2nd talk. The Mott transition

Mott physics 

2nd Talk 

E. Bascones 

Instituto de Ciencia de Materiales de Madrid (ICMM-CSIC)

Summary I 

Independent electrons: Odd number of electrons/unit cell = metal 

Interactions in many metals can be described following Fermi liquid 

theory: 

Description in k-space. Fermi surface and energy bands are 

meaningful quantities. Rigid band shift 

There are elementary excitations called quasiparticles with 

charge e and spin ½ 

Quasiparticle have finite lifetime & renormalized energy 

dispersion (heavier mass). Better defined close to Fermi level & low T 

Quasiparticle weight Z , it also gives mass renormalization m* 

Increasing correlations: smaller Z. m* (and Z) can be estimated 

from ARPES bandwidth, resistivity, specific heat and susceptibility 

~ 0 + A T 2 

A ~ m* 2 

C ~ T ~ 

~ m* ~ m*

Summary I-b 

Interactions are more important in f and d electrons and decrease 

with increasing principal number (U 3d > U 4d …) . 

With interactions energy states depend on occupancy: non-rigid 

band shift 

In one orbital systems with one electron per atom (half-filling) onsite 

interactions can induce a metal insulator transition : Mott 

transition. 

In Mott insulators : description in real space (opposed to k-space) 

Mott insulators are associated with avoiding double occupancy not 

with magnetism (Slater insulators) 

Magnetism: 

Weakly correlated metals: Fermi surface instability 

Mott insulators: Magnetic exchange (t 2 /U). Spin models

Outline II: The Mott transition in single band systems 

The Mott-Hubbard transtion. Hubbard bands. Mott and 

charge transfer insulators 

The correlated metallic state. Brinkman-Rice transition 

The DMFT description of the Mott transition 

Finite temperatures

The Mott transition. Paramagnetic state 

Paramagnetic 

Mott 

Insulator 

Metal-Insulator 

transition with 

decreasing pressure 

Antiferromagnetism 

Increasing Pressure: decreasing U/W 

McWhan et al, PRB 7, 1920 (1973)

The Mott transition. Paramagnetic state 

Atomic lattice single orbital per site and average occupancy 1: half filling 

Hopping 

saves energy t 

Double occupancy 

costs energy U 

Small U/t 

Metal 

Increasing U/t 

Mott transition 

Large U/t 

Insulator 

W 

Double 

occupancy 

Single 

occupancy 

U

Hubbard model. Kinetic and On-site interaction Energy 

Hopping 

saves energy t 

Kinetic energy 

Intra-orbital 

repulsion 

Double occupancy 

costs energy U 

Atomic lattice with a single orbital per site and average occupancy 1 (half filling) 

E 

Hopping restricted to first nearest neighbors: Electron-hole symmetry 

Tight-binding (hopping) 

Intra-orbital 

repulsion

Mott insulator. Paramagnetic state: Hubbard bands 

Double 

electron 

occupancy 

Single 

electron 

occupancy 

U


Double 

electron 

occupancy 

U 

Single 

electron 

occupancy 

Remove an electron 

(as in photoemission) 

Empty 

state


Double 

electron 

occupancy 

U 

Single 

electron 

occupancy 

Remove an electron 

(as in photoemission) 

Empty 

state 

Empty state is free to move


t 

t


t 

t 

Double 

occupancy 

U 

U 

Single 

occupancy 

W 

Lower Hubbard band


t 

t 

Upper Hubbard Band 

W 

U 

Lower Hubbard Band 

W


Doubly occupied states 

Upper Hubbard Band 

W 

U 

Lower Hubbard Band 

W 

Non-degenerate bands 

Singly occupied states

The Mott-Hubbard transition. Paramagnetic state 

Increasing U 

U=0 

W 

W 

Double 

degenerate 

band (spin) 

Gap 

U- W 

W 

U 

Non-degenerate 

bands

The Mott-Hubbard transition. Paramagnetic state 

Increasing U 

U=0 

W 

Double 

degenerate 

band (spin) 

W 

W 


Uc= W 

Gap 

U- W 

W 

W 

U 


bands 

Gap opens at the Fermi level at Uc

Mott vs charge transfer insulators 

3d oxides 

U=0 

4s band 

3d narrow band 

2p oxygen band


3d oxides 

U=0 

4s band 

W 

U 


W 

2p oxygen band


3d oxides 

U=0 

Mott insulator 

4s band 

Lowest excitation 

energy d-type (Mott) 


Charge transfer 

insulator 

2p oxygen band 

Lowest excitation 

energy p-type


Cuprates are 

charge transfer insulators

The Brinkman-Rice transition from the metallic state. 

