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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

slide2

Contents

I. Introduction

II. Brief Review of What we have done

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

IV. Summary

slide3

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.

slide4

Bethe ansatz

Critical phenomena

Quantum

Many-body systems

Exact

diagonalization

Quantum Phase

transitions

Group Methods

Methods used

slide5

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.

slide6

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.

slide7

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)

slide9

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

slide10

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

slide11

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.

slide13

Nearest Level Pairing Hamiltonian can be

written as

which is equivalent to the hard-core

Bose-Hubbard model in condensed

matter physics

slide14

Eigenstates for k-pair excitation can be expressed as

The excitation energy is

2n dimensional

n

slide15

232-239U

227-233Th

Binding Energies in MeV

238-243Pu

slide16

232-238U

227-232Th

First and second 0+ excited energy levels in MeV

238-243Pu

slide17

230-233Th

238-243Pu

odd-even mass differences

in MeV

234-239U

slide18

226-232Th

230-238U

236-242Pu

Moment of Inertia Calculated in the NLPM

slide21

Exact solution

Bethe Ansatz Wavefunction:

w

M

k

slide25

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

slide27

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:

slide28

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

slide29

Gaudin-Bose and Gaudin Fermi algebras

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.

slide31

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

slide32

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

slide33

Extensions for fermions and hard-core bosons:

GB or GF algebras

normalization

Commutation relation

slide34

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

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

with

slide35

Let

(C)

slide36

(C)

(D)

slide40

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.

slide45

SU(1,1) type

Nucl. Phys. A636 (1998)156

slide47

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

Phys. Rev. C66 (2002) 044134

slide49

Bose-Hubbard model

Phys. Lett. A339(2005)403

slide52

SU(2) and SU(1,1) mixed type

Phys. Lett. A341(2005)94

slide53

Eigen-energy:

Bethe Ansatz Equation:

slide54

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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