Valery Pokrovsky Dept. of Physics, Texas A&M University and Landau Institute for Theoretical Physics

Transcription

1 Landau and Theory of Phase Transitions Valery Pokrovsky Dept. of Physics, Texas A&M University and Landau Institute for Theoretical Physics

2 Scirus 2008 Landau phenomenon Fermi liquids 45,000 Phase transitions -- 30,000 Landau 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

3 Extremely general and simple notions: Density matrix Spontaneous symmetry breaking Fermi liquid Quasiparticles Simple and effective formalism Phase transitions Landau levels Neutron stars Unique view of entire physics

4 Landau theory of phase transitions: History Predecessors Curie-Weiss theory of self-consistent field in ferromagnets Ehrenfest theory of higher order transitions Landau: Contribution to theory of specific heat anomaly Phys. Zs. Sowjet., 8, 113 (1935) 2 3 Φ=Φ + ξ + ξ + ξ ξ - degree of order 0 a b c ( ) a= α T T c

5 3 articles published in 1937 in ZhETF and Phys. Zs. Sowjet. Theory of phase transitions I Concept of spontaneous symmetry violation. Ordered phase is characterized by some irreducible Only one IR representation appears near of transition the initial point symmetry group T T η = i δρ = cniϕ c 4 c aη bη ; a α Φ= + = η ni i Tc ni, Specific heat has a finite jump at transition Theory of phase transitions II Scattering of X-rays in crystals near the Curie point T T 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) 2 4 Φ= a + b + c ( ) 2 η η η I q 2 T η = q a + cq 2 c

6 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. Yuri Rumer, Landau s friend and coworker, jailed by the same affair. He was released in Landau in jail, 1938

7 Developments of mean-field Landau theory Crystal reconstruction First group-theoretical calculation of a crystal phase transition - E.M. Lifshitz, 1941 Recent reviews: Y.A. Izymov, V.N. Syromyatnikov, Phase Transitions and Crystal Symmetry. Kluwer, Boston, 1990 P.. 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, 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).

8 Ferroelectrics: Ginzburg-Devonshire theory, Experiments and technological applications in Ralph C. Smith, Smart Material Systems. Model Development. SIAM, Frontiers in Applied Mathematics, Superconductivity: Ginzburg-Landau theory, 1950 Superfluidity: Ginzburg-Pitaevski, 1958 Vitali Ginzburg Gross-Pitaevski, 1961 Liquid crystals: Isotropic liquid-nematic and nematic-smectic transitions De Gennes, 1970-th Lev Pitaevsky P.-G. de Gennes

9 Phase Transitions in the range of developed fluctuations Onsager solution of 2d Ising model Singularities of thermodynamic values: ln ; C T T c m ( T T) 1/8 ; C const; m T T; c c χ χ T T c T T c 7/4 1 Lars Onsager Experiment: Buckingham, Fairbank and Kellers, 1961 λ-point of He C ln T Tc Voronel, Bagatski and Gusak, 1962 Critical point of Ar C ln T Tc Alexander Voronel

10 Levanyuk-Ginzburg criterion ( ): fluctuations are small if Gi 2/ T Tc 1 ( ) ( 4 D ) 2 D/ 4 Tb c r 0 Gi = T D/4 ( D) = c αc ξ0 ( D) r0 - interaction radius; ξ0 - correlation length far from transition General Theory of Phase Transitions: Landau, end of 1950-th 1 Z T a b c d x D T c Mesoscopic description, universality D ( ) = exp η + η + ( η) η( ) x Search of the most essential graphs Alexander Patashinskii and VP, 1964: all graphs are of the same order Scaling Sasha Patashinskii

11 Scaling theories A ( ) A( ) x λx A λx λ Δ x 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 Leo Kadanoff Magnetization vs magnetic field, Kouvel and Rodbell, 1964 Argon near critical point Anisimov et al. 1974

12 Introduction and calculation of critical exponents Specific heat Michael Fisher, 1959 ; C τ α β ; η ( τ) ; ξ τ ν ; Correlation length Susceptibility χ τ γ Numerical calculations of critical Exponents from high-temperature Series by Pade method: Cyril Domb and his group at Kings College, London h η δ External field τ = ( ) / T T T c c α + 2β + γ = 2; α = 2 Dν δ = β + γ Michael Fisher

13 Algebra of fluctuating fields: L. Kadanoff, A. Polyakov, 1969: A A ( λx) λ Δ A( x) ( ) ( ) ( ) = λ ( ) ( + ) A x A x x x A x x i 1 k 2 ikl 1 2 l 1 2 /2 l Sasha Polyakov

14 Renormalization group Wilson 1971 Wilson and Fisher 1972 Wilson 1972 Dimension 4 ε Ken Wilson Precursors: Larkin and Khmelnitskii 1969 Dimension 4 Tolya Larkin Di Castro and Jona-Lasinio 1970 Carlo Di Castro

15 Critical dynamics Landau and Khalatnikov, 1954 Ferrel, Menyhard, Schmidt, Schwabl, Sepfalusi z Halperin and Hohenberg ω = q f ( qξ ) q Critical fluctuations qξ =1 Bert Halperin Hydrodynamic Diffusion Helium near λ-point z = 3/2 Archibald, Mochel and Weaver, 1968 T Pierre Hohenberg

16 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, , Kosterlitz and Thouless, 1972 No phase transition in the 2d Heisenberg magnet: A. Polyakov, 1975.

17 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