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

**The**re 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

**Mott** insulator. Paramagnetic state: Hubbard bands

Double

electron

occupancy

U

Single

electron

occupancy

Remove an electron

(as in photoemission)

Empty

state

**Mott** insulator. Paramagnetic state: Hubbard bands

Double

electron

occupancy

U

Single

electron

occupancy

Remove an electron

(as in photoemission)

Empty

state

Empty state is free to move

**Mott** insulator. Paramagnetic state: Hubbard bands

t

t

**Mott** insulator. Paramagnetic state: Hubbard bands

t

t

Double

occupancy

U

U

Single

occupancy

W

Lower Hubbard band

**Mott** insulator. Paramagnetic state: Hubbard bands

t

t

Upper Hubbard Band

W

U

Lower Hubbard Band

W

**Mott** insulator. Paramagnetic state: Hubbard bands

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

**Mott** **transition**

Uc= W

Gap

U- W

W

W

U

Non-degenerate

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

**Mott** vs charge transfer insulators

3d oxides

U=0

4s band

W

U

3d narrow band

W

2p oxygen band

**Mott** vs charge transfer insulators

3d oxides

U=0

**Mott** insulator

4s band

Lowest excitation

energy d-type (**Mott**)

3d narrow band

Charge transfer

insulator

2p oxygen band

Lowest excitation

energy p-type

**Mott** vs charge transfer insulators

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** 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

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** Brinkman-Rice **transition** from the metallic state.

**The** uncorrelated metallic state: **The** Fermi sea 1FS>

E=K+U

=U/4

=D/2

**The** Brinkman-Rice **transition** from the metallic state.

**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

**The** Brinkman-Rice **transition** from the metallic state.

**The** correlated metallic state: Gutzwiller wave function

Uncorrelated

Correlated

**The** Brinkman-Rice **transition** from the metallic state.

**The** correlated metallic state: Gutzwiller wave function

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

Heavy quasiparticle

(reduced Kinetic Energy)

Quasiparticle disappears

Reduced

quasiparticle residue

Quasiparticle disappears

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

Heavy quasiparticle

(reduced K.E.)

Reduced quasiparticle residue

F* ~Z F

Quasiparticle disappears

**The** Brinkman-Rice **transition** from the metallic state.

**The** correlated metallic state: Gutzwiller wave function

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

**The** correlated metallic state: Gutzwiller wave function

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

Heavy quasiparticle

(reduced K.E.)

Reduced quasiparticle residue

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**

Georges et al , RMP 68, 13 (1996)

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

Georges et al , RMP 68, 13 (1996)

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

Georges et al , RMP 68, 13 (1996)

In the region between the dotted lines both

a metallic and an insulator solution exist

A gap between

Hubbard bands

opens at U c1

**Mott** **transition**

**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** **Mott** **transition**. Finite temperatures

**The** system becomes

insulating with

increasing temperature

First order **transition**

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

**The** **Mott** **transition**. Finite temperatures

Also in liquid gas **transition**

Critical point:

No distinction of

what it is a metal

and what an insulator

at higher temperatures

**The** **Mott** **transition**. Finite temperatures

Critical point

Histeresis

First order

Limelette et al, Science 302, 89 (2003)

**The** **Mott** **transition**. Finite temperatures

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

**The** **Mott** **transition**. Finite temperatures

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

**The** **Mott** **transition**. Finite temperatures

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

**The** **Mott** **transition**. Finite temperatures

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

with interactions

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

**The** **Mott** **transition**. Finite temperatures

2.85

3

2.65

2.45

2.25

2

U/D=1

**The** slope of the linear T

dependence increases

with interactions

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

**The** **Mott** **transition**. Finite temperatures

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

Non-degenerate

Charge transfer insulators: Lowest excitation with different orbital

character than the one which opens the gap

Summary II-b: **The** **Mott** **transition**.

Half-filling. Zero T . Paramagnetic state

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

Heavy quasiparticle

(reduced K.E.)

Reduced quasiparticle residue

F* ~Z F

Quasiparticle disappears

Z as an order parameter for the **transition**

Summary II-c: **The** **Mott** **transition**.

Half-filling. Zero T . Paramagnetic state

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