Phase separation, interfaces and wetting in two dimensions. Exact - PowerPoint PPT Presentation

Mathematical Statistical Physics 29 July - 3 August 2013, YITP, Kyoto Phase separation, interfaces and wetting in two dimensions. Exact results from field theory Gesualdo Delfino SISSA, Trieste

Based on : GD, J. Viti, Phase separation and interface structure in two dimensions from field theory, J. Stat. Mech. (2012) P10009 [arXiv:1206.4959] GD, A. Squarcini, Interfaces and wetting transition on the half plane. Exact results from field theory, J. Stat. Mech. (2013) P05010 [arXiv:1303.1938] GD, A. Squarcini, Multiple interfaces, to appear

phase separation classical topic of statistical mechanics empha- sizing role of boundary conditions and notion of interface − − − − − − − + + + + + + + − − − + + − − − + + − − + + − + + + + + − + − + + − − + − − + + + − + + + + + − − − − − − + − − − − + + − − + − + − + − − − − − + − + + + + + − − − − − − − + + + + + + + exact analytic results for bulk magnetization have been available for 2D Ising issues in 2D: • general results • role of integrability • universality answers provided by field theory

Pure phases and kinks ferromagnet with spin σ taking discrete values, and 2nd order transition at T c scaling limit ↔ Euclidean field theory Ω 2 below T c : degenerate vacua | Ω a � Κ 12 Ω Κ 1 23 elementary excitations in 2D : kinks | K ab ( θ ) � connecting Ω 3 | Ω a � and | Ω b � ( e, p ) = ( m ab cosh θ, m ab sinh θ ) | Ω a � , | Ω b � non-adjacent if connected by | K ac 1 ( θ 1 ) K c 1 c 2 ( θ 2 ) . . . K c j − 1 b ( θ j ) � with j > 1 only a lim R →∞ : pure phase a � σ � a ≡ � Ω a | σ ( x, y ) | Ω a � R a

Phase separation (adjacent phases) y a b surface tension : R/2 R ln Z ab ( R ) 0 x Σ ab = − lim R →∞ 1 Z a ( R ) −R/2 a b boundary states : � | K ac K cb � + . . . 2 H �� dθ = e ± R | B ab ( ± R � 2 ) � = 2 π f ( θ ) | K ab ( θ ) � + � c a b � | K ac K ca � + . . . ] = e ± R 2 H [ | Ω a � + � | B a ( ± R 2 ) � = c a | f (0) | 2  Z ab ( R ) = � B ab ( R 2 ) | B ab ( − R 2 πm ab R e − m ab R √ 2 ) � ∼     = ⇒ Σ ab = m ab  Z a ( R ) = � B a ( R 2 ) | B a ( − R  2 ) � ∼ � Ω a | Ω a � = 1  

magnetization profile : Z ab � B ab ( R 1 2 ) | σ ( x, 0) | B ab ( − R � σ ( x, 0) � ab = 2 ) � θ 12 ≡ θ 1 − θ 2 θ 2 θ 2 ∼ | f (0) | 2 � dθ 1 1 2 dθ 2 ab ( θ 1 | θ 2 ) e − m [(1+ 4 + 4 ) R − iθ 12 x ] 2 π F σ mR ≫ 1 Z ab 2 π a b a b F σ ab ( θ 1 | θ 2 ) ≡ � K ab ( θ 1 ) | σ (0 , 0) | K ab ( θ 2 ) � a b σ σ σ = + = i � σ � a −� σ � b + � ∞ n =0 c n θ n 12 + 2 π δ ( θ 12 ) � σ � a θ 12 − iǫ [Berg, Karowski, Weisz, ’78; Smirnov, 80’s; GD, Cardy, ’98] Does not require integrability � 2 m � σ ( x, 0) � ab = 1 2 [ � σ � a + � σ � b ] − 1 2 [ � σ � a − � σ � b ] erf( R x ) ⇒ � z πmR e − 2 mx 2 /R + . . . 0 dt e − t 2 � 2 2 + c 0 erf ( z ) ≡ √ π

