exactly solvable gl(m/n) bose-fermi systems feng pan, lianrong dai, and j. p. draayer
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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
Recent Advances in Quantum Integrable Systems,Sept. 6-9,05 Annecy, France
Dedicated to Dr. Daniel Arnaudon
Louisiana State Univ. Baton Rouge 70803 USA
I. Introduction
II. Brief Review of What we have done
III. Algebraic solutions of a gl(m/n) Bose-Fermi Model
IV. Summary
Contents
Introduction: Research Trends1) 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.
Exact diagonalization
Group Methods
Bethe ansatz
Quantum
Many-body systems
Methods used
Quantum Phase
transitions
Critical phenomena
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.
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)
(3) 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)
(5) Quantum critical behavior of two coupled BEC (2005)
(6) QPT in interacting boson systems (2005)
II. Brief Review of What we have done
(7) An extended Dicke model (2005)
General Pairing Problem
)()()(2'
'0 jSjScjSH
j jjjjjj
jj
j j
jmm
mjmj
mjm
jmmj
aajS
aajS
0
0
)()(
)()(21 jj
)ˆ(2
1)1(
2
1)(
0
0jjmjmjjm
mjm NaaaajS
Some Special Cases
'jjc {G'jj
cc', jj
constant pairing
separable strength pairing
cij=A ij + Ae-B(i-
i-1)2 ij+1 + A e
-B(i-
i+1)2 ij-1
nearest level pairing
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
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 + 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
AG
Gt
Gtt
ii
iiiii
iiiiii
2111
Nilsson s.p.
ii
i
iii
aab
aab
jijji
jijji
iijji
bbN
bbN
Nbb
,
)21( ,
,
)(2
1
ii
iii aaaaN
AG
Gt
Gtt
ii
iiiii
iiiiii
2111
PbbPtH jji
iiji
i
,
'
Nearest Level Pairing Hamiltonian can be
written as
which is equivalent to the hard-core
Bose-Hubbard model in condensed
matter physics
),...,,(... ),...,,(,;2121
21
2121...
)(... fjjjiii
iiiiiifjjj nnnnbbbCnnnnk
rk
k
kr
k
k
kk
k
k
iii
iii
iii
ggg
ggg
ggg
...
...
...
21
22
2
2
1
11
2
1
1
k
jjjk
jEE1
)(')(
ppp
ijj
ij gEgt )(~
Eigenstates for k-pair excitation can be expressed as
The excitation energy is
AG
Gt
Gtt
ii
iiiii
iiiiii
2111
2n dimensional n
Binding Energies in MeV
227-233Th232-239U
238-243Pu
227-232Th 232-238U
238-243Pu
First and second 0+ excited energy levels in MeV
230-233Th 238-243Pu
234-239U
odd-even mass differences
in MeV
226-232Th 230-238U
236-242Pu
Moment of Inertia Calculated in the NLPM
Solvable mean-field plus Solvable mean-field plus extended pairing modelextended pairing model
2)!(
1'
1 '2
ˆ
GaaGnH j
p
j jjjjj
2212
221
1......
...iiiii
iiii aaaaaa
Different pair-hopping structures in the constant pairing and the extended pairing models
0,...,,| 21 mi jjja
miii
piiiiiim jjjkaaaCjjjk
k
k
k,...,,;,|...,,,;,| 21
...1
)(...21 21
21
21
k
ik
xiiiC
1
)(211
1)(...
Bethe Ansatz Wavefunction:
Exact solution
Mkw
)0|...0;,(|0;,|21
21
)(
...1
2
k
k
iiipiii
xj
jj aaakkn
0;,|)1(0|...
0;,|......
...1
)(...
...1
...1)!(
1
21
2121
21
221
221
212
)(
)(
kkaaaC
kaaaaaaaa
k
k
k
k
ipiii
iiiiipiii
iiiiiii
iij
jj
22121
221
2 ......,)!(
1
,1 iiiiii
iiij
jii aaaaaaVaaV
totalV
VR
Higher Order Terms
Ratios: R = <V> / < Vtotal>
P(A) =E(A)+E(A-2)- 2E(A-1) for 154-171Yb
Theory
Experiment
“Figure 3”
Even A
Odd A
Even-Odd Mass Differences
66
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:
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
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 Oj=bj or fj for Gaudin-Bose or Gaudin-Fermi algebra,
and x is a complex parameter.
These operators satisfy the following relations:
(A)
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
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
Extensions for fermions and hard-core bosons:
GB or GF algebras
normalization
Commutation relation
Using the normalized operators, we may construct a set of commutative pairwise operators,
Let S be the permutation group operating among the indices.
with
Let
(C)
(C)
(D)
Similarly, we have
The k-pair excitation energies are given by
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.
Thank You !
Phys. Lett. B422(1998)1
SU(2) type
Phys. Lett. B422(1998)1
Nucl. Phys. A636 (1998)156
SU(1,1) type
Nucl. Phys. A636 (1998)156
Phys. Rev. C66 (2002) 044134
Sp(4) Gaudin algebra with complicated Bethe ansatz Equations to determine the roots.
Phys. Rev. C66 (2002) 044134
Phys. Lett. A339(2005)403
Bose-Hubbard model
Phys. Lett. A339(2005)403
Phys. Lett. A341(2005)291
Phys. Lett. A341(2005)94
SU(2) and SU(1,1) mixed typePhys. Lett. A341(2005)94
)1(2
)()( kG
xEk
0
1
)(21
)(
1...1
2
k
ikx
G
piiix
miii
piiiiiim jjjkaaaCjjjk
k
k
k,...,,;,|...,...,,;,| 21
...1
)(...21 21
21
21
Eigen-energy:
Bethe Ansatz Equation:
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=5.0066=6.969
7=7.262
8=8.6879=9.89910=10.20
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