Chapter 5 Ising model and phase transitions · 5.2.1 Mean ﬁeld phase diagram mean field phase...

Chapter 5

Ising model and phase transitions

©2015 by Alessandro Codello

5.1 Equilibrium statistical mechanicsgibbs–

boltzmann

distribu-

tion and

partition

function

A spin system is described by placing a spin variable !i ! {"1, 1} at everysite i of a given lattice. A microstate is defined by a spin configuration {!i},which specifies the values of the spins of every lattice site. The (canonical)probability to observe the system in the spin configuration {!i} is given bythe Gibbs–Boltzmann weight:

p{!i} =e!"E{!i}

Z, (5.1)

where E{!i} is the energy of the given configuration. The (canonical) partitionfunction Z appearing in (5.1) is defined by:

Z(", B) =!

{!i}

e!"E{!i} , (5.2)

and in general depends on the inverse temperature1 " = 1/T and the externalmagnetic field B. Averages are computed as usual:

#O$ =!

{!i}

p{!i}O{!i} .

1We will measure the temperature in units of the Boltzmann constant, which can bereintroduced at any moment by the replacement T % kBT .

62

CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 63

As we will see in a moment, the knowledge of the partition function is thekey to all thermodynamical quantities.

We need now to specify the form of E{!i} for a given spin configuration. definition

of the

ising

model

Nearest neighbor interactions are assumed2 so that the energy of a givenspin configuration is given by

E{!i} = "!

"i,j#

Jij!i!j "!

i

!iBi . (5.3)

Jij is the spin interaction matrix strength and Bi is the external appliedmagnetic field (it most cases it can be considered constant Bi = B). If J > 0the interaction is ferromagnetic while it is antiferromagnetic if J < 0. Theenergy (5.3) defines the model known as Ising model.

The free energy F can be obtained from the partition function (5.2) using free

energy,

internal

energy,

entropy,

specific

heat,

magne-

tization

and

suscepti-

bility

the following relation:

F (", B) = "1

"logZ(", B) . (5.4)

The internal energy E and the entropy S are related to the free energy bythe usual thermodynamic relation:

F = E " TS . (5.5)

In particular, the internal energy and the specific heat can be obtained di-rectly from the partition function (5.2) by employing the following relations:

E = "#

#"logZ =

#

#"("F ) (5.6)

C =#E

#T= ""2#E

#". (5.7)

where used ##T = ""2 #

#" . The total magnetization

#M$ =!

i

#!i$ = N #!i$ & Nm

2They are actually the more general, if we add a constant term, since the spins arefermions!


can also be computed from the partition function:

#M$ = "#F

#B, (5.8)

while the susceptibility is determined/defined by:

$ =1

N

# #M$#B

= "1

N

#2F

#B2. (5.9)

We will work with intensive variables, or equivalently, per spin variables:

f =F

Nm =

#M$N

% =E

Nc =

C

N, (5.10)

for which we have f = " 1"N logZ and

% =#

#"("f) c = ""2 #%

#"m = "

#f

#B$ =

#m

#B. (5.11)

From (5.5) one can find the entropy per spin s = $!fT . Using #

#" = "T 2 ##T

we can also write the internal energy and the specific heat as:

% = "T 2 #

#T

"1

Tf

#

c =#%

#T. (5.12)

The computation of f, %, c,m,$ and s is the main goal of the study of equi-librium spin systems.

5.1.1 Fluctuations and correlationssusceptibility

and spe-

cific

heat as a

measure

of fluc-

tuations

It’s useful to define the generating function

W (", B) = logZ(", B) = ""F (", B) ,

from which we find:

#2W

#B2= "2

$%

M2&

" #M$2'

= ""#2F

#2B= ""N

#2f

#B2.

This last quantity is proportional to the magnetic susceptibility,

$ = "#2f

#B2.


which plays thus the role of magnetic variance3 since:

$T =1

N

$%

M2&

" #M$2'

. (5.13)

Thus the susceptibility measures the relative size of magnetization fluctua-tions around the average. The specific heat is instead the energy variance:

#2W

#"2=

%

E2&

" #E$2 = "#

#"#E$ =

1

"2C ,

which measures the size of internal energy fluctuations around the average:

c T 2 =1

N

$%

E2&

" #E$2'

. (5.14)

From relation (5.13) and (5.14) we expect that at a phase transition, wherefluctuations are large and on all scales, both the susceptibility and the spe-cific heat will diverge. We will soon see that this is indeed the case.

The two–point spin connected correlation function is: correlation

functionCij = #!i!j$ " #!i$ #!j$ = #!i!j$ "m2 = #(!i " #!i$) (!j " #!j$)$ . (5.15)

This correlation function can be obtained from the generating function W ifwe promote the magnetic field to be space dependent B % Bi:

Cij =1

"2

#2W

#Bi#Bj. (5.16)

Obviously W then becomes the generating function of spin connected corre-lation functions.

