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Lecture VI 16

Lecture VI: Tight-binding and the Mott transition

According to band picture of non-interacting electrons, a 1/2-filled band of states is metal-lic. But strong Coulomb interaction of electrons can lead to a condensation or crystallisa-tion of the electron gas into a solid, magnetic, insulating phase Mott transition. Herewe employ the second quantisation to explore the basis of this phenomenon.

Atomic Limit of crystalHow do atomic orbitals broaden into band states? Transparencies

0

1

0

1

V(x)

x

Es=1

s=0 0A

E

0B

1A

1B

n1 a

E a

E

k0

a

x

n+1 ana

Weak overlap of tightly bound states narrow band:Bloch states |ks, band index s, k [/a, /a]

Bloch states can be used to define Wannier basiscf. discrete Fourier decomposition

|ns 1N

B.Z.k[/a,/a]

eikna|ks, |ks 1N

Nn=1

eikna|ns, k = 2Na

m

n0

(x)

(n1)a

x

(n+1)ana

In atomic limit, Wannier states |ns mirror atomic orbital |s on site n

Lecture Notes October 2005

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Lecture VI 17

Field operators associated with Wannier basis:

cns

|

|ns = dxc(x)

|

|xns(x) x|ns

cns

dx ns(x)c(x)

and using completeness (exercise)

ns ns(x

)ns(x) = (x x)

c(x) =ns

ns(x)cns, [cns, c

ns ]+ = nnss

i.e. operators cns/cns create/annihilate electrons at site n in band s with spin

In atomic limit, bands are well-separated in energy. If electron densities are low,one may project onto lowest band s = 0

Transforming to Wannier basis, interacting electron Hamiltonian takes form

H =mn

tmncmcn +

mnrs

Umnrscmc

ncrcs

where hopping matrix elements: tmn = m|H(0)|n = tnmand interaction parameters

Umnrs =1

2

dx

dxm(x)n(x

)e2

|x x|r(x)s(x)

(For lowest band) representation is exact:but, in atomic limit, matrix elements decay exponentially with separation

(i) Tight-binding approximation:

tmn =

m = nt mn neighbours0 otherwise

, H(0) n

cncn tn

cn+1cn + h.c.

In discrete Fourier basis: cn =1N

B.Z.k[/a,/a]

eiknack

tNn

cn+1cn + h.c.

= t

kk

kk 1

N

n

ei(kk)na eikackck + h.c. = 2t

k

cos(ka) ckck

H(0) =

k( 2t cos ka)ckck

As expected, as k 0, spectrum becomes free electron-like:k 2t + t(ka)2 + (with m = 1/2a2t)

Lecture Notes October 2005

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Lecture VI 18

/a

(k)

/a k-

B.Z.

(ii) Interaction

Focusing on lattice sites m = n:1. Direct terms Umnnm

Vmn couple to density fluctuations: m=n Vmnnmnn

potential for charge density wave instabilities

2. Exchange coupling JFmn Umnmn (exercise see lecture handout)m=n,

Umnmncmc

ncmcn = 2

m=n

JFmn

Sm Sn + 1

4nmnn

, Sm =

1

2cmcm

i.e. weak ferromagnetic coupling (JF > 0) cf. Hunds rule in atoms

spin alignment symmetric spin state and asymmetric spatial state lowers p.e.

But, in atomic limit, both tmn and JFmn exponentially small in separation |m n|a

On-site Coulomb or Hubbard interactionn

Unnnncnc

ncn cn = U

n

nnnn, U 2Unnnn

Minimal model for strong interaction: Hubbard Hamiltonian

H = tn

(cn+1cn + h.c.) + Un

nnnn

...could have been guessed on phenomenological grounds

Transparencies on Mott-Insulators and the Magnetic State

Lecture Notes October 2005