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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
Recent Advances in Quantum Integrable Systems,Sept. 6-9,05 Annecy, France
Dedicated to Dr. Daniel Arnaudon
Contents
I. Introduction
II. Brief Review of What we have done
III. Algebraic solutions of a gl(m/n) Bose-Fermi Model
IV. Summary
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.
Bethe ansatz
Exact
diagonalization
Group Methods
Critical phenomena
Quantum
Many-body systems
Quantum Phase
transitions
Methods used
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 nonperturbative and highly non-linear. In
such cases, Approximation approaches
fail to provide useful information. Thus,
exact treatment is in demand.
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)
General Pairing Problem
H j j 2 j S ( j ) c jj ' S ( j ) S ( j )
0
j
j
S ( j ) ( )
j m
S ( j ) ( )
j m
m 0
m 0
jj '
jm
a a
j
j m
a j m a jm
j j 1 2
1
1 ˆ
S ( j ) (a jma jm a j m a j m 1) ( N j j )
2 m 0
2
0
Some Special Cases
constant pairing
G j ,
c jj ' c c
j j'
{
j'
separable strength
pairing
2
2
-B(
-
)
-B(
-
)
cij=A ij + Ae i i-1 ij+1 + A e i i+1 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:
2
2
-B(
-
)
-B(
-
)
cij=Gij=A ij + Ae i i-1 ij+1 + A e i i+1 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
tii 1 ti 1i Gii 1
Nilsson s.p.
tii 2 i Gii
Gii A
1
N i (ai ai a a )
i i
2
b , b
j
i
N
i
N
i,
,b
j
bj
ij (1 2 N i )
ij b
j
ij b j
i
b a a
i
bi a ai
tii 1 ti 1i Gii 1
tii 2 i Gii
Gii A
i
i
Nearest Level Pairing Hamiltonian can be
written as
H i Pt b b j P
ij i
'
i
i, j
which is equivalent to the hard-core
Bose-Hubbard model in condensed
matter physics
Eigenstates for k-pair excitation can be expressed as
k ; , (n j1 , n j2 ,..., n jr )n f
( )
C
b
b
...
b
i1i2 ...ik i1 i2 ik (n j1 , n j2 ,..., n jr )n f
i1 i2 ... ik
The excitation energy is
( )
k
E
k
j E
'
j
g i11 g i21 ...g ik1
( j )
g i1 2 g i22 ...g ik2
j 1
g i1 k g i2k ...g ikk
tii 1 ti 1i Gii 1
tii 2 i Gii
Gii A
~
t
j
2n dimensional
n
ij
g
p
j
E
( p )
g
p
i
227-233Th
232-239U
Binding Energies in MeV
238-243Pu
227-232Th
232-238U
First and second 0+ excited
energy levels in MeV
238-243Pu
230-233Th
238-243Pu
odd-even mass differences
234-239U
in MeV
226-232Th
230-238U
236-242Pu
Moment of Inertia Calculated in the NLPM
Solvable mean-field plus
extended pairing model
p
1
ˆ
H j n j G a j a j ' G ( !)2
j 1
jj '
a
i1 i2
i1 i2 ... i2
i
2
a ...a ai 1 ai 2 ...ai2
Different pair-hopping structures in the constant
pairing and the extended pairing models
Exact solution
Bethe Ansatz Wavefunction:
| k , ; j1, j2 ,, jm
( )
C
a
a
...
a
kkw
, ; j1, j2 ,..., jm
i1i2 ...ik i1 i2 ik |M
1i1 i2 ...ik p
ai | j1 , j2 ,..., jm 0
( )
i1i2 ...ik
C
1
1 x ( )
k
i
1
2
n
|
k
,
;
0
( ) (| k , ;0
jj
x
ai1 ai2 ... aik | 0 )
1i1 i2 ...ik p
j
( a a
j
j
j
(
1
1
( !) 2
1i1 i2 ...ik p
)
i1 i2 ...i2
( )
i1i2 ...ik
C
a
a
...
a
i1 i2 i ai1 ai2 ...ai2 | k , ;0
) a a ...a
i1 i2
1i1 i2 ...ik p
ik
| 0 (k 1) | k , ;0
Higher Order Terms
V1 ai a j , V ( 1!)2
i, j
R
i1 i2 i2
V
ai1 ai2 ...ai ai 1 ai 2 ...ai2
Vtotal
Ratios: R = <V> / < Vtotal>
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
6
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
where
m+n, satisfying the graded commutation relations
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
Phys. Lett. B422(1998)1
SU(2) type
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 type
Phys. Lett. A341(2005)94
E
Eigen-energy:
( )
k
2
( ) G (k 1)
x
Bethe Ansatz Equation:
2
x
( )
G
( )
1
x
1i1 i2 ...ik p
| k , ; j1 , j2 ,..., jm
k
i
0
1
( )
C
a
a
...
a
i1i2 ...ik i1 i2 ik | k, ; j1, j2 ,..., jm
1i1 i2 ...ik p
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
Energies as functions of G for k=5 with p=10 levels