XY model in 2D and 3D

Transcription

1 XY model in 2D and 3D Gabriele Sicuro PhD school Galileo Galilei University of Pisa September 18, 2012

2 The XY model XY model in 2D and 3D; vortex loop expansion Part I The XY model, duality and loop expansion

3 The XY model XY model in 2D and 3D; vortex loop expansion Why the XY model? An important model used, for example, in superconductivity. It presents a particular phase transition involving topological excitations (vortices). It is dual to other interesting models (Coulomb gas model, sos model... )

4 The XY model XY model in 2D and 3D; vortex loop expansion The XY model in d dimensions Towards a dual model The Hamiltonian of the XY model in d dimensions is given by βh = β [ cos( µθ i) Z = δθ exp β ] cos( µθ i), θ i ( π, π], µ,i µ;i where µθ i = θ i θ i ˆµ is the discrete derivative in the µ direction. It is invariant under the transformation µθ i µθ i + 2πn µ,i, n µ,i Z. We can expand the hamiltonian using the identity e a cos b = + n= I n(a) e inb, I n(x) modified Bessel function. Then for k µ;i Z, substituting and integrating in θ, Z = δθ e β cos( µθi) = µ;i e ln I k µ;i (β) δ ki,0 }{{} µ,i {k µ;i } null divergence k i = (k µ;i) µ=1,...d

5 The XY model XY model in 2D and 3D; vortex loop expansion Using Einstein s notation µk µ;i = 0 so XY model in 2D and 3D d = 2 d = 3 k µ;i = ɛ µν νφ i k µ;i = ɛ µνλ νφ λ,i φ i Z φ i Z 3 the new field is located on the dual lattice Z = ( ) exp ln I ɛµν ν φ i (β) Z = ( ) exp ln I ɛµνλ ν φ λ,i (β) {φi} {φi } µ;i ( ) exp V({ɛ φ}) {φi} µ;i From now on we will work on the dual lattice.

6 The XY model XY model in 2D and 3D; vortex loop expansion Poisson summation formula We can use now the Poisson summation formula d = 2 must be invariant under φ i φ i +φ 0, φ 0 R {}}{ e V({ɛ φ}) = δφ e V({ɛ φ})+2πi i m iφ i, {mi} {φ} δφ = def d φ i i }{{} now φ is a continuous field! In the previous formula m i Z. Imposing the invariance i mi = 0. d = 3 must be invariant under φ µ;i φ µ;i + µρ i {}}{ e V({ɛ φ}) = δφ e V({ɛ φ})+2πi i m i φ i, {mi} {φ} δφ def = µ;i d φ µ;i In the previous formula m i Z 3. Imposing the invariance m i = 0.

7 The XY model XY model in 2D and 3D; vortex loop expansion Low temperature expansion Now expanding in n, considering β 1 (low temperature behaviour) and omitting multiplicative constants, we can substitute ln I n(β) 1 2β n2 Z δφ {mi} d = 2 d = 3 ( µφj ) 2 +2πi m j φ j Z δφ {m} e 1 2β e 1 2β (ɛµνλ ν φ λ,j ) 2 +2πim j φ j By a gaussian integration over φ i we obtain Z {mi} e πβ r r m(r)v2(r r )m(r ) Z {m} e πβ r r m(r) m(r )V 3(r r ) where V d (r) is the Green function on the lattice.

8 The XY model XY model in 2D and 3D; vortex loop expansion 2D XY model for low temperature The Coulomb gas For d = 2, after some calculations we obtain, removing the divergence V 2(r) Ṽ 2(r) = def V 2(r) V 2(0) = ln r c, c R + Z = {mi} e πβ r r m(r)m(r ) ln r πcβ r m2 (r) i.e., the system is equivalent to a neutral ( r m(r) = 0) Coulomb gas in 2D! The variables m(r) are called vortex variables (topological excitations), because it can be shown GO! that m(r) 1 δθ, 2π γ γ walk surrounding r on the lattice.

