Fluctuation-induced forces near critical points

Transcription

1 Fluctuation-induced forces near critical points From Casimir to anisotropic critical systems H. W. Diehl Fakultät für Physik, U. Duisburg-Essen October 20, 2010 Third International Conference Models in Quantum Field Theory dedicated to A. N. Vassiliev Collaborators: Matthias Burgsmüller, U. Duisburg-Essen Felix M. Schmidt, U. Duisburg-Essen Mykola A. Shpot, ICMP Lviv (Ukraine) Work supported in part by: DFG, grant # Die-378/5

2 Fluctuation-induced forces effective force F 12 induced by fluctuations in medium examples: van der Waals Casimir, Casimir-Polder, dispersion forces, (near-)critical Casimir forces critical Casimir forces in anisotropic critical systems

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: q ω q,ν = 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 X ω q,ν = C Λ V + CΛ s A+ π2 2 {z } 720 q,ν=1,2 infinities {z } universal BC c L A L 2 (fluctuation induced) force: F C (L) = E0 L = π2 240 c L 4 A

4 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: q ω q,ν = 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 X ω q,ν = C Λ V + CΛ s A+ π2 2 {z } 720 q,ν=1,2 infinities {z } universal BC (fluctuation induced) force: c L A L 2 F C (L) = E0 L = (L/µm) 4 dyn cm 2 A

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

6 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

7 Microelectromechanical Systems (MEMS) Attractive Casimir forces may cause STICTION. Interest in repulsive Casimir forces.

8 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: (D,D) = 2 d π d/2 Γ(d/2)ζ(d) < 0 Krech & Dietrich 91: (N,N) = (D,D) < 0 (PBC) = 2 d (D,D) < 0 (D,N) = (2 1 d 1) (D,D) > 0 (APB) = 2 d (D,N) > 0 Theorems for attractive CFs: Kenneth & I. Klich 2007 ( Gauss ) Bachas 2007 (beyond Gauss: reflection positivity )

9 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 F L,A (T) k B T = LAf b(t) {z } bulk contribution + A[f s,1(t,...) + f s,2(t,...)] {z } surface contributions + Af res(t,l,...) {z } residual

10 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 F L,A (T) k B T = LAf b(t) {z } bulk contribution + A[f s,1(t,...) + f s,2(t,...)] {z } surface contributions + Af res(t,l,...) {z } residual Casimir force per area: F(T, L,...)/A = k B T fres L finite size scaling (only short-range interactions): f res(t,l,...) 1 L d 1 Θ {z} universal (L/ξ,...) at T c, : f res 1 L d 1 {z} Casimir amplitude

11 Universality Casimir amplitudes and scaling functions Θ are universal, depend on: 1 gross features of the medium (bulk universality class) 2 gross features of boundaries (surface universality classes) 3 geometry

12 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 c 1 n V n c 2 H = Z R d 1 [0,L] d d x L b {z} bulk density + 2X Z da L j j=1 B j {z } boundary planes z = 0 and z = L L b = 1 2 ( φ)2 + τ 2 φ2 + ů 4! φ4 ; L j = 1 cj φ2 2 mesoscopic BC: n φ = c j φ

13 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 c 1,h 1 n V n c 2,h 2 H = Z R d 1 [0,L] d d x L b {z} bulk density + 2X Z da L j j=1 B j {z } boundary planes z = 0 and z = L L b = 1 2 ( φ)2 + τ 2 φ2 + ů 4! φ4 ; L j = 1 2 cj φ2 h jφ mesoscopic BC: n φ = c j φ h j

14 Standard φ 4 model for slab B 1 B 2 c 1 n V n c 2 H = Z R d 1 [0,L] d d x L b {z} bulk density + 2X Z da L j j=1 B j {z } boundary planes z = 0 and z = L L b = 1 2 ( φ)2 + τ 2 φ2 + ů 4! φ4 ; L j = 1 cj φ2 2 mesoscopic BC: n φ = c j φ

15 Standard φ 4 model for slab B 1 B 2 c 1 n V n c 2 H = Z R d 1 [0,L] d d x L b {z} bulk density + 2X Z da L j j=1 B j {z } boundary planes z = 0 and z = L L b = 1 2 ( φ)2 + τ 2 φ2 + ů 4! φ4 ; L j = 1 cj φ2 2 mesoscopic BC: n φ = c j φ, c j c j c sp SD/BD T SD/BD Dirichlet SO/BD c j = c j SO/BO c j = 0 c sp SO/BO enhancement c j =

