Effective forces due to confined nearly critical fluctuations

Transcription

1 Effective forces due to confined nearly critical fluctuations H. W. Diehl Fachbereich Physik, U. Duisburg-Essen June 24, 2009 Collaborators: Daniel Grüneberg, U. Duisburg-Essen F. M. Schmidt, U. Duisburg-Essen M. A. Shpot, ICMP Lviv (Ukraine) Work supported in part by: DFG, grant # Die-378/5

2 Casimir Effect in QED July 15, 1909 May 4, : vacuum fluctuations L vacuum bound. cond. 1 conducting plates bound. cond. 2 normal modes of electromagnetic field between plates: ω q = c q = c qx 2 + q2 y + q2 z ; q = (q x, q y, q z = m π/l), m N ground-state energy: E 0 (L) = 1 ω q = C Λ V + CΛ s A + 1,2 QED 2 }{{} (d) ca }{{} L d, D,D π2 QED (3) = 720 q,ν=1,2 infinities universal (fluctuation induced) force: F C (L) = E L = A c L d 1,2 d+1 QED (d)

3 Casimir Effect in QED July 15, 1909 May 4, : vacuum fluctuations L vacuum bound. cond. 1 conducting plates bound. cond. 2 normal modes of electromagnetic field between plates: ω q = c q = c qx 2 + qy 2 + qz; 2 q = (q x, q y, q z = m π/l), m N ground-state energy: E 0 (L) = 1 ω q = C Λ V + CΛ s A + 1,2 QED 2 }{{} (d) ca }{{} L d, D,D π2 QED (3) = 720 q,ν=1,2 infinities universal (fluctuation induced) force: F C (L) = E L = (L/µm) 4 dyn cm 2 A

4 Experimental Verification S. Lamoreaux, PRL 87, 5 (1997); U. Mohideen & A. Roy, PRL 81, 4549 (1998); plates: G. Bressi et al, PRL 99, (2002) polystyrene sphere ( 196 µm) and sapphire plate coated with Au plate-sphere separations from 0.1 to 0.9 µm

5 Experimental Verification S. Lamoreaux, PRL 87, 5 (1997); U. Mohideen & A. Roy, PRL 81, 4549 (1998); plates: G. Bressi et al, PRL 99, (2002) polystyrene sphere ( 196 µm) and sapphire plate coated with Au plate-sphere separations from 0.1 to 0.9 µm solid line: Casimir force for plate-sphere geometry including corrections due to finite conductivity surface roughness finite temperatures Proximity Force (Derjaguin) Approximation

6 Signs: Gaussian Results for Parallel Plates Massless Euclidean field theory in d space dimensions H = d d x 1 2 ( φ)2 R d 1 [0,L] Symanzik 81: Krech & Dietrich 91: (D,D) C = 2 d π d/2 Γ(d/2)ζ(d) < 0 (N,N) C = (D,D) C < 0 (per) C = 2 d (D,D) C < 0 (D,N) C = (2 1 d 1) (D,D) C > 0 (antiper) C = 2 d (D,N) C > 0

7 Signs: Gaussian Results for Parallel Plates Massless Euclidean field theory in d space dimensions H = d d x 1 2 ( φ)2 Symanzik 81: Krech & Dietrich 91: Theorems for attractive CFs: R d 1 [0,L] (D,D) C = 2 d π d/2 Γ(d/2)ζ(d) < 0 (N,N) C = (D,D) C < 0 (per) C = 2 d (D,D) C < 0 (D,N) C = (2 1 d 1) (D,D) C > 0 (antiper) C = 2 d (D,N) C > 0 If A = θ(b) = mirror image of B, then E AB (L)/ L > 0. O. Kenneth & I. Klich, PRL 97 (2006) ( Gauß ) C. P. Bachas: J. Phys. A 49 (2007) 9089 (beyond Gauss: reflection positivity )

8 Important Properties of Casimir Force (QED) Casimir force (QED) is independent of microscopic details ( universal ) depends on gross features of medium: space dimension d, dispersion relation, scalar / vector field, geometry,... boundaries: boundary conditions, geometry, curvature,... usually is described by noninteracting (effective Gaussian) field theory coupling to matter field: only through boundary conditions

9 Important Properties of Casimir Force (QED) Casimir force (QED) is independent of microscopic details ( universal ) depends on gross features of medium: space dimension d, dispersion relation, scalar / vector field, geometry,... boundaries: boundary conditions, geometry, curvature,... usually is described by noninteracting (effective Gaussian) field theory coupling to matter field: only through boundary conditions Interacting Field Theories?

