Phase Separation, Interfaces and Wetting in Two Dimensions Gesualdo - PowerPoint PPT Presentation

Lattice Models: Exact Methods and Combinatorics 18 – 22 May 2015, GGI Arcetri, Florence Phase Separation, Interfaces and Wetting in Two Dimensions 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 GD, A. Squarcini, Interfaces and wetting transition on the half plane. Exact results from field theory, J. Stat. Mech. (2013) P05010 GD, A. Squarcini, Exact theory of intermediate phases in two dimensions, Annals of Physics 342 (2014) 171 GD, A. Squarcini, Phase separation in a wedge. Exact results, PRL 113 (2014) 066101 GD, Order parameter profiles in presence of topological defect lines, J. Phys. A 47 (2014) 132001

Introduction − − − − − − − + + + + + + + − − − + + − − − + + − − + + − + + + + + − + − + + − − + − − + + + − + + + + + − − − R − − − + − − − − + + − + − + − + − − − − − + − + + + + + − − − − − − − + + + + + + + Ising ferromagnet: phase separation emerges when T < T c , R ≫ ξ exact magnetization profile [Abraham, ’81] Issues : – role of integrability – other universality classes – structure of the interfacial region – different geometries

Introduction − − − − − − − + + + + + + + − − − + + − − − + + − − + + − + + + + + − + − + + − − + − − + + + − + + + + + − − − R − − − + − − − − + + − + − + − + − − − − − + − + + + + + − − − − − − − + + + + + + + Ising ferromagnet: phase separation emerges when T < T c , R ≫ ξ exact magnetization profile [Abraham, ’81] Issues : – role of integrability – other universality classes – structure of the interfacial region – different geometries field theory yields exact answers and suggests applications in D > 2

Pure phases and kinks bulk system at a spontaneous symmetry breaking point scaling limit ↔ Euclidean field theory ↔ QFT with imaginary time Ω 2 coexisting phases ↔ 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 interfacial free energy : 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  

order parameter 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 � dθ 1 ∼ | f (0) | 2 1 2 dθ 2 2 π F σ ( θ 1 | θ 2 ) e − m [(1+ 4 + 4 ) R − iθ 12 x ] mR ≫ 1 2 π Z ab a b a b F σ ( 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 kinematical pole at θ 12 =0 accounts for phase separation in 2D

� 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 � 2 m Ising: � σ � + = −� σ � − , c 0 = 0 ⇒ � σ � − + ∼ � σ � + erf( R x ) matches lattice result [Abraham, ’81] q-state Potts ( q ≤ 4) : 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

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 the passage probability density becomes � p ( x ; y ) = 1 2 m πR e − χ 2 κ � � 2 m x 1 − 4 y 2 /R 2 κ ( y ) ≡ χ ≡ R κ p � x;y � 10 0.3 5 0.2 my 0 0.1 � 5 0 � 10 � 10 � 5 0 5 10 mx

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 a b c a b

q -state Potts : the order of the transition changes at q = 4 1 1 5 2 q=3, T<T c q=5, T=T c 0 3 2 4 3 q → 4 + , T = T c : field theory gives 1 0 2 π e − 2 z 2 − 2 z √ π erf( z ) e − z 2 + erf 2 ( z ) � � � � σ 1 ( x, 0) � 12 ∼ � σ 1 � 1 q − 2 1 − 2 2 2( q − 1) √ π e − z 2 − erf( z ) � �� + q � 2 m z z ≡ R x q − 1 πR ( z 1 − z 2 ) 2 e − ( z 2 1 + z 2 2 ) p ( x 1 , x 2 ) = 2 m ⇒ passage probability mutually avoiding interfaces

Wetting transition c a b a b c c 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 ∀ J 4 (−−) (−+) πβ 2  2 m sin 2(8 π − β 2 ) , J 4 > 0  Σ (++)( −− ) = 2 m , J 4 ≤ 0  J 4 <0 −+ −− ++ +− 4 π β 2 = 1 − 2 tanh 2 J 4 π arcsin( tanh 2 J 4 − 1 ) on square lattice

Boundary wetting a θ 0 b B phenomenological description in terms of contact angle θ 0 wetting transition for θ 0 = 0 equilibrium condition at contact points (Young’s law, 1805): Σ Ba = Σ Bb + Σ ab cos θ 0

field theory : y B a boundary condition selecting a µ (y) the vacuum | Ω a � 0 whit energy E 0 0 x ab θ b µ ab ( y ) switches from B a to B b 0 � Ω a | µ ab ( y ) | K ba ( θ ) � 0 = e − ym cosh θ F µ 0 ( θ ) forbid the particle to stay on the boundary ⇒ F µ 0 ( θ ) = c θ + O ( θ 2 ) Lorentz boost B Λ sends θ → θ + Λ y α B − iα rotates by an angle α : F µ 0 ( θ ) = F µ α ( θ − iα ) a F µ b α ( θ ) ≃ c ( θ + iα ) for θ, α small

interface in a wedge : � σ ( x, y ) � W aba = α � Ω a | µ ab (0 , R 2 ) σ ( x,y ) µ ba (0 , − R 2 ) | Ω a � − α y ∼ α � Ω a | µ ab (0 , R 2 ) µ ba (0 , − R α 2 ) | Ω a � − α R/2 − α ( θ 2 ) e − m 2 [( R 1+( R � + ∞ 2 − y ) θ 2 2 + y ) θ 2 dθ 1 dθ 2 (2 π )2 F µ α ( θ 1 ) F σ ( θ 1 | θ 2 ) F µ 2]+ imx ( θ 1 − θ 2) −∞ b a − α ( θ ) e − mRθ 2 � ∞ 2 π F µ α ( θ ) F µ dθ 0 x 2 0 √ 2 mR α −R/2 χ + 1+ mRα 2 e − χ 2 � � 2 κ ∼ � σ � b + ( � σ � a − � σ � b ) erf( χ ) − √ π α � � 2 m x 1 − 4 y 2 /R 2 κ ≡ χ ≡ R κ passage probability density: � 2 � √ x + Rα − ( αy ) 2 � 3 / 2 e − χ 2 2 p ( x ; y ) ∼ 8 2 � m √ π κ 3 1+ mRα 2 R

wedge wetting : b b α = 0: for T < T 0 < T c boundary bound state | Ω ′ a � 0 with energy a a θ∼ i θ 0 E ′ b 0 = E 0 + m cos θ 0 Young’s law! θ 0 b resonance angle θ 0 = contact angle wetting transition = kink unbinding : θ 0 ( T 0 ) = 0

wedge wetting : b b α = 0: for T < T 0 < T c boundary bound a state | Ω ′ a a � 0 with energy θ∼ i θ 0 b E ′ θ 0 0 = E 0 + m cos θ 0 Young’s law! b resonance angle θ 0 = contact angle wetting transition = kink unbinding : θ 0 ( T 0 ) = 0 E ′ α � = 0: α = E α + m cos( θ 0 − α ) wedge wetting at T α such that θ 0 ( T α ) = α (θ −α) i 0 b condition known phenomenologically [Hauge, ’92] a ”wedge covariance” actually is relativistic covariance (θ −α) b −i 0

Higher dimensions What done so far relies on the fact that 2D interfaces are tra- jectories of topological particles (kinks) 2D Ising: kink