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Liaoning Normal Univ. Dalian 116029 China. Louisiana State Univ. Baton Rouge 70803 USA. Exactly Solvable gl(m/n) Bose-Fermi Systems Feng Pan, Lianrong Dai, and J. P. Draayer. Recent Advances in Quantum Integrable Systems,Sept. 6-9,05 Annecy, France

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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

Exactly Solvable gl(m/n) Bose-Fermi Systems

Feng Pan, Lianrong Dai, andJ. P. Draayer

Recent Advances in Quantum Integrable Systems,Sept. 6-9,05 Annecy, France

Dedicated to Dr. Daniel Arnaudon


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Contents

I. Introduction

II. Brief Review of What we have done

III.Algebraic solutions of a gl(m/n) Bose-Fermi Model

IV. Summary


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

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.


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Bethe ansatz

Critical phenomena

Quantum

Many-body systems

Exact

diagonalization

Quantum Phase

transitions

Group Methods

Methods used


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

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.


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

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.


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

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)



Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Some Special Cases

constantpairing

separable strength pairing

cij=A ij+ Ae-B(i-i-1)2ij+1 + Ae-B(i-i+1)2ij-1

nearestlevel pairing


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

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


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

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 + Ae-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

Nilsson s.p.


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Nearest Level Pairing Hamiltonian can be

written as

which is equivalent to the hard-core

Bose-Hubbard model in condensed

matter physics


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Eigenstates for k-pair excitation can be expressed as

The excitation energy is

2n dimensional

n


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

232-239U

227-233Th

Binding Energies in MeV

238-243Pu


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

232-238U

227-232Th

First and second 0+ excited energy levels in MeV

238-243Pu


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

230-233Th

238-243Pu

odd-even mass differences

in MeV

234-239U


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

226-232Th

230-238U

236-242Pu

Moment of Inertia Calculated in the NLPM




Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Exact solution and the extended pairing models

Bethe Ansatz Wavefunction:

w

M

k


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Higher Order Terms and the extended pairing models


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Ratios: and the extended pairing modelsR = <V> / < Vtotal>


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Even-Odd Mass Differences and the extended pairing models

Odd A

Theory

Experiment

“Figure 3”

Even A

P(A) =E(A)+E(A-2)- 2E(A-1) for 154-171Yb


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

6 and the extended pairing models


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

III. Algebraic solutions of a and the extended pairing modelsgl(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:


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

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


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Gaudin-Bose and Gaudin Fermi algebras Lie superalgebra gl(m/n) with

Let

be a set of independent real parameters with

and

for

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.


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

These operators satisfy the following relations: Lie superalgebra gl(m/n) with

(A)


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Using (A) one can prove that the Hamiltonian Lie superalgebra gl(m/n) with

(B)

where G is a real parameter, is exactly diagonalized under the Bethe ansatz waefunction

The energy eigenvalues are given by

BAEs


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Next, we assume that there are Lie superalgebra gl(m/n) withm 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


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Extensions for fermions and hard-core bosons: Lie superalgebra gl(m/n) with

GB or GF algebras

normalization

Commutation relation


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Using the normalized operators, we may construct a set of commutative pairwise operators,

Let Sbe the permutation group operating among the indices.

with


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Let commutative pairwise operators,

(C)


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

(C) commutative pairwise operators,

(D)


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Similarly, we have commutative pairwise operators,


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

The k-pair excitation energies are given by commutative pairwise operators,


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

In summary commutative pairwise operators,

(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.


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Thank You ! commutative pairwise operators,


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Phys. Lett. B422(1998)1 commutative pairwise operators,


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Phys. Lett. B422(1998)1 commutative pairwise operators,

SU(2) type


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Nucl. Phys. A636 (1998)156 commutative pairwise operators,


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

SU(1,1) type commutative pairwise operators,

Nucl. Phys. A636 (1998)156


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Phys. Rev. C66 (2002) 044134 commutative pairwise operators,


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Sp(4) Gaudin algebra with complicated Bethe ansatz Equations to determine the roots.

Phys. Rev. C66 (2002) 044134


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Phys. Lett. A339(2005)403 to determine the roots.


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Bose-Hubbard model to determine the roots.

Phys. Lett. A339(2005)403


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Phys. Lett. A341(2005)291 to determine the roots.


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Phys. Lett. A341(2005)94 to determine the roots.


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

SU(2) and SU(1,1) mixed type to determine the roots.

Phys. Lett. A341(2005)94


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

Eigen-energy: to determine the roots.

Bethe Ansatz Equation:


Exactly solvable gl m n bose fermi systems feng pan lianrong dai and j p draayer

to determine the roots.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