• Introduction

Landau Theory

Many phase transitions exhibit similar behaviors: critical temperature, order parameter…

Can one find a rather simple “unifying theory” that gives a general “phenomenological” overview of phase transitions ?

Molecular field (Weiss ~1925): solve the Schrödinger equation for a one particle system but with an effective interaction potential :

Several approaches :

Microscopic model (Ising 1924): solve the Schrödinger equation for “pseudo spins” on a lattice with effective interaction Hamiltonian restricted to first neighbors

eff

2

V2m

pH

j,

io σσ2

1HH

ji

J

• Introduction

Landau Theory

– Express a thermo dynamical potential as a function of the order parameter (), its conjugated external field (h) and temperature.

Landau Theory :

– Keep close to a stable state minimum of energy power series expansion, eg. like:

h6

C

4

B

2

A)( 642

o

– Find and discuss minima of versus temperature and external field.

– Look at thermodynamics’ properties (latent heat, specific heat, susceptibility, etc.) in order to classify phase transitions

• Broken symmetry

Landau Theory

a simple 1D mechanical illustration :

d

0 x

l

2222pot d

2

1

2

1E )lx(k)l(lk oo

let go with d > lo : equilibrium position (minimum energy) x = 0

• Broken symmetry

Landau Theory

a simple 1D mechanical illustration :

d

0 x

l

2222pot d

2

1

2

1E )lx(k)l(lk oo

let go with d < lo : equilibrium position (minimum energy) x = xo 0Order parameter

@ critical value dc= lo spontaneous symmetry breaking

dcd

xo

Only irreversible microscopic events will make the system settle at +xo or –xo when the system slowly exchanges energy with external world

• Broken symmetry

Landau Theory

a simple 1D mechanical illustration :

d

0 x

l

2222pot d

2

1

2

1E )lx(k)l(lk oo

Taylor expansion of potential (elastic) energy

2

22

2

2

d2

11d

d1d

xxl

2

2

22

2d)d()(

xlll oo

4d2

1

2d)d(2

2

1)d(

2

1)(

2

1E

4222

pot

xk

xlklkllk ooo

• Broken symmetry

Landau Theory

a simple 1D mechanical illustration :

d

0 x

l

2222pot d

2

1

2

1E )lx(k)l(lk oo

Taylor expansion of potential (elastic) energy )(B4

1A

2

1EE 642

opot xOxx

)dd(d

A(d)A ck

Change sign at d=dc !!!

0B Does not change sign

h=0

•Second Order Phase Transitions

Landau Theory

hB4

1A

2

1)( 42

oh

)T(TA ca

0aT >>Tc

T <<Tc

T =Tc

=0 stable above Tc , unstable below Tc

•Second Order Phase Transitions

Landau Theory

hB4

1A

2

1)( 42

oh

)T(TA ca

0a Stationary solution : 0&02

2

B

T)-(T

B

A

0

0BA

c2o

o2oo

ao

T Tc

T Tc

TTc

o

•Second Order Phase Transitions

Landau Theory

2c

2ooh

TTB2

0

)(

a

Free energy :

T Tc

T Tc

TTc

(o) - o

Entropy : c

2oh TT

B2

)(

a

TTS

T

TcS(o) - S o

No Latent Heat: TcS = 0

•Second Order Phase Transitions

Landau Theory

Specific heat :T Tc

T Tc

TTc

cp - co

B2

T

c

TT

T

STc

2

o

2

2

p

a

•Second Order Phase Transitions

Landau Theory

Susceptibility :

hB4

1A

2

1)( 42

oh

h

h1

h)(

0)( 0hh

eq

2

0h2

1 )(h

20h

21 )(h

T Tc

T TcT)(T22A-

)T(TA

3BA

c

c2o

1

a

a

Curie law

TTc

-1

h

•Second Order Phase Transitions

Landau Theory

field hysteresis : 30h BA)(

h

T Tc

A 0

T Tc

A 0

h

•Second Order Phase Transitions SUMMARY

Landau Theory

One critical temperature Tc

No discontinuity of , , S (no latent heat) at Tc

Jump of Cp at Tc

Divergence of and at Tcc

Field hysteresis

One critical temperature Tc

No discontinuity of , , S (no latent heat) at Tc

Jump of Cp at Tc

Divergence of and at Tcc

Field hysteresis

)T(TA ca

0a

0C 0B

•First Order Phase Transitions:

Landau Theory 642

oh C6

1B

4

1A

2

1)(

)(co TTat0 order2from nd

T > T1 : o=0 stable

T1 > T > To : o=0 stable

o0 metastable

To > T > Tc : o=0 metastable

o0 stable

Tc > T : o 0 stable

T>>T1

T>T1

T=T1

T1>T>T

oT=T

oTo>T>T

cT=T

cTc>T

T

equ.

