The document summarizes key aspects of Mott physics and the Mott transition. It discusses how interactions in metals can be described using Fermi liquid theory with quasiparticles. It then covers how the Mott transition occurs in single band systems at half filling from a metal to an insulator as the ratio of on-site interaction to bandwidth (U/W) increases. Specifically: 1) The Mott-Hubbard approach views the insulator as the starting point, with the opening of a gap U-W between upper and lower Hubbard bands at the transition point Uc=W. 2) The Brinkman-Rice approach views the metal as the starting point, with quasiparticles

1. Mott physics
2nd Talk
E. Bascones
Instituto de Ciencia de Materiales de Madrid (ICMM-CSIC)

2. 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 T2
A ~ m*2
C ~ T
~ m*
~
~ m*

3. Summary I-b
 Interactions are more important in f and d electrons and decrease
with increasing principal number (U3d > U4d …) .
 With interactions energy states depend on occupancy: non-rigid
band shift
In one orbital systems with one electron per atom (half-filling) on-
site 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 (t2/U). Spin models

4. 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

5. The Mott transition. Paramagnetic state
Paramagnetic
Mott
Insulator
Metal-Insulator
transition with
decreasing pressure
Increasing Pressure: decreasing U/W
Antiferromagnetism
McWhan et al, PRB 7, 1920 (1973)

6. 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
Large U/t
Insulator
Increasing U/t
Mott transition
W
Single
occupancy
Double
occupancy
U

7. Hubbard model. Kinetic and On-site interaction Energy
Tight-binding (hopping)
Intra-orbital
repulsion
Kinetic energy
Intra-orbital
repulsion
E
Atomic lattice with a single orbital per site and average occupancy 1 (half filling)
Hopping
saves energy t
Double occupancy
costs energy U
Hopping restricted to first nearest neighbors: Electron-hole symmetry

8. Single
electron
occupancy
Double
electron
occupancy
U
Mott insulator. Paramagnetic state: Hubbard bands

9. U
Remove an electron
(as in photoemission)
Mott insulator. Paramagnetic state: Hubbard bands
Empty
state
Single
electron
occupancy
Double
electron
occupancy

10. Mott insulator. Paramagnetic state: Hubbard bands
U
Empty
state
Empty state is free to move
Remove an electron
(as in photoemission)
Single
electron
occupancy
Double
electron
occupancy

11. t
t
Mott insulator. Paramagnetic state: Hubbard bands

12. Single
occupancy
Double
occupancy
U
t
t
U
W
Lower Hubbard band
Mott insulator. Paramagnetic state: Hubbard bands

13. t
t
U
W
WLower Hubbard Band
Upper Hubbard Band
Mott insulator. Paramagnetic state: Hubbard bands

14. Mott insulator. Paramagnetic state: Hubbard bands
U
W
WLower Hubbard Band
Upper Hubbard Band
Singly occupied states
Doubly occupied states
Non-degenerate bands

15. The Mott-Hubbard transition. Paramagnetic state
U
W
W
W
Double
degenerate
band (spin)
Increasing U
U=0
Non-degenerate
bands
Gap
U-W

16. The Mott-Hubbard transition. Paramagnetic state
U
W
W
W
Double
degenerate
band (spin)
Increasing U
W
W
U=0
Non-degenerate
bands
Gap
U-W
Mott transition
Uc=W
Gap opens at the Fermi level at Uc

17. Mott vs charge transfer insulators
U=0
3d oxides
3d narrow band
2p oxygen band
4s band

18. Mott vs charge transfer insulators
U=0
3d oxides
3d narrow band
2p oxygen band
4s band
U
W
W

19. Mott vs charge transfer insulators
U=0
3d oxides
3d narrow band
2p oxygen band
4s band
Lowest excitation
energy p-type
Lowest excitation
energy d-type (Mott)
Mott insulator
Charge transfer
insulator

20. Mott vs charge transfer insulators
Cuprates are
charge transfer insulators

21. 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

22. 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
Cost in interaction energy per particle
Probability in real space: ¼ per the 4 possible states (half filling)
Kinetic energy gain per particle
(constant DOS)
<U>=U/4
<K>=W/4=D/2
The Brinkman-Rice transition from the metallic state.

23. The uncorrelated metallic state: The Fermi sea 1FS>
<U/D>
<K/D>
E=K+U
<E/D>
<U>=U/4
<K>=D/2
The Brinkman-Rice transition from the metallic state.

24. The correlated metallic state: Gutzwiller wave function
| >=j[ 1-(1- )njnj]1FS>
Variational Parameter
=1 U=0
=0 U=
Gutzwiller Approximation. Constant DOS
uniformly diminishes
the concentration of
doubly occupied sