Landau Centenary, APS March Meeting, March 18, 2009.

Valery PokrovskyDept. of Physics, Texas A&M University

andLandau Institute for Theoretical Physics

Landau and Theory of Phase Transitions

Landau Centenary, APS March Meeting, March 18, 2009.

Landau phenomenonScirus2008 Fermi liquids – 45,000

Phase transitions -- 30,000Landau levels -- 75,500

Landau-Lifshitz Equation -- 23,000

5 Nobel prizes based on Landau works:K. Wilson; P.-G. De Gennes; A.A. Abrikosov, V.L. Ginzburg, A. Legget; K. von Klitzing;D.Tsui, H. Störmer, R.B. Loughlin.

Longevity of the LL Course 1940-20..

Landau Centenary, APS March Meeting, March 18, 2009.

Extremely general and simple notions:

Density matrixSpontaneous symmetry breaking

Fermi liquid

Quasiparticles

Simple and effective formalismPhase transitions

Landau levels

Neutron stars

Unique view of entire physics

Landau Centenary, APS March Meeting, March 18, 2009.

Curie-Weiss theory of self-consistent field in ferromagnets

Ehrenfest theory of higher order transitions

Predecessors

Landau: Contribution to theory of specific heat anomalyPhys. Zs. Sowjet., 8, 113 (1935)

2 30 a b cξ ξ ξΦ = Φ + + + ξ - degree of order

( )ca T Tα= −

Landau theory of phase transitions: History

Landau Centenary, APS March Meeting, March 18, 2009.

3 articles published in 1937 in ZhETFand Phys. Zs. Sowjet.

Theory of phase transitions I

Theory of phase transitions II

Scattering of X-rays in crystals near the Curie point

Concept of spontaneous symmetry violation. Ordered phase is characterized by some irreducible representation of the initial symmetry group.

,ni ni

n icδρ ϕ=∑

2 2i

i

cη =∑

Only one IR appears near transition point2 4; c

c

T Ta b aT

η η α⎛ ⎞−

Φ = + = ⎜ ⎟⎝ ⎠

cT Tη⇒ ∝ −

Specific heat has a finite jump at transition

1. Instability of smectics in 3 dimensions.Instability of a state with violated continuous symmetry in 2d (Peierls,1936)2. Transition from liquid to crystal is always of the first order. (Cubic invariants)

( )22 4a b cη η ηΦ = + + ∇2

2

TIa cq

η∝ =+q q

Landau Centenary, APS March Meeting, March 18, 2009.

Purges of 1937 and arrest

Lev Vasilyevich Shubnikov, outstanding experimentalist, Landau’s friend and colleague; arrested and shot in1937.

Landau visits Kapitza at his home confinement, end 1940-th.

Landau in jail, 1938

Yuri Rumer, Landau’s friend and coworker, jailed by the same affair. He was released in 1953.

Landau Centenary, APS March Meeting, March 18, 2009.

Developments of mean-field Landau theoryDevelopments of mean-field Landau theory

First group-theoretical calculation of a crystal phase transition - E.M. Lifshitz, 1941

Crystal reconstruction

Y.A. Izymov, V.N. Syromyatnikov, Phase Transitions and Crystal Symmetry. Kluwer, Boston, 1990P.. Toledano and V. Dmitriev, Reconstructive Phase Transitions. World Scientific, Singapur, 1996

Magnetic symmetries in crystals:

Color groups: J.N. Kotsev, V.A. Koptsik and K.A. Rustamov in “Group Theoretical Methods in Physics, Vol. 3, Eds. M.A. Markov, V.I. Man’ko, A.E. Shabad, Harwood, 1987.

Recent reviews:

Exchange groups: A.F. Andreev, V.I. Marchenko, Usp. Fiz. Nauk 130 (1980). 39 (Sov. Phys. Usp. 23 (1980) 21).

Weak ferromagnetism: I.E. Dzyaloshinskii, J. Phys. Chem. Solids 4, 241 (1958).А.S. Borovik-Romanov, ZhETF 36, 766 (1959).T. Moria, Phys. Rev. 120, 91 (1960).

Landau Centenary, APS March Meeting, March 18, 2009.

Ferroelectrics: Ginzburg-Devonshire theory, 1959-61

Superconductivity: Ginzburg-Landau theory, 1950

Experiments and technological applications inRalph C. Smith, Smart Material Systems. Model Development. SIAM,

Frontiers in Applied Mathematics, 2005.

Superfluidity: Ginzburg-Pitaevski, 1958

Gross-Pitaevski, 1961

Liquid crystals: Isotropic liquid-nematic and nematic-smectic transitions

De Gennes, 1970-th

Vitali Ginzburg

Lev Pitaevsky

P.-G. de Gennes

Landau Centenary, APS March Meeting, March 18, 2009.