The two–point connected correlation is obviously related to the susceptibilitysince both are derived from W by derivatives with respect to the magneticfield. This relation is obtained as follows:

!

ij

Cij =!

ij

[#!i!j$ " #!i$ #!j$]

=

(

!

i

!i

!

j

!j

)

"

(

!

i

!i

)(

!

j

!j

)

=%

M2&

" #M$2

= N T $ . (5.17)

3The magnetic kurtosis is the so–called Binder cumulant.


Changing variables to r = |i" j| gives*

ij Cij = N*

r C(r) and we find thesum rule:

$ =1

T

!

r

C(r) . (5.18)

We will use relations (5.13) and (5.18) in the following to make precise quan-titative statements about the strength/weakness of fluctuations and to derivescaling relations between critical exponents.

5.2 Mean field theorymean

field

theory

& weak

fluctua-

tions

The mean field approach consists in discarding fluctuations, and thus, as wewill see it is quantitatively correct only above the upper critical dimension.The assumption of the mean field approach is that fluctuations aroundthe average are small. Still it often gives a qualitatively correct pictureof the phase diagram. In the Ising model, the mean field approximation isimplemented by linearizing the interaction between two near spins:

!i!j = [!i " #!i$+ #!i$] [!j " #!j$+ #!j$]= #!i$ #!j$+ [!i " #!i$] #!j$+ [!j " #!j$] #!i$+

[!i " #!i$] [!j " #!j$]+ ,- .

small

% mimj +mi [!j "m] +mj [!i "mi]

= "mimj + !imj +mi!j , (5.19)

where #!i$ = mi and we discarded the quadratic fluctuation term since weare assuming that the fluctuation are small. If the external magnetic field isconstant then mi & m for every i due to translation invariance and the theenergy of a spin configuration becomes simply:

EMF{!i} = "J

!

"i,j#

$

"m2 +m (!i + !j)'

" B!

i

!i

= m2JNz

2" (Jzm +B)

!

i

!i , (5.20)

where we used the fact that there are Nz2 nearest neighbors on a lattice (z is

the coordination number) and that*

"i,j# !i =z2

*

i !i. For a square lattice


we have z = 2d and in general the coordination number is proportional tothe dimension. In this approximation the net e!ect of the spin interaction isto shift the external magnetic field to the value Jmz +B.

The partition function is easily evaluated: mean

field

free

energy

ZMF =!

{!i}

e!"EMF{!i}

= e!"m2J Nz2

!

{!i}

e"(Jzm+B)!

i !i

= e!"m2J Nz2

!

{!i}

/

i

e"(Jzm+B)!i

= e!"m2J Nz2 {2 cosh [" (Jzm +B)]}N .

Thus the free energy per spin fMF = " 1"N logZMF is:

fMF = m2Jz

2"

1

"log 2 cosh [" (Jzm +B)] . (5.21)

Note that this relation is valid for any lattice in arbitrary dimension.

5.2.1 Mean field phase diagrammean

field

phase

diagram:

phase

transi-

tion

To find the equilibrium value for the magnetization per spin we need to findthe minimum of the free energy:

#fMF

#m= 0 ' m(T,B) = tanh

0Tc

Tm(T,B) +

B

T

1

, (5.22)

where we defined the mean field critical temperature Tc = Jz. For B = 0is easy to see that there is a phase transition at T = Tc by studying thesolutions of:

m = tanhTc

Tm , (5.23)

which can be found graphically as shown in Figure 5.1. Near Tc the magne-tization is small and we can expand equation (5.20) to obtain4

m =Tc

Tm"

T 3c

3T 3m3 +O

2

m43

,

4tanhx = x" x3

3 + ...


Figure 5.1: Graphical solution to equation (5.23) at B = 0. From top T < Tc,T = Tc and T > Tc. For T < Tc the system has a spontaneous magnetization.

and so

m± =

4

5

6

0 T ( Tc

±7

38

TTc

93:TcT " 1 ) ±

*3("t)1/2 T % T!

c

where we defined the reduced temperature t = TTc

" 1. The order parameteris m and at the second order phase transition behaves thus as:

m± ) ±("t)1/2 T % T!c t % 0! . (5.24)

Thus the magnetization critical exponent is "MF = 12 in the mean field ap-

proximation.

At criticality the magnetization as a function of the external magnetic fieldcan be written in the following way:

m(Tc, B) = m(Tc, B) +B

Tc"

1

3

"

m(Tc, B) +B

Tc

#3

+ ...

= m+B

Tc"

1

3m3 + ...


Figure 5.2: Mean field per spin magnetization for the Ising model.


from which we find:

m(Tc, B) = sign(B)

"3|B|Tc

# 13

) sign(B)|B|1" . (5.25)

This gives the mean field magnetic exponent &MF = 3.

It’s easy now to compute the susceptibility:

$(T,B) =#m

#B=

1

T

1

cosh22

mTcT + B

T

3

" TcT

. (5.26)

Around the critical point we find using (5.24):

$±(t, 0) =

;

4 1T!Tc

T % T+c

2 1Tc!T T % T!

c

thus$±(t, 0) ) $±|t|!1 t % 0± . (5.27)

This gives the mean field susceptibility critical exponent 'MF = 1 and themean field amplitude ratio $+/$! = 2. Note that the fact that the suscepti-bility diverges at the critical point means that the magnetic variance diverges.

