Mott insulators - edu.itp.phys.ethz.chedu.itp.phys.ethz.ch/fs12/sst/Mott-insulator.pdf · Mott...

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Atomic limit - view electrons in real space lattice of H-Atoms: a B << d H 2a B d Virtual system: Mott insulators

Transcript of Mott insulators - edu.itp.phys.ethz.chedu.itp.phys.ethz.ch/fs12/sst/Mott-insulator.pdf · Mott...

Page 1: Mott insulators - edu.itp.phys.ethz.chedu.itp.phys.ethz.ch/fs12/sst/Mott-insulator.pdf · Mott insulators Gutzwiller approximation. Metal-insulator transition from the metallic side

Atomic limit - view electrons in real space

lattice of H-Atoms: aB << d

H

2aB d

Virtual system:

Mott insulators

Page 2: Mott insulators - edu.itp.phys.ethz.chedu.itp.phys.ethz.ch/fs12/sst/Mott-insulator.pdf · Mott insulators Gutzwiller approximation. Metal-insulator transition from the metallic side

Atomic limit - view electrons in real space

lattice of H-Atoms: aB << d

H +H -

H

2aB d

-t

Virtual system:

hopping - electron transfer

ionization energy

delocalization localization

kinetic energy charge excitation energy

metal Mott insulator

Mott insulators

Page 3: Mott insulators - edu.itp.phys.ethz.chedu.itp.phys.ethz.ch/fs12/sst/Mott-insulator.pdf · Mott insulators Gutzwiller approximation. Metal-insulator transition from the metallic side

Atomic limit - view electrons in real space

lattice of H-Atoms: aB << d

H

2aB d

-t

Virtual system:

hopping - electron transfer

ionization energy

S = 1/2

Mott isolator

no charge fluctuation only spin fluctuation

effective low-energy modellow-energy physics

Mott insulators

Page 4: Mott insulators - edu.itp.phys.ethz.chedu.itp.phys.ethz.ch/fs12/sst/Mott-insulator.pdf · Mott insulators Gutzwiller approximation. Metal-insulator transition from the metallic side

Metal-insulator transition from the insulating sideHubbard-model:

n.n. hopping onsite repulsion

density: n=1

charge excitation

„ground state“

dh

E

U

t = 0

Mott insulators

Page 5: Mott insulators - edu.itp.phys.ethz.chedu.itp.phys.ethz.ch/fs12/sst/Mott-insulator.pdf · Mott insulators Gutzwiller approximation. Metal-insulator transition from the metallic side

Metal-insulator transition from the insulating sideHubbard-model:

n.n. hopping onsite repulsion

density: n=1

charge excitation

„ground state“

dh

Mott insulators

E

U

t > 0

W = 4d t

metal-insulator transition: Uc = 4dt

Page 6: Mott insulators - edu.itp.phys.ethz.chedu.itp.phys.ethz.ch/fs12/sst/Mott-insulator.pdf · Mott insulators Gutzwiller approximation. Metal-insulator transition from the metallic side

Metal-insulator transition from the metallic side

Mott insulators

tight-binding model

half-filledconduction

band

dh

empty sites h = 1/4 doubly occupied sites d = 1/4singly occupied sites s = 1/2

density

reducing mobility

Page 7: Mott insulators - edu.itp.phys.ethz.chedu.itp.phys.ethz.ch/fs12/sst/Mott-insulator.pdf · Mott insulators Gutzwiller approximation. Metal-insulator transition from the metallic side

Metal-insulator transition from the metallic side

Mott insulators Gutzwiller approximation

Gutzwiller-approach: variational

variationalgroundstate

density of doublyoccupied sites

renormalized hopping

uncorrelatedstate

diminish double-occupancy

Page 8: Mott insulators - edu.itp.phys.ethz.chedu.itp.phys.ethz.ch/fs12/sst/Mott-insulator.pdf · Mott insulators Gutzwiller approximation. Metal-insulator transition from the metallic side

Metal-insulator transition from the metallic side

densities:

1 electron density

density of singly occupied sites with spin

density of singly occupied sites with spin d density of doubly occupied sites h density of empty sites

no spin polarization half-filling

sum rule

Mott insulators Gutzwiller approximation

Page 9: Mott insulators - edu.itp.phys.ethz.chedu.itp.phys.ethz.ch/fs12/sst/Mott-insulator.pdf · Mott insulators Gutzwiller approximation. Metal-insulator transition from the metallic side

Metal-insulator transition from the metallic sidehopping probability:

correlated uncorrelatedrenormalizationfactor

sector of fixed d

analogous with same gt

d changed, not in sector

Mott insulators Gutzwiller approximation

Page 10: Mott insulators - edu.itp.phys.ethz.chedu.itp.phys.ethz.ch/fs12/sst/Mott-insulator.pdf · Mott insulators Gutzwiller approximation. Metal-insulator transition from the metallic side

Metal-insulator transition from the metallic sidecorrelated uncorrelatedrenormalization

factor

minimize w.r.t. d

Brinkmann-Rice (1970)

per site

Mott insulators Gutzwiller approximation

Page 11: Mott insulators - edu.itp.phys.ethz.chedu.itp.phys.ethz.ch/fs12/sst/Mott-insulator.pdf · Mott insulators Gutzwiller approximation. Metal-insulator transition from the metallic side

optical conductivity:

Drude weight

Mott insulators Gutzwiller approximation

quasiparticle weight:

at MIT

variational

renormalized hopping

single particle spectral function

E

E

µ

A

A

lowerHubbard band

upperHubbard band

coherent quasiparticles metallic

insulating

Page 12: Mott insulators - edu.itp.phys.ethz.chedu.itp.phys.ethz.ch/fs12/sst/Mott-insulator.pdf · Mott insulators Gutzwiller approximation. Metal-insulator transition from the metallic side

Fermi liquid properties within Gutzwiller approximation:

kkF

nk

Z kkF

nk

Znin nout

renormalizedmomentumdistribution

Gutzwiller

half-filling

renormalized hopping

particle-hole symmetry

Mott insulators Gutzwiller approximation

Page 13: Mott insulators - edu.itp.phys.ethz.chedu.itp.phys.ethz.ch/fs12/sst/Mott-insulator.pdf · Mott insulators Gutzwiller approximation. Metal-insulator transition from the metallic side

Fermi liquid properties within Gutzwiller approximation:

kkF

nk

Z kkF

nk

Znin nout

renormalizedmomentumdistribution

Gutzwiller

half-filling

renormalized hopping

particle-hole symmetry

Mott insulators Gutzwiller approximation

half-filling

renormalized hopping

particle-hole symmetry

compressibilityquasiparticle weight