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