� 2 m � σ ( x, 0) � ab = 1 2 [ � σ � a + � σ � b ] − 1 2 [ � σ � a − � σ � b ] erf( R x ) πmR e − 2 mx 2 /R + . . . � 2 + c 0 � Ising: � σ � + = −� σ � − , c 0 = 0 (by parity); 2 m � σ � − + ∼ � σ � + erf( R x ) matches lattice result [Abraham, ’81] q-state Potts: 1.0 —– � σ 1 � 12 /M q = 3 0.8 σ c ( x ) = δ s ( x ) ,c − 1 /q , c = 1 , . . . , q 0.6 —– � σ 3 � 12 /M mR = 10 � σ c � a = ( qδ ac − 1) M 0.4 q − 1 0.2 c ab,c = [2 − q ( δ ac + δ bc )] B ( q ) 0 mx � 10 � 5 5 10 � 0.2 M M B (3) = 3 , B (4) = √ √ 4 3 3 � 0.4 • non-local (erf) term amounts to sharp separation between pure phases • local (gaussian) term sensitive to interface structure

percolation: sites randomly occupied with probability p on the plane: infinite cluster for p > p c P =prob. site ∈ infinite cluster maps on q → 1 Potts on the strip, take only configurations s without clusters connecting left 0 x and right parts of the boundary P s ( x, 0)=prob. ( x, 0) ∈ cluster spanning at x < 0 ( p > p c ) � � πmR e − 2 mx 2 /R + · · · � 2 m � = P 2 1 − erf( R x ) − γ 2

Passage probability and interface structure � + ∞ a b � σ ( x, 0) � ab = −∞ du σ ab ( x | u ) p ( u ) . . x u p ( u ) du = passage probability in ( u, u + du ) a b σ ab ( x | u ) = Θ( u − x ) � σ � a +Θ( x − u ) � σ � b + A 0 δ ( x − u )+ A 1 δ ′ ( x − u )+ . . . � 1 , x ≥ 0 Θ( x ) ≡ 0 , x < 0 πR e − 2 mu 2 /R , � 2 m A 0 = c 0 matches field theory for p ( u ) = m • local terms account for branching c a b

for y � = 0 field theory leads to πR e − 2 mu 2 /Rρ 2 ( y ) � 2 m 1 ∀ | y | < R p ( u ; y ) = 2 as R → ∞ ρ ( y ) � 1 − ( y R/ 2 ) 2 ρ ( y ) = = ⇒ the interface behaves as a brownian bridge • brownian bridge property rigorously known for Ising and Potts [Greenberg, Joffe, ’05; Campanino, Joffe, Velenik, ’08] • field theory says that it holds for any interface between adja- cent phases

Wetting is the ability of a phase to maintain contact with a surface phenomenological description in terms of contact angle θ c 0 < θ c < π : partial wetting θ c = 0 : complete wetting equilibrium condition at contact points known as Young’s law

half plane : y B a = boundary condition at x = 0 breaking the symmetry in direction a 0 x � σ ( x, y ) � B a = B a � Ω | σ ( x, y ) | Ω � B a → � σ � a , x → ∞ H B a | Ω � B a = E B | Ω � B a H B a | K ba ( θ ) � B a = ( E B + m cosh θ ) | K ba ( θ ) � B a boundary condition changing fields : µ ab ( y ) switches from B a to B b a θ B a � Ω | µ ab ( y ) | K ba ( θ ) � B a = e − ym cosh θ F µ ( θ ) µ (y) ab b F µ ( θ ) = a θ + O ( θ 2 )

