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Liaoning Normal Univ. Dalian 116029 China. Louisiana State Univ. Baton Rouge 70803 USA. On Solutions of the one-dimensional Holstein Model Feng Pan and J. P. Draayer. 23rd International Conference on DGM in Theoretical Physics, Aug. 20-26,05 Tianjin. Contents. I. Introduction - PowerPoint PPT Presentation
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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
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