Renormalized Wannier functions

at the border of Mott localization

Jozef Spałek

Jagiellonian Universityand AGH University of Science and Technology

PL - 30-059 KRAKÓW

Jan Kurzyk – Tech. Univ., Krakow

Robert Podsiadły – Jag. Univ., Krakow

Włodek Wójcik – Tech. Univ., Krakow

Collaboration:

Plan

1. From atoms to metals

2. Wave function readjustment in thecorrelated electron state

3. Example: exact solution for nanosystemsand Hubbard chain

4. Quantum critical behavior of the wavefunction (size)

a) Metal

b) Mott-Hubbard insulator

Plane waves(Bloch states)

Atomic states

Delocalized versus localized

J. S. et al., PRL 59, 728 (1987)– orbitally nondegenerate;A. Klejnberg & J. S., PRB 57, 12 041 (1998)– degenerate.

G. Lonzarich, Nature (2005)

Universal scaling

inverse susceptibility→→→→(A. Ślebarski, JS, PRL (2005)

resi

stiv

ity →→ →→

Resistivity data ρρρρ(T; x)

A.Ś. & J.S.PRL (2005)

Tk

CB

VB

orelse:

cond. el.

Stot ≡≡≡≡ 0Sce

χ→χ→χ→χ→0 for T→→→→0

x = 0.13

x = 0.14

II. Metallization of magnetite

•Ferrimagnetic material:

TC=860 K, M=4.1 Bohr magnetons

S=5/2 S=5/2 S=2

Fe3+ Fe3+ Fe2+

Tetra site Octa sites

•Verwey transition: TV = 122 K ±±±± 1 K(at p = 0)

N. Mori et al.Physica B, 2002

Z. Kąkol, A. Kozłowski, Z. Tarnawski,…J.M. Honig

J.S., A. Kozłowski, Z. Tarnawski, Z.Kąkol,, Y.Fukami, F.Ono, R. Zach, L.J. Spałek, andJ.M. Honig,Phys. Rev.B 78, 100401 (R) (2008)

Localization criterion: Mott

Kinetic energy in e- gas/particle

32

32

VN

3m25

353 2

2

F ρ

π=∈=∈ ∗ ~

h

31

2e

de

21 2

ee

2

ee ρε

=ε

=∈−

−

31

NV

d ee

=−

ee−=∈∈ gas instability

( )170

3

135

a

em 32

31

2c2

2

B

.≅π

=ρ

ε∗

43421

h

20170a 31

cB .~.≅ρ⋅

⇒⇒⇒⇒ Fermi - sphere collapse

In one dimension:

aB ρρρρC ≅≅≅≅ 1 ⇒⇒⇒⇒ RC ≅≅≅≅ aB

Microscopic many-particle

Hamiltonian for a nanosystem

The microscopic parameters t, U, K should be calculated together with the H diagonalization -> wave function optimization in the correlated state

(for extended system no phase factor in t)

The main ingredient:interelectronic correlations

and wave function treated on

the same footing

The main result:evolution of the many-atom

system as a function of

interatomic spacing

J. Spałek, R. Podsiadły, W. Wójcik, and A. Rycerz, Phys.Rev. B 61, 15676 (2000);PRB (2001-2002)

EDABI

Momentum distribution:Fermi-Dirac vs continuous

J. S. & A. Rycerz, PRB-R (2001-2004); review: 2007;Didactical: J.S., in Encyclopedia of Condensed MatterPhysics, Elsevier, vol. 3, pp. 126-136 (2005)

Renormalized band energies:

even and odd

Extended systems : EDABID=1

Supplement: Infinite Hubbard chain vs. nanochain

( ) ( )( )[ ] ω−∈= ∫

∞

ω+ωωω d4

02/exp1

JJeffaN

E 10

tUt

0n1n ii ≡−≡δ

Ground state energy functional:

Periodic bound cond.

( )( )

( )( )( ) ( )∑λ=⋅∇−

≥∇δ

µ−δ

δ

µ−δ∗∗

jijijw

NE

w

NE wi

e

i

e rrr

Renormalized wave equation:

Adjustable Slater or STO-3G basis forms a trial Wannier

function obtained variationally

j1i wHwt =2i12

2i wVwU =

Atomic functions:

( ) ( ) ( )ii Rrr 21

−α−απ=Φ exp3

ijji S=ΦΦ

Wannier functions (wave functions):

ijji δ=ww

( ) ( )rr jj

iji Ψβ= ∑w

Square lattice

3 Dimensions: Gutzwillerapproach

Body Centered Cubic lattice

1. Method allows for study of the electronstate evolution as a funtion of interatomicspacing

2. The evolution of the wave function in thecorrelated state through the Mott threshold: from atoms to solid state (or vice versa)

3. Scaling and critical behavior of the wavefunction and a critical behavior

4. Future: Bose Hubbardd-orbitals

Outlook