Short historical notes Dislocations: V. Volterra (1905) K. Kondo - PowerPoint PPT Presentation

Short historical notes Dislocations: V. Volterra (1905) K. Kondo (1952) J. F. Nye (1953) B. Bilby, R. Bullough, E. Smith (1955) E. Kröner, I. Dzyaloshinskii, G. Volovik, M. Kléman, I. Kunin, J. Madore, A. Kadi ć , D. Edelen, H. Kleinert, E. Bezerra de Mello, F. Moraes, C. Malyshev, M. Lazar, ... Disclinations: F. Frank (1958) I. Dzyaloshinskii, G. Volovik (1978) J. Hertz (1978) Torsion: E. Cartan (1922) Torsion in gravity: A. Einstein, E. Schrödinger, H. Weyl, T. Kibble, D. Sciama, R. Finkelstein, F.Hehl, P. von der Heyde, G. Kerlich, J. Nester, Y. Ne’eman, J. Nitsch, J. McCrea, J. Mielke, Yu. Obukhov M. Blagojeci ć, I. Nikolić, M. Vasilić, K. Hayashi, T. Shirafuji, E. Sezgin, P. van Nieuwenhuizen, I. Shapiro, ... 1

Chern-Simons term in the Geometric Theory of Defects M. O. Katanaev, Steklov Mathematical Institute, Moscow J. Zanelli, Centro de Estudios Cienificos, Valdivia, Chile Katanaev, Volovich Ann . Phys. 216(1992)1; ibid. 271(1999)203 Katanaev Theor.Math.Phys.135(2003)733; ibid. 138(2004)163 Phisics – Uspekhi 48(2005)675. Notation  3 - continuous elastic media = Euclidean three-dimensional space = i i x y , i 1,2,3 - Cartesian coordinates δ - Euclidean metric ij i u x ( ) - displacement vector field ε = ∂ + ∂ Elasticity theory of small deformations 1 ( u u ) - strain tensor ij i j j i 2 ∂ σ + = ij j f 0 - Newton’s law σ ij i - stress tensor σ = λδ ε + µε ij ij k ij 2 - Hooke’s law k i f = i f ( ) x ( 0) - density of nonelastic forces λ µ , - Lame coefficients 2

Differential geometry of elastic deformations → i i  →  3 3 x y x y ( ) - diffeomorphism: i i  y x = + y i i i x y u x ( ) δ g ij ij ∂ ∂ k l y y = δ ≈ δ − ∂ − ∂ = δ − ε g ( ) x u u 2 (*) - induced metric ij ∂ ∂ kl ij i j j i ij ij i j x x  Γ = ∂ + ∂ − ∂ ≠ 1 2 ( g g g ) 0 - Christoffel’s symbols ijk i jk j ik k ij     = ∂ Γ − Γ Γ − ↔ = l l m l R ( i j ) 0 - curvature tensor ijk i jk ik jm  = −Γ  i i   j k x x x - extremals (geodesics) jk = l (*) R 0 - Saint-Venant integrability conditions of ijk   = Γ − Γ = k k k T 0 - torsion tensor ij ij ji 3

Dislocations Linear defects: 2 x 2 x b 1 x 1 x b 3 3 x x Edge dislocation Screw dislocation b - Burgers vector Point defects: Vacancy  is continuous = elastic deformations i  u x ( ) is not continuous = dislocations  4

Edge dislocation 2 x ∫ µ ∫ µ ∂ = − ∂ = −   i i i (*) dx u dx y b µ µ b C C x µ µ = 1 , 1,2,3 - arbitrary curvilinear coordinates x i y x ( ) - is not continuous ! C ∂ i  y - outside the cut µ =  i - triad field e ( ) x µ ∂ i (continuous on the cut)  lim y  - on the cut µ ∫ µ ∫∫ µ ν (*) ⇒ =  = ∧ ∂ − ∂ i i i i b dx e dx dx ( e e ) - Burgers vector in elasticity µ µ ν ν µ C S = ∂ − ω − µ ↔ ν i i ij T e e ( ) - torsion ω = − ω µν µ ν µ ν ij ji j µ µ = ∂ ω − ω ω − µ ↔ ν ij ij ik j R ( ) - curvature SO(3)-connection µν µ ν µ ν k ∫∫ µ ν = ∧ i i b dx dx T - definition of the Burgers vector µν in the geometric theory = ω → ij ij R µν 0 0 then Back to elasticity: if 5 µ

Disclinations i i n n x ( ) - unit vector field - fixed unit vector Ferromagnets 0 = ω i j i n n S ( ) 0 j ∈  j S (3) - orthogonal matrix i ω = − ω ∈ so ij ji (3) - Lie algebra element (spin structure) 1 ω = ε ω jk - rotational angle i ijk 2 ε = ε ( 1) - totally antisymmetric tensor 123 ijk Examples 2 ∫ dx µ 2 x Ω =  ∂ µ ω x ij ij C Θ = ε Ω 1 jk 1 x x - Frank vector i ijk (total angle of rotation) C C Θ = Θ Θ i Θ = 2 π Θ = 4 π i 6