Next we compute the energy per spin; we find simply:

% =#("f)

#"= "

Jz

2m2 .

The specific heat per spin is not well defined in the mean field approxima-tion (in particular it is not divergent at the critical point), but it usuallyassumed/defined that (MF = 0.

5.2.2 Correlation functioncomputing

the mean

corre-

lation

function

If we want to compute the correlation function we need to relax the as-sumption of constant magnetic field in order to use (5.16). In this case thepartition function turns out to be:

ZMF =!

{!i}

e"J!

!i,j"[!mimj+!imj+mi!j ]+"!

i Bi!i

= e!"J!

!i,j" mimj!

{!i}

e"J!

!i,j"[!imj+mi!j ]+"!

i Bi!i . (5.28)


We need now to evaluate the sums:!

{!i}

e"J!

!i,j"[!imj+mi!j ]+"B!

i !i =!

{!i}

e2"J!

!i,j" !imj+"!

Bi!i

=!

{!i}

e"!

i !i[J!

j(i) mj+Bi]

=/

i

2 cosh"

<

=J!

j(i)

mj +Bi

>

? ,

where j(i) are the nearest neighbors of i. The free energy is thus:

fMF =J

N

!

"i,j#

mimj "1

"N

!

i

log 2 cosh

@

A"

<

=J!

j(i)

mj +Bi

>

?

B

C . (5.29)

Minimizing the free energy leads to:

mi = tanh

@

A"

<

=J!

j(i)

mj +Bi

>

?

B

C , (5.30)

where we used: !

j(i)

&jk = &ik

and#

#mk

!

"i,j#

mimj = 2!

"i,j#

mi&jk =!

i

mi

!

j(i)

&jk = mk .

The average magnetic field felt by a spin !i is now J*

j(i)mj +Bi; equation(5.30) reduces to (5.22) when mi is constant since

*

j(i) 1 = z. To computethe correlation function using (5.16) we need to determine the solutions of(5.30) to linear order in the external magnetic field. Since we are interestedin the critical region, we can assume that also the magnetization is small andexpand the hyperbolic tangent to first order:

""J!

j(i)

mj +mi = "Bi . (5.31)


To solve (5.31) we perform a Fourier transform:

""Jmk

!

a

2 cos ka +mk = "Bk , (5.32)

where a = 1, ..., d runs over the basis vectors of the lattice. We find:

mk =1

T " J*

a 2 cos kaBk ,

which gives5:

mi = !

D

BZ

ddk

(2))de!ixi·k

T " J*

a 2 cos kaBk .

This expression is IR divergent when T = Tc since:

J!

a

2 cos ka = Jz + J!

a

k2a + ... = Tc + J

!

a

k2a + ...

Finally, since Cij =#mi#Bj

, the correlator near the critical point can be writtenas (on a hyper-cubical lattice):

C(r = |r|) =1

Tc

Dddk

(2))de!ir·k

t+ 1z

*

a 2(1" cos ka), (5.33)

where t = TTc

" 1 is the reduced temperature. At large r we can write6:

C(r) +1

Tc

Dddk

(2))de!ir·k

t+ k2

=1

Tc

1

(2))d

"*t

r

#d!2

K d2!1(

*tr)

+1

Tc

1

(2))d

"*t

r

#d!2 7)

2

1E*

tre!

$tr . (5.34)

The correlation length is the inverse lattice mass * = 1/m and thus from(5.34) we see that at the critical point it diverges like:

* = t!1/2 & t!% . (5.35)

5mi & m(xi).6Kn(x) is a Bessel function.


The mean field correlation length critical exponent is thus +MF = 12 .

Exactly at the critical point we find instead:

C(r) +1

rd!2&

1

rd!2+&, (5.36)

telling us that the mean field anomalous dimension is ,MF = 0.

Finally the mean field critical exponents are: mean

field

critical

expo-

nents

(MF = 0 "MF =1

2'MF = 1

&MF = 3 +MF =1

2,MF = 0 . (5.37)

To understand the quantitative validity of the mean field approximation weneed to understand the quantitative meaning of weak fluctuations assump-tion.

5.2.3 Ginzburg criterionthe

ginzburg

crite-

rion and

upper

critical

dimen-

sion

The Ginzburg criterion establishes when the mean field analysis is valid, i.e.under which conditions fluctuations are small. We start from:

:

#M2$ " #M$2 , #M$ ,

which is equivalent toNT$ , #M$2 .

At criticality fluctuations are on all scales and we have (for t % 0!):

* ) |t|!% #M$2 ) ("t)2"L2d N$ ) ("t)!'Ld .

Thus, setting * = L, we find:

T ("t)!'*d , ("t)2"*2d ,

orT ("t)!'!2"+%d , 1 ,


which may happen when "' " 2" + +d > 0 or:

' + 2"

+< d ,

which is the Ginzburg criterion. If we insert the mean field exponents wefind d > 4. Thus the mean field predictions are valid in dimensions greaterthan four, which for this reason is called upper critical dimension dc = 4.

