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On Solutions of the one-dimensional Holstein Model
Feng Pan and J. P. Draayer
Liaoning Normal Univ. Dalian 116029 China
Louisiana State Univ. Baton Rouge 70803 USA
23rd International Conference on DGM in
Theoretical Physics, Aug. 20-26,05 Tianjin
Contents
I. Introduction
II. Brief Review of What we have done
III. Algebraic solutions the one-dimensional Holstein Model
IV. Summary
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.
Bethe ansatz
Exact
diagonalization
Group Methods
Critical phenomena
Quantum
Many-body systems
Quantum Phase
transitions
Methods used
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 nonperturbative 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)
Origin of the Pairing interaction
Seniority 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

H    j  j  2  j S ( j )   c jj ' S ( j ) S ( j )

0
j
j
S ( j )   ( )
j m
S ( j )   ( )
j m

m 0

m 0

jj '

jm
a a
j

j m
a j m a jm
 j  j 1 2
1
1 ˆ


S ( j )   (a jma jm  a j m a j m  1)  ( N j   j )
2 m 0
2
0
Some Special Cases
constant pairing
G j ,
c jj '  c c
j j'
{
j'
separable strength
pairing
2
2
-B(
-
)
-B(
-
)
cij=A ij + Ae i i-1 ij+1 + A e i i+1 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?
Solvable mean-field plus
extended pairing model

p

1
ˆ
H    j n j  G a j a j '  G  (  !)2 
j 1
jj '
a
 
i1 i2
i1  i2 ... i2 

i
 2
a ...a ai 1 ai 2 ...ai2 
Different pair-hopping structures in the constant
pairing and the extended pairing models
Exact solution
Bethe Ansatz Wavefunction:
| k ,  ; j1, j2 ,, jm  
( )
 

C
a
a
...
a
kkw
,  ; j1, j2 ,..., jm 
 i1i2 ...ik i1 i2 ik |M
1i1 i2 ...ik  p
ai | j1 , j2 ,..., jm  0
( )
i1i2 ...ik
C

1
1 x (  )
k
 i
 1
2

n
|
k
,

;
0


(  ) (| k ,  ;0  
jj
x

ai1 ai2 ... aik | 0 )
1i1 i2 ...ik  p
j

( a a  

j
j
j
( 
 1
1
(  !) 2
1i1 i2 ...ik  p
)
i1 i2 ...i2 
( )
i1i2 ...ik
C
 

a
a
...
a
 i1 i2 i ai1 ai2 ...ai2  | k ,  ;0 
)  a a ...a
 
i1 i2
1i1 i2 ...ik  p

ik
| 0  (k  1) | k ,  ;0 
E
Eigen-energy:
( )
k
2
 ( )  G (k  1)
x
Bethe Ansatz Equation:
2
x
( )


G
( )
1

x
1i1 i2 ...ik  p
| k ,  ; j1 , j2 ,..., jm  
k
 i
0
 1
( )
 

C
a
a
...
a
 i1i2 ...ik i1 i2 ik | k, ; j1, j2 ,..., jm 
1i1 i2 ...ik  p
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
Higher Order Terms
V1   ai a j , V  ( 1!)2
i, j
R 

i1 i2 i2 
 V 
ai1 ai2 ...ai ai 1 ai 2 ...ai2 
 Vtotal 
Ratios: R = <V> / < Vtotal>
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
6
Nearest Level Pairing Interaction for
deformed nuclei
In the nearest level pairing interaction model:
2
2
-B(
-
)
-B(
-
)
cij=Gij=A ij + Ae i i-1 ij+1 + A e i i+1 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
tii 1  ti 1i  Gii 1
Nilsson s.p.
tii  2 i  Gii
Gii  A
1 
N i  (ai ai  a a )
i i
2
b , b 

j
i
N
i
N
i,
,b

j
bj

  ij (1  2 N i )

  ij b

j
   ij b j

i

b  a a
i
bi  a ai
tii 1  ti 1i  Gii 1
tii  2 i  Gii
Gii  A

i
i
Nearest Level Pairing Hamiltonian can be
written as

H    i   Pt b b j P

ij i
'
i
i, j
which is equivalent to the hard-core
Bose-Hubbard model in condensed
matter physics
Eigenstates for k-pair excitation can be expressed as
k ; , (n j1 , n j2 ,..., n jr )n f 
( )
 

C
b
b
...
b
 i1i2 ...ik i1 i2 ik (n j1 , n j2 ,..., n jr )n f
i1  i2 ...  ik
The excitation energy is
( )
k
E
k
  j E
'
j
g i11 g i21 ...g ik1
( j )
g i1 2 g i22 ...g ik2
j 1
g i1 k g i2k ...g ikk
tii 1  ti 1i  Gii 1
tii  2 i  Gii
Gii  A
~
t
j
2n dimensional
n
ij
g
p
j
E
( p )
g
p
i
227-233Th
232-239U
Binding Energies in MeV
238-243Pu
227-232Th
232-238U
First and second 0+ excited
energy levels in MeV
238-243Pu
230-233Th
238-243Pu
odd-even mass differences
234-239U
in MeV
226-232Th
230-238U
236-242Pu
Moment of Inertia Calculated in the NLPM
III. Algebraic solutions the one-dimensional Holstein Model
Models of interacting electrons with phonons have been attracting
much attention as they are helpful in understanding superconductivity
in 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).
The model
Here we present a study of the one- dimensional Holstein model of
spinless fermions with an algebraic approach.
The Hamiltonian is
(1)
Analogue
(3)
(4)
(5)
Solutions
Let us introduce the differential realization for the boson
operators with
(7)
For i=1,2,…,p. Then, the Hamiltonian (1) is mapped into
(8)
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 eigenequation 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 FermiHubbard 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
Phys. Lett. B422(1998)1
SU(2) type
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 type
Phys. Lett. A341(2005)94