Integrable Systems in Contemporary Physics

Transcription

1 Integrable Systems in Contemporary Physics Angela Foerster Universidade Federal do Rio Grande do Sul Instituto de Física talk presented in honour of Michael Karowski and Robert Schrader at the FU-Berlin Integrable Systems in Contemporary Physics p.1/62

2 OUTLINE Brief review of Michael Karowskis work Integrable models of Bose Einstein condensates Integrable spin ladder systems Integrable Systems in Contemporary Physics p.2/62

3 BRIEF REVIEW EDUCATION: Diplom (1967) Hamburg (Prof. H. Lehmann) PHd (1970) Hamburg (Prof. H. Lehmann) INSTITUTIONS: Institut für Theoretische Physik Hamburg Max-Planck Institut München University of Dortmund University of Göttingen LPTHP University, Paris ETH Zürich FU- Berlin Integrable Systems in Contemporary Physics p.3/62

4 MAIN COLLABORATORS: H. Babujian B. Berg R. Schrader B. Schroer H. J. Thun T. T. Truong H. J. de Vega P. Weisz PhD STUDENTS: A. Foerster A. Zapletal G. Jüttner T. Quella (Diplom) Integrable Systems in Contemporary Physics p.4/62

5 CONTRIBUTION TO THE DEVELOPMENT: S-matrices Exact form factors Yang-Baxter algebras Bethe ansatz methods Finite size corrections Integrable quantum field theories * Monte Carlo simulations for QFT * Combinatorial approach to topological QFT ( Robert Schrader) Quantum groups Integrable systems... Integrable Systems in Contemporary Physics p.5/62

6 TOP CITED PAPERS 1) Karowski M, Thun HJ, Truong TT, Weisz P Uniqueness of a purely elastic S-matrix in (1+1) dim. Physics Letters B 67; ) Foerster A, Karowski M Algebraic properties of the Bethe ansatz for an SPL(2,1)-susy t-j model Nuclear Physics B396; ) Karowski M, Weisz P Exact form-factors in (1+1)-dimensional field theoretic models with soliton behaviour Nuclear Physics B139; Integrable Systems in Contemporary Physics p.6/62

7 4) Foerster A, Karowski M The susy t-j model with quantum group invariance Nuclear Physics B408; ) Berg B, Karowski M, Weisz P Construction of Green-functions from an exact S-matrix Physical Review D19; ) de Vega HJ, Karowski M Conformal-invariance and integrable theories Nuclear Physics B285; AUG Total number of citations : 1490 (Web of Science) Integrable Systems in Contemporary Physics p.7/62

8 INTEGRABLE SYSTEMS Statistical Physics Quantum Field Theory *** Condensed Matter *** Atomic and Molecular Physics Integrable Systems in Contemporary Physics p.8/62

9 INTEGRABLE MODELS OF BEC INTRODUCTION THE MODEL FOR TWO COUPLED BEC The Hamiltonian for two coupled BEC Integrability and exact solution Definition of the regimes CLASSICAL DYNAMICS Equations of motion QUANTUM DYNAMICS Temporal evolution for different regimes OPEN PROBLEMS Integrable Systems in Contemporary Physics p.9/62

10 I - INTRODUCTION A. Einstein (1925) theoretical prediction of the existence of the condensate - all particles are in the same quantum state (coherent state) Integrable Systems in Contemporary Physics p.10/62

11 D.S. Durfee and W. Ketterle, Optics Express 2 (1998) 299 Integrable Systems in Contemporary Physics p.11/62

12 Since 1935: improving exp. techniques Ketterle et. al.(1995) experim. observ. D.S. Durfee and W. Ketterle, Optics Express 2 (1998) 299 Integrable Systems in Contemporary Physics p.12/62

13 Integrable Systems in Contemporary Physics p.13/62 D.S. Durfee and W. Ketterle, Optics Express 2 (1998) 299

14 JOSEPHSON TUNNELING AND SELF- TRAPPING M. Albiez, R. Gati et. al., cond/mat/ (2004) Integrable Systems in Contemporary Physics p.14/62

15 COLAPSE AND REVIVAL PATTERN M. R. Mathews, B.P. Anderson et al, Phys. Rev. Lett. 83 (1999) 3358 Integrable Systems in Contemporary Physics p.15/62

16 II - A MODEL FOR TWO COUPLED BEC H = k 8 (N 1 N 2 ) 2 µ 2 (N 1 N 2 ) E J 2 (a 1 a 2+a 2 a 1) N 1 = a 1 a 1: number of atoms in the well 1 N 2 = a 2 a 2: number of atoms in the well 2 k: atom-atom interaction term µ: chemical potential E J : tunneling strength G. Milburn et al, Phys. Rev.A55 (1997) 4318 A. Leggett, Rev. Mod. Phys.73 (2001) 307 S. Kohler and F. Sols, Phys. Rev. Lett.89 (2002) Integrable Systems in Contemporary Physics p.16/62

17 Schematic representation k/8 2 1 k/8 external potencial 1 2 external potential = 0 H = k 8 (N 1 N 2 ) 2 µ 2 (N 1 N 2 ) E J 2 (a 1 a 2 +a 2 a 1) Integrable Systems in Contemporary Physics p.17/62

18 Conserved quantities N = N 1 + N 2 [H,N] = 0 Symmetries E J E J ; a 1 a 1 ; a 2 a 2 µ µ ; a 1 a 2 If µ = 0 N 1 N 2 ; N 2 N 1 Integrable Systems in Contemporary Physics p.18/62

19 2. Integrability and exact solution: R-matrix: R(u) = b(u) c(u) 0 0 c(u) b(u) 0, b(u) = u u + η R-matrix satisfies the YBE c(u) = η u + η R 12 (x y)r 13 (x)r 23 (y) = R 23 (y)r 13 (x)r 12 (x y) Integrable Systems in Contemporary Physics p.19/62

20 Monodromy matrix: T(u) = ( ) A(u) B(u) C(u) D(u) R 12 (u v)t 13 (u)t 23 (v) = T 23 (v)t 13 (u)r 12 (u v) Integrable Systems in Contemporary Physics p.20/62

21 Suppose we have a realization: L(u) = π(t(u)) = L 1 (u + w)l 2 (v w) L i (u) = ( ) u + ηni a i a i η 1 i = 1, 2 R 12 (u v)l 1 (u)l 2 (v) = L 2 (v)l 1 (u)r 12 (u v) Integrable Systems in Contemporary Physics p.21/62

22 Transfer matrix τ(u) = π(tr(t(u))) = π(a(u) + D(u)) Integrability [τ(u),τ(v)] = 0 [H,τ(v)] = 0 H = κ (τ(u) 14 (τ (0)) 2 uτ (0) η 2 + w 2 u 2 ) Integrable Systems in Contemporary Physics p.22/62

23 with the identification: k 4 = κη2 2, µ 2 = κηw, E J 2 = κ H = k 8 (N 1 N 2 ) 2 µ 2 (N 1 N 2 ) E J 2 (a 1 a 2 +a 2 a 1) Integrable Systems in Contemporary Physics p.23/62

24 Applying the algebraic Bethe ansatz method: Energy: N ( E = κ(η η ) η2 N 2 uηn v i=1 i u 4 N ( u 2 η 2 + w 2 + (u 2 w 2 ) 1 η ) v i u ) ) i=1 Integrable Systems in Contemporary Physics p.24/62

25 u = w E = κ(η 2 N i=1 η 2 (v i w + η)(v i + w) η2 N 2 4 ηwn η 2 ) Integrable Systems in Contemporary Physics p.25/62

26 Bethe Ansatz Equations: η 2 (v 2 i w2 ) = N j i v i v j η v i v j + η Integrable Systems in Contemporary Physics p.26/62

