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Valery Pokrovsky Landau and Theory of Phase Transitions Landau and Theory of Phase Transitions Valery Pokrovsky Dept. of Physics, Texas A&M University and Landau Institute for Theoretical Physics Landau Centenary, APS March Meeting, March 18, 2009. Landau phenomenon Scirus Fermi liquids – 45,000 2008 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 1940-20.. Landau Centenary, APS March Meeting, March 18, 2009. 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 Landau Centenary, APS March Meeting, March 18, 2009. 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) 23 - degree of order Φ=Φ0 +abξ +ξξ + c ξ aTT=−α ( c ) Landau Centenary, APS March Meeting, March 18, 2009. 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 irreducibleOnly one IR representation appears near oftransition the initial point symmetry group. 24 ⎛⎞TT− c η 22= c Φ=abaηη +; = α ⇒∝η TT − δρϕ= c ∑ i ⎜⎟ c ∑ ni ni i ⎝⎠Tc ni, Specific heat has a finite jump at transition Theory of phase transitions II 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) Scattering of X-rays in crystals near the Curie point 2 T 24 2 Φ=abcη +ηη + ∇ Iqq∝=η 2 () acq+ 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. Yuri Rumer, Landau’s friend and coworker, jailed by the same affair. He was released in Landau Centenary, APS March Landau in jail, 1938 1953. Meeting, March 18, 2009. 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, 1987. 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 Experiments and technological applications in Ralph C. Smith, Smart Material Systems. Model Development. SIAM, Frontiers in Applied Mathematics, 2005. Superconductivity: Ginzburg-Landau theory, 1950 Vitali Ginzburg Superfluidity: Ginzburg-Pitaevski, 1958 Gross-Pitaevski, 1961 Lev Pitaevsky Liquid crystals: Isotropic liquid-nematic and nematic-smectic transitions De Gennes, 1970-th Landau Centenary, APS March Meeting, March 18, 2009. P.-G. de Gennes Phase Transitions in the range of developed fluctuations Onsager solution of 2d Ising model 1942-1944 Singularities of thermodynamic values: 1/8 −7/4 CTT∝−ln − c ; mTT∝−()c ; χ ∝−TTc χ ∝−TT−1 C→ const; mTT∝ c − ; c Lars Onsager Experiment: Buckingham, Fairbank and Kellers, 1961 λ-point of He CTT∝ −−ln c Voronel, Bagatski and Gusak, 1962 Critical point of Ar CTT∝ −−ln c Alexander Voronel Landau Centenary, APS March Meeting, March 18, 2009. Levanyuk-Ginzburg criterion (1949-50): fluctuations are small if 2/() 4−D 2/4DD( − ) TT− c ()Tb ⎛⎞r Gi 1 Gi ==c 0 T DD/4()− ⎜⎟ c αc ⎝⎠ξ0 r0 - interaction radius; ξ0 - correlation length far from transition General Theory of Phase Transitions: Landau, end of 1950-th ⎧⎫1 2 ZT =−exp ⎡⎤aηη24 ++∇ b c η dxDD ηx () ∫∫⎨⎬⎣⎦() () ⎩⎭Tc Mesoscopic description, universality Search of the most essential graphs Alexander Patashinskii and VP, 1964: all graphs are of the same order Scaling Sasha Patashinskii Landau Centenary, APS March Meeting, March 18, 2009. Scaling theories xx→⇒λλλAA( x) →−ΔA ( 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 Argon near critical point Magnetization vs magnetic field, Anisimov et al. 1974 Kouvel and Rodbell, 1964 Introduction and calculation of critical exponents Michael Fisher, 1959 −α β −γ C ∼τ ; ητ∼ ()− ; χτ∼ ; τ =−(TTcc) / T Specific Susceptibility heat δ ξτ∼ −ν ; h ∼ η Correlation length External field Michael Fisher Numerical calculations of critical α + 22;βγ+= Exponents from high-temperature α = 2 − Dν Series by Pade method: δ = βγ+ Cyril Domb and his group at Kings College, London Algebra of fluctuating fields: L. Kadanoff , A. Polyakov, 1969: AA(λλxx) → −ΔA ( ) Sasha Polyakov AxAik()12 () x=−+∑λ ikll ( x 1212 xA )() ( x x )/2 l Landau Centenary, APS March Meeting, March 18, 2009. Renormalization group Wilson 1971 Wilson and Fisher 1972 Dimension 4 −ε Ken Wilson Wilson 1972 Precursors: Larkin and Khmelnitskii 1969 Tolya Larkin Dimension 4 Di Castro and Jona-Lasinio 1970 Carlo Di Castro Critical dynamics Landau and Khalatnikov, 1954 Ferrel, Menyhard, Schmidt, Schwabl, Sepfalusi z Halperin and Hohenberg ω = q f ()qξ q Critical fluctuations Bert Halperin qξ =1 Hydrodynamic Diffusion T Helium near λ-point z =3/2 Archibald, Mochel and Weaver, 1968 Pierre Hohenberg 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. Landau Centenary, APS March Meeting, March 18, 2009. 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.
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