5.3 Solving the Ising model

In this section we explore the Ising model in dimensions below the uppercritical dimension dc = 4, where the Ginzburg criterion shows that the meanfield approach is unable to describe the phase transition quantitatively. For-tunately in both d = 1 and d = 2 it is possible to solve the Ising modelexactly. These solutions are invaluable models characterized by non–meanfield phase transitions.

5.3.1 1d Ising modelthe

transfer

matrix

approach

In one dimension the partition function for the Ising model can be rewrittenas follows

Z =!

{!i}

e"J!N

i=1 !i!i+1+"B!N

i=112 (!i+!i+1)

=!

{!i}

N/

i=1

e"[J!i!i+1+B2 (!i+!i+1)]

=!

{!i}

T!1!2 ...T!N!1

= trTN , (5.38)

where we introduced the transfer matrix

T =

"

e"(J+B) e!"J

e!"J e"(J!B)

#

. (5.39)

By diagonalizing (5.39) we can write the partition function in terms of theeigenvalues of the transfer matrix:

Z = tr

"

-+ 00 -!

#N

= -N+ + -N

! ,


Figure 5.3: Ising spins on a one dimensional infinite lattice.

where

-± = e"J"

cosh "B ±:

sinh2 "B + e!4"J

#

.

Note that we have -+ > -!. The free energy is thus given by the followingexpression:

F = "1

"logZ = "

1

"

;

N log -+ + log

F

1 +

"-!

-+

#NGH

,

in the thermodynamic limit N % - only the contribution from -+ survivesand we find the following free energy per spin:

f(T,B) = "J " T log

I

coshB

T+

7

sinh2 B

T+ e!4J/T

J

.

In the case of zero magnetic field we have simply

f(T, 0) = "J + T log2

1 + e!2J/T3

= "T log 2" T log cosh2

T.

We can easily calculate the magnetization per spin:

m(T,B) = "#f

#B=

sinh BT

:

sinh2 BT + e!4J/T

.

Since the hyperbolic sine is zero in the origin there is no spontaneous mag-netization at B = 0. The susceptibility is

$(T,B) =#m

#B=

e!4J/T cosh BT

T2

sinh2 BT + e!4J/T

33/2,

while the energy per spin and the specific heat are:

%(T,B) =#("f)

#"= "J tanh

J

T


Figure 5.4: Exact (per spin) thermodynamic functions for the 1d Ising model.From top: f, %, c, s,m,$; left column as a function of T , right column as afunction of B.


c(T,B) = "1

T 2

#%

#"=

J2sech2 JT

T 2.

We can also calculate correlation functions. The magnetization is:

#!i$ =1

Z

!

{!i}

!ie"E{!i}

=1

Z

!

{!i}

T!1!2 · · ·T!i#1!i!iT!i!i+1 · · ·T!N!1

=1

ZtrTi!1!zT

N!i+1

=1

ZtrTN!z ,

where

!z =

"

1 00 "1

#

.

When B = 0 the diagonalizing matrix is U =

"

1 11 "1

#

and we find:

#!i$ =1

Ztr

"

-N+ 00 -N

!

#"

0 11 0

#

=1

Ztr

"

0 -N+

-N! 0

#

= 0 ,

as expected. The correlation function can be calculated in a similar way:

#!i!i+r$ =1

ZtrTN!r!zT

r!z

=1

Ztr

"

-N!r+ 00 -N!r

!

#"

0 11 0

#"

-r+ 00 -r

!

#"

0 11 0

#

=-N!r+ -r

! + -r+-N!r

!

-N+ + -N

!

%"

-!

-+

#r

.

Thus we have:C(r) = #!i!i+r$ = e!r/( ,

where the correlation length is:

*(T, 0) =1

log tanh"J.


Figure 5.5: Exact (per spin) thermodynamic functions for the 1d Ising model.From top: f, %, c, s,m,$; the color code is consistent with the previous figure.


We can also consider the limit r % - where

C(r) % 0 ,

showing again that there is no spontaneous magnetization.

The critical exponents for the one dimensional Ising model are: critical

expo-

nents

for the

d = 1

ising

model

(d=1 = 1 "d=1 = 0 'd=1 = 1

&d=1 = - +d=1 = 1 ,d=1 = 1 .

These clearly di!er from the mean field ones. Note that the Ginzburg in-equality is violated but saturated since '+2"

% = 1.

5.3.2 2d Ising modelonsager’s

solutionThe exact zero–field free energy for the square lattice Ising model was foundfor the first time by L.Onsager using algebraic methods. The zero–field freeenergy can be exactly computed for square, triangular and hexagonal lattices;the result is:

f(T, 0) = "Tp log 2"T

2

Dd2k

(2))2logP (T,k) , (5.40)

where the argument of the logarithms is:

PS(T,k) = cosh2 2J

T" sinh

2J

T(cos k1 + cos k2)

PT (T,k) = cosh3 2J

T+ sinh3 2J

T" sinh

2J

T(cos k1 + cos k2 + cos k3)

PH(T,k) =1

2

0

1 + cosh3 2J

T" sinh2 2J

T(cos k1 + cos k2 + cos k3)

1

,

for, respectively, square, triangular and hexagonal lattices. p is the numberof sites in a unit cell of the lattice, which is one for the square and triangularlattices and two for the hexagonal lattice.