27 3. Definition of regimes: k/e J 0 1 RR JR FR Rabi regime (RR) k/e j << 1/N Josephson reg.(jr) 1/N << k/e j << N Fock regime (FR) k/e j >> N. Integrable Systems in Contemporary Physics p.27/62

28 II - CLASSICAL DYNAMICS Replacing the operators a 1 and a 2 in H : Defining the variables: a 1 N 1 exp ( iφ 1 ) a 2 N 2 exp ( iφ 2 ) z = (N 1 N 2 )/N φ = N(φ 1 φ 2 )/2 Integrable Systems in Contemporary Physics p.28/62

29 New (rescaled) Hamiltonian ( µ = 0) H(z,φ) = E JN 2 ( λ 2 z2 ) 1 z 2 cos(2φ/n) λ = kn 2E J (z,φ) are canonically conjugate variables Integrable Systems in Contemporary Physics p.29/62

30 Equations of motion: φ = H z = E JN 2 ż = H φ = E J ( λz + ) z cos(2φ/n) 1 z 2 ( 1 z2 sin(2φ/n)) Integrable Systems in Contemporary Physics p.30/62

31 1 0,5 z 0-0,5 λ=1.9 λ= t There is a critical value for λ = 2 λ = kn 2E J = 2 k E J = 4 N Integrable Systems in Contemporary Physics p.31/62

32 II - QUANTUM DYNAMICS Standard procedure time evolution of any physical quantity is determined by the Temporal operator U {λ n } ; { ψ n } H ψ n = λ n ψ n Numerical analysis U = N n=0 e iλ nt ψ n ψ n Integrable Systems in Contemporary Physics p.32/62

33 Temporal evolution of any state ψ(t) = U φ = N n=0 a ne iλnt ψ n, a n = ψ n φ Expectation value of any operator A A = ψ(t) A ψ(t) Imbalance population A = (N 1 N 2 )/N Plot the time evolution of the expectation value of the imbalance population for different ratios of the coupling k/e J Integrable Systems in Contemporary Physics p.33/62

34 k/e J 0 1 RR JR FR k/e J = 1/N 2, 1/N, 1,N,N 2 initial state = φ = N, 0 Integrable Systems in Contemporary Physics p.34/62

35 k/e J 0 4/N 1 RR JR FR on the left: k/e J = 1/N, 2/N, 3/N, 4/N on the right: k/e J = 5/N, 10/N, 50/N, 1 Integrable Systems in Contemporary Physics p.35/62

36 CONCLUSIONS Threshold point k/e J 0 4/N Delocalised phase Self-trapped phase Integrable Systems in Contemporary Physics p.36/62

37 Open problems Quantum fluctuation; Entanglement; Energy gap for µ 0 AB model model for atom-molecule BEC Integrable Systems in Contemporary Physics p.37/62

38 A MODEL FOR ATOM- MOLECULE BEC: 1. The Hamiltonian: H = U a N 2 a + U b N 2 b + U ab N a N b + µ a N a + µ b N b + Ω(a a b + b aa) N a = a a: number of atoms; N b = b b: number of molecules; N = N a + 2N b : total atom number U a : atom-atom interaction strength; U b : molecule-molecule interaction strength; U ab : atom-molecule interaction strength; µ a,µ b : external potentials; Ω: amplitude for interconversion of atoms and molecules Integrable Systems in Contemporary Physics p.38/62

39 Collaborators - Dr. Jon Links, UQ-QLD-Australia - Dr. Arlei Prestes Tonel, UFRGS-RS-Brazil - Gilberto Nascimento Filho, UFRGS-RS-Brazil Integrable Systems in Contemporary Physics p.39/62

