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