16 Standard φ 4 model for slab B 1 B 2 c 1 n V n c 2 H = Z R d 1 [0,L] d d x L b {z} bulk density + 2X Z da L j j=1 B j {z } boundary planes z = 0 and z = L L b = 1 2 ( φ)2 + τ 2 φ2 + ů 4! φ4 ; L j = 1 cj φ2 2 mesoscopic BC: n φ = c j φ, c j c j c sp SD/BD T SD/BD Dirichlet Neumann c j = 0 SO/BD c j = c j SO/BO c j = 0 c sp SO/BO enhancement c j =

17 Consequences T Dirichlet Neumann c j = 0 SO/BD c j = c j SO/BO c j = 0 c sp SO/BO enhancement c j = BC = scale-dependent! assume c 1, c 2 0 (non-supercritical enhancement), then: microscopic BC: free mesoscopic BC: Robin BC nφ = c jφ large-scale asymptotic BC: D-D, D-sp, sp-sp (sp = special, critically enhanced)

18 Consequences T Dirichlet Neumann c j = 0 SO/BD c j = c j SO/BO c j = 0 c sp SO/BO enhancement c j = BC = scale-dependent! assume c 1, c 2 0 (non-supercritical enhancement), then: microscopic BC: free mesoscopic BC: Robin BC nφ = c jφ large-scale asymptotic BC: D-D, D-sp, sp-sp (sp = special, critically enhanced) f res,crit(l)/n D`c 1L Φ/ν,c 2L Φ/ν L (d 1) Φ/ν 0.8

19 Scaling Function: Critical Casimir Force F nak B T L d D ( c 1 L Φ/ν, c 2 L Φ/ν) Φ/ν 0.8 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 )

20 Scaling Function: Critical Casimir Force F nak B T L d D ( c 1 L Φ/ν, c 2 L Φ/ν) 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 )

21 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 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

22 Universality Casimir amplitudes and scaling functions Θ are universal, depend on: 1 gross features of the medium (bulk universality class) 2 gross features of boundaries (surface universality classes) surface coupling subcritically enhanced: ordinary (O) transition surface coupling critically enhanced: special (sp) transition fixed boundary magnetizations / h 1 = ±, h 2 = ± 3 geometry (O,O), (O,sp), (sp,sp), (+,+), (+, ), PBC, APB

23 Universality Casimir amplitudes and scaling functions Θ are universal, depend on: 1 gross features of the medium (bulk universality class) 2 gross features of boundaries (surface universality classes) surface coupling subcritically enhanced: ordinary (O) transition surface coupling critically enhanced: special (sp) transition fixed boundary magnetizations / h 1 = ±, h 2 = ± 3 geometry (O,O), (O,sp), (sp,sp), (+,+), (+, ), PBC, APB experimental verifications: a) indirect: thinning of 4 He films (Garcia & Chan 1999) b) direct measurements (binary fluids): Hertlein et al 2008 theoretical results: many, distinct methods Nightingale & Indekeu, Krech & Dietrich 1990,..., HWD s group,..., Hucht, Vasilyev et al, Hasenbusch,..., Abrahams, Maciolek, Dantchev, Dohm,...,Kardar et al,... reviews: M. Krech: Casimir effect in Critical Systems (World Scientific, 1994); Brankov, Tonchev, & Dantchev (World Scientific, 2000); Gabassi 2009

24 Isotropic versus strongly anisotropic critical behavior isotropic strongly anisotropic ξ = ξ (0) ± τ ν ν α ν β, θ = ν α /ν β 1 ξ α = ξ (0) α,± τ να ξ β = ξ (0) β,± τ ν β

25 Isotropic versus strongly anisotropic critical behavior isotropic strongly anisotropic ξ = ξ (0) ± τ ν ν α ν β, θ = ν α /ν β 1 ξ α = ξ (0) α,± τ να ξ β = ξ (0) β,± τ ν β

26 Isotropic versus strongly anisotropic critical behavior isotropic strongly anisotropic ξ = ξ (0) ± τ ν ν α ν β, θ = ν α /ν β 1 ξ α = ξ (0) α,± τ να ξ β = ξ (0) β,± τ ν β f res,crit BC CP L (d 1) f res,crit E }{{} nonuniversal BC L ζ

27 φ 4 model for m-axial Lifshitz point Z H b = h 1 d d x 2 dx β=m+1 ( β φ) 2 + ρ 2 mx ( αφ) 2 + σ 2 α=1 mx ( αφ) τ 2 φ2 + ů 4! φ4i α=1 LP in Landau theory at τ = ρ = 0. Full two-loop RG analysis in d = 4 + m 2 ǫ dimensions: HWD & MA Shpot 2000, MA Shpot & HWD 2001; HWD, MA Shpot & RKP Zia 2003; work on boundary critical behavior: HWD with Gerwinski, Rutkevic, Shpot, Prudnikov