40 Publications A. Foerster, J. Links and H. Q. Zhou, "Exact Solvability in Contemporary Physics (2003), in Classical and Quantum nonlinear integrable systems, IOP-Publishing, edited by A. Kundu A.P. Tonel, J. Links and A. Foerster, J. Phys. A 38 (2005) 1235; A.P. Tonel, J. Links and A. Foerster, J. Phys. A 38 (2005) 6879; G. Santos, A.P. Tonel, A. Foerster and J. Links, "Classical and quantum dynamics of atom-molecule Bose-Einstein condensate", cond-mat/ (2005) Integrable Systems in Contemporary Physics p.40/62

41 INTEGRABLE SPIN LADDER SYSTEMS INTRODUCTION THE SU(4)-SPIN LADDER MODEL The Hamiltonian Integrability Exact solution and energy gap THERMODYNAMICAL PROPERTIES Partition function Comparison with experiments Integrable Systems in Contemporary Physics p.41/62

42 I - IMPORTANCE OF THE STUDY OF SPIN LADDERS: Some compounds have been realized experimentally with a ladder structure SrCu 2 O 3 La 1 x Sr x CuO 2, 5 Sr 14 x Ca x Cu 2 4O 41 Cu 2 (C 5 H 12 N 2 )Cl 4 CaV 2 O 5 KCuCl 3 Integrable Systems in Contemporary Physics p.42/62

43 Experiments that report on the spin gap: (magnetic susceptibility, NMR techniques) In some of these compounds superconductivity has been detected upon the introduction of hole carriers (chemical substitution) Integrable Systems in Contemporary Physics p.43/62

44 DEFINITION: Spin ladder structure: Example: number of legs L=3 rung leg n o of legs (L) << n o of rungs Integrable Systems in Contemporary Physics p.44/62 interpolates between 1 and 2 dimensions

45 Schematic Representation: L=2: SrCu 2 O 3 L = 3 : Sr 2 Cu 3 O 5 Azuma et al, PRL 94 Integrable Systems in Contemporary Physics p.45/62

46 2-leg ladder: Shows an exponential decay caused by the gap: χ = C e T T ; T 0 Integrable Systems in Contemporary Physics p.46/62

47 3-leg ladder: There is NO exponential decay: χ(t) tends to a finite number as T 0 Troyer et al, PRL54 (1996); Azuma et al, PRL73 (1994) Integrable Systems in Contemporary Physics p.47/62

48 II - AN INTEGRABLE SPIN LADDER MODEL H = L j=1 J l h j,j+1 + J r 2 L j=1 ( σ j. τ j 1). h j,j+1 = 1 4 (1 + σ j. σ j+1 ) (1 + τ j. τ j+1 ) σ j+1 σ j τ j+1 τ j Y.Wang. PRB60 (1999) Integrable Systems in Contemporary Physics p.48/62

49 INTEGRABILITY: for J r = 0, H can be derived from an R matrix that obeys YBA [J r -term,h] = 0 Integrable Systems in Contemporary Physics p.49/62

50 EXACT SOLUTION AND ENERGY GAP: Bethe Ansatz method : Energy eigenvalues: E = M 1 j=1 ( ) J l λ 2 j + 1/4 2J r (J l 2J r ) L with {λ j } solutions of the BAE Integrable Systems in Contemporary Physics p.50/62

51 BAE ( ) L λj i/2 = λ j + i/2 M 2 α=1 M 1 l j λ j λ l i λ j λ l + i λ j µ α + i/2 λ j µ α i/2 M 2 β α M1 µ α µ β i µ α µ β + i = j=1 M 3 δ=1 µ α λ j i/2 µ α λ j + i/2 µ α ν δ i/2 µ α ν δ + i/2 Integrable Systems in Contemporary Physics p.51/62

52 Energy Gap: M 3 γ δ M2 ν δ ν γ i ν δ ν γ + i = α=1 ν δ µ α i/2 ν δ µ α + i/2 Ground state E 0 : corresponds to the product of rung singlets reference state in the BAE {M 1 = M 2 = M 3 = 0} First elementary excitation E 1 : characterized by M 1 = 1,M 2 = M 3 = 0 corresponds to the solution λ = 0 in the BAE Gap: = E 1 E 0 = 2J r 4J l Integrable Systems in Contemporary Physics p.52/62

