Statistical physics of polymerized membranes D. Mouhanna LPTMC - - PowerPoint PPT Presentation

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach Statistical physics of polymerized membranes D. Mouhanna LPTMC - UPMC - Paris 6 J.-P. Kownacki (LPTM - Univ. Cergy-Pontoise, France) K. Essafi (OIST, Okinawa, Japan) O. Coquand (LPTMC- Univ. Paris 6) Functional Renormalization - from quantum gravity and dark energy to ultracold atoms and condensed matter IWH Heidelberg, March 7-10 2017 1 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach Outline Introduction 1 Fluid vs polymerized membranes 2 Perturbative approaches 3 FRG approach 4 2 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach Introduction membranes: D-dimensional extended objects embedded in a d-dimensional space subject to quantum and/or thermal fluctuations fluctuating membranes / random surfaces occur in several domains: 3 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach chemical physics / biology : (Aronovitz - Lubensky, Helfrich, David - Guitter, Le Doussal - Radzihovsky, Nelson - Peliti,’70’s- 90’s) = ⇒ structures made of amphiphile molecules (ex: phospholipid) one hydrophilic head hydrophobic tails = ⇒ bilayers: 4 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach condensed matter physics: graphene, silicene, phosphorene . . . uni-layers of atoms located on a honeycomb lattice striking properties: high electronic mobility, transmittance, conductivity, . . . 5 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach mechanical properties: both extremely strong and soft material: = ⇒ example of genuine 2D fluctuating membrane 6 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach Fluid vs polymerized membranes Properties of fluid membranes very weak interaction between molecules = ⇒ free diffusion inside the membrane plane = ⇒ no shear modulus very small compressibility and elasticity = ⇒ main contribution to the energy: bending energy 7 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach Energy: point of the surface described by the embedding: σ = ( σ 1 , σ 2 ) → r ( σ 1 , σ 2 ) ∈ I R d r : σ σ σ 1 z σ 2 e 1 e 2 n r y • ( σ 1 , σ 2 ) ≡ local coordinates on the membrane • tangent vectors e a = ∂ r ∂σ a = ∂ a r a = 1 , 2 x n = e 1 × e 2 • a unit norm vector normal to ( e 1 , e 2 ) : ˆ | e 1 × e 2 | 8 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach curvature tensor K : K ab = − ˆ n . ∂ b e a = e a . ∂ b ˆ n K ab can be locally diagonalized with eigenvalues K 1 and K 2 mean or extrinsic curvature: H = 1 2( K 1 + K 2 ) = 1 2 Tr K Gaussian or intrinsic curvature: K = K 1 K 2 = det K b a ⇒ no role in fixed topology (Gauss-Bonnet theorem) = ⇒ bending energy: F = κ σ √ g H 2 � d 2 σ σ 2 g µν = ∂ µ r .∂ ν r ≡ metric induced by the embedding r ( σ σ σ ) √ g ensures reparametrization invariance of F 9 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach Low-temperature fluctuations in fluid membranes n = K ab e b one has: a remark: with ∂ a ˆ F = − κ ′ F = κ � d 2 σ n ) 2 � σ σ ( ∂ a ˆ or n i . ˆ ˆ n j 2 2 � i,j � where ˆ n i is a unit normal vector on the plaquette i very close to a O ( N ) nonlinear σ -model / Heisenberg spin system: - with (rigidity) coupling constant κ - with ”spins” living on a fluctuating surface - with d playing the role of the number of components N 10 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach Low temperature: Monge parametrization x = σ 1 , y = σ 2 and z = h ( x, y ) with h height, capillary mode r ( x, y ) = ( x, y, h ( x, y )) n ( x, y ) = ( − ∂ x h, − ∂ y h, 1) ˆ � 1 + |∇ ∇ ∇ h | 2 1 n ( x, y ) . e z = cos θ ( x, y ) = ˆ � ∇ 1 + |∇ ∇ h | 2 11 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach Free energy: F ≃ κ � d 2 x (∆ h ) 2 + O ( h 4 ) 2 flat phase ? = ⇒ fluctuations of θ ( x, y ) ? � κ q 2 ≃ k B T 1 � L � � θ ( x, y ) 2 � = k B T d 2 q ln → ∞ κ a = ⇒ no long-range order between the normals 12 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach At next order in h , κ is renormalized and decreased at long distances.: � 1 κ R ( q ) = κ − 3 k B T � d � � ln 2 π 2 qa ⇒ divergence of � θ ( x, y ) 2 � : worse = = ⇒ strong analogy with 2D-NL σ model: n ( 0 ) � ∼ e − r/ξ correlations: � ˆ n ( r ) . ˆ ξ ≃ a e 4 πκ/ 3 k B Td correlation length – mass gap: d/ 2 = ⇒ N − 2 nothing really new . . . 