9 The XY model XY model in 2D and 3D; vortex loop expansion 3D XY model for low temperature QED analogy and smoke rings In 3D the partition function looks as the generating function for free photons with φ(r) A(r): Z δφ e 2β 1 j (ɛ µνλ ν φ λ,j ) 2 Ignoring gauge issues V 3(r) 1 r Biot-Savart law Moreover m(r) = 0 so currents m(r) generate closed loops (smoke rings). Introduce a new current loop variable L: Z = {m L } e πβ L,L r r ml (r) m L (r )V 3(r r )

10 Scaling and renormalization in 2D Scaling and renormalization in 3D Conclusions Bibliography Part II Scaling and Renormalization group

11 Scaling and renormalization in 2D Scaling and renormalization in 3D Conclusions Bibliography Phase transition in the 2D model? Mermin Wagner Hohenberg theorem and Kosterlitz Thouless argument Mermin Wagner Hoenberg theorem There cannot be any spontaneous breaking of a continuous symmetry in a system in d 2 dimensions. but 1. suppose that vortices are separated by a distance L 0 in a lattice with step a; 2. by analogy with ( electromagnetic theory, they have energy βe = β m ) 2 d 2 r = πm 2 β ln L0 ; 2 r a L0 a 3. there are ( L 0 ) 2 a different position for them { βf (πm 2 β 2) ln L0 a β > 2 f > 0 free vortices suppressed π β < 2 f < 0 free vortices with m = ±1 proliferate π There is a transition regarding vortices for β 2 π! No spontaneous magnetization appears for β +.

12 Scaling and renormalization in 2D Scaling and renormalization in 3D Conclusions Bibliography The fugacity: a control parameter for vorticity Z = exp πβ m(r) ln r a m(r ) πcβ {mi} r r r m 2 (r) exp πβ m(r) ln r a m(r ) y r m2 (r), {mi} r r with y = e πcβ fugacity, y 0 for β +.

13 Scaling and renormalization in 2D Scaling and renormalization in 3D Conclusions Bibliography Renormalization Group procedure First step: integrate We consider near the transition only vortex pairs m(r 1) + m(r 2) = 0, m(r) = ±1 (lowest energy excitations). Integration on pairs at distances a < r 1 r 2 < a(1 + δl) so: Z = Z + + δz 1. Z + = sum of configurations with vortex separations greater than a(1 + δl); 2. δz interaction between one pair of separation between a and a(1 + δl) and the others pairs δz = e βh + {m} δz Z + = 1 + y2 m(r 1),m(r 2)=±1 a< r 1 r 2 <a(1+δl) m(r 1)=±1 r 1 r 2 [a,a(1+δl)] y 2 e πβ r m(r) [ d 2 r 1 d 2 r 2 a 2 a 2 m(r 1) ln r 1 r +m(r a 2) ln r ] 2 r a r r 1,r 2 e πβm(r1) ln r1 r r2 r m(r) e y2 δl 2πL2 a }{{ 2 } e π4 y 2 β 2 δl r r m(r ) ln r scaling law for f r m(r ) a

14 Scaling and renormalization in 2D Scaling and renormalization in 3D Conclusions Bibliography Renormalization Group procedure Second step: rescale Rescaling to the new minimum scale a a(1 + δl) we obtain two contributions to fugacity ln r a = ln So finally r + δl e πβδl r m2 (r) a(1 + δl) d 2 r a 2 Z = e y2 δl 2πL2 a 2 = (1 + 2δl) {m i } d 2 r a 2 (1 + δl) ( δl) r m 2 (r) e 2δl r m2 (r) e π(β π3 y 2 β 2 δl) ( r r m(r) ln r a m(r ) y e (2 πβδ)l) r m2 (r) up to an infinite constant e πβ r r m(r) ln r a m(r ) y r m2 (r) {mi}