10 Important Properties of Casimir Force (QED) Casimir force (QED) is independent of microscopic details ( universal ) depends on gross features of medium: space dimension d, dispersion relation, scalar / vector field, geometry,... boundaries: boundary conditions, geometry, curvature,... usually is described by noninteracting (effective Gaussian) field theory coupling to matter field: only through boundary conditions Interacting Field Theories? Yes, for condensed matter systems at critical points! space dimension d < 4: Ginzburg criterion fails as T T c!

11 Thermodynamic Casimir Effect pressure 0 0 K solid liquid gas critical point temperature M.E. Fisher & P.-G. de Gennes (1978): large-λ modes massless consider confined nearly critical systems B 1 : area A nearly critical fluid B 2 : area A L partition sum: Z = φ e H[φ] = Dφe H[φ] = exp [ F L,A (T)/k B T] F L,A (T) k B T = LAf bk(t) }{{} bulk contribution + A[f s,1 (T,...) + f s,2 (T,...)] }{{} surface contributions + Af res (T, L,...) }{{} residual

12 Thermodynamic Casimir Effect pressure 0 0 K solid liquid gas critical point temperature F L,A (T) k B T = LAf bk(t) {z } bulk contribution M.E. Fisher & P.-G. de Gennes (1978): large-λ modes massless consider confined nearly critical systems B 1 : area A nearly critical fluid B 2 : area A + A[f s,1(t,...) + f s,2(t,...)] {z } surface contributions L + Af res(t,l,...) {z } residual Casimir force per area: F C (T, L,...)/A = k B T fres L finite size scaling (only short-range interactions): f res(t,l,...) 1 L d 1 Y {z} universal (L/ξ,...) at T c, : f res 1 L d 1 C {z} Casimir amplitude

13 Experimentall Verifications indirect: wetting experiments Garcia & Chan (Helium) Fukuto, Yano & Pershan (binary liquids) Rafai, Bonn & Meunier (binary liquids) direct: Hertlein, Helden, Gambassi, Dietrich & Bechinger (binary liquids): Nature 451, 172 (2008)

14 Experimentall Verification: 4 He wetting films copper plate He vapor L liquid He mgh }{{} gravitation = γ vdw L 3 }{{} van der Waals L/L 1/2 }{{} retardation + v k B T Ξ(L/ξ) L 3 }{{} Casimir

15 Thinning of 4 He films near T λ Experiment: Garcia & Chan, PRL 83, 1187 (1999) Theory: still in somewhat modest state Krech & Dietrich 1991/92: T T λ Li & Kardar 1991: T T λ (Goldstone modes) Zandi, Rudnick & Kardar 2004: interface fluctuations Monte Carlo simulations: A. Hucht (PRL 2007); Vasilyev, Gambassi, Macio lek & Dietrich (EPL 2007); see Poster 103

16 Thinning of 4 He films near T λ Experiment: Garcia & Chan, PRL 83, 1187 (1999) Theory: still in somewhat modest state Krech & Dietrich 1991/92: T T λ Li & Kardar 1991: T T λ (Goldstone modes) Zandi, Rudnick & Kardar 2004: interface fluctuations Monte Carlo simulations: A. Hucht (PRL 2007); Vasilyev, Gambassi, Macio lek & Dietrich (EPL 2007); see Poster 103

17 Thinning of 4 He films near T λ Experiment: Garcia & Chan, PRL 83, 1187 (1999) Theory: still in somewhat modest state Krech & Dietrich 1991/92: T T λ Li & Kardar 1991: T T λ (Goldstone modes) Zandi, Rudnick & Kardar 2004: interface fluctuations Monte Carlo simulations: A. Hucht (PRL 2007); Vasilyev, Gambassi, Macio lek & Dietrich (EPL 2007); see Poster 103

18 Monte Carlo Results A. Hucht (U. Duisburg-Essen): PRL 99, (2007) cf. Poster 103, Vasilyev et al

19 Relevant Issues bulk critical behavior at T c, as L confined critical fluctuations, boundary field theory finite size and boundary effects pseudo-critical or critical behavior in slab at T c (L) < T c, when L < dimensional crossover low-t Casimir force from confined Goldstone modes interface fluctuations

20 Relevant Issues HERE: Restriction to a) disordered phases, b) T T c,, c) generic non symmetry breaking boundary conditions focus directly on fluctuation-induced forces! Synopsis: (i) breakdown of ǫ = 4 d expansion at T c, for some boundary conditions; fractional powers ǫ k/2 mod ln m ǫ (ii) boundary conditions = scale-dependent properties! (iii) Neumann boundary condition L Dirichlet bc! (iv) crossover attractive repulsive attractive Casimir forces

21 Lattice and Continuum Models free bc K K 1 X H lat = K s is j i,j / B 1 B 2 X K 1 s is j K 2 i,j B 1 X i,j B 2 s is j K 2 free bc B 1 B 2 n n V Z H[φ] = V» d d 1 x 2 ( φ)2 + τ 2 φ2 + ů 4! φ4 + c j = 1 2(d 1)[K j/k 1]/a 2X Z c j d d 1 r φ 2 2 B j j=1

