Phonons and the Isotopically Induced Mott transition - Physics

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Phonons and the Isotopically Induced Mott transition - Physics

( 1 2 )

s 1

µ

= a 2

µ

+ a

µ

b 1

µ

= ( a1µ − a2µ

).

2

The electron-phonon term also has two contributions

where g

+ = g / 2 .

µ µ


µ

µ


( 1 2 )( )

s

H = hΩ g n + n s + s

ep


+

µ µ µ µ


( 1 2 )( ).

b

H = hΩ g n − n b + b

Displaced oscillator transformation

ep

+

µ µ µ µ

(A.5)

As there are two different boson operators - s and b - to perform the displaced

oscillator transformation two generators R s and R b are required. These are defined

as

and


µ


( 1 2 )( )

R = δ n − n b − b (A.6)

b


µ

µ µ µ


( 1 2 )( )

R = η n + n s − s (A.7)

s

µ µ µ

where ηµ

and δ µ

are 2N as yet undetermined parameters where N is the total number

if vibrational modes.

Electronic transformation

The electron operators can be transformed using the R b generator where we let


( )

B = b − b . The transformed operator can be rewritten as

µ µ µ

c

= e c e

Rb

− Rb



Using the fermionic anticommutator relation { , }

c c = δ gives n c = c ( n − ).

α β α , β


1 1σ


1

1

R

2 2

1 1 {

bj −Rbj

σ

= ⎡

σ

1− µ µ

+

µ µ 1


2

+

µ µ 1


2

− 1 + ... ⎤

∏ ⎢⎣

⎥⎦

µ

δ δ ( ) 1 ( δ

2 ) ( )

2 2

( 1 2 ) 1 ( δ

2 ) ( 1 2 ) ⎤

⎥}

e c e c B B n n B n n

× ⎡


1 −δ

µ

Bµ n − n +

µ


n − n + ...




( δ B )

( )

( δ ) ( ) 1 ( ) ( ) 1 ( )

= c ⎡1 − B + B n − n + n − n − n − n + n − n − 1 + ... ⎤

⎥⎦

2 2 2 2

1σ ⎢

δµ µ µ µ 1 2

2

1 2 1 2

2

1 2


µ


( δ )

2

= c ⎡1 − δ + + ... ⎤


B B


⎥⎦

1σ µ µ µ µ

µ


= c exp −

1σ µ µ

µ

Applying the same principle to the other Fermi operators gives


( δ )

c = c exp − B

1σ 1σ µ µ

µ


( δ )

c = c exp B

† †

1σ 1σ µ µ

µ


( δ )

c = c exp B

2σ 2σ µ µ

µ


( δ )

c = c exp − B

† †

2σ 2σ µ µ

µ

(A.8)

These operators can now be applied to the terms in (A.2) to give the transformed

electronic Hamiltonian. It can be seen from the transformed operators in (A.8) that

the number operators remain unchanged i.e.

so the only noninvariant term in (A.2) is

n = n = c c

(A.9)


iσ iσ iσ iσ

c c ⎛

+ c c = c c ∏exp 2 B + c c exp −2

B


( δ ) ∏ ( δ )

† † † †

1σ 2σ 2σ 1σ ⎜ 1σ 2σ µ µ 2σ 1σ µ µ

µ µ

The transformed electronic component is therefore




Using this identity gives

and hence


R

e b

c ⎡ B n n B n n ⎤c

( ) 1 ( δ ) ( )

2 2

1σ = 1 + δµ µ 1


2

+

1 2

...

2

µ µ

− +


⎣⎢

⎥⎦

µ


2 2

= c ⎡1 + δ ( − − 1) + 1 ( δ ) ( − − 1 ) + ... ⎤


B n n B n n


2

⎥⎦

1σ µ µ 1 2 µ µ 1 2

µ



H ( ) ( ) † ( )


el

= µ

m∑ n1 σ

+ n2 σ

− tm

⎜∑∏exp 2δ µ

Bµ c1 c2 + ∏exp −2δ

µ


c2c1


σ ⎝ σ µ µ


+ U n n + V n n


m

i=

1,2

i↑

i↓

m 1 2

Free Phonon and electron-phonon transformations

Using R b , R s and the Campbell-Baker-Hausdorff theorem

(A.10)

67

68

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