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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