53 THERMODYNAMICAL PROPERTIES Hamiltonian + external field: H = L [ J l h j,j+1 + J r 2 ( σ j. τ j 1) h 2 j=1 ( σ z j + τ z j ) ] Partition function: Z = conf e βe Integrable Systems in Contemporary Physics p.53/62

54 Magnetization: M = 1 2 L i=1 < σ z i + τ z i >= 1 2Lβ h Ln(Z) Magnetic susceptibility χ = h M h=0 Integrable Systems in Contemporary Physics p.54/62

55 Comparison with experimental curves: Integrable Systems in Contemporary Physics p.55/62

56 (5IAP 2 CuBr 4 2H 2 O) C. Landee, PRB63 (2001); M.B., X.G, N.O., Z.T, A.F, PRL (2003) Integrable Systems in Contemporary Physics p.56/62

57 Cu 2 (C 5 H 12 N 2 ) 2 Cl 4 G. Chaboussant et al, PRL80 (1998); M.B., X.G, N.O., Z.T, A.F, PRL (2003) Integrable Systems in Contemporary Physics p.57/62

58 Cu 2 (C 5 H 12 N 2 ) 2 Cl 4 : specific heat M. Hagiwara et al, PRB62(2000); M.B., X.G, N.O., Z.T, A.F, PRL (2003) Integrable Systems in Contemporary Physics p.58/62

59 Collaborators - Dr. Murray Batchelor, ANU Canberra, Australia - Dr. Xiwen-Guan, ANU Canberra, Australia - Norman Oelkers, ANU Canberra, Australia - Dr. Jon Links, UQ, Australia - Dr. Mark Gould, UQ, Australia - Dr. Katrina Hibberd, UQ, Australia - Dr. Arlei Prestes Tonel, UFRGS-RS, Brasil - Dr. Silvio Dahmen, UFRGS-RS, Brasil - Dr. Itzhak Roditi, CBPF, Brasil - Dr. Andre Malvezzi, U. Bauru, SP, Brasil - Dr. K. Sakai, University of Tokyo, Japan - Dr. Zengo Tsuboi, University of Tokyo, Japan Integrable Systems in Contemporary Physics p.59/62

60 Publications J. Links and A. Foerster, Physical Review B62 (2000) 65 A. Foerster, K. Hibberd, J. Links, I. Roditi, Journal of Physics A34 (2001) L25-L29 A. P Tonel, A. Foerster, J. Links, A. Malvezzi, Physical Review B64 (2001) A. P. Tonel, S. Dahmen, A. Foerster, A. Malvezzi, Europhysics Letters64 (2003) 111 A. P. Tonel, A. Foerster, X. W. Guan, J. Links, Journal of Physics A36 (2003)359 M. Batchelor, X. W. Guan, A. Foerster, A. Tonel, H. Q. Zhou, Nuclear Physics B669 (2003) 385 M. Batchelor, X. W. Guan, N. Oelkers, A. Foerster, New Journal of Physics 5, (2003) 107 Integrable Systems in Contemporary Physics p.60/62

61 M. Batchelor, X. W. Guan, N. Oelkers, K. Z. Tsuboi, A. Foerster, Physical Review Letters 41 (2003) 67 Z. Ying, A. Foerster, X. W. Guan, I. Roditi, EPJB38, (2004) 535 Z. Ying, A. Foerster, X. W. Guan, I. Roditi, EPJB41, (2004) 67 Integrable Systems in Contemporary Physics p.61/62

62 " Integrable systems are relevant and can be solved, so why not do so and see what they tell us?" R. Baxter Integrable Systems in Contemporary Physics p.62/62