The argument of the logarithm is always positive except for k = 0 in which a critical

tempera-

ture

singularity occurs. This singularity corresponds to the phase transition, thusthe critical temperature is the solution of:

P (Tc, 0) = 0 .


We find, respectively, the following equations:

cosh2 2J

T Sc

" 2 sinh2J

T Sc

= 0

cosh3 2J

T Tc

+ sinh3 2J

T Tc

" 3 sinh2J

T Tc

= 0

1 + cosh3 2J

THc

" 3 sinh2 2J

THc

= 0 ,

which give the following critical temperatures:

T Sc = 2

log(1+$2)J = 2.26919 J (5.41)

T Tc = 2

log$3J = 3.64096 J (5.42)

THc = 2

log(2+$3)J = 1.51865 J , (5.43)

which may be compared with the mean field result TMFc = z J . The critical

temperature (5.41) was first found by Kramer and Wannier using dualityarguments. Since the free energy diverges logarithmically we have (d=2 = 0in all three cases.

We can now compute the per spin internal energy and the specific heat using internal

energy

and

specific

heat

equation (5.12):

%(T, 0) =T 2

2

Dd2k

(2))21

P (T,k)

#P (T,k)

#T(5.44)

and

c(T, 0) =%(T, 0)

T 2+

T 2

2

Dd2k

(2))21

P (T,k)

0#2P (T,k)

#T 2"

1

P (T,k)

#P (T,k)

#T

1

.

(5.45)The three thermodynamic functions f, %, c are shown in Figure 5.6.

Onsager and Yang were also able to determine the spontaneous magneti- magnetization

zation exactly:


0 2 4 6 8!4

!3

!2

!1

0

1

2

3Square Lattice

0 2 4 6 8!4

!3

!2

!1

0

1

2

3Triangular Lattice

0 2 4 6 8!4

!3

!2

!1

0

1

2

3Hexagonal Lattice

Figure 5.6: Per spin free energy (bottom), internal energy (middle) and spe-cific heat (top) for the 2d Ising model on a square, triangular and hexagonallattices.


0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

1.2

T

m

Figure 5.7: Spontaneous magnetization per spin for the 2d Ising model on asquare (middle), triangular (right) and hexagonal (left) lattices.

mS(T, 0) =

F2

1 + e4J/T32 2

1" 6e4J/T + e8J/T3

(1" e4J/T )4

G 18

mT (T, 0) =

F2

3" e4J/T3 2

1 + e4J/T33

(1" e4J/T )3 (3 + e4J/T )

G 18

mH(T, 0) =

F2

1 + e4J/T33 2

1 + e4J/T " 4e2J/T3

(1" e2J/T )6 (1 + e2J/T )2

G 18

.

These expressions are valid for T < Tc and are shown in Figure 5.7. In thecase of the square lattice we also have the equivalent simple form:

mS(T, 0) =

F

1"1

sinh4 2JT

G 18

, (5.46)

first derived by Yang.

The correlation length is the inverse of the lattice mass. P (T,k) plays the correlation

lengthrole of the lattice propagator (it is the argument of the log in the free energy)


0 1 2 3 4 5 60

20

40

60

80

100

T

"

Figure 5.8: Correlation length for the 2d Ising model on a square (middle),triangular (right) and hexagonal (left) lattices.

as in the mean field case, thus the mass squared can be defined by writing:

P (T,k) = Z(T )

F

m2(T ) +1

z

!

a

2(1" cos ka)

G

, (5.47)

where Z(T ) is a wave function renormalization. From this we can write:

m2(T ) =P (T, 0, 0)

P (T, )/2, )/2, )" P (T, 0, 0)(5.48)

or

*(T ) =

K

P (T, )/2, )/2, )" P (T, 0, 0)

P (T, 0, 0). (5.49)

For the lattices we are considering we find the following results:

*S(T, 0) =1

log$2!1

1#e#2J/T

1+e#2J/T

.

Note that the argument of the log is e!2KD tanh 2K (= 1 at the critical point)or something like this. At the critical point the correlation length diverges


in the following way:

*S2

T Sc (1 + t), 0

3

=1

4 log2

1 +*23 |t|!1 + regular

. 0.321825 |t|!1 + regular .

*T2

T Tc (1 + t), 0

3

=1

72Arctanh2

2"*33 |t|!1 + regular

. 0.184119 |t|!1 + regular

*H2

THc (1 + t), 0

3

=1

24Arctanh 1$3

|t|!1 + regular

. 0.0960959 |t|!1 + regular ,

giving +d=2 = 1.

We can expand the magnetization around the critical point finding:

mS

2

T Sc (1 + t), 0

3

=L

28*

2" 19M 1

4L8

4 + 3*29

log8

1 +*29M 1

8("t)

18 + regular

. 1.22241 ("t)18 + regular .

mT

2

T Tc (1 + t), 0

3

=*2L

Arctanh8

2"*39M 1

8("t)

18 + regular

. 1.20327 ("t)18 + regular

mH

2

THc (1 + t), 0

3

=*28

2 +*39 1

4

0"7*3" 4

#

Arccoth*3

1 18

("t)18 + regular

. 1.25318 ("t)18 + regular .

We thus see that independently of the lattice the order parameter criticalexponent is "d=2 = 1

8 . This is an instance of universality.

The knowledge of the two critical exponents ( and " allows the determi- critical

expo-

nents

for the

d = 2

ising

model

nation of all the other critical exponents if we assume the validity of thescaling relations (to be proved in the next section):

(d=2 = 0 "d=2 =1

8'd=2 =

7

4

&d=2 = 15 +d=2 = 1 ,d=2 =1

4.


0 1 2 3 4 5 6!4

!3

!2

!1

0

1

AFM vs FM

Figure 5.9: Comparison between ferromagnetic and antiferromagneticthermodynamic quantities for the triangular lattice Ising model.

1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8Entropy

Figure 5.10: The frustrated antiferromagnetic ground state of the triangularlattice Ising model has non–zero entropy at T = 0.


These clearly di!er from the mean field ones, in accord with the fact thatthe Ginzburg inequality is violated (but saturated) since '+2"

% = 2.

5.3.3 Anti–ferromagnetism and frustrationanti–

ferromagnet

and frus-

tration

For antiferromagnetic coupling J < 0 the triangular lattice Ising model be-comes very interesting since it represents the simplest example of frustratedsystem. There is no phase transition at finite temperature and the model isalways disordered, even at T = 0 where the ground state is frustrated andthe system has a non–zero entropy. The thermodynamic quantities and theentropy per spin are shown in Figure 5.10.

5.4 Scaling hypothesishomogeneous

func-

tions

A function of one variable is said to be homogeneous if there exist a scalingexponent a such that

f(-ax) = -f(x) - > 0 .

It is easy to see that a necessary and su"cient condition for f(x) to be anhomogeneous functions that it is a power law. Taking - = |x|!1/a we find:

f(x) = f(±1)|x|1/a ,

as discussed in the first lecture.

A function of n variables x, y, z, ... is homogeneous if there exist scaling ex-ponents a, b, c, .. such that:

f(-ax,-by,-cz, ...) = -f(x, y, z, ...) .

In this case we can eliminate one variable by introducing the scaling functionsF± defined as before by setting - = |x|!1/a:

f(x, y, z, ...) = |x|1/af"

±1,y

|x|b/a,

z

|x|c/a, ...

#

& |x|1/aF±

"y

|x|b/a,

z

|x|c/a, ...

#

.


In the case of a two variable function the scaling function is a function of one data col-

lapserescaled variable: by plotting |x|!1/af(x, y) as a function of y/|x|b/a all data“collapses” to the graph of the scaling function F±.

Near the critical point the various thermodynamic functions exhibit a scaling scaling

hypothe-

sis

behavior. The scaling hypothesis assumes that near to the phase transitionthe (non-analytical part of the) free energy per spin is an homogeneous func-tion:

f(t, b) = |t|2!)F±

"b

|t|!

#

t % 0± b % 0 , (5.50)

where t = T!TcTc

is the reduced temperature, b = "B and the exponents ( < 2and " are the first two critical exponents. Since derivatives of homogeneousfunctions are again homogeneous, the scaling ansatz (5.50) implies that allthermodynamic quantities are so near the phase transition. The magnetiza-tion per spin scales as:

m(t, b) = ""#f

#b

= ""|t|2!)!!F %±

"b

|t|!

#

& |t|2!)!!M±

"b

|t|!

#

. (5.51)

The exponents " and & where defined by the relations:

m(t, 0) ) ±|t|" m(0, b) ) sign(b)|b|1/* . (5.52)

Comparing (5.51) to (5.52) gives the first two scaling relations:

" = 2" ( "" " = "& . (5.53)

The susceptibility scales as follows:

$(t, b) = "#m

#b

= ""2|t|2!)!2!F %%±

"b

|t|!

#

& |t|2!)!2!$±

"b

|t|!

#

. (5.54)


The exponent ' is defined by the relation $(t, 0) ) $±|t|!' and from (5.54)we find the scaling relation:

' = 2" ( " 2" (5.55)

and the relation $± = $±(0). The singular part of the specific heat scales as:

c(t, b) = "T#2f

#T 2

= "T

Tc

(2" ()(1" ()

Tc|t|!)F±

"b

|t|!

#

& |t|!)C±"

b

|t|!

#

; (5.56)

thus at the phase transition the specific heat diverges like c(t, 0) ) C±|t|!)

with C± = C±(0) .

