Loading in 2 Seconds...

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

Loading in 2 Seconds...

- 66 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' Exactly Solvable gl(m/n) Bose-Fermi Systems Feng Pan, Lianrong Dai, and J. P. Draayer' - blithe

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

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

Odd A

Theory

Experiment

“Figure 3”

Even A

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

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

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.

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

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

Let Sbe the permutation group operating among the indices.

with

(C)

(D)

(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.

SU(2) type

Nucl. Phys. A636 (1998)156

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

Phys. Rev. C66 (2002) 044134

Phys. Lett. A339(2005)403

Phys. Lett. A341(2005)94

Bethe Ansatz Equation:

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

Download Presentation

Connecting to Server..