ToTc T1

)T(TA ca

0a

0C 0B

•First Order Phase Transitions:

Landau Theory 642

oh C6

1B

4

1A

2

1)(

T > T1 : o=0 stable

T1 > T > To : o=0 stable

o0 metastable

To > T > Tc : o=0 metastable

o0 stable

Tc > T : o 0 stable

Thermal hysteresis

)T(TA ca

0a

0C 0B

•First Order Phase Transitions:

Landau Theory 642

oh C6

1B

4

1A

2

1)(

0CBA 4o

2oo

o

02C

AC4BB ?22o

Steady state :

T Tc0o

T Tc0o

0AC4B2

+

4aC

BTT0AC4Bwhenlimit

2

c12

2C

B)T( 1

2o

C

B)T( c

2o

)T(TA ca

0a

0C 0B

•First Order Phase Transitions:

Landau Theory 642

oh C6

1B

4

1A

2

1)(

0C

3

2B

2

1

0C6

1B

4

1A

2

1

0CBA

4o

2o

4o

2o

2oo

4o

2oo

Steady state : T = To

4aC

B

4

3TT

4C

B3)T(

2

coo2o

)T(TA ca

0a

0C 0B

•First Order Phase Transitions:

Landau Theory 642

oh C6

1B

4

1A

2

1)(

Entropy : T = To

6o

4o

2oo C

6

1B

4

1A

2

1

TS

TS A and 2 depend on T !

T

CBA2

1

T

A

2

1SS-

2o4

o2o

2oo

= 0

0TC

B

8

3STheatLatent oTo

o

a

)T(TA ca

0a

0C 0B

•First Order Phase Transitions:

Landau Theory 642

oh C6

1B

4

1A

2

1)(

Specific heat : T T1

T

A

2

1SS 2

oo

2

22o

2o

op T

A

2

1

TT

A

2

1cc

2C

AC4BB 22o

= 0

c2

22o

op

TTB

C41

1

B2TT

A

2

1cc

a

acp

T1

co

)T(TA ca

0a

0C 0B

•First Order Phase Transitions:

Landau Theory 642

oh C6

1B

4

1A

2

1)(

Susceptibility : 4o

2o2

21 5C3BA

T

c1

o TT:TT a o= stable until T down to To

4o

2oc

1o 5C3BTT:TT a

2C

T-TaC4BB c2

2o

-1

ToTc T1

•First Order Phase Transitions SUMMARY

Landau Theory

Existence of metastable phases

Temperature domain (Tc T1) for coexistence of high

and low temperature phases

at To (Tc < To < T1) both high and low teperature

phases are stable

Temperature hysteresis

Discontinuity of , , S (latent heat), Cp, at Tc

Existence of metastable phases

Temperature domain (Tc T1) for coexistence of high

and low temperature phases

at To (Tc < To < T1) both high and low teperature

phases are stable

Temperature hysteresis

Discontinuity of , , S (latent heat), Cp, at Tc

•Tricritical point

Landau Theory

In the formalism of first order phase transitions, it can happen that B parameter changes sign under the effect of an external field. Then there is a point, which is called tricritical point, where B=0. The Landau expansion then takes the following form:

62o C

6

1A

2

1)(

Equilibrium conditions : 0CA 4oo

o

C

A-

0

4o

o

0C5A 4o2

2

o

C

A-pourA4

0pourA

4o

o

c

c

TTpour0

TTpour0

Landau Theory

Potential :

6o

2oo C

6

1A

2

1

•Tricritical point

T>Tc: =00

T>Tc: 0 2321

C

AC

6

1

C

AA

2

1

T

Tc

23

21

23

TcTC

a

3

1

Landau Theory

Entropy :

6o

2oo C

6

1A

2

1

TS

TS A and 2 depend on T !

T

CA2

1

T

A

2

1S-SS

2o4

o2oo

•Tricritical point

T>Tc: =0 0S

T>Tc: 0TC

ACA

2

1

T

A

2

1S

2o2

o

21

c21

232o TT

C

a

2

1

T

A

2

1S

T

TcS

Landau Theory

Specific heat :

T

A

2

1

TT

T

STCp 2

o

•Tricritical point

T>Tc: =0 0Cp

T>Tc: 0

2

22o

2o

T

A

T

A

TT

2

1Cp

0a

T2

1T

2

1Cp

4o214

o

21

c21

23

T-TTC

a

4

1Cp

TTc

Cp

Landau Theory

Susceptibility :

•Tricritical point

T>Tc: =0

T>Tc: 0

4o2

21 5CA

T

Tc-TaA1

Tc-Ta4A41

TTc

-1

TTc