pinned interfaces : µ (R/2) Z B aba = B a � Ω | µ ab ( R 2 ) µ ba ( − R a 2 ) | Ω � B a ab b | a | 2 e − mR � dθ 2 π |F µ ( θ ) | 2 e − mR cosh θ ∼ ∼ √ 2 π ( mR ) 3 / 2 2 µ (−R/2) ba � σ ( x, 0) � B aba = Z − 1 B aba B a � Ω | µ ab ( R 2 ) σ ( x, 0) µ ba ( − R 2 ) | Ω � B a θ 2 θ 2 � dθ 1 1 2 dθ 2 µ ( θ 2 ) e − m [(1+ 4 + 4 ) R − iθ 12 x ] 1 2 π F µ ( θ 1 ) F σ ab ( θ 1 | θ 2 ) F ∗ ∼ Z Baba 2 π � πR x e − 2 m R x 2 � � 2 m � 8 m ∼ � σ � b +( � σ � a −� σ � b ) erf( R x ) − mR, mx ≫ 1 , √ � � σ � a , x → ∞ � σ ( x, 0) � B aba → wall-interface distance ∼ R � σ � b , R → ∞ Ising: � σ � + = −� σ � − ; matches lattice result [Abraham, ’80]

passage probability : � x � ∞ � σ ( x, 0) � B aba ∼ � σ � a 0 du p ( u ) + � σ � b du p ( u ) , mx ≫ 1 x matches field theory for p � x � —– mR=2500 0.020 � 3 / 2 x 2 e − 2 mx 2 /R 4 � 2 m —– mR=5000 p ( x ) = √ π R 0.015 0.010 0.005 general result provided : mx 50 100 150 200 i) adjacent phases ii) no boundary bound states

boundary bound states : b b g a a kink-boundary amplitude has pole at θ = iu θ ∼ iu g b u b | K ba ( θ ≃ iu ) � B a ∼ | Ω � B ′ a µ a µ E B ′ = E B + m cos u , 0 < u < π ab ab a b g ig F µ ( θ ≃ iu ) ≃ θ − iu B a � Ω | µ ab (0) | Ω � B ′ b a | Ω � B ′ a now leading as R → ∞ µ (R/2) 2 e − mR cos u ab � � Z B aba ∼ � B a � Ω | µ ab (0) | Ω � B ′ � � a � a µ a + O ( e − mR (1 − cos u ) ) (−R/2) � σ ( x, 0) � B aba = � σ ( x, 0) � B ′ ba � σ ( x ≫ 1 ⇒ mR diverges faster than 1 /u 2 m , 0) � B aba → � σ � a ∀ R

field theory ← → wetting phenomenology dictionary : splitting and recombination of B ′ a ← → partial wetting u = contact angle a b E B ′ = E B + m cos u ← → Young’s condition u m (cos u − 1) = spreading coefficient u = 0 ← → complete wetting � dy φ (0 , y ) boundary interaction : u = u ( ( T c − T ) (1 − xφ ) ν λ ) λ u = 0 determines wetting transition temperature T w ( λ ) < T c

Double interfaces suppose going from | Ω a � to | Ω b � requires two kinks Ω c Ω a Ω b � dθ 1 dθ 2 f acb ( θ 1 , θ 2 ) | K ac ( θ 1 ) K cb ( θ 2 ) � + . . . ] 2 ) � = e ± R 2 H [ | B ab ( ± R � c a b a b

Ashkin-Teller σ 1 , σ 2 = ± 1 � H = − { J [ σ 1 ( x 1 ) σ 1 ( x 2 ) + σ 2 ( x 1 ) σ 2 ( x 2 )] + J 4 σ 1 ( x 1 ) σ 1 ( x 2 ) σ 2 ( x 1 ) σ 2 ( x 2 ) } � x 1 x 2 � (+−) (++) 4 degenerate vacua below T c scaling limit → sine-Gordon Σ (++)(+ − ) = m (−−) (−+) πβ 2  2 m sin J 4 > 0 2(8 π − β 2 ) ,  Σ (++)( −− ) = 2 m , J 4 ≤ 0  4 π β 2 = 1 − 2 tanh 2 J 4 π arcsin( tanh 2 J 4 − 1 ) on square lattice −+ −− ++ +− double interface between ( −− ) and (++) for J 4 ≤ 0