22 Lattice and Continuum Models free bc K K 1 X H lat = K s is j i,j / B 1 B 2 X K 1 s is j K 2 i,j B 1 X i,j B 2 s is j B 1 K 2 free bc Z n H[φ] = V V» 1 d d x 2 ( φ)2 + τ 2 φ2 + ů 4! φ4 + 2X Z c j d d 1 r φ 2 2 B j j=1 B 2 n c j = 1 2(d 1)[K j/k 1]/a (fluctuating) boundary condition: nφ = c j φ ( : Dirichlet c j = 0 : Neumann

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30 Boundary Conditions? Universality various scales 1 microscopic: lattice (a min = a) free bc; K j K in general 2 mesoscopic: continuum theory (a min π/λ) nφ = c j φ; d = 3 : c j = Dirichlet c j a = 1 4(K j /K 1) (nonuniversal fctn) c j = 0 Neumann K j /K = 5/4 c > 0 subcritical! c j = c sp (MC, d = 3) K j /K 1.5

31 Boundary Conditions? Universality various scales 1 microscopic: lattice (a min = a) free bc; K j K in general 2 mesoscopic: continuum theory (a min π/λ) nφ = c j φ; d = 3 : c j = Dirichlet c j a = 1 4(K j /K 1) (nonuniversal fctn) c j = 0 Neumann K j /K = 5/4 c > 0 subcritical! c j = c sp (MC, d = 3) K j /K large length scales: a z ξ a) c > 0 (subcritical, ordinary trans.): φ(x) C ord(z j) nφ(x j) Dirichlet! {z} {z } {z } ξ β/ν z (βord 1 β)/ν ξ βord 1 /ν

32 Boundary Conditions? Universality various scales 1 microscopic: lattice (a min = a) free bc; K j K in general 2 mesoscopic: continuum theory (a min π/λ) nφ = c j φ; d = 3 : c j = Dirichlet c j a = 1 4(K j /K 1) (nonuniversal fctn) c j = 0 Neumann K j /K = 5/4 c > 0 subcritical! c j = c sp (MC, d = 3) K j /K large length scales: a z ξ a) c > 0 (subcritical, ordinary trans.): φ(x) C ord(z j) nφ(x j) Dirichlet! {z} {z } {z } ξ β/ν z (βord 1 β)/ν ξ βord 1 /ν b) c = 0 (critical, special transition): φ(x) C sp(z j) φ Bj {z } z (β βsp 1 )/ν j c) Neumann bc: no (stable or unstable) fixed pt associated!

33 Previous Field Theory RG Results RG analysis in d = 4 ǫ dimensions: Symanzik 1981: (bc) C (ǫ,n)/n = a(bc) 0 + a (bc) 1 (n)ǫ for Dirichlet-Dirichlet (D-D) boundary conditions ( c 1 = c 2 = ) Krech & Dietrich 1991, 1992: (bc) C (ǫ,n) and Y (bc) for T T c, to O(ǫ) free bc z } { for bc = periodic, antiperiodic, D-D, D-sp, sp-sp (bc) C ( < 0 bc = per, D-D, sp-sp > 0 bc = aper, D-sp c = 0 c sp = c sp(λ) 0 sp = special = critically enhanced

34 Problems: for periodic and sp-sp bc Θ (per) (L)/n n = n = 2 n = 3 n d = 3 n (exact) L/ξ L Comparsion of O(ǫ) extrapolation (Krech & Dietrich) with exact d = 3 spherical model result (Dantchev) for periodic BC

35 Problems: for periodic and sp-sp bc Loop expansion: F k B T = + + O(3-loops) Z d d 1 p X (2π) d 1 = G (bc) L (x;x ) = m z m m z eip (r r ) p 2 + km 2 + τ periodic and sp-sp BC involve k 0 = 0 zero mode at T c,! theory ill-defined beyond 2 loops! = = infrared singular at T c,

36 Problems: for periodic and sp-sp bc Loop expansion: F k B T = + + O(3-loops) Z d d 1 p X (2π) d 1 = G (bc) L (x;x ) = m z m m z eip (r r ) p 2 + km 2 + τ periodic and sp-sp BC involve k 0 = 0 zero mode at T c,! theory ill-defined beyond 2 loops! = = infrared singular at T c, remedy: φ(r,z) = ϕ(r) +ψ(r,z) ; e Heff[ϕ] = Tr ψ e H[φ] {z} {z } k m=0 k m 0 ϕ ϕ gives shift τ τ (L) bc = τ + δ τ (L) bc HWD, D. Grüneberg & M.A. Shpot: EPL 75, 241 (2006); D. Grüneberg & HWD, PRB 77, (2008)

37 Extrapolated ǫ-expansion Results: periodic boundary conditions 0 0 Ä ½ Ü Ò n = 1 (Ising), d = Ä ½ Ä ½ Ü Ò n = 2 (XY), d = Ä ½ Krech & Dietrich 91 Grüneberg & HWD: PRB 77, (2008) cond-mat/ per C n = π π2 ǫ 180 π2 9 6» 1 γ lnπ + 2ζ (4) n + 2 n + 8 «3/2 ǫ 3/2 + O(ǫ 2 ) ζ(4) + 5 n n + 8 cf. Poster 103 by Vasilyev et al for comparison with Monte Carlo results!