15 Scaling and renormalization in 2D Scaling and renormalization in 3D Conclusions Bibliography Kosterlitz Thouless scaling equations dβ dl = π3 β 2 y 2, dy dl = (2 πβ)y TKT = π 2 For T < T TK the model can have thermally generated topological excitations in pairs of vortices with m = ±1 Show me! : G(r) 1. r πβ 1 For T > T TK pairs are separated and Show me! G(r) e r ξ with ξ e (T T KT) 12. y Note that ξ (T T KT) ν

16 Scaling and renormalization in 2D Scaling and renormalization in 3D Conclusions Bibliography Scaling and renormalization in 3D Preparation to the analysis Let us rewrite the partition function for the 3D XY model as Z = y (L) exp πβ J (L) (r ) J (L ) (r )U(r r ) L L {J (L) } L where y (L) is the fugacity for each loop L of diameter a L and core dimension a c: r r y (L) = exp πβ J (L) (r) J (L) (r )U(r r Warning! Approximation in progress ) J µ(r)=0,±1,circular loop of diameter a L r r a La [ ] exp 2π 2 β al al ln = e 2π2 β ln ac a a a c }{{} e 2π2 β a L a ln a L a y Z = y a L a e 2π 2 β a L a ln a L a exp πβ J (L) (r) J (L ) (r )U(r r ) L L L r r {J (L) }

17 Scaling and renormalization in 2D Scaling and renormalization in 3D Conclusions Bibliography RG procedure: a sketch 1. For N loops we can explicit the sum over loop configurations assuming J µ(r) = 0, ±1 (low energy configurations), circular loops of center R so that r (L) = R (L) + ρ (L) : {J} 1 N! }{{} N N=0 J (L) L=1 µ =0,±1 shuffling centers d 3 R (L) a 3 d 3 ρ (L) a 3 2. We can repeat the Kosterlitz procedure for d = 2, decomposing Z = Z + + δz integrating over loops of radius ρ ( a 2, a 2 (1 + δl)). 3. Rescaling all explicit scale dependences: a L a = a L a(1 + δl) (1 + δl), U(r) 1 r = 1 (1 + δl). a(1 + δl)r Finally we can extract the renormalization equation that leave invariant the partition function.

18 Scaling and renormalization in 2D Scaling and renormalization in 3D Conclusions Bibliography Scaling equation in the 3D XY model dβ dl = β 2π3 3 yβ2, ( dy dl = y 6 π2 2 β ln a ) + 1 a c }{{} h self energy β is not marginal. Trivial fixed point: β = y = 0. High temperature: β = 0 y = y 0 e 6l + i.e., vorticity proliferates. Low temperature: y = 0, β = β 0 e l. Nontrivial fixed point y β = 12 π 2 h, y = h 0.0 8π

19 Scaling and renormalization in 2D Scaling and renormalization in 3D Conclusions Bibliography Linearized solutions Writing y = y (1 + δy) and β = β (1 + δβ) { dβ = β dl Ayβ2, = y(6 Bβ) dy dl d ( ) ( δy 6 Bβ = dl δβ { β = 6 B being d y = B dl 6A Eigenvalues and eigenvectors: λ + = 2, v + = 1 ( ) 3, 10 1 }{{} relevant Bβ 1 2Ay β Aβ y ( ) δy = δβ ) ( ) δy δβ ( ) }{{} A, B indipendent ) λ = 3, v = 1 ( }{{} irrelevant ( ) δy δβ We assume that the relevant axis is the temperature axis, i.e., λ + is the temperature eigenvalue, so, from general theory loop size ξ (T T c) ν with ν = 1 λ + = 1 2

20 Scaling and renormalization in 2D Scaling and renormalization in 3D Conclusions Bibliography Conclusions We have studied the XY model in 2D and 3D in absence of external fields. We have obtained the following results. 2D model In 2D there is no spontaneous magnetization at T 0 but at T = T KT the system undergoes to a Kosterlitz Thouless transition: for T < T KT free vortices are suppressed but there are pairs vortex antivortex; ξ = + for T > T KT free vortices proliferate; ξ finite. 3D model A duality transformation shows that the model is equivalent to an abelian gauge theory on a lattice (e.g., QED). Vortex lines are found as topological excitations. A phase transition occurs for a certain T c between a phase where linear vortices are suppressed (T < T c) and a phase with linear vortices are favoured (T > T c). It can be shown that spontaneous magnetization for T 0 is present.

