1. Exactly Solvable gl(m/n) Bose-Fermi Systems Feng Pan, Lianrong Dai, and J. P. Draayer Liaoning Normal Univ. Dalian 116029 China Recent Advances in Quantum Integrable Systems,Sept. 6-9,05 Annecy, France Dedicated to Dr. Daniel Arnaudon Louisiana State Univ. Baton Rouge 70803 USA

2. I. Introduction II. Brief Review of What we have done III. Algebraic solutions of a gl(m/n) Bose-Fermi Model IV. Summary Contents

3. Introduction: Research Trends 1) Large Scale Computation (NP problems) Specialized computers (hardware & software), quantum computer? 2)Search for New Symmetries Relationship to critical phenomena, a longtime signature of significant physical phenomena. 3) Quest for Exact Solutions To reveal non-perturbative and non-linear phenomena in understanding QPT as well as entanglement in finite (mesoscopic) quantum many-body systems.

4. Exact diagonalization Group Methods Bethe ansatz Quantum Many-body systems Methods used Quantum Phase transitions Critical phenomena

5. Goals: 1) Excitation energies; wave-functions; spectra; correlation functions; fractional occupation probabilities; etc. 2) Quantum phase transitions, critical behaviors in mesoscopic systems, such as nuclei. 3) (a) Spin chains; (b) Hubbard models, (c) Cavity QED systems, (d) Bose-Einstein Condensates, (e) t-J models for high Tc superconductors; (f) Holstein models.

6. All these model calculations are non- perturbative and highly non-linear. In such cases, Approximation approaches fail to provide useful information. Thus, exact treatment is in demand.

7. (1) Exact solutions of the generalized pairing (1998)Exact solutions of the generalized pairing (1998) (3) Exact solutions of the SO(5) T=1 pairing (2002)Exact solutions of the SO(5) T=1 pairing (2002) (2) Exact solutions of the U(5)-O(6) transition (1998) (4) Exact solutions of the extended pairing (2004)Exact solutions of the extended pairing (2004) (5) Quantum critical behavior of two coupled BEC (2005) Quantum critical behavior of two coupled BEC (2005) (6) QPT in interacting boson systems (2005)QPT in interacting boson systems (2005) II. Brief Review of What we have done (7) An extended Dicke model (2005)An extended Dicke model (2005)

8. General Pairing Problem

9. Some Special Cases constant pairing separable strength pairing c ij =A  ij + Ae -B(  i -  i-1 ) 2  ij+1 + A e -B(  i -  i+1 ) 2  ij-1 nearest level pairing

10. Exact solution for Constant Pairing Interaction [1] Richardson R W 1963 Phys. Lett. 5 82 [2] Feng Pan and Draayer J P 1999 Ann. Phys. (NY) 271 120

11. Nearest Level Pairing Interaction for deformed nuclei In the nearest level pairing interaction model: c ij =G ij =A  ij + Ae -B(  i -  i-1 ) 2  ij+1 + A e -B(  i -  i+1 ) 2  ij-1 [9] Feng Pan and J. P. Draayer, J. Phys. A33 (2000) 9095 [10] Y. Y. Chen, Feng Pan, G. S. Stoitcheva, and J. P. Draayer, Int. J. Mod. Phys. B16 (2002) 2071 Nilsson s.p.

13. Nearest Level Pairing Hamiltonian can be written as which is equivalent to the hard-core Bose-Hubbard model in condensed matter physics

14. Eigenstates for k-pair excitation can be expressed as The excitation energy is 2 n dimensional n

15. Binding Energies in MeV 227-233 Th 232-239 U 238-243 Pu

16. 227-232 Th 232-238 U 238-243 Pu First and second 0 + excited energy levels in MeV

17. 230-233 Th 238-243 Pu 234-239 U odd-even mass differences in MeV

18. 226-232 Th 230-238 U 236-242 Pu Moment of Inertia Calculated in the NLPM

19. Solvable mean-field plus extended pairing model

20. Different pair-hopping structures in the constant pairing and the extended pairing models

21. Bethe Ansatz Wavefunction: Exact solution Mk w

23. Higher Order Terms

24. Ratios: R  = /

25. P(A) =E(A)+E(A-2)- 2E(A-1) for 154-171 Yb Theory Experiment “Figure 3” Even A Odd A Even-Odd Mass Differences

26. 6 6

27. III. Algebraic solutions of a gl(m/n) Bose-Fermi Model Let and A i be operator of creating and annihilating a boson or a fermion in i-th level. For simplicity, we assume where b i, f i satify the following commutation [.,.] - or anti-commutation [.,.] + relations:

28. Using these operators, one can construct generators of the Lie superalgebra gl(m/n) with for 1 i, j m+n, satisfying the graded commutation relations where and

29. Gaudin-Bose and Gaudin Fermi algebras Let be a set of independent real parameters with for and One can construct the following Gaudin-Bose or Gaudin-Fermi algebra with where O j =b j or f j for Gaudin-Bose or Gaudin-Fermi algebra, and x is a complex parameter.

30. These operators satisfy the following relations: (A)

31. Using (A) one can prove that the Hamiltonian (B) where G is a real parameter, is exactly diagonalized under the Bethe ansatz waefunction The energy eigenvalues are given by BAEs

32. Next, we assume that there are m non-degenerate boson levels  i (i = 1; 2,..,m) and n non-degenerate fermion levels with energies  i (i = m + 1,m + 2,…,m + n). Using the same procedure, one can prove that a Hamiltonian constructed by using the generators E ij with is also solvable with BAEs

33. Extensions for fermions and hard-core bosons: GB or GF algebras normalization Commutation relation

34. Using the normalized operators, we may construct a set of commutative pairwise operators, Let S  be the permutation group operating among the indices. with

35. Let (C)

36. (D)

38. Similarly, we have

39. The k-pair excitation energies are given by

40. In summary (1) it is shown that a simple gl(m/n) Bose-Fermi Hamiltonian and a class of hard-core gl(m/n) Bose-Fermi Hamiltonians with high order interaction terms are exactly solvable. (2) Excitation energies and corresponding wavefunctions can be obtained by using a simple algebraic Bethe ansatz, which provide with new classes of solvable models with dynamical SUSY. (3) The results should be helpful in searching for other exactly solvable SUSY quantum many-body models and understanding the nature of the exactly or quasi-exactly solvability. It is obvious that such Hamiltonians with only Bose or Fermi sectors are also exactly solvable by using the same approach.

41. Thank You !

42. Phys. Lett. B422(1998)1

43. SU(2) type Phys. Lett. B422(1998)1

44. Nucl. Phys. A636 (1998)156

45. SU(1,1) type Nucl. Phys. A636 (1998)156

46. Phys. Rev. C66 (2002) 044134

47. Sp(4) Gaudin algebra with complicated Bethe ansatz Equations to determine the roots. Phys. Rev. C66 (2002) 044134

48. Phys. Lett. A339(2005)403

49. Bose-Hubbard model Phys. Lett. A339(2005)403

50. Phys. Lett. A341(2005)291

51. Phys. Lett. A341(2005)94

52. SU(2) and SU(1,1) mixed type Phys. Lett. A341(2005)94

53. Eigen-energy: Bethe Ansatz Equation:

54. Energies as functions of G for k=5 with p=10 levels  1 =1.179  2 =2.650  3 =3.162  4 =4.588  5 =5.006  6 =6.969  7 =7.262  8 =8.687  9 =9.899  10 =10.20