28 Orientation Matters! 2 fundamentally distinct orientations ς: parallel ( ) perpendicular ( ) y 1 y 2 y 1 y 2 L n n L n n z = β-direction z = α-direction z l 1 H = V dv L b + δ BC,free z l θ da L, s B 1 B 2 1 BC 2 BC large-scale BC O φ = 0 φ = 0, n φ = 0

29 Methods of Investigation 1 parallel orientation: Gauß theory (1-loop) for general d,m RG-improved perturbation theory in d = 4 + m 2 ǫ dimensions (2-loops, O(ǫ)) PBC: modified RG-improved perturbation theory (zero-mode): e Heff[ϕ] = exp( H[ ϕ + ψ ]) {z} {z} q=0 modes q 0 modes CP case: HWD, Grüneberg &Shpot 2006, Grüneberg & HWD 2008, 2009 n limit (PBC) 2 perpendicular orientation: Gauß theory (1-loop), general d,m, for (O,O) and PBC PBC: modified RG-improved perturbation theory in d = 4 + m 2 ǫ dimensions (2-loops), failed n limit (PBC) M. Burgsmüller, HWD & MA Shpot, Jstat (to appear); arxiv:

30 General Results: Non-/Universality fres;lp BC (L; σ, µ) L nonuniv. µ m(1 2θ) 2 ( µ (d m)(2θ 1) 2θ {}}{ E σ σ) m 4 BC L ζ, ς =, ( Eσ σ) d m 4θ BC }{{} L ζ, ς =, nonuniv. ζ = d m + θ m 1 ζ = d m θ + m 1. BC, definable as universal ratios! M. Burgsmüller, HWD & MA Shpot, Jstat (to appear); arxiv:

31 Results: Orientation 1 Gauß: BC = PBC and (O, O) BC,G(d,m,n) = C m BC CP,G(d m 2,n), Cm = π(2 m)/4 2 m Γ( m+2 4 ) (1) PBC,G (d,m,n) = 2 d m/2 (O,O),G (d,m, n) = n Cm Γ( d 2 m 4 ) ζ(d m 2 ) π (d m/2)/2 2 small-ǫ expansion: (1) prevails to calculated order (O,O) /n = C m[a 0 + a 1(n)ǫ] + O(ǫ 2 ) PBC /n = C m[b 0 + b 1(n)ǫ + b 3/2 (n)ǫ 3/2 ] + o(ǫ 3/2 ) 3 large-n limit: (1) prevails lim n PBC,(O,O) /n PBC,(O,O), PBC: consistency with small-ǫ expansion! (d, m) = C }{{} m PBC,(O,O) CP, (d m 2 ) }{{} 3,1 5/2

32 Results: Orientation d = 3, m = n = 1 PBC,G = PBC, = (n ) PBC,small-ǫ = (O,O),G = (O,O),small-ǫ = Critical Casimir force is attractive!

33 Orientation, PBC,G (d, m, n)/n = Γ(d m 22d 2m+1 2 )ζ(2d m) π (d 1)/2 Γ( m d 1 PBC PBC,G (3, 1, 1) = +12ζ(5) = π PBC, (3, 1, 1) (n ) Critical Casimir force is repulsive! 2 )

34 Orientation, (O,O) BC Y Y φ = 0, n φ = 0 Tr φ e H b[φ] δ(φ)δ( nφ) δ(φ)δ( nφ) x 1 B 1 x 2 B 2 φ = 0, n φ = 0 use Fadeev-Popov trick, introduce 4 n-component auxiliary fields χ 1,..., χ 4 integrate out φ lim A F G Ak B T = n 2 Z d d 1 p ln det M(p) (2π) d 1 {z } 4 4 (O,O),G (d, 1, n)/n = K d 12 1 d dq q 2d 3 ln[2e q (coshq + cosq 2)] 0 (O,O),G (3, 1, 1) Critical Casimir force is attractive!

35 Summary & Conclusions fluctuation-induced critical forces depend on type of orientation! distinct ζ, and BC,! universal ratios BC, definable! attraction and repulsion possible! simulation results for ANNNI model desirable experiments?

36 Summary & Conclusions fluctuation-induced critical forces depend on type of orientation! distinct ζ, and BC,! universal ratios BC, definable! attraction and repulsion possible! simulation results for ANNNI model desirable experiments? Thank you for your attention!