13 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach Polymerized membranes ex: - organic: red blood cell, . . . - inorganic: graphene, phosphorene, . . . made of molecules linked by V ( | r i − r j | ) = ⇒ free energy built from both bending and elastic energy 14 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach Free energy and low-temperature fluctuations in polymerized membranes reference configuration: r 0 ( x, y ) = ( x, y, z = 0) fluctuations: r ( x, y ) = r 0 + u x e 1 + u y e 2 + h ˆ n z y r(x ) α u x u y x h(x,y) 15 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach stress tensor: u ab = 1 2 ( ∂ a r .∂ b r − ∂ a r 0 .∂ b r 0 ) = 1 2 ( ∂ a r .∂ b r − δ ab ) ⇒ u ab = 1 = 2 [ ∂ a u b + ∂ b u a + ∂ a u .∂ b u + ∂ a h ∂ b h ] u ν describes the longitudinal – phonon-like – degrees of freedom h describes height, capillary – degrees of freedom free energy: � κ 2(∆ h ) 2 + µ ( u ab ) 2 + λ � � d 2 x 2 ( u ab ) 2 F ≃ κ ≡ bending rigidity λ , µ ≡ elastic coupling constants non-trivial coupling between longitudinal - in plane - and height fluctuations = ⇒ frustration of height fluctuations 16 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach Gaussian approximation on phonon fields: u ab ≃ 1 2 [ ∂ a u b + ∂ b u a + ∂ a h ∂ b h ] integrate over u : F eff = κ � d 2 x (∆ h ) 2 + K � � 2 d 2 x P T � ab ∂ a h ∂ b h 2 8 P T ab = δ ab − ∂ a ∂ b / ∇ 2 κ bending, rigidity coupling constant K = 4 µ ( λ + µ ) / (2 µ + λ ) : Young elasticity modulus 17 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach Self-consistent screening approximation (SCSA) ∼ Schwinger-Dyson equation closed at large d � 2 � q a P T ˆ ab ˆ q b � d 2 k κ eff ( q ) = κ + k B T K κ eff ( q + k ) | q + k | 4 √ k B T K = ⇒ κ eff ( q ) ∼ rigidity increased by fluctuations ! q normal fluctuations: � 1 � θ ( x, y ) 2 � = k B T d 2 q κ eff ( q ) q 2 < ∞ ! = ⇒ Long-range order between normals even in D = 2 ! 18 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach polymerized membranes = ⇒ possibility of spontaneous symmetry breaking in D = 2 and even in D < 2 = ⇒ low-temperature - flat - phase with non-trivial correlations in the I.R.  G hh ( q ) ∼ q − (4 − η )  G uu ( q ) ∼ q − (6 − D − 2 η )  with η � = 0 = ⇒ associated e.g. to stable sheet of graphene 19 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach Perturbative approach of the flat phase (Aronovitz and Lubensky’88) Field theory of the flat phase: � κ 2(∆ h ) 2 + µ ( u ab ) 2 + λ � � d 2 x 2 ( u aa ) 2 F ≃ λ ≡ λ/κ 2 and ¯ µ ≡ µ/κ 2 in ⇒ perturbative expansion in ¯ = D uc = 4 − ǫ non-trivial fixed point governs the flat phase increasing rigidity κ eff ( q ) ∼ q − η = ⇒ orientational order ր decreasing elasticity K eff ( q ) ∼ q η = ⇒ positional disorder ց ≃ ripples formation 20 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach However: flat phase properties: very poorly determined in D = 2 because D uc = 4 SCSA or weak-coupling tedious beyond leading order due to derivative interaction multiplicity of fields: h , u propagator structure: Capillary modes: G αβ ( q 2 ) = δ αβ κ q 4 � � δ ij − q i q j + G 2 ( q 2 ) q i q j Phonon modes: G ij ( q 2 ) = G 1 ( q 2 ) q 2 q 2 with: 1 G 1 ( q 2 ) = κ q 4 + ζ 2 µ q 2 1 G 2 ( q 2 ) = κ q 4 + ζ 2 (2 µ + λ ) q 2 21 / 34

Introduction Fluid vs polymerized membranes Perturbative approaches FRG approach FRG approach to polymerized membranes (Kownacki and D.M.’08, Essafi, Kownacki and D.M.’14, Coquand and D.M.’16) Effective action: Γ k [ ∂ µ r ] expanded around the flat phase configuration: D � r ( x ) = ζ x α e α α =1 � d D x Z 2 ( ∂ α ∂ α r ) 2 + Γ k [ ∂ µ r ] = � 2 + u 2 ∂ α r .∂ β r − ζ 2 δ αβ ∂ α r .∂ α r − D ζ 2 � 2 � � + u 1 + . . . + . . . ∂ α r .∂ β r − ζ 2 δ αβ ∂ β r .∂ γ r − ζ 2 δ βγ � � � � + u 10 × ∂ γ r .∂ δ r − ζ 2 δ γδ ∂ δ r .∂ α r − ζ 2 δ δα � � � � 22 / 34