Renormalization group analysis for the 3D XY model Statistical Physics PhD Course Statistical physics of Cold atoms 5/04/2012
The XY model Duality transformation H = J i,j cos(ϑ i ϑ j ) Plan of the seminar: Duality transformation for low temperature and RG flow Comparison with 2D XY
Duality transformation Vortex loop espansion Comparison with 2D XY Steps of the procedure: 1 start from the spin variables 2 expansion of the partition function 3 integration over the spin variables gives a constraint 4 satisfy constraint as an identity 5 expression in terms of vortex variables (Poisson summation formula)
Vortex loop espansion Comparison with 2D XY The partition function of the 3D XY model is Z = N k=1 π π (notation: µϑ i = ϑ i ϑ i+µ ) Expansion in terms of Bessel functions: expk 0 cos(ϑ i ϑ j )] = Z = {n} = {n µ,i } j,µ ] dϑ k 2π exp K 0 cos( µϑ i ) n= N I nµ,j (K 0) e V ({n µ,i }) exp (notation: n i = (n i,µ ) µ=1,2,3) k=1 µ,i I n(k 0) expın(ϑ i ϑ j )] π π ı j dϑ k 2π exp ϑ j n j ] ı j ϑ j n j ]
Vortex loop espansion Comparison with 2D XY Integrate over ϑ i : Z = e V ({n µ,i }) i {n µ,i } δ n i,0 Satisfy the constraint as an identity: n µ,i = ɛ µνλ νn λ,i N i : integer valued vectors on the side of a face of the dual lattice. Using the Poisson summation formula: Z = e V ({N µ,i }) {N µ,i } = µ,i dφ µ,i e V ( µφ µ,i ) exp 2πı ] J µ,i φ µ,i µ,i {J µ} with the constraint (assumed verified from now on): µj µ,i = 0 i µ
Vortex loop espansion Comparison with 2D XY Z = µ,i dφ µ,i e V ( µφ µ,i ) exp 2πı µ,i {J µ} J µ,i φ µ,i ] Low temperature expansion (expanding I n = I µφ µ,i ): ] e V ({ µφ µ,i }) exp 1 ( µφ µ,i ) 2 2K 0 Integrating φ µ,i (gaussian integral): Z = {Jµ,i } e πk 0 µ,i 2 exp J µ(r)u(r r )J µ(r ) µ,r,r U(r) is the 3D lattice Green s function (asymptotically U(r) a 0 r 1 ) Note: Partition function expressed in terms of topological excitations.
Vortex loop espansion Comparison with 2D XY Separating the contributions of different loops: Label each loop with the index L Consider only J µ = 0, ±1 (low temperature) Split interaction into self energy (L L) and different loop (L L ) parts Consider only circular loops for simplicity Self energy part: πk0 2 r r L J (L) µ (r)u(r r )J (L) µ (r ) = π 2 K 0 L a L log a ] L a c + c 0
Vortex loop espansion Comparison with 2D XY Definitions: a c short distance cutoff a c = a c e c 0 ( a 0 minimum loop size) K 0a K l, with l = log a a 0 Ũ(r) U(r) = 1 r At a general cutoff scale a: Z = exp {J (L) µ } πk l 2 L yl a La J µ (L) (r)u(r r )J (L ) µ (r ) L L µ,r,r ] exp π 2 a L K l a log a L a L, with y l = e π2 K l log a ac
Vortex loop espansion Comparison with 2D XY Comparison with the result of 2D XY model: Z = exp {J (L) µ } πk l 2 L yl a La J µ (L) (r)u(r r )J (L ) µ (r ) L L µ,r,r ] exp π 2 a L K l a log a L a In 2D XY: Coulomb gas Z = exp πk l m(r)u(r r )m(r ) yl r r {m(r)} L, with y l = e π2 K l log a ac r m2 (r) with y 0 = e π2 K l and for low temperatures.
