Exactly Solvable gl(m/n) Bose-Fermi Systemslapth.cnrs.fr/conferences/RAQIS/RAQIS05/TALKS/Pan.pdf ·...
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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 longtimesignature of significant physical phenomena.
3) Quest for Exact Solutions
To reveal non-perturbative and non-linear phenomenain understanding QPT as well as entanglement infinite (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 occupationprobabilities; 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-EinsteinCondensates, (e) t-J models for high Tcsuperconductors; (f) Holstein models.
All these model calculations are non-perturbative and highly non-linear. Insuch cases, Approximation approachesfail 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
−+∧
∑ ∑∑ −+Ω= εε
jε jε
jmm
mjmj
mjm
jmmj
aajS
aajS
∑
∑
>−
−−
+−
>
+−+
−=
−=
0
0
)()(
)()(21+=Ω jj
)ˆ(2
1)1(
2
1)(
0
0jjmjmjjm
mjm NaaaajS Ω−=−+= −
+−
>
+∑
Some Special Cases
='jjc {G 'jjcc
', jj∀
constant pairing
separablestrength pairing
cij=A δij + Ae-B(εi-ε
i-1)2 δij+1 + A e-B(ε
i-ε
i+1)2 δij-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 fordeformed nuclei
In the nearest level pairing interaction model:
cij=Gij=A δij + Ae-B(εi-ε
i-1)2 δij+1 + A e-B(ε
i-ε
i+1)2 δij-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
=
+=
== +++
ε2111
Nilsson s.p.
ii
i
iii
aab
aab
−
−
=
= +++[ ]
[ ]
[ ] jijji
jijji
iijji
bbN
bbN
Nbb
δ
δ
δ
−=
=
−=
++
+
,
)21( ,
,
)(2
1−−
++ +=ii
iii aaaaN
AG
Gt
Gtt
ii
iiiii
iiiiii
=
+=
== +++
ε2111
PbbPtH jji
iiji
i ∑∑ +∧
+=,
' ε
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
+++
〈〈〈∑= ξξ
k
k
kk
k
k
iii
iii
iii
ggg
ggg
ggg
ξξξ
ξξξ
ξξξ
...
...
...
21
22
2
2
1
11
2
1
1
∑∑=
+=k
jjjk
jEE1
)(')( ξξ ε
ppp
ijj
ij gEgt ξξξ )(~
=∑
Eigenstates for k-pair excitation can be expressed as
The excitation energy is
AG
Gt
Gtt
ii
iiiii
iiiiii
=
+=
== +++
ε2111
2n dimensional n
Binding Energies in MeV
227-233Th 232-239U
238-243Pu
227-232Th 232-238U
238-243Pu
First and second 0+ excitedenergy 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 plusSolvable mean-field plusextended pairing modelextended pairing model
×−−= ∑∑ ∑∞
==
+
2)!(1
'1 '
2ˆ
µµ
ε GaaGnH j
p
j jjjjj
µµµµ
µ
2212
221
1......
...iiiii
iiii aaaaaa
++
++
≠≠≠
+∑
Different pair-hopping structures in the constantpairing and the extended pairing models
0,...,,| 21 >=mi jjja
>=> +++
≤≤≤≤≤∑ miii
piiiiiim jjjkaaaCjjjk
k
k
k,...,,;,|...,,,;,| 21
...1
)(...21 21
21
21ςς ςL
∑=
=
−k
ik
xiiiC
1
)(211
1)(...
µµ
ς ε
ς
Bethe Ansatz Wavefunction:
Exact solution
Mkw
)0|...0;,(|0;,|21
21
)(
...1
2 >−>=> +++
≤≤≤≤≤∑∑ k
k
iiipiii
xj
jj aaakkn ςςε ς
>−+>
>+
+
≤≤≤≤≤
++
≤≤≤≤≤
+
≠≠≠
++∞
=
+
∑∑
∑∑ ∑
=
++
0;,|)1(0|...
0;,|......
...1
)(...
...1
...1)!(1
21
2121
21
221
221
212
)(
)(
ς
ς
ς
µµ µµµµ
µ
kkaaaC
kaaaaaaaa
k
k
k
k
ipiiiiiiii
piii
iiiiiii
iij
jj
µµµµ
µ
µµ 22121
221
2 ......,)!(1
,1 iiiiii
iiij
jii aaaaaaVaaV
++
+++
≠≠≠
+ ∑∑ ==L
><><
=totalV
VR µµ
Higher Order Terms
Ratios: Rµ = <Vµ> / < 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. Forsimplicity, we assume
where bi, fi satify the following commutation [.,.]- oranti-commutation [.,.]+ relations:
Using these operators, one can construct generators of the Liesuperalgebra 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-Fermialgebra 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 underthe Bethe ansatz waefunction
The energy eigenvalues are given by
BAEs
Next, we assume that there are m non-degenerate boson levelsεi (i = 1; 2,..,m) and n non-degenerate fermion levels withenergies εi (i = m + 1,m + 2,…,m + n). Using the sameprocedure, one can prove that a Hamiltonian constructed byusing 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 ofcommutative pairwise operators,
Let Sτ be the permutation group operating among theindices.
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-FermiHamiltonian and a class of hard-core gl(m/n) Bose-FermiHamiltonians with high order interaction terms are exactlysolvable.
(2) Excitation energies and corresponding wavefunctions canbe obtained by using a simple algebraic Bethe ansatz, whichprovide with new classes of solvable models with dynamicalSUSY. (3) The results should be helpful in searching for other exactlysolvable SUSY quantum many-body models andunderstanding the nature of the exactly or quasi-exactlysolvability. It is obvious that such Hamiltonians with only Boseor Fermi sectors are also exactly solvable by using the sameapproach.
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 todetermine 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)(
)( −−= kGx
Ek ςς
0
1
)(21
)(
1...1
2 =∑
+
=
−≤≤≤≤≤∑ k
ikx
G
piiix
µµ
ςς
ε
>=> +++
≤≤≤≤≤∑ miii
piiiiiim jjjkaaaCjjjk
k
k
k,...,,;,|...,...,,;,| 21
...1
)(...21 21
21
21ςς ς
Eigen-energy:
Bethe Ansatz Equation:
Energies as functions of G for k=5 with p=10 levels
ε1=1.179
ε2=2.650
ε3=3.162
ε4=4.588
ε5=5.006ε6=6.969
ε7=7.262
ε8=8.687ε9=9.899ε10=10.20