38 Two-Loop Calculations: 0 c j < F Loop expansion: k B T = + + O(3-loops) Z = G (bc) d L (x;x d 1 p X z m m z ) = eip (r r ) (2π) d 1 p 2 + km 2 + τ 2 z m = k 2 m m z m = A m(l, c 1, c 2)cos[k m(l, c 1, c 2)z + ϑ m(k m, c 1)] k m from bc: R(k m;l, c 1, c 2)! = 0 transcendental equation! m

39 Two-Loop Calculations: 0 c j < F Loop expansion: k B T = + + O(3-loops) Z = G (bc) d L (x;x d 1 p X z m m z ) = eip (r r ) (2π) d 1 p 2 + km 2 + τ 2 z m = k 2 m m z m = A m(l, c 1, c 2)cos[k m(l, c 1, c 2)z + ϑ m(k m, c 1)] k m from bc: R(k m;l, c 1, c 2)! = 0 transcendental equation! evaluation of sums: complex integration (Abel-Plana techniques) g(k m )(km 2 + b) a = Res[g(k)(k 2 + b) a f (k)] k=km m m dz = 2πi g(k)(k2 + b) a f (k) m C 1 deform contour

40 Scaling Function: Residual Free Energy f res (L)/n L (d 1) D ( c 1 L Φ/ν, c 2 L Φ/ν) D(c1, c2) c 1 c c 2 c 2+1

41 Scaling Function: Residual Free Energy f res (L)/n L (d 1) D ( c 1 L Φ/ν, c 2 L Φ/ν) c 2 c c 1 c 1+1

42 Scaling Function: Critical Casimir Force F C nak B T L d D ( c 1 L Φ/ν, c 2 L Φ/ν) D(c1, c2) c 1 c c 2 c 2+1 D(c 1, c 2 ) = [ d 1 + (Φ/ν) ( c 1 c1 + c 2 c2 )] D(c1, c 2 )

43 Scaling Function: Critical Casimir Force F C nak B T L d D ( c 1 L Φ/ν, c 2 L Φ/ν) c2 c c1 c1+1 D(c 1, c 2 ) = [ d 1 + (Φ/ν) ( c 1 c1 + c 2 c2 )] D(c1, c 2 )

44 Crossover repulsive attractive 0.02 D(c1L Φ/ν, c2l Φ/ν ) (c 1, c 2 ) = (0, 0.1) (c 1, c 2 ) = (10, 0.1) L Scaled Casimir force F C nak B T Ld = D ( c 1 L Φ/ν, c 2 L Φ/ν) F. M. Schmidt & HWD: PRL 101, (2008); arxiv: , and to be published

45 Summary / Conclusions breakdown of ǫ = 4 d expansion for BC involving zero modes! BC = scale dependent properties Neumann BC does not correspond to fixed point! Casimir force can be attractive or repulsive Crossover attractive repulsive Casimir forces Observable? in Monte Carlo simulations? yes! in experiments? requires tuning of K js and non symmetry breaking boundaries

46 Summary / Conclusions breakdown of ǫ = 4 d expansion for BC involving zero modes! BC = scale dependent properties Neumann BC does not correspond to fixed point! Casimir force can be attractive or repulsive Crossover attractive repulsive Casimir forces Observable? in Monte Carlo simulations? yes! in experiments? requires tuning of K js and non symmetry breaking boundaries Analogous sign-changing crossover for binary liquid mixtures can use different choices of mixtures (varying h j) can use different choices of substrates h js can be changed by chemically modifying the surfaces

47 Summary / Conclusions breakdown of ǫ = 4 d expansion for BC involving zero modes! BC = scale dependent properties Neumann BC does not correspond to fixed point! Casimir force can be attractive or repulsive Crossover attractive repulsive Casimir forces Observable? in Monte Carlo simulations? yes! in experiments? requires tuning of K js and non symmetry breaking boundaries Analogous sign-changing crossover for binary liquid mixtures can use different choices of mixtures (varying h j) can use different choices of substrates h js can be changed by chemically modifying the surfaces effect of quadratic symmetry breaking boundary terms: HWD & D. Grüneberg, arxiv:0905:3113