Critical Behavior I: Phenomenology, Universality & Scaling

Critical Behavior I: Phenomenology, Universality & Scaling H. W. Diehl Fachbereich Physik, Universität Duisburg-Essen, Campus Essen 1

Goals recall basic facts about (static equilibrium) critical behavior consider examples discuss qualitative and quantitative universality universality in mean-field / Landau theory the role of fluctuations the phenomenological theory of scaling set the stage for RG theory (lecture II)

Assumptions some background in thermodynamics and equilibrium statistical physics some familiarity with phase transitions mean field theory concept of order parameter etc (previous lectures) 3

Phase Diagram (e.g. CO ) pressure P solid liquid critical point 0 0 triple point gas temperature [K] 4

Phase Diagram (e.g. CO ) melting pressure P solid freezing liquid critical point 0 0 triple point gas temperature [K] 4

Phase Diagram (e.g. CO ) melting pressure P 0 0 solid triple point freezing liquid vaporization condensation gas critical point temperature [K] 4

Phase Diagram (e.g. CO ) melting pressure P 0 0 solid liquid vaporization triple condensation point deposition sublimation freezing gas critical point temperature [K] 4

Phase diagram: PV space s + l liquid l + g solid gas s + g 5

D Fields & Densities conventional distinction: extensive: Q = V,(internal) energy E, S, M, intensive: q =, p, µ, H,, s = S / N, e = E / N, m = M / N, Q N, q N 0 more appropriate (Griffiths & Wheeler): D fields : ζ =, p, µ, H, same ( α ) ( β ) values ζ = ζ at α- β coexistence D densities : ρ = ρ, υ = V / N, s, e, m, i ( α ) ( β ) normally different values ρi ρi at α- β coexistence, p α β coex 7

Comparison: Fluid-Magnet p or µ fluid ferromagnet H ρ m m ( ) 0 length of tie lines OP: φ = β φ τ ; β 0.3 ( 3D) c τ = ( ) / c c φ = ( ρ ρ ) / φ = m0( ) l g = m(, H= 0 + ) 9

Comparison: Fluid-Magnet p or µ fluid ferromagnet H critical behavior ρ m m ( ) 0 length of tie lines OP: φ = β φ τ ; β 0.3 ( 3D) c τ = ( ) / c c φ = ( ρ ρ ) / φ = m0( ) l g = m(, H= 0 + ) 9

Static Bulk Critical Behavior order parameter φ φ susceptibility χ compressibility κ Τ φ τ β χ τ C γ sing sing V, CH = ; β 0.330 (3D I) 0.366 (3D H) ; γ 1.4 (3D I) τ 1.39 (3D H) α 0 ; α 0.11 0.1 OP vanishes continuously as τ anomalies! correlation length ξ diverges! 0 10

Static Bulk Critical Behavior order parameter φ φ susceptibility χ compressibility κ Τ φ τ β χ τ C γ sing sing V, CH = ; β 0.330 (3D I) 0.366 (3D H) ; γ 1.4 (3D I) τ 1.39 (3D H) α 0 ; α 0.11 0.1 qualitative and quantitative universality! 11

D Ising Model: =1.4 c ξ 1

D Ising Model: = c ξ 13

D Ising Model: =0.8 c 14

Pair Correlation Function G( x, ) φ( x + x ) φ( x ) φ( x + x ) φ( x ) 0 0 0 0 0 0 cum φ( x + x ) φ( x ) ("cumulant") Power x exp( x / ξ ) for c ; ( d + η ) x for = c ; Fourier transform: G( ɶ q, ) c q ( η ) 15

Definition of Critical Exponents (i) C, V H τ κ, χ τ C ξ τ α γ ν = c (ii) p (i) (ii) δ p δυ H m (iii) C τ δ α δ (iii) V = c κ, χ φ τ τ γ β V=V c (iv) Gɶ ( q, ) c = φ φ q q q ( η ) f x x ln f ( x) λ ( ), if lim x 0+ ln x = λ 16

Qualitative Universality analogous theory for phase transitions and critical behavior in distinct systems, must identify: order parameter conjugate field thermodynamic paths 17

Fluid-Magnet Analogy p or µ fluid ferromagnet H ρ? µ H ρ m m m ( ) 0 c corresponding paths: asymptotically parallel to coexistence curve 18

Fluid-Magnet Analogy p or µ fluid ferromagnet H H = 0, ρ? µ H ρ m m m ( ) 0 + 0 c c corresponding paths: asymptotically parallel to coexistence curve 18

