Phonons and the Isotopically Induced Mott transition - Physics

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Phonons and the Isotopically Induced Mott transition - Physics

method on a cluster of up to four dimers. Powell et. al. 7 obtain their value of U m from

an application of density functional theory DTF. These values are summarized along

with the associated method in Table 4.1.

Because the values of V m are of the same order of magnitude as U m the offsite

Coulomb repulsion cannot be neglected, as McKenzie has done (c.f. Eq. (4.21)),

when expressing U d as a function of the monomer parameters. Choosing reasonable

parameters values of U m = 13t m and V m = 8t m (c.f. Table 4.1) in Eq. (4.20) we obtain

≈ 1 + in contrast to the 2t 2

m calculated by McKenzie.

a value of U ( U V )

d m m

Parameter Group Method Value (eV)

t m Fortunelli et. al. RHF-SCF 0.272

Fortunelli et. al. Extended Huckel 0.224

T. Komatsu et. al Extended Huckel 0.244

t b2 Fortunelli et. al. RHF-SCF 0.085

Fortunelli et. al. Extended Huckel 0.071

T. Komatsu et. al Extended Huckel 0.092

t q Fortunelli et. al. RHF-SCF 0.040

Fortunelli et. al. Extended Huckel 0.040

T. Komatsu et. al Extended Huckel 0.034

t p Fortunelli et. al. RHF-SCF 0.130

Fortunelli et. al. Extended Huckel 0.094

T. Komatsu et. al Extended Huckel 0.101

U m Castet et. al. AM1 3.90

Fortunelli et. al. RHF-SCF 3.56

Powell et. al. DFT 3.60

V m Ducasse et. al. Mixed Valence Bond/Hartree Fock 2.70

Castet et. al. AM1 ~2.30

Fortunelli et. al. RHF-SCF 3.56

Table 4.1 A summary of inter- and intra- dimer hopping terms and Hubbard parameters with the

associated methods of calculation

Table 4.2 Eigenvalues and eigenvectors of the two-site Hubbard Hamiltonian for HOMO fillings

of n=1,2,3,4. The labels a, b denote antibonding and bonding states respectively, whilst S and T

correspond to singlet and triplet states and tanθ = −4 t / U − V + ( U − V ) + 16t

Filling and energy Degeneracy Eigenvectors

(n=1)

E

E

= ε − t

1 0

= ε + t

2 0

(n=2)

0

( ) 2 2

E = 2ε

+ U + V − U − V + 16t

ET

sb

ECT

= 2ε

+ V

0

= 2ε

+ U

0

0

( ) 2 2

E = 2ε

+ U + V + U − V + 16t

sb

(n=3)

E = 3ε

+ U + 2V − t

3b

0

E = 3ε

+ U + 2V + t

3a

0

(n=4)

E = 4ε

+ 2U + 4V

1

4 0

2

2

1

3

1

1

2

2

2 2

( )

† †

( 1↑

2↑

)

1 b, ↑ = 1 c + c 0,0

2

† †

( 1↓

2↓

)

1 b, ↓ = 1 c + c 0,0

2

† †

( −

1↑

2↑

)

1 a, ↑ = 1 c c 0,0

2

† †

( −

1↓

2↓

)

1 a, ↓ = 1 c c 0,0

2

† † † †

( 1↑ 1↓ 2↑ 2↓

)

Sa = 1 ⎡sinθ

c c + c c

2 ⎣

T,1 = c c 0,0

† † † †

( c c c c

1↑ 2↓ 2↑ 1↓

)

− cosθ

+ ⎤


0,0

† †

1↑

2↑

T, − 1 = c c 0,0

† †

1↓

2↓

† † † †

T,0 = 1 ⎡c c − c c ⎤ 0,0

2↓ 1↑ 1↓ 2↑

2 ⎣


† † † †

CT = 1 ⎡c c − c c ⎤ 0,0

1↑ 1↓ 2↑ 2↓

2 ⎣ ⎦

† † † †

( 1↑ 1↓ 2↑ 2↓

)

Sb = 1 ⎡cosθ

c c + c c

2 ⎣

† † † †

( c c c c

1↑ 2↓ 2↑ 1↓

)

+ sinθ

+ ⎤


0,0

† † † † † †

3 b, ↑ = 1 ⎡c c c + c c c ⎤ 0,0

1↑ 1↓ 2↑ 2↑ 2↓ 1↑

2 ⎣


† † † † † †

3 b, ↓ = 1 ⎡c c c + c c c ⎤ 0,0

1↑ 1↓ 2↓ 2↑ 2↓ 1↓

2 ⎣


† † † † † †

3 a, ↑ = 1 ⎡c c c − c c c ⎤ 0,0

1↑ 1↓ 2↑ 2↑ 2↓ 1↑

2 ⎣


† † † † † †

3 a, ↓ = 1 ⎡c c c − c c c ⎤ 0,0

1↑ 1↓ 2↓ 2↑ 2↓ 1↓

2 ⎣


4 = c c c c 0,0

† † † †

1↑ 1↓ 2↑ 2↓

27

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