2[G(i , j ) + si s j ]

Above T c we have Pi j =1

2[1 + G(i , j )]

Around T c

At T=0



Pair correlation functionFor a translationally invariant system we have

The result (homework exercise, see e.g., Goldenfeld) is

Then, obviously G(i)=1.This denes perfect long-range order.

How about the other limit, T-> 0?

Denition for the correlation length:

For the 1D Ising model we have

Around T c

At T=0

G(i , i + j ) = G(i + j − i) = G(i)

G (i) = ( tanh K ) i

G (i) = e − i / ξ

ξ = [ ln (coth K )]− 1

As T->0, ξ e 2 K → ∞

This is not a power law but an essential singularity!

Denition, the correlation function exponent :G (i) i 2 − d − η

η

G (i) = e − i / ξ = 1 −

i

ξ+ ... constantNow,



2D magnetization



Phase equilibrium

Tc

T(~1/J)

Hcoexistence line

disordered

Phase diagram of the Ising model at nite temperature (d>1):

domain wall

system size, L

TcT

L-dM H=0



Landau theory of phase transitions



Reminder: phase transitions

Change of a system from one phase (state) toanother at a minute change in the externalphysical conditions.

They are divided into two classes:

1) First-order transitions2) Continuous transitions





Digression: Lev Davidovich Landau

• Graduated from Leningrad University at the age of 19. Hestarted at the age of 14! After graduating from Leningrad hespent time in Denmark with Bohr. Collaborated andinteracted also with Pauli, Peierls and Teller. For his travels hegot a Rockefeller fellowship!

• His work covers basically all of theoretical physics fromuids to quantum eld theory.

• Was imprisoned by Stalin for a year after being accused tobe a German spy. Was freed after Piotr Kapitza threatenedto stop his own work unless Landau was released

• On Jan. 7 1962 he suffered a major car accident and wasunable to continue his work. For the same reason he wasnot able to attend the Nobel Prize seremonies.

Lev Davidovich Landau, Jan. 22 1908 in Baku – Apr. 1 1968 MoscowNobel Prize 1962 for pioneering theories in condensed matter physics

Akhiezer, Recollections of Lev Davidovich Landau, Physics Today 47, 35-42 (1994).Ginzburg, Landau's attitude towards physics and physicists, Physics Today 42 , 54-61 (1989).

Khalatnikov, Reminiscences of Landau, Physics Today 42 , 34-41 (1989).

More reading:



Landau and Lifshitz started in 1930’s and the 10volume series was completed in 1979 byLifshitz.They received the 1962 Lenin Prize for the“Course of Theoretical Physics”.



Landau’s revolutionary ideas

• Superuidity:Landau considered the quantized states of the motion

of the whole liquid instead of single atoms. That was a revolutionaryidea and using it Landau was able to explain superuidity.

• Superconductivity: Even before the BCS theory, Ginzburg andLandau suggested a phenomenological theory of superconductivity based

on Landau's earlier theory of continuous phase transitions. When it waspublished, the GL theory received only limited attention. This changeddramatically in 1959, when L.P. Gorkov showed rigorously that close toTc the GL theory and the BCS theory become equivalent. Furthermore,two years before Gorkov, A. Abrikosov predicted the possibility of twodifferent kinds of superconductors by using the GL theory!

• Landau’s theory of phase transitions.

• If we sum up the leading ideas we end up with two things: theimportance of symmetry and symmetry breaking, and theexistence of an order parameter.



The phenomenological Landau theory of continuous phase transitions stresses

the importance of overall general symmetry properties and analyticityover microscopic details in determining the macroscopic properties of a system.

Those generic properties were also used in the superconductivityand superuidity problems!

The Landau theory is based on the following assumptions:

Landau theory



Symmetry

Ψ

Ψ = 0

The order parameter characterizes the system thefollowing way:

in the disordered state (above T c),is small and nite in the ordered state (T<T c).



system order parameterliquid-gas density

ferromagnetic magnetization

superconducting condensate wave function

liquid crystal degree of molecular alignemnt

binary mixture (methanol-n-hexane) concentration of either substancehelix-coil number of helix base pairs

XY-model magnetization (Mx,My)

BaTiO 3 polarization

crystal density wave

liquid crystal director

Order parameters



Symmetry

F (Ψ ) =∞

∑n = 0

a 2 n Ψ 2 n

expansion coefcients are phenomenologicalparameters that depend on T and microscopics

free energy order parameter. must be small

Close to T c the free energy can be expanded in powers of the order parameter

Order parameter must be small for the expansion to converge.

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Ising Landau 1