The uncorrelated metallic state: The Fermi sea |FS> 

W 

Spin degenerate 

Energy states are filled 

according to their kinetic energy. 

States are well defined in k-space


The uncorrelated metallic state: The Fermi sea |FS> 

W 

Spin degenerate 

Energy states are filled 

according to their kinetic energy. 

States are well defined in k-space 

Probability in real space: ¼ per the 4 possible states (half filling) 

Cost in interaction energy per particle 

=U/4 

Kinetic energy gain per particle 

(constant DOS) 

=W/4=D/2


The uncorrelated metallic state: The Fermi sea 1FS> 

E=K+U 

 

=U/4 

 

 

=D/2


The correlated metallic state: Gutzwiller wave function 

| >= j [ 1-(1- )n j n j ]1FS> 

Variational Parameter 

=1 U=0 

Uncorrelated 

=0 U= 

uniformly diminishes 

the concentration of 

doubly occupied sites 

Correlated 

Gutzwiller Approximation. Constant DOS



Uncorrelated 

Correlated



uncorrelated 

correlated 

Average potential energy 

reduced due to reduced 

double occupancy 

correlated 

uncorrelated 

Kinetic Energy 

is reduced

The Brinkman-Rice transition 

Correlated metallic state 

W 

Heavy quasiparticle 

(reduced Kinetic Energy) 

Quasiparticle disappears 

U

The Brinkman-Rice transition 

Correlated metallic state. Fermi liquid like aproach 

W 


(reduced Kinetic Energy) 


Reduced 

quasiparticle residue 


at the Mott transition

Mott-Hubbard vs Brinkman-Rice transition 

The Mott-Hubbard transition (insulator) Uc=W 

W 

W 

W 

Gap 

U- W 

U 

The Brinkman-Rice transition (metallic) Uc=2W 

W 


(reduced K.E.) 

Reduced quasiparticle residue 

F* ~Z F 

Quasiparticle disappears



uncorrelated 

correlated 

Transition happens when 

double occupancy 

dissapears 

correlated 

uncorrelated 

Energy of 

independent 

localized electrons

Large U limit. The Insulator. Magnetic exchange 

Antiferromagnetic interactions 

between the localized spins 

(not always ordering) 

Effective exchange interactions 

J ~t 2 /U 

Antiferromagnetic correlations/ordering can reduce the energy 

of the localized spins 

Double occupancy is not zero

Transition between correlated metal and insulator 


Uncorrelated 

insulator 

Uncorrelated 

Metal 

Correlated 

Insulator 

t 2 /U 

Correlated 

Metal 

Transition happens 

with non vanishing 

double occupancy

Mott-Hubbard vs Brinkman-Rice transition 

The Mott-Hubbard transition (insulator) 

W 

W 

W 

Gap 

U- W 

U 

The Brinkman-Rice transition (metallic) 

W 




F* ~Z F 

Quasiparticle disappears

Mott-Hubbard + Brinkman-Rice transition 

Heavy quasiparticle which disappears when 

F* vanishes at U c2 > U c1 

U 

F* ~Z F 

Gap U- W 

between the 

Hubbard bands 

opens at 

U c1 =W=2D 

- Density of States: Quasiparticle and Hubbard 

Bands three peak structure. 

- Two energy scales: F* and the gap between 

the Hubbard bands

Mott transition. Paramagnetic state. DMFT picture 

Infinite dimensions 

Georges et al , RMP 68, 13 (1996) 

U/D=1 

Three peak structure 

U/D=2 

U/D=2.5 

F* 

Heavy quasiparticles 

(coherent) 

U/D=3 

U/D=4 

Hubbard bands 

(incoherent) 

Two energy scales: F* and the gap between the Hubbard bands 

F* Fermi liquid, 

F* Non-Fermi liquid

Mott transition. Paramagnetic state. DMFT picture 

Infinite dimensions 

U/D=1 

U/D=2 

Transfer of spectral weight 

from the quasiparticle peak 

to the Hubbard bands 

U/D=2.5 

U/D=3 

U/D=4 

The gap between the 

Hubbard bands 

opens in the metallic state 

Quasiparticles disappear at the Mott transition 

Georges et al , RMP 68, 13 (1996)

The Mott transition. 

Quasiparticle weight vanishes 

at the Mott transition 

Best order parameter for the 

transition 


Quasiparticle weight : Step at Fermi surface 

Luttinger theorem 

(original version): 

At the Mott transition 

the Fermi surface disappears 

Fermi surface volume 

proportional 

to carrier density 

Localization 

In real space 

Delocalization 

in momentum space

The Mott transition. Paramagnetic state. DMFT picture 

Quasiparticle weight vanishes at the Mott transition 

but double occupancy does not 


DMFT numerical results can depend on the a 

approximation used to solve the impurity problem

The Mott transition. Paramagnetic state. 