21 Scaling and renormalization in 2D Scaling and renormalization in 3D Conclusions Bibliography Thank you for your attention!

22 Scaling and renormalization in 2D Scaling and renormalization in 3D Conclusions Bibliography Bibliography CARDY, J., Scaling and Renormalization in Statistical Physics, Cambridge Lecture Notes in Physics 5 (1996). CHATTOPADHYAY, B., Anisotropic 3D XY model and vortex loops, PhD dissertation, DODGSON, M., Vortex-unbinding transition in the 2D-XY model, lecture notes in Statistical Physics at MIT. KOSTERLITZ, J.M., The critical properties of the two-dimensional xy model, J. Phys. C: Solid State Phys., 7, , (1974). SAVIT, R., Topological Excitations in U(1)-Invariant Theories, Phys. Rev. Lett., 30(2), 55 58, (1977). SAVIT, R., Vortices and the low-temperature structure of the x-y model, Phys. Rev. B, 17(3), , (1979). SAVIT, R., Duality in field theory and statistical systems, Rev. Mod. Phys., 52(2), , (1980). SHENOY, S.R., Notes on Josephson Junction Arrays (1989). SHENOY, S.R., Vortex-loop scaling in the three-dimensional XY ferromagnet, Phys. Rev. B, 40(7), , (1980).

23 Meaning of m fields clarified Correlation length in 2D XY model Part III Appendix

24 Meaning of m fields clarified Correlation length in 2D XY model Additional notes: meaning of φ and m in 2D XY model The Villain approximation At low temperature in a lattice of N sites µθ i 2πn µ;i, n µ;i Z. To take into account the periodicity, we consider θ i R. In this case n µ;i n µ;i + µρ i, ρ i Z, is a redefinition of θ i and we have to fix a gauge to do not overcount the configurations. Z = (2β e 2β ) N {n} = (4πβ e 2β ) N {n} + δθ = (4πβ e 2β ) N {n} + δk µ exp [ k2 µ;j 2β δk µ exp [ k2 µ;j 2β 2πikµ;jnµ;j + ikµ;j( µθj 2πnµ;j) ] i δ ki,0 ] null divergence condition k µ;i =ɛ µν ν φ i δφ exp [ 1 ] 2β ( µφj)2 + 2πiφ jɛ µν µn ν;j i.e, m i = ɛ µν µn ν;i. If µθ i = µθ i 2πn µ;i then for a loop γ µθ = 2π m s, s dual sites in γ γ s Back to the 3D model!

25 Meaning of m fields clarified Correlation length in 2D XY model The correlation lenght in the 2D XY model Below the critical temperature Below the critical temperature we suppose θ small; in the continuum limit So βh = β ij cos(θ i θ j) constant β ( θ) 2 d 2 r 2 G(r) G(0) 1 ln r + constant 2πβ s(r) s(0) R e i(θr θ0) = e (θr θ 0) 2 2 = e G(r) G(0) = 1 r 1 2πβ η = 1 2πβ Back!

26 Meaning of m fields clarified Correlation length in 2D XY model The correlation lenght in the 2D XY model Above the critical temperature Above the critical temperature, being ξ e l, we denote { dx dl Ay2 dy dl = xy l = l 0 d l = x(l) x(0) { x = 2 πβ t = T T KT T KT dy dx = x Ay Ay2 x 2 = constant 2Ay(0) 2 t near the critical line d x Ay 2 = x(l) x(0) d x x 2 + 2Ay(0) 2 t + d x x 2 + 2Ay(0) 2 t = so π 2Ay(0)2 t ξ e constant t. Back!