Fluid-Magnet Analogy p or µ fluid g = aδµ + bτ = h + 0 c 0 ferromagnet H H = 0, ρ? µ H ρ m m m ( ) 0 + 0 c c corresponding paths: asymptotically parallel to coexistence curve 18

Binary Mixed Fluids α (gas) 1phase βγ βγ homogen. mixture c β phase β + γ 0 1 composition x γ α : gas β : A-poor γ : A-rich γ β OP: composition φ = ( xa xa ) / examples: aniline & cyclohexan, CCl 4 & perfluoroheptane 19

Fe 3 Al: Phase Diagram OP = sublattice occupation n or n A Fe n B Fe Fe n Al 0

Fe 3 Al: Phase Diagram Al/Fe A phase OP = sublattice occupation n or n A Fe n B Fe Fe n Al 0

Fe 3 Al: Phase Diagram Al/Fe Al/Fe Fe B phase A phase OP = sublattice occupation n or n A Fe n B Fe Fe n Al 0

Fe 3 Al: Phase Diagram Al/Fe Al/Fe Fe B phase A phase Fe Al Fe OP = sublattice occupation n or n A Fe n B Fe Fe n Al DO 3 phase 0

Phase Diagram 4 He solid 4 He pressure [atm] superfluid 4 He l line normalfluid 4 He liquid-gas critical point = 5.0 K, p =.64 atm temperature [K]

Phase Diagram 4 He solid 4 He upper critical endpoint = 1.76 K, P = 9.8 atm pressure [atm] superfluid 4 He l line normalfluid 4 He liquid-gas critical point = 5.0 K, p =.64 atm temperature [K]

Phase Diagram 4 He solid 4 He upper critical endpoint = 1.76 K, P = 9.8 atm pressure [atm] superfluid 4 He l line normalfluid 4 He liquid-gas critical point = 5.0 K, p =.64 atm temperature [K] lower critical endpoint =.17 K, p = 0.0497 atm

Specific Heat at λ ransition M.J. Buckingham and W.M. Fairbank (1961); taken from H. E. Stanley, Introduction to Phase ransitions and Critical Phenomena (Oxford UP, Oxford 1971), p. 0 3

C p of 4 He at λ ransition exp. result: J.A. Lipa et al, cond-mat/0310163 α = 0.017 ± 0.0003 http://spaceresearch.hamptonu.edu/lpe/lpe.html 4

Quantitative Universality critical behavior ( universal quantities ) = quantitatively the same for whole classes of microscopically distinct systems nonuniversal quantities: c, amplitudes ( metric factors ) 5

Law of Corresponding States c E. A. Guggenheim, J. Chem. Phys. 13, 53 (1945) r / ρ ρ c 6

Universal Quantities a) critical exponents: α, β, γ, δ,, ν, η, b) certain ratios of nonuniversal amplitudes : ξ (0) (0) + C ξ χ C [ m ] (0) (0) (0) + + c) scaling functions: C (0) (0) + ξ D h τ f m B τ β χ χ (0) (0) + [ C ] (0) (0) 1/ d + + ξ ξ χ χ C h C = 0 m m equation of state (0) ± (0) ± (0) ± (0) τ τ τ τ ν γ α β ( ξ ) cum ( d + η ) 1 φ x1 φ x = const 1 Ξ 1 G( x ) ( ) ( ) x x will see: only independent critical exponents: α + β + γ =, α = dν (hyperscaling),... 7

Static Bulk Universality Classes bulk critical behavior depends on: d = space dimensionality of system n = # OP components gross features of the interaction: short-ranged / long-ranged / dipolar StBUC: D Ising model (n = 1) with short-range interactions 3D Heisenberg model (n = 3) with short-range interactions 3D uniaxial dipolar ferromagnet 8

Order Parameters symmetry system order parameter φ conjugate field physical example -- liquid-gas δρ δµ CO,, many Z /Ising uniaxial FM unaxial AFM disorder-order displacive m z A mz n u z A m n B B z h z A hz f z h B z A B µ µ Rb NiF 4, K MnF 4 β-brass, FeCo O(), U(1) easy-plane FM easy-plane AFM superfluid m = ( mx, my ) h = ( h x, h y ) A B A B m m h h ψ h ψ 4 He O(3) O(0) Heisenberg FM Heisenberg AFM SAW m m = ( mx, my, mz ) B m A h h A h B EuS, EuO, Fe, Ni RbMnF 3 polymer superconductors, liquid crystals, 3 He,. 9

classical lattice n-vector model K ij K = J k B periodic boundary conditions E 1 = = s s h s Hlat Kij i j i i kb i j i s i =1 30