Analogy between Mott transition & liquid-gas transition 

Metal: liquid 

First order phase transition 

(some exception could exist) 

Insulator: gas 

(larger entropy) 

The particles in the gas 

are the doubly occupied 

sites. Density is smaller 

in the insulator (gas)

The Mott transition. Finite temperatures. DMFT 


In the region between the dotted lines both 

a metallic and an insulator solution exist 

A gap between 

Hubbard bands 

opens at U c1 


The quasiparticle peak 

disappears at U c2 

At zero temperature the Mott transition happens at U c2 

when the quasiparticle peak disappears

The Mott transition. Finite temperatures 

The system becomes 

insulating with 

increasing temperature 

First order transition 

Georges et al , RMP 68, 13 (1996)


The system becomes 

insulating with 

increasing temperature 

First order transition 

Georges et al , RMP 68, 13 (1996) McWhan et al, PRB 7, 1920 (1973)


Also in liquid gas transition 

Critical point: 

No distinction of 

what it is a metal 

and what an insulator 

at higher temperatures


Critical point 

Histeresis 

First order 

Limelette et al, Science 302, 89 (2003)


U/D=2.5 

T=0.05 D 

T=0.03 D 

T=0.08 D 

T=0.10 D 

The quasiparticle weight Z decreases with increasing temperature


U/D=2.4 

Resistivity decreases 

with temperature 

(insulator) 

Resistivity increases 

with temperature 

(metal) 

Change from metallic to insulating 

like behavior at a given temperature 

Georges et al, J. de Physique IV 114, 165 (2004), arXiv:0311520


Not so clear distinction between a metal and an insulator at finite temperatures 

Georges et al, J. de Physique IV 114, 165 (2004), arXiv:0311520


2.85 

3 

2.65 2.45 

2.25 

2 

The slope of the linear T 

dependence increases 

with interactions 

3.1 

Fermi liquid: Specific heat 

linear with temperature 

C ~ T ~ m* 

Mass enhanced 


DMFT Georges et al , RMP 68, 13 (1996)


2.85 

3 

2.65 

2.45 

2.25 

2 

U/D=1 

The slope of the linear T 

dependence increases 


3.1 

Fermi liquid: Specific heat 

linear with temperature 

C ~ T ~ m* 

Linearity is lost at a temperature which 

decreases with increasing interactions 

DMFT 

Mass enhanced 

with interactions


T-linear dependence 

at low temperatures 

(Metallic) 

U/D=4 

Change to insulating 

Like behavior at high 

temperatures 

U/D=2 

Activated behavior at low temperatures 

(Insulating) 

DMFT Georges et al , RMP 68, 13 (1996)

Summary II: The Mott transition. 

Half-filling. Zero T . Paramagnetic state 

At half filling and zero temperature. Hubbard model (only on-site 

interactions) Mott transtion: Metal-insulator transition at a given U/W 

Mott-Hubbard approach: Insulator as starting point. A hole or a 

doubly occupied state is able to move. Non-degenerate lower and 

upper Hubbard bands (width W). Gap U-W. Transition Uc=W 

U=0 Degenerate 

W 

W 

W 

Gap 

U- W 

U 


Charge transfer insulators: Lowest excitation with different orbital 

character than the one which opens the gap

Summary II-b: The Mott transition. 


Brinkmann-Rice approach: Metal as starting point. The correlated 

metal avoids double occupancy (Gutzwiller). Quasiparticles with 

larger mass, renormalized Fermi energy, reduced quasiparticle weight 

Z. Transition U ~2 W when Z=0 

W 




F* ~Z F 


Z as an order parameter for the transition

Summary II-c: The Mott transition. 


U/D=1 

U/D=2 

DMFT: 

3-peak spectral function Hubbard 

bands+ quasiparticle peak 

U/D=2.5 

U/D=3 

U/D=4 

2 energy scales: * F 

Gap: U-W 

Z dies at the transition, Gap 

opens at smaller U 

Similarity with liquid-gas 

transition: number of particles in 

the gas is the number of doubly 

occupied states

Summary II-d: The Mott transition. Finite temperatures 

First order transition & critical point 

U/D=4 

U/D=2 

For intermediate U/t 

The metallic character decreases with temperature and eventually can become 

insulator. Change from Fermi liquid behavior at low temperature to insulating 

behavior at higher temperatures 

T=0.03 D 

Incoherence increases with increasing 

temperature & quasiparticles can 

disappear 

T=0.08 D 

T=0.05 D 

T=0.10 D

2nd talk. The Mott transition

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?