More examples Nematic liquid crystals 2 2 x x 1 1 x x − i i  n n C C Θ = π Θ = 3 π Model for a spin structure: i x ω ∈ so ( ) (3) - basic variable ω ε ω ω k j j = δ ω + ω + − ω ∈  ω = ω ω j j i ki i S cos sin (1 cos ) (3), i i ω i ω 2 − = ∂ j 1 k j l ( S ) S - trivial SO(3)-connection (pure gauge) µ µ i i k µ ∂ = ij l 0 - principal chiral SO(3)-model µ 7

Frank vector ij x ω ( ) - is not continuous ! ∂ ω ij  - outside the cut µ ω =  - SO(3)-connection ij ( ) x µ ∂ ω (continuous on the cut) ij  lim - on the cut  µ ∫ µ ∫∫ µ ν Ω = ω = ∧ ∂ ω − ∂ ω  ij ij ij ij dx dx dx ( ) - the Frank vector µ µ ν ν µ = ∂ ω − ω ω − µ ↔ ν ij ij ik j R µν ( ) - curvature µ ν µ ν k ∫∫ µ ν Ω = ∧ ij ij - definition of the Frank vector dx dx R µν in the geometric theory n ∈   →  2 (3) (2) Back to the spin structure: if then 8

Summary of the geometric approach (physical interpretation) Media with dislocations and disclinations = =  3 with a given Riemann-Cartan geometry  i  e µ - triad field  Independent variables ω ij   - SO(3)-connection µ = ∂ − ω − µ ↔ ν i i ij T e e ( ) - torsion (surface density of µν µ ν µ ν j the Burgers vector) = ∂ ω − ω ω − µ ↔ ν ij ij ik j R ( ) - curvature (surface density of µν µ ν µ ν k the Frank vector) = = ij i R 0, T 0 Elastic deformations: µν µν = ≠ ij i R 0, T 0 Dislocations: µν µν ≠ = ij i R 0, T 0 Disclinations: µν µν ≠ ≠ ij i R 0, T 0 Dislocations and disclinations: µν µν 9

The free energy [ ] [ ] = + ω S S e S - the total free energy HE CS ( ) = δ = δ i i → g e S 0 - Euclidean metric No dislocations: µν µν µ µ HE Three dimensions: ω = ω ε ω = ω ε 1 ij kij ij , : , -SO(3) connection µ µ µ µ k k ijk 2 (1-form, the only variable) µ ν = = , ,... 1,2,3; i j , ,... 1,2,3 - indices ( ) ω = ε = ∂ ω − ∂ ω + ω ω ε ij i j R ( ): R 2 - curvature µν µν µ ν ν µ µ ν k ijk k k ijk ( ) ∫ = ω ω + ε ω ω ω − ω 1 1 j i i k i S ∧ d ∧ ∧ ∧ J - free energy for disclinations CS i ijk i 2 3  3 i J - the source term (2-form) = k k R J - equations of equilibrium µν µν 10

One linear disclination ( ) µ ∈  ∈  3 , q t t - the core of disclination ∫ µ ∫ µ = ω = ω =  i i S dq J dtq J - the interaction term µ µ int i i µ  q ( ) ( ) ∫ µ ∫ = ω δ − = ω δ −  3 i 3 3 i 2 dtd xq J x q d x J x q µ µ i i  3 q µ δ  S q ( ) = δ − = i 2 1 2 int x q J x ( x x , ) - the source term δω  3 q µ i 2 3 x x One disclination along axis = = = + 1 2 x x , x y , z : x iy Notation: 1 x ω ω 3 3 , - the only nontrivial components x y C ω = 1 ω − i ω 3 3 3 - one complex component z x y 2 2 ( ) = ∂ ω − ∂ ω 3 3 3 R 2 - the curvature tensor zz z z z z 11

One straight linear disclination ( ) = π δ ∈  3 R 4 iA z , A - new kind of defect Fixing the source term: zz iA 1 ( ) ω = − ∂ = πδ 3 z The solution: - important formula z z z z 2 Ay 2 Ax ω = − ω = 3 3 , - real components x + y + 2 2 2 2 x y x y ( ) ω x Rotational angle field 2 Ay 2 Ax ∂ ω = − ∂ ω = , x y + + 2 2 2 2 x y x y ∂ ω = ∂ ω The integrability conditions are fulflled xy yx arctan x ω = − + 2 A C - a general solution y ω x = π ⇒ ϕ ϕ = C A tan = , : - polar angle y 2 A 12

One straight linear disclination ( ) ω x y , To make the rotational angle field well defined, we must impose the quantization condition: n = ∈  ⇒ ω = ϕ ฀ A , n n 2 ny ω = − = − ϕ 12 n sin , SO(3)-connection: x + 2 2 x y nx ω = = ϕ 12 n cos . y + 2 2 x y y x C 13

Conclusion 1) The first example of disclination is described within the geometric theory of defects. 2) The Chern-Simons term is well suited for disclinations in the geometric theory of defects. 3) Linear disclinations correspond to a new type of defects of the SO(3) connection (the spin structure). 14