Duality transformation Kosterlitz-Thouless procedure and RG flow Kosterlitz Thouless procedure: 1 identify the simplest, neutral excitation; 2 integrate the interactions of excitations between scales a and a + da (neglecting interaction at far distances); 3 rescale all explicit scale dependence and absorption of scales into renormalized values of couplings and fugacity; 4 recover the scaling equations. For 3D XY model: 1 vortex loop L = 1 is the simplest excitation 2 Z > vortices > a + da, δz vortices between a and a + da. Then ( Z = Z > 1 + δz ) Z >
Kosterlitz-Thouless procedure and RG flow Note: the sum over configurations is {J (L) µ } N vorteces=0 J (L) µ =0,±1 1 N vorteces! N vorteces L=1 d 3 R a 3 d 3 ϱ a 3 Then (no interaction between vortices farther than a + da): δz exp πk l Z > 2 {J (L) µ } L L µ a< r r <a+ da J (L) µ (r)u(r r )J (L ) µ (r ) Considering L = 1 as the simplest neutral configuration and expanding the exponential: δz Z > J (1) µ =±1 d 3 R a 3 a + da a d 3 ϱ a 3 r,l 1 y l 1 πk l 2 J(1) µ (r)u(r r )J (L ) µ (r ) +... ] y l }{{} from a L =a
Kosterlitz-Thouless procedure and RG flow Note: in a circular loop J µ(ϱ ϱ 1) = J µ( ϱ ϱ 2). Then J µ (1) (r)u(r r ) 1 2 J(1) µ (ϱ) J µ (1) ( ϱ)]u(ϱ+r r ) J µ (1) (ϱ)2ϱ U(ϱ+R r ) There are only quadratic contributions in the product (due to the neutrality): ϱ Uϱ U U(ϱ ) 2 U Note: 2 U(r) = 4πa 3 δ(r) Define dl = da a. Integrating: δz = π dl Z > a y 3 l + 4π 3 Kl 2 y l dl U(r r )J µ (L ) (r )J (L ) µ (r ) r r a+ da d 3 ϱ r r 2 +... a a 3 In the end: δz = π dl Z > a y 3 l π 4π 3 2 3 K l 2 y l dl U(r r )J µ (L ) (r )J (L ) µ (r ) +... r r 1 + δz = e π dl a Z 3 y 4π3 l 2π e > 3 K l 2 y l dl r r U(r r )J (L ) µ (r )J (L ) µ (r )
Kosterlitz-Thouless procedure and RG flow 1 + δz Z > = e π dl a 3 y l e π 2 4π3 {J (L) µ } 3 K l 2 y l dl r r U(r r )J (L ) µ (r )J (L ) µ (r ) Absorb scale dependent quantities in Z > and compare with the starting expression: Z = exp πk l µ (r ) 2 L y l a La exp L L J µ (L) (r)u(r r )J (L ) µ,r,r π 2 K l L a L a log a L a Note: in the self energy a L a = a L a+ da (1 + dl) rescale K l by (1 + dl); in the integration d3 R d 3 ϱ = d3 R d 3 ϱ (1 + 6 dl). a 6 (a+ da) 6 K l+ dl = K l + K l dl 4π3 3 y lkl 2 dl y l+ dl = y l + (6 π 2 K l L l )y l dl with L l = log a a c + 1 F l+ dl = F l πy l dl a 3 ]
Kosterlitz-Thouless procedure and RG flow Summing up: dk l dl dy l dl df l dl = K l 4π3 3 y lkl 2 = (6 π 2 K l L l )y l = πy l a 3 Fixed points? Note: at high temperature K l = 0, y l = y 0 e 6l (free vortices) at low temperature K l = K 0 e l, y l = y 0 e ξ l 0 where ξ = (π 2 K 0L 0) 1 (no free vortices above ξ) Then it exists a nontrivial fixed point: K l = 6 π 2 L l, y l = 3 4π 3 K l Linearize around the fixed point: define K l K L(1 + K l ), y l = y l (1 + ỹ l )
Kosterlitz-Thouless procedure and RG flow Substituting into the scaling equations, at first order: dỹ l dl = 6 K l d K l dl = ỹ l K l Eigenvalues: λ = 2, λ = 3 Note: From scaling Z(K 0, y 0) = e (F l F 0 )L 3 Z ( A + e λ+l, A e ) λ l Relevant field temperature: A + = A = A ɛ T Tc T c Define l log ξ a 0 (ξ size scale of the vortices, ξ if ɛ 0) Z(A ɛ e λ+l, A e λ l ) = Z ( A ɛ ξ Then for Z to be well defined for l a 0 ] λ+, 0) ξ = a 0 ɛ ν, ν = λ 1 + = 1 2
Kosterlitz-Thouless procedure and RG flow RG flow:
Comparison with 2D XY model Kosterlitz-Thouless procedure and RG flow : dk l dl = 4π 3 y l Kl 2 dy l dl = (4 2πK l )y l df l = 2πy l dl a 2 Nontrivial fixed point: RG flow: K = 2 π, y = 0
Thank you for your attention References: 1 Shenoy, Vortex loop scaling in three dimensional XY ferromagnet, Physical Review B, volume 40, number 7, 5056 (1989) 2 Savit, Vortices and the low.temperature structure of the x y model, Physical Review B, volume 17, number 3, 1340 (1978) 3 Shenoy, Notes on Josephson Junction Arrays (1989) 4 Biplab Chattopadhyay, Anisotropic 3D XY model and vortex loops, PhD thesis (1994)