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Exactly Solvable gl(m/n) Bose-Fermi Systems Feng Pan, Lianrong Dai, and J. P. Draayer

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

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Exactly Solvable gl(m/n) Bose-Fermi Systems Feng Pan, Lianrong Dai, and J. P. Draayer

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

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

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

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

  8. General Pairing Problem

  9. Some Special Cases constantpairing separable strength pairing cij=A ij+ Ae-B(i-i-1)2ij+1 + Ae-B(i-i+1)2ij-1 nearestlevel 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: cij=Gij=A ij+ Ae-B(i-i-1)2ij+1 + Ae-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.

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

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

  14. 232-239U 227-233Th Binding Energies in MeV 238-243Pu

  15. 232-238U 227-232Th First and second 0+ excited energy levels in MeV 238-243Pu

  16. 230-233Th 238-243Pu odd-even mass differences in MeV 234-239U

  17. 226-232Th 230-238U 236-242Pu Moment of Inertia Calculated in the NLPM

  18. Solvable mean-field plus extended pairing model

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

  20. Exact solution Bethe Ansatz Wavefunction: w M k

  21. Higher Order Terms

  22. Ratios: R = <V> / < Vtotal>

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

  24. 6

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

  26. 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

  27. Gaudin-Bose and Gaudin Fermi algebras Let be a set of independent real parameters with and for One can construct the following Gaudin-Bose or Gaudin-Fermi algebra with where Oj=bj or fj for Gaudin-Bose or Gaudin-Fermi algebra, and x is a complex parameter.

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

  29. 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

  30. 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 Eij with is also solvable with BAEs

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

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

  33. Let (C)

  34. (C) (D)

  35. Similarly, we have

  36. The k-pair excitation energies are given by

  37. 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.

  38. Thank You !

  39. Phys. Lett. B422(1998)1

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

  41. Nucl. Phys. A636 (1998)156

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

  43. Phys. Rev. C66 (2002) 044134

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

  45. Phys. Lett. A339(2005)403

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

  47. Phys. Lett. A341(2005)291

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