5.4.1 Scaling relationsscaling

rela-

tions

Combining the relations (5.53) with (5.55) we can eliminate " to obtain

( + 2" + ' = 2

" + ' = "& . (5.57)

One also finds the scaling relation: anomalous

dimen-

sion

scaling

relation

' = +(2 " ,) . (5.58)

To derive (5.58) we start from (5.18) and transform the sum in an integral:

$ )D L

0

dr rd!1C(r) )D L

0

dr rd!1r!(d!2+&) =

D L

0

dr r1!& =L2!&

2" ,, (5.59)

where we used (5.36) to express the correlation function at criticality. At thecritical point * = L and thus $ ) *2!&. Recalling the scaling behavior of boththe susceptibility and the correlation length this implies |t|!' ) |t|!%(2!&),which immediately gives (5.58). Note that this scaling relation is the samewe found when we studied percolation. Note also that since $ > 0 we musthave , < 2, which is thus a restriction on the possible values assumed by theanomalous dimension.


The hyper-scaling relation is: hyper-

scaling

relation2" ( = +d , (5.60)

which depends explicitly on d. To see why (5.60) holds consider that the freeenergy per spin is an intensive variable and thus scales as f(t, 0) ) N!1 )L!d. But from the scaling ansatz (5.50) we also know that f(t, 0) ) |t|2!),thus at criticality where * = L we must have |t|!%d ) |t|2!)and so weget(5.60). Note that we have already met this relation when we studiedthe random walk; we called ( = 3

2 the Hurst exponent and we had d = 1 anddf = 1

% = 2.

Note that the mean field critical exponents (5.37) and the one and two di-mensional Ising model critical exponents indeed satisfy the scaling relations.Note that in the mean field case the hyper-scaling relation implies 2 = d/2or d = 4.

Combining the previous relations we find: there

are only

two

inde-

pendent

critical

expo-

nents

( = 2" +d

" = +d " 2 + ,

2' = +(2 " ,)

& =d+ 2" ,

d" 2 + ,, (5.61)

with only + and , independent.

5.5 RG for lattice models

We discuss the RG approach to lattice models developed mostly by Kadano!and Wilson.

5.5.1 RG for the 1d Ising model

We construct the RG transformation by grouping the spins in pairs and byapplying the decimation procedure where we sum over the second spin in


C

S

R

Figure 5.11: RG on a one dimensional lattice.

each pair.

By defining the coupling K1 = "B and K2 = "J we have that the parti- rg

trans-

forma-

tion for

the d = 1

ising

model

tion function becomes:

Z =!

{!i}

eK1!

i !i+K2!

i !i!i+1 . (5.62)

To perform the decimation, consider the sum over the second spin:

e12K1(!1+!3)

!

!2=±

eK1!2+K2!2(!1+!3) = e12K1(!1+!2)2 cosh [K1 +K2(!1 + !3)] .

& eK$0+

12K

$1(!1+!3)+K $

2!1!3 , (5.63)

where we defined the renormalized couplings so to preserve the functionalform of the partition function. From (5.63) we can extract three independentrelations by evaluating it for the various configuration of the two spins, i.e.for !1 = !3 = 1, for !1 = "!3 = 1 and for !1 = !3 = "1, which giverespectively:

eK12 cosh (K1 + 2K2) = eK$0+K $

1+K $2

2 coshK1 = eK$0!K $

2

e!K12 cosh (K1 " 2K2) = eK$0!K $

1+K $2 . (5.64)

Equation (5.64) defines (implicitly) the RG transformation

R

<

=

K0

K1

K2

>

? =

<

=

K %0

K %1

K %2

>

? (5.65)


0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

Figure 5.12: The RG flow for the coupling K2.

in the three dimensional theory space of all the free energies of the form:

""F (K0, K1, K2) = K0 +K1

!

i

!i +K2

!

i

!i!i+1 .

The fact that theory space is finite dimensional is a peculiarity of the onedimensional model, note that this implies that (5.65) is an exact RG.

Let’s consider for a moment the case where the magnetic field is zero. Thusthere are only parity invariant interactions, i.e. only the couplings K0 andK2:

K %0 = log 2 +

1

2log cosh 2K2

K %2 =

1

2log cosh 2K2 . (5.66)

The RG transformation (5.66) has a fixed point for K&2 = 0 and K&

2 = -.The second is attractive and corresponds to the ordered phase transitionwhile the first represent the disordered high temperature phase. Note thatK0 converges in the IR to the free energy, since at every RG step we areintegrating out some degree of freedom (spins).

In terms of the variables u = e!K2 , v = e!K1, w = e!K0 the solution of


0.0 0.2 0.4 0.6 0.8 1.0u#e!$J0.0

0.2

0.4

0.6

0.8

1.0v#e!$B

Figure 5.13: RG flow for the 1d Ising model in the (u, v) plane.

(5.64) can be written as:

u% =

2

v + 1v

31/2

2

u4 + 1u4 + v2 + 1

v2

31/4

v% =

"u4 + v2

u4 + 1v2

#1/2

w% =

"

v +1

v

#1/2 "

u4 +1

u4+ v2 +

1

v2

#1/4

, (5.67)

and the exact RG transformation becomes:

R

<

=

uvw

>

? =

<

=

u%

v%

w%

>

? .

In the (u, v) plane there is a repulsive fixed point (0, B), a mixed fixed point(0,-) and a line of fixed points (-, B), as can be seen in Figure 1.13.