Phase Transitions in the range of developed fluctuations

Onsager solution of 2d Ising model 1942-1944

Singularities of thermodynamic values:ln ;cC T T∝ − − ( )1/8 ;cm T T∝ − 7/4

cT Tχ −∝ −

;C const→ ;cm T T∝ −1

cT Tχ −∝ −

Experiment: Buckingham, Fairbank and Kellers, 1961

λ-point of He ln cC T T∝ − −

Voronel, Bagatski and Gusak, 1962

Critical point of Ar ln cC T T∝ − −

Lars Onsager

Alexander Voronel

Landau Centenary, APS March Meeting, March 18, 2009.

Levanyuk-Ginzburg criterion (1949-50): fluctuations are small if

1c

c

T TGiT− ( ) ( )

( )

( )2 / 42/ 40

/ 40

D DDc

D D

T b rGic ξα

−−

−

⎛ ⎞= = ⎜ ⎟

⎝ ⎠

0r - interaction radius; 0ξ - correlation length far from transition

General Theory of Phase Transitions: Landau, end of 1950-th

( ) ( ) ( )22 41exp D

c

Z T a b c d x DT

η η η η⎧ ⎫⎡ ⎤= − + + ∇⎨ ⎬⎣ ⎦⎩ ⎭

∫ ∫ x

Mesoscopic description, universalitySearch of the most essential graphs

Alexander Patashinskii and VP, 1964: all graphs are of the same order

ScalingSasha Patashinskii

Scaling theories

B. Widom 1965 Hypothesis about scaling equation of state

C. Domb and D. Hunter 1965 Arguments based on high-temperature expansion

A. Patashinskii and VP 1966 Mesoscopic picture based on scaling of all correlations

L. Kadanoff 1966 Scaling at critical point + idea of renormalization

Magnetization vs magnetic field, Kouvel and Rodbell, 1964

Argon near critical pointAnisimov et al. 1974

( ) ( )AA Aλ λ λ−Δ→ ⇒ →x x x x

Leo Kadanoff

Introduction and calculation of critical exponents

Michael Fisher, 1959

Numerical calculations of criticalExponents from high-temperature

Series by Pade method:

Cyril Domb and his groupat Kings College, London

;C ατ −∼ ( ) ;βη τ−∼ ;γχ τ −∼Specific heat Susceptibility

( ) /c cT T Tτ = −

;νξ τ −∼ h δη∼

Correlation length External field

2 2;α β γ+ + =

2 Dα ν= −

δ β γ= +

Michael Fisher

Landau Centenary, APS March Meeting, March 18, 2009.

Algebra of fluctuating fields:

L. Kadanoff , A. Polyakov, 1969:

( ) ( )AA Aλ λ−Δ→x x

( ) ( ) ( ) ( )( )1 2 1 2 1 2 / 2i k ikl ll

A x A x x x A x xλ= − +∑

Sasha Polyakov

Renormalization group

Wilson 1971Wilson and Fisher 1972

Wilson 1972

Precursors:

Di Castro and Jona-Lasinio 1970

Larkin and Khmelnitskii 1969

4Dimension ε−

4Dimension

Ken Wilson

Tolya Larkin

Carlo Di Castro

Critical dynamics

Ferrel, Menyhard, Schmidt, Schwabl, Sepfalusi

Halperin and Hohenberg ( )zq f qω ξ=

Hydrodynamic Diffusion

Critical fluctuations

T

q

1qξ =

Helium near λ-point 3/2z=Archibald, Mochel and Weaver, 1968

Landau and Khalatnikov, 1954

Bert Halperin

Pierre Hohenberg

Landau Centenary, APS March Meeting, March 18, 2009.

2-dimensional systems with continuous symmetry,smectics

Proof of absence of the LRO in 2d superfluids and magnets: P. Hohenberg; N.D. Mermin and H. Wagner, 1966

, Algebraic order, vortices Berezinskii, 1970-71, Kosterlitz and Thouless, 1972

No phase transition in the 2d Heisenberg magnet:A. Polyakov, 1975.

Experimenters contributing to study of phase transition

Neutron scattering: Passel, Shapiro, Shirane, Als-Nielsen, Jacrot, Cribier

Light scattering: Cummins, Fabelinski

Thermodynamic measurements: Voronel, Ahlers, Anisimov, Levels-Sengert, Sengert, Litster

Magnetic measurements: Benedeck, Kouvel, Karimov

Measurements of superfluid density: Tyson, Douglas, Reppy