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Exactly Solvable gl(m/n) Bose-Fermi Systems Feng Pan, Lianrong Dai, and J. P. DraayerPowerPoint Presentation

Exactly Solvable gl(m/n) Bose-Fermi Systems Feng Pan, Lianrong Dai, and J. P. Draayer

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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, andJ. P. Draayer

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

Dedicated to Dr. Daniel Arnaudon

I. Introduction

II. Brief Review of What we have done

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

IV. Summary

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.

Critical phenomena

Quantum

Many-body systems

Exact

diagonalization

Quantum Phase

transitions

Group Methods

Methods used

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.

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)

constantpairing

separable strength pairing

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

nearestlevel 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)2ij+1 + Ae-B(i-i+1)2ij-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.

Nearest Level Pairing Hamiltonian can be

written as

which is equivalent to the hard-core

Bose-Hubbard model in condensed

matter physics

Different pair-hopping structures in the constant pairing and the extended pairing models

Higher Order Terms and the extended pairing models

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

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

6 and the extended pairing models

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:

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 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.

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

(A)

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

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

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

GB or GF algebras

normalization

Commutation relation

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

Let Sbe the permutation group operating among the indices.

with

Let commutative pairwise operators,

(C)

(C) commutative pairwise operators,

(D)

Similarly, we have commutative pairwise operators,

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

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.

Thank You ! commutative pairwise operators,

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

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

SU(2) type

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

SU(1,1) type commutative pairwise operators,

Nucl. Phys. A636 (1998)156

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

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

Phys. Rev. C66 (2002) 044134

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

Bose-Hubbard model to determine the roots.

Phys. Lett. A339(2005)403

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

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

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

Phys. Lett. A341(2005)94

Eigen-energy: to determine the roots.

Bethe Ansatz Equation:

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

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