effective field: H. W. Diehl (Essen): Critical Behavior I: Phenomenology, Universality & Scaling variational principle: MF theory h h K m = + eff i i ij j j i m i6 m i5 F e = Z = r e = Z e Z e m i1 m i4 ( H H ) H ( H H ) tr tr tr m i m i3 H H tr tr tr tr tr F F + H H tr tr tr 31

graphical solution (MF theory) ( ) B ij m = tanh Km ; K J / k h = 0 j x / K = tanh( x) K( c ) = 1 1 > c -0.5 0.5 0 < c x = Km 0 0-1 -3 - -1 0 1 3 x = Km 3

graphical solution (MF theory) ( h) B ij j m = tanh Km + ; K J / k ( x h) / K = tanh( x) K( c ) = 1 h 0 1 > c 0.5 0 < c -0.5-1 -3 - -1 0 1 3 x = Km + h 33

MF Equation of State h = 0.77 c = 1.43 c = c h 1 1+ m = Km + ln 1 m m = ( ) + + 3 3 4 1 K m O( m ) m 0 () m 0 () m 35

MF Equation of State h = 0.77 c = 1.43 c = c h 1 1+ m = Km + ln 1 m m = ( ) + + 3 3 4 1 K m O( m ) δ MF = 3 m 0 () m 0 () m 35

MF Equation of State h = 0.77 c = 1.43 c = c β = MF 1 h 1 1+ m = Km + ln 1 m m = ( ) + + 3 3 4 1 K m O( m ) δ MF = 3 m 0 () m 0 () m 35

= c H. W. Diehl (Essen): Critical Behavior I: Phenomenology, Universality & Scaling Helmholtz Free Energy A(, m) / V = [ A(, m) A(,0)]/ V > c < c m 0 m 0 m A(, m ) 1 1 1 1 4 6 = K m + ln ( 1 m ) + m ln + m = ( 1 K ) m + m + O( m ) V 1 m 1 36

Ornstein-Zernike approximation for correlation function m = tanh i Kilml + hi l ( 1 ) h j Gij = mi KilGlj + δij l ( 1 )( ɶ ɶ 1 ) Gɶ = m K G + q q q 1 1 Gɶ = ɶ q K + + + q m K q O q 1 m ( ) ( 4 τ ) 0 1 ɶ 1/ K Gq = q + ξ ± 1 ɶ MF G q > 1 MF c 1 > c ν η = 1 = 0 ξ K 1/ τ >, τ, ± 1/ < c c ( ξ (0) q ) + ξ (0) (0) + ξ = 37

Specific Heat etc FN 1 / = ln cosh ( + ) Km h Km ch= 0 0; > = 3 kb ; < c ɶ c c h = 0 φ exp C h=0 MF C h=0 MF 1 κ α β γ α β γ MF MF MF 3D-Ising 3D-Ising 3D-Ising = 0 = 1/ = 1 = 0.11(0) = 0.3(7) = 1.3(9) MF MF c MF c 38

Specific Heat etc FN 1 / = ln cosh ( + ) Km h Km ch= 0 0; > = 3 kb ; < c ɶ c c h = 0 φ exp C h=0 MF C h=0 MF 1 κ α β γ α β γ MF MF MF 3D-Ising 3D-Ising 3D-Ising = 0 = 1/ = 1 = 0.11(0) = 0.3(7) = 1.3(9) MF c MF c MF α β γ D-Ising D-Ising D-Ising = 1/ 8 = 7 / 4 sing C h = 0 = 0; lnτ 38

Landau heory (Magnets) assumptions: a) expandability b) analyticity 1 1 4 A (, φ ) / V = A0 ( ) + A ( ) φ + A4 ( ) φ + 4 A ( ) = A + A + O τ ( τ ),0,1 = 0 A ( ) = A + O 4 4,0 ( τ ) > 0 O(n) invariant (n = 1) h A χ ( ) 3 φ φ 1 = = = A + A4,0 φ 39

Defects of MF / Landau heory wrong values of critical exponents for d = 3 and d = wrong universal amplitudes, equations of state etc critical expo s are independent of d and n hyperscaling broken for d < 4 α MF = dν ( MF ) transitions at c > 0 K for 1, Ising; d d* = lcd =, O( n) symmetry ( qualitatively wrong! ) Why? 40

Ginzburg Criterion correlation volume V ξ x x 0 δφ( x) = φ( x) φ( x) fluctuations negligible if: V ξ d d x ξ δφ( x ) δφ( x + x) V φ 0 0 ξ 41

Ginzburg Criterion correlation volume V ξ x x 0 δφ( x) = φ( x) φ( x) fluctuations negligible if: ξ δφ( ) δφ VV ξ ξφ φ d x Ξ( x / ) Ad ξ ( B β ξ τ ) d d x δφ( x 0 ) δφ( x 0 + x ) V ξ ( d + η) 41

Ginzburg Criterion correlation volume V ξ x x 0 δφ( x) = φ( x) φ( x) fluctuations negligible if: ξ = ξ τ (0) ν 1 MF MF ; MF ν = β = η = 0 ξ δφ( ) δφ VV ξ ξφ φ d x Ξ( x / ) Ad ξ ( B β ξ τ ) d d x δφ( x 0 ) δφ( x 0 + x ) V ξ ( d + η) S d d 1 η 1 η d β dr r Ξ( r) Sd B 0 ξ ξ τ τ ( ξ ) (4 d )/ (0) N d d C 41

Lower Critical Dimension I: Ising case Ising chain, = 0: D stable for > 0? low-energy excitations: kinks! F = E S E = J S = ln L ln L F J L L < > 0 const ln 0 if 0 ordered state D unstable for > 0 d * = 1 4

Lower Critical Dimension II: O(n) case U ( φ ) A τ A = φ +,1 4,0 φ 4 L E L L φ ( x) cos( q x) φ q q φq E q k B 43

Lower Critical Dimension II: O(n) case U ( φ ) A τ A = φ +,1 4,0 φ 4 L E L L φ ( x) cos( q x) φ q q φq E q k B φ ( x) 1 k d d q B q Λ q = for d 43

Lower Critical Dimension II: O(n) case U ( φ ) A τ A = φ +,1 4,0 φ 4 L E L L φ ( x) cos( q x) φ q q φq E q k B φ ( x) 1 k d d q B q Λ q = for d no long-range order for d (Mermin-Wagner theorem) d * = 43

owards Phenomenological Scaling ( ),1 τ 4,0 h = m A + A m ( 1/ ),1τ 4, 0 h = m A + A m β m0 β τ 44

owards Phenomenological Scaling ( ),1 τ 4,0 h = m A + A m ( 1/ ),1τ 4, 0 h = m A + A m β ( γ 1/ β ),1τ 4,0 h = m A + A m m0 β τ nonanalytic behavior for τ = 0 if m 0, h 0 44

owards Phenomenological Scaling ( ),1 τ 4,0 h = m A + A m ( 1/ ),1τ 4, 0 h = m A + A m β ( γ 1/ β ),1τ 4,0 h = m A + A m m0 β τ nonanalytic behavior for τ = 0 if m 0, h 0 no! 44

owards Phenomenological Scaling ( ),1 τ 4,0 h = m A + A m ( 1/ ),1τ 4, 0 h = m A + A m β ( γ 1/ β ),1τ 4,0 h = m A + A m m0 β τ nonanalytic behavior for τ = 0 if m 0, h 0 no! B. Widom D h m m = ± 1 β + β τ B τ B τ 1/ β MF 3 = 44

Matching condition ( D h ) m( τ, h) B τ M τ β ± β = δ M± ( y) const 1/ y δ M ( y) m analytic for h 0! M+ ( y) y 45

Scaling Plot data must collapse on branches of a curve m < c = c m τ β M ( y) M+ ( y) > c h h τ 46

Scaling Ansatz for Gibbs Free Energy F k V B F (, h) V reg = = f + f sing matching conditions! f sing α (, ) ( τ h A Y ) τ τ ± Ah h τ 47

Scaling Ansatz for Gibbs Free Energy F k V B F (, h) V reg = = f + f sing matching conditions! α (, ) ( ) τ h Aτ τ Y± Ah h τ sing f g g g g magnet τ τ gτ = τ + c,0τ + c1, τ h + g c hτ c τ h h h h 3 = h + 1,1 + 1,3 + nonlinear scaling fields fluid ( δµ ) ( ) τ τ τ τ τ τ + c0,1 δµ 1,1 τ δµ,0 τ 1, τ δµ g = + c + c + c + g = δµ + c τ + c τ δµ + c + h h h h 1, 0 1,1 0, 48

Consequences f sing α (, ) ( τ h A Y ) τ τ ± Ah h τ 1) two independent D critical exponents: 0 h= 0 α m = f h τ Y ( 0) β = α α χ = f h τ Y h= 0 ± ( 0) γ = α + ) hyperscaling: f single relevant length if h = 0 f d ( τ,0) ξ τ d ν α = dν 3) two-scale-factor universality: ( ) A, A h = nonuniversal metric factors just! τ 49