Interface Profiles in Field Theory

Transcription

1 Florian König Institut für Theoretische Physik Universität Münster January 10, 2011 / Forschungsseminar Quantenfeldtheorie

2 Outline φ 4 -Theory in Statistical Physics Critical Phenomena and Order Parameter Landau Theory Statistical Field Theory Intrinsic Profile and Capillary Waves Mean Field Profile Capillary Wave Model Convolution Approximation First Principles Approach Separation of Fluctuations One-Loop-Diagram Boundary Conditions and K 1

3 Outline φ 4 -Theory in Statistical Physics Critical Phenomena and Order Parameter Landau Theory Statistical Field Theory Intrinsic Profile and Capillary Waves Mean Field Profile Capillary Wave Model Convolution Approximation First Principles Approach Separation of Fluctuations One-Loop-Diagram Boundary Conditions and K 1

4 Outline φ 4 -Theory in Statistical Physics Critical Phenomena and Order Parameter Landau Theory Statistical Field Theory Intrinsic Profile and Capillary Waves Mean Field Profile Capillary Wave Model Convolution Approximation First Principles Approach Separation of Fluctuations One-Loop-Diagram Boundary Conditions and K 1

5 Critical Phenomena and Order Parameter Criticality T < T c : 2 phases separated by an interface T T c : Long range fluctuations (critical opalescence) T > T c : One supercritical phase

6 Critical Phenomena and Order Parameter Critical Opalescence

7 Critical Phenomena and Order Parameter Order Parameter Describe continuous phase transitions by a quantity which is nonzero in a phase of higher order (T < T c ) zero in the phase of higher symmetry Example Ising model: φ(x) a D S i W a(x) S i Binary fluid, constituents A and B: φ(x) = ρ A (x) ρ B (x) ( ρ A,c (x) ρ B,c (x) )

8 Critical Phenomena and Order Parameter Order Parameter Describe continuous phase transitions by a quantity which is nonzero in a phase of higher order (T < T c ) zero in the phase of higher symmetry Example Ising model: φ(x) a D S i W a(x) S i Binary fluid, constituents A and B: φ(x) = ρ A (x) ρ B (x) ( ρ A,c (x) ρ B,c (x) )

9 Landau Theory Energy Functional Landau s idea: Thermodynamic potential as a functional of φ(x) φ 0 as T T c Expand functional respecting the symmetries of the system Ginzburg-Landau-Hamiltonian H(x) = 1 2 ( φ(x))2 + µ2 2 φ(x)2 + g 4! φ(x)4 Symmetry: φ φ Only interested in long wavelength fluctuation neglect higher orders of derivatives

10 Landau Theory Energy Functional Landau s idea: Thermodynamic potential as a functional of φ(x) φ 0 as T T c Expand functional respecting the symmetries of the system Ginzburg-Landau-Hamiltonian H(x) = 1 2 ( φ(x))2 + µ2 2 φ(x)2 + g 4! φ(x)4 Symmetry: φ φ Only interested in long wavelength fluctuation neglect higher orders of derivatives

11 Landau Theory Free Energy Interpretation of H[φ] = d D xh(x) as a free energy Minimize H Statistical physics: free energy via F = β 1 ln(z ) Landau-Theory is a mean field approximation, i.e. neglects fluctuations Incorporate fluctuations per Z = Dφ exp ( H[φ]) (β = 1)

12 Statistical Field Theory φ 4 -Theory Description of the critical system through a full-fledged euclidean φ 4 -Theory Exploit the apparatus of QFT to calculate expectation values

13 Statistical Field Theory Partition Function Moment-generating functional Z [J] = ( Dφ exp H[φ] + ) d D x J(x)φ(x) Correlation functions: φ(x 1 )...φ(x n ) = 1 Z [0] δ n Z δj(x 1 )...δj(x n ) J=0 Wick s Theorem: corresponding to diagrams without vacuum-fluctuations

14 Statistical Field Theory Free Energy Cumulant-generating functional W [J] = ln(z [J]) Connected correlation functions: δ n W φ(x 1 )...φ(x n ) c = δj(x 1 )...δj(x n ) Connected diagrams + J=0

15 Statistical Field Theory Gibbs Free Energy Generating functional Γ[φ c ] = W [J] d D x J(x)φ c (x) with φ c (x) = δw δφ(x) Proper vertices δ n Γ δφ c (x 1 )...δφ c (x n ) φc= φ +

16 Statistical Field Theory Field Expectation Value Since φ c (x) J=0 = φ(x) and δφ = J(x) c(x) δγ δφ c (x) = 0 φc(x)= φ(x) Analogon to Euler-Lagrange-Equations allowing for fluctuations Solve differential equation get φ(x) δγ

