Critical Behavior I: Phenomenology, Universality & Scaling

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

2 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)

3 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

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

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

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

7 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

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

9 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

10 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

11 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

12 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

13 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 (3D I) (3D H) ; γ 1.4 (3D I) τ 1.39 (3D H) α 0 ; α OP vanishes continuously as τ anomalies! correlation length ξ diverges! 0 10

14 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 (3D I) (3D H) ; γ 1.4 (3D I) τ 1.39 (3D H) α 0 ; α qualitative and quantitative universality! 11

15 D Ising Model: =1.4 c ξ 1

16 D Ising Model: = c ξ 13

17 D Ising Model: =0.8 c 14

18 Pair Correlation Function G( x, ) φ( x + x ) φ( x ) φ( x + x ) φ( x ) 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

19 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

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

21 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

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

23 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 ( ) c c corresponding paths: asymptotically parallel to coexistence curve 18

24 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

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

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

27 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

28 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

29 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]

30 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]

31 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 = atm

32 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

33 C p of 4 He at λ ransition exp. result: J.A. Lipa et al, cond-mat/ α = ±

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

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

36 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

37 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

38 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

39 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

40 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

41 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 choose optimal eff H = tr hi si i 31

42 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 choose optimal eff H = tr hi si i result (Ising case): m = tanh + i Kijm j hi j h eff i 31

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

44 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 < c x = Km + h 33

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

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

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

48 = 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 ) = K m + ln ( 1 m ) + m ln + m = ( 1 K ) m + m + O( m ) V 1 m 1 36

49 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

50 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

51 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

52 Landau heory (Magnets) assumptions: a) expandability b) analyticity 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

53 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

54 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? MF theory neglects fluctuations! 40

55 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

56 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

57 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

58 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 * ( (0) ) d d < d 4 : not satisfied for τ< τ Gi = N ɶ d ξ C /(4 d ) 41

59 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

60 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

61 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

62 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

63 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

64 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

65 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

66 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

67 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

68 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

69 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

70 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

71 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

Critical Behavior II: Renormalization Group Theory

Critical Behavior II: Renormalization Group Theor H. W. Diehl Fachbereich Phsik, Universität Duisburg-Essen, Campus Essen 1 What the Theor should Accomplish Theor should ield & explain: scaling laws #