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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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
II. Brief Review of What we have done
III.Algebraic solutions of a gl(m/n) Bose-Fermi Model
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.
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.
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.
(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)
separable strength pairing
cij=A ij+ Ae-B(i-i-1)2ij+1 + Ae-B(i-i+1)2ij-1
 Richardson R W 1963 Phys. Lett.5 82
 Feng Pan and Draayer J P 1999 Ann. Phys. (NY) 271 120
In the nearest level pairing interaction model:
cij=Gij=A ij+ Ae-B(i-i-1)2ij+1 + Ae-B(i-i+1)2ij-1
 Feng Pan and J. P. Draayer, J. Phys. A33 (2000) 9095
 Y. Y. Chen, Feng Pan, G. S. Stoitcheva, and J. P. Draayer,
Int. J. Mod. Phys. B16 (2002) 2071
which is equivalent to the hard-core
Bose-Hubbard model in condensed
The excitation energy is
Binding Energies in MeV
First and second 0+ excited energy levels in MeV
odd-even mass differences
Moment of Inertia Calculated in the NLPM
Different pair-hopping structures in the constant pairing and the extended pairing models
Exact solution and the extended pairing models
Bethe Ansatz Wavefunction:
Higher Order Terms and the extended pairing models
Ratios: and the extended pairing modelsR = <V> / < Vtotal>
Even-Odd Mass Differences and the extended pairing models
P(A) =E(A)+E(A-2)- 2E(A-1) for 154-171Yb
6 and the extended pairing models
III. Algebraic solutions of a and the extended pairing modelsgl(m/n) Bose-Fermi Model
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:
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
Gaudin-Bose and Gaudin Fermi algebras Lie superalgebra gl(m/n) with
be a set of independent real parameters with
construct the following Gaudin-Bose or Gaudin-Fermi
where Oj=bj or fj for Gaudin-Bose or Gaudin-Fermi algebra,
and x is a complex parameter.
These operators satisfy the following relations: Lie superalgebra gl(m/n) with
Using (A) one can prove that the Hamiltonian Lie superalgebra gl(m/n) with
where G is a real parameter, is exactly diagonalized under the Bethe ansatz waefunction
The energy eigenvalues are given by
Next, we assume that there are Lie superalgebra gl(m/n) withm 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
Extensions for fermions and hard-core bosons: Lie superalgebra gl(m/n) with
GB or GF algebras
Using the normalized operators, we may construct a set of commutative pairwise operators,
Let Sbe the permutation group operating among the indices.
Let commutative pairwise operators,
(C) commutative pairwise operators,
Similarly, we have commutative pairwise operators,
The k-pair excitation energies are given by commutative pairwise operators,
In summary commutative pairwise operators,
(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.
Thank You ! commutative pairwise operators,
Phys. Lett. B422(1998)1 commutative pairwise operators,
Phys. Lett. B422(1998)1 commutative pairwise operators,
Nucl. Phys. A636 (1998)156 commutative pairwise operators,
SU(1,1) type commutative pairwise operators,
Nucl. Phys. A636 (1998)156
Phys. Rev. C66 (2002) 044134 commutative pairwise operators,
Sp(4) Gaudin algebra with complicated Bethe ansatz Equations to determine the roots.
Phys. Rev. C66 (2002) 044134
Phys. Lett. A339(2005)403 to determine the roots.
Bose-Hubbard model to determine the roots.
Phys. Lett. A339(2005)403
Phys. Lett. A341(2005)291 to determine the roots.
Phys. Lett. A341(2005)94 to determine the roots.
SU(2) and SU(1,1) mixed type to determine the roots.
Phys. Lett. A341(2005)94
Eigen-energy: to determine the roots.
Bethe Ansatz Equation:
to determine the roots.1=1.179
Energies as functions of G for k=5 with p=10 levels