17 Mean Field Profile Starting Point φ ±v 0 as z ±

18 Mean Field Profile Van der Waals Ansatz Free energy density given by H = 1 2 ( φ (z) ) 2 + V (φ(z)) First term: Energy corresponding to fluctuations Second term: Potential stationary at equilibrium values of φ

19 Mean Field Profile Double-Well-Potential V (φ) = m2 0 4 φ2 + g 0 4! φ4 ( + g 0v 4 0 4! ) with v 2 0 = 3m2 0 g 0 φ 4 -Theory in a phase of broken symmetry

20 Mean Field Profile Cahn-Hilliard-Profile Classical equation of motion: δh δφ = 0 2 φ(x) m2 0 2 φ(x) + g 0 3! φ3 (x) = 0 Cahn-Hilliard-Profile ( m0 ) φ(x) = v 0 tanh 2 (z a) Free parameter a: Interface position

21 Mean Field Profile Cahn-Hilliard-Profile Classical equation of motion: δh δφ = 0 2 φ(x) m2 0 2 φ(x) + g 0 3! φ3 (x) = 0 Cahn-Hilliard-Profile ( m0 ) φ(x) = v 0 tanh 2 (z a) Free parameter a: Interface position

22 Mean Field Profile Intrinsic Profile MFA provides intrinsic interface structure Only fluctuations λ < ξ = m 1 are taken into account by CH-Profile

23 Capillary Wave Model Long-Range Fluctuations Complementing the intrinsic profile: Capillary Waves Interface as a sharp discontinuity between phases Deformations against interfacial tension σ (and gravity)

24 Capillary Wave Model CW-Hamiltonian Energy of deviations h from the plane (disregarding gravity): [ ] H = σ dx dy 1 + ( h(x, y)) 2 1 Interested in long-range fluctuations (λ > ξ) ( h) 2 is small Expansion yields: H[h] σ 2 dx dy ( h) 2

25 Capillary Wave Model CW-Hamiltonian Energy of deviations h from the plane (disregarding gravity): [ ] H = σ dx dy 1 + ( h(x, y)) 2 1 Interested in long-range fluctuations (λ > ξ) ( h) 2 is small Expansion yields: H[h] σ 2 dx dy ( h) 2

26 Capillary Wave Model Free Field Theory Periodic boundary conditions on [0, L] [0, L] 2D-Field Theory via Z = Dh exp ( H[h]) Propagator 1 k 2 Interface width: h 2 = 1 2πσ dq 1 q

27 Capillary Wave Model Cut-Offs is IR- and UV-divergent! h 2 = 1 2πσ dq 1 q Lower cut-off: 2π L since λ L 2π Upper cut-off: B cw since small fluctuations where excluded Intrinsic and capillary wave profile should be complementary: B cw ξ

28 Capillary Wave Model Cut-Offs is IR- and UV-divergent! h 2 = 1 2πσ dq 1 q Lower cut-off: 2π L since λ L 2π Upper cut-off: B cw since small fluctuations where excluded Intrinsic and capillary wave profile should be complementary: B cw ξ

29 Capillary Wave Model Cut-Offs is IR- and UV-divergent! h 2 = 1 2πσ dq 1 q Lower cut-off: 2π L since λ L 2π Upper cut-off: B cw since small fluctuations where excluded Intrinsic and capillary wave profile should be complementary: B cw ξ

30 Convolution Approximation Convolution Approximation Intrinsic profile: φ(z) Propability to find interface at h: P(h) Emerging profile: m(z) = dh φ(z h)p(h)

31 Convolution Approximation Interface Width Müller, 2004: Define the width by w 2 = dz z 2 m (z) dz m (z) It follows: w 2 = wint 2 + w cw 2 w 2 = const. ξ ( ) 2πσ ln L const. ξ

32 Convolution Approximation Interface Width Müller, 2004: Define the width by w 2 = dz z 2 m (z) dz m (z) It follows: w 2 = wint 2 + w cw 2 w 2 = const. ξ ( ) 2πσ ln L const. ξ

33 Convolution Approximation CW introduce a dependence on L Profile is infinitely smeared out for L Distinction between intrinsic profile and capillary waves seems artificial All the more the introduction of cut-offs

34 Convolution Approximation CW introduce a dependence on L Profile is infinitely smeared out for L Distinction between intrinsic profile and capillary waves seems artificial All the more the introduction of cut-offs

35 Separation of Fluctuations Holistic Approach Start from φ 4 -Theory H(φ) = 1 2 ( φ(x))2 m2 0 4 φ2 + g 0 4! φ4 Fluctuations about CH-Profile ( ) φ (a) z a 0 (z) = v 0 tanh 2ξ Z = Dϕ exp ( H[φ 0 + ϕ]) ϕ(x) obeys periodic boundary conditions

36 Separation of Fluctuations Holistic Approach Start from φ 4 -Theory H(φ) = 1 2 ( φ(x))2 m2 0 4 φ2 + g 0 4! φ4 Fluctuations about CH-Profile ( ) φ (a) z a 0 (z) = v 0 tanh 2ξ Z = Dϕ exp ( H[φ 0 + ϕ]) ϕ(x) obeys periodic boundary conditions

37 Separation of Fluctuations Holistic Approach Start from φ 4 -Theory H(φ) = 1 2 ( φ(x))2 m2 0 4 φ2 + g 0 4! φ4 Fluctuations about CH-Profile ( ) φ (a) z a 0 (z) = v 0 tanh 2ξ Z = Dϕ exp ( H[φ 0 + ϕ]) ϕ(x) obeys periodic boundary conditions

38 Separation of Fluctuations Hamiltonian H(φ 0 + ϕ) = H(φ 0 ) + 1 ϕ(x)k ϕ(x) + g 0 } 2 {{} 3! φ 0(x)ϕ 3 (x) + g 0 4! ϕ4 (x) }{{} gaussian interaction with the fluctuation operator Two interaction vertices K = m g 0 2 φ2 0 (x) g 0 φ 0 (x) g 0

39 Separation of Fluctuations Interface Translations For obeying it follows φ (a+ε) 0 = φ (a) 0 + δϕ (ε) δh δφ φ=φ (a+ε) 0 K δϕ (ε) = 0 = 0 δϕ (ε) is an eigenvector to eigenvalue 0

40 Separation of Fluctuations Gaussian part (0 th order) of the partition functional would diverge: ( ) 1 Dϕ exp 2 dd xϕ(x)k ϕ(x) = (det K ) 1 2 Choosing an interface position breaks translational invariance Interface has to be fixed to examine its properties

41 Separation of Fluctuations Method of Collective Coordinates Separate interface translation from interesting fluctuations {f m } ONB in function space ϕ = m c mf m Write measure Dϕ m dc m Transform to ONB containing δϕ (ε) Dϕ da dc m

42 Separation of Fluctuations Method of Collective Coordinates Separate interface translation from interesting fluctuations {f m } ONB in function space ϕ = m c mf m Write measure Dϕ m dc m Transform to ONB containing δϕ (ε) Dϕ da dc m

43 Separation of Fluctuations Method of Collective Coordinates Separate interface translation from interesting fluctuations {f m } ONB in function space ϕ = m c mf m Write measure Dϕ m dc m Transform to ONB containing δϕ (ε) Dϕ da dc m

44 Separation of Fluctuations Dϕ da dc m m N Dϕ }{{} Separation works because of translational invariance of H a is called a collective coordinate Done by an insertion of 1 : 1 = dc 0 (a) δ (c 0 (a)) with c 0 (a) = ( ϕ (a), δϕ (ε))

45 Separation of Fluctuations Dϕ da dc m m N Dϕ }{{} Separation works because of translational invariance of H a is called a collective coordinate Done by an insertion of 1 : 1 = dc 0 (a) δ (c 0 (a)) with c 0 (a) = ( ϕ (a), δϕ (ε))

46 One-Loop-Diagram Modified Theory ( Z [J] = Dϕ exp H + N ) d D x J(x)ϕ(x) { Z [J] = N exp { 1 exp 2 [ ( ) d D g 0 δ 3 x 3! φ 0(x) + g ( ) ]} 0 δ 4 δj(x) 4! δj(x) } d D x d D y J(x)K 1 (x, y)j(y) with K 1 (x, y) the kernel of the inverse to K = K N

47 One-Loop-Diagram Modified Theory ( Z [J] = Dϕ exp H + N ) d D x J(x)ϕ(x) { Z [J] = N exp { 1 exp 2 [ ( ) d D g 0 δ 3 x 3! φ 0(x) + g ( ) ]} 0 δ 4 δj(x) 4! δj(x) } d D x d D y J(x)K 1 (x, y)j(y) with K 1 (x, y) the kernel of the inverse to K = K N

48 One-Loop-Diagram 1-Loop-Order Loop-expansion gives ϕ(x) = Alternative: Solve = g 0 2 d D x K 1 (x, x )K 1 (x, x )φ 0 (x ) δγ δφ c(x) φc(x)= φ(x) = 0 to this order

49 One-Loop-Diagram 1-Loop-Order Loop-expansion gives ϕ(x) = Alternative: Solve = g 0 2 d D x K 1 (x, x )K 1 (x, x )φ 0 (x ) δγ δφ c(x) φc(x)= φ(x) = 0 to this order

50 Boundary Conditions and K 1 Spectrum of K ( ) K = x 2 + y 2 }{{} free particle in 2D + [ z 2 m g ( ] 0 2 v 0 2 m ) 0 tanh2 2 z }{{} particle in potential Free particle spectrum determined by boundary conditions Particle in potential -spectrum is also known:

51 Boundary Conditions and K 1 Boundary Conditions of the Free Operator Köpf, 2008: Periodic boundary conditions Eigenfunctions χ n exp(i 2π L n x) with n Z 2 Zero-mode exists combines to zero-mode of K Drees, 2010: Fixed boundary conditions φ(0, y, z) = φ(l, y, z) = φ(x, 0, z) = φ(x, L, z) = v 0 tanh ( m 0 2 z) Eigenfunctions χ n sin ( n 1 π L x) sin ( n 2 π L y) with n N 2 \{0} No zero-mode! Interface is fixed

52 Boundary Conditions and K 1 Boundary Conditions of the Free Operator Köpf, 2008: Periodic boundary conditions Eigenfunctions χ n exp(i 2π L n x) with n Z 2 Zero-mode exists combines to zero-mode of K Drees, 2010: Fixed boundary conditions φ(0, y, z) = φ(l, y, z) = φ(x, 0, z) = φ(x, L, z) = v 0 tanh ( m 0 2 z) Eigenfunctions χ n sin ( n 1 π L x) sin ( n 2 π L y) with n N 2 \{0} No zero-mode! Interface is fixed

53 Boundary Conditions and K 1 Calculation of K 1 Representation in terms of eigenvalues and -functions: K 1 = Ψ λ (x)ψ λ (x) 1 λ while K Ψ λ (x) = λψ λ (x) Periodic boundary conditions: Exclude λ = 0 λ Calculation amounts to evaluation of infinite series and divergent integrals ( renormalization)

54 Boundary Conditions and K 1 Periodic Boundary Conditions Profile calculation done to 1-loop-order δγ solving = 0 by variation of parameters with φ(x) δφ c(x) K 1 = C 0 + (C 2 + C 3 ) sech 2 ( m 0 2 z ) + + (C 1 + C 4 C 2 ) sech 4 ( m 0 2 z )

55 Boundary Conditions and K 1 C 0 = 1 2π dp n C 1 = 3m 0 8 C 2 = 3m 0 4 n 1 4π 2 n 2 + (m p2 )L 2 n 0 1 4π 2 n 2 1 4π 2 n m2 0 L2 C 3 = 3m0 2 1 m0 2 dp + p2 2π(4p 4 + 5p 2 m0 2 + m4 0 }{{ ) 4π 2 n 2 + (m 2 } n 0 + p2 )L 2 C 4 = 9m4 0 dp Np π 2 n 2 + (m0 2 + p2 )L 2 n

56 Boundary Conditions and K 1 Solution after renormalization: g R 8πm R φ R (z) = v R {tanh ( mr 2 z ) + [ 1 ( 2 [α + ln (m mr ) RL)] tanh 2 z η ( mr 2 z )] sech 2 ( m R 2 z )}

57 Boundary Conditions and K 1 Interface Width Width given by resulting profile: ( ) w 2 = const. incl. wint 2 + const. ln (m R L) Includes results of the convolution approximation Especially L-dependence endorses the capillary wave picture

58 Boundary Conditions and K 1 Fixed Boundary Conditions Expression of K 1 in terms of sums already given by Drees Even though no zero-modes: Those sums might bring further complications The (numerical?) evaluation has yet to be done

59 Boundary Conditions and K 1 K 1 = n 1,n 2 >0 m 1,m 2 >0 A n1,2m 1 1A n2,2m 2 1 ( ) ( ) (2m1 1)π (2m2 1)π sin x 1 sin x 2 M n1,n L L 2 with A ni,2m i 1 = 4n 2 i (2m i 1)(4n 2 i (2m i 1) 2 )π { M n1,n 2 = 4 3m0 L 2 L 2 8 sech 4 ( m 0 π 2 n 2 2 z) + + 3m tanh 2 ( m 0 π 2 n 2 L m2 2 z) sech 2 ( m 0 2 z) dp Ψλp (z) } 2 π 2 n 2 L 2 +m2 0 +p2

60 Resume The convolution approximation and φ 4 -theory deliver consistent results In the latter different difficulties concerning boundary conditions arise Next task: Find the profile for fixed boundary conditions

61 Thank you for your attention!

62 Resources M. Köpf, Rauhigkeit von Grenzflächen in der Ising-Universalitätsklasse, Diploma Thesis (2008) M. Köpf and G. Münster, Interfacial roughening in field theory, J. Stat. Phys. 132 (2008) 417 B. Drees, Intrinsische Fluktuationen und Rauigkeit kritischer Grenzflächen, Diploma Thesis (2010) M. Müller, Fluktuationen kritischer Grenzflächen auf verschiedenen Größenskalen, Diploma Thesis (2004) M. Müller and G. Münster, Profile and width of rough interfaces, J. Stat. Phys. 118 (2005) 669.