5.5.2 General RG analysis

The general case is much more complicated. The coarse–graining is stillimplemented by decimation but for d > 1 this procedure introduces new


C S

R

Figure 5.14: RG transformation for the two dimensional Ising model. Thecoarse–graining is performed by summing over any second spin, while thesimilarity has scale factor - = 1$

2and a rotation of +

4 .

couplings at every iteration of the RG transformation.

The full complexity is already present in the two dimensional square lat- rg for

the d = 2

ising

model

tice case. The RG transformation is illustrated in the Figure. Since now

!

!5=±1

eK2!5(!1+!2+!3+!4) = 2 cosh [K2 (!1 + !2 + !3 + !4)]

= eK$0+

12K

$2(!1!2+!1!4+!2!3+!3!4)+K $

3(!1!3+!2!4)+K $4!1!2!3!4 , (5.68)

new next–nearest–neighbor interactions K3 and quadruple interactions K4

are generated already after the first RG step. At every iteration the num-ber of coupling constants grows and finally the RG flow takes place inthe infinite dimensional theory space parametrized by the couplings K =(K0, K1, K2, K3, K4, ...):

RK = K% . (5.69)

The most general free energy has now the form:

""F (K) = K0+K1

!

i

!i+K2

!

"i,j#

!i!j+K3

!

""i,j##

!i!j+K4

!

!

!i!j!k!l+ ...

(5.70)We go on with the theory.

Phase transitions correspond to fixed points: fixed

pointsRK& = K& , (5.71)


and their domain of attraction define a universality class.

Linear stability. We can linearize the RG transformation around a fixed linear

stabilitypoint:RK = RK& + L(K"K&) + ... (5.72)

where the linear RG transformation is now the stability matrix:

Lij =#K %

i

#Kj

NNNN&. (5.73)

In general there is no reason whyL should be symmetric, but for ease ofdiscussion we will assume it. Thus the eigenvalues and eigenvectors are allreal:

Lvn = -nvn -n = -!,n , (5.74)

where we introduced, as customary, the RG eigenvalues .n. The couplingsin the direction of the eigenvectors are called the scaling fields or couplingsand denoted with si. Since we are interested in the Ising fixed point whichhas two relevant directions we will set t = s1 and b = s2.

We have seen the the Ginzburg criterium tells us that the mean field critical deriving

the scal-

ing form

and the

critical

expo-

nents

exponents are correct in d > 4. We have the critical exponents in d = 1, 2from the exact solutions. We now see that we can use the RG to compute thecritical exponents for every d. Note that by the RG decimation procedure weare not changing the partition function (since we are considering the mostgeneral for the energy or free energy):

Z(K) =!

{!i}

e!"E{!i}(K)

=!

{!i}

!

{!deci }

e!"E{!i,!dec

i }(K)

=!

{!i}

e!"E{!i}(K$)

= Z(K%) , (5.75)


thus the free energy transforms as:

f(K) = "1

"NlogZ(K)

= "1

"NlogZ(K%)

=N %

Nf(K%) . (5.76)

Since N % = -dN we find:

f(K) = -df(RK) . (5.77)

As we said, the coupling K0 converges to the total free energy since at everystep we integrate out half of the spins. Around a fixed point we can diag-onalize the stability matrix and express the coupling vector in terms of thescaling fieldsK = (t, b, s3, s4, ...) and we find:

f(K) = -ndf(RnK)

= -ndf(-!n,tt,-!n,bb,-!n,3s3,-!n,4s4, ...)

%n'(

-ndf(-!n,tt,-!n,bb, 0, 0, ...) , (5.78)

if we now choose - = |t|1/n,t , we find the scaling form for the free energy:

f(t, b) ) |t|d/,tf"

±,b

|t|,b/,t, 0, 0, ...

#

t % 0 b % 0 , (5.79)

with:

2" ( =d

.t" =

.b

.tF±(x) = f(±1, x, ...) . (5.80)

The RG analysis has thus proved the scaling hypothesis and showed that thecritical exponents are related to the RG eigenvalues! The first relation is justthe hyperscaling relation (5.60) and thus

+ =1

.t, (5.81)

which immediately tells us how to compute the correlation length criticalexponent. Spin block transformations as defined in this section are not anexact way to implement the coarse–grain C and are thus of of limited prac-tical/quantitative use, even if conceptually very illuminating. In the nextlecture we will develop a better formalism to construct the transformation Rexactly.

Bibliography

[1] H. Maris and L. Kadano!, Teaching the Renormalization Group, Am.J. Phys. 46(6) (1978) 652.

[2] K. Wilson, The Renormalization Group and Critical Phenomena. Rev.Mod. Phys. 55(3) (1983) 583.

[3] E. Stanley, Scaling, universality, and renormalization: Three pillars ofmodern critical phenomena, Rev. Mod. Phys. 71(2) (1999) S358–S366.

96

Chapter 5 Ising model and phase transitions · 5.2.1 Mean ﬁeld phase diagram mean field phase...

Documents

Transcript of Chapter 5 Ising model and phase transitions · 5.2.1 Mean ﬁeld phase diagram mean field phase...