On Solutions of the one-dimensional Holstein Model

Feng Pan and J. P. Draayer

Liaoning Normal Univ. Dalian 116029 China

23rd International Conference on DGM in Theoretical Physics, Aug. 20-26,05 Tianjin

Louisiana State Univ. Baton Rouge 70803 USA

I. Introduction

II. Brief Review of What we have doneIII. Algebraic solutions the one-dimensional Holstein

Model

IV. Summary

Contents

Introduction: Research Trends1) Large Scale Computation (NP problems)

Specialized computers (hardware & software), quantum computer?

2) Search for New SymmetriesRelationship 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)

Origin of the Pairing interactionSeniority scheme for atoms (Racah) (Phys. Rev. 62 (1942) 438)

BCS theory for superconductors (Phys. Rev. 108 (1957) 1175)

Applied BCS theory to nuclei (Balyaev) (Mat. Fys. Medd. 31(1959) 11

Constant pairing / exact solution (Richardson) (Phys. Lett. 3 (1963) 277; ibid 5 (1963) 82;

Nucl. Phys. 52 (1964) 221)

General Pairing Problem

)()()(2'

'0 jSjScjSH

j jjjjjj

jj

j j

jmm

mjmj

mjm

jmmj

aajS

aajS

0

0

)()(

)()(21 jj

)ˆ(21)1(

21)(

0

0jjmjmjjm

mjm NaaaajS

Some Special Cases

'jjc {G'jjcc

', 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

Next Breakthrough?Next Breakthrough?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

)1(2)(

)( kGx

Ek

01

)(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=1.1792=2.6503=3.1624=4.5885=5.0066=6.9697=7.2628=8.6879=9.89910=10.20

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

6

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

AGGt

Gtt

ii

iiiii

iiiiii

2111

Nilsson s.p.

ii

i

iii

aab

aab

jijji

jijji

iijji

bbN

bbN

Nbb

,

)21( ,

,

)(21

iiiii aaaaN

AGGt

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

)(')(

pppij

jij gEgt )(~

Eigenstates for k-pair excitation can be expressed as

The excitation energy is

AGGt

Gtt

ii

iiiii

iiiiii

2111

2n dimensional n

Binding Energies in MeV

227-233Th 232-239U

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

Models of interacting electrons with phonons have been attracting much attention as they are helpful in understanding superconductivityin many aspects, such as in fullerenes, bismuth oxides, and the high-Tc superconductors.

Many theoretical treatments assume the adiabatic limit and treat the phonons in a mean-field approximation. However, it has been argued that in many CDW materials the quantum lattice fluctuations are important.

[1]A. S. Alexandrov and N. Mott, Polarons and Bipolarons (World Scientific, Singapore, 1995).[2] R. H. McKenzie, C.J. Hamer and D.W. Murray, PRB 53, 9676 (96).[3] R. H. McKenzie and J. W. Wilkins, PRL 69, 1085 (92).

III. Algebraic solutions the one-dimensional Holstein Model

Here we present a study of the one- dimensional Holstein model of spinless fermions with an algebraic approach.

The Hamiltonian is

The model

(1)

Analogue

(3)

(4)

(5)

Let us introduce the differential realization for the bosonoperators with

(7)

For i=1,2,…,p. Then, the Hamiltonian (1) is mapped into

(8)

Solutions

According to the diagonalization procedure used to solve the eigenvalue problem (2), the one-fermion excitation states can be assumed to be the following ansatz form:

(9)

Where |0> is the fermion vacuum and

By using the expressions (8) and (9), the energy eigen-equation becomes

(11)

which results in the following set of the extended Bethe ansatz equations:

for ¹ = 1, 2, …, p , which is a set of coupled rank-1 Partial Differential Equations (PDE’s), which completely determine the eigenenergies E and the coefficients .Though we still don’t know whether the above PDE’s are exactly solvable or not, we can show there are a large set of quasi-exactly solutions in polynomial forms. The results will be reported elsewhere.

Once the above PDEs are solved for one-fermion excitation, according to the procedure used for solving the hard-core Fermi-Hubbard model, the k-fermion excitation wavefunction can be orgainzed into the following from:(13)

with

(14)

The corresponding k-fermion excitation energy is given by

(15)

In summary

(1) General solutions of the 1-dim Holstein model is derived based on an algebraic approach similar to that used in solving 1-dim hard-core Fermi-Hubbard model.

(2) A set of the extended Bethe ansatz equations are coupled rank-1 Partial Differential Equations (PDE’s), which completely determine the eigenenergies and the corresponding wavefunctions of the model. (3) Though we still don’t know whether the PDE’s are exactly solvable or not, at least, these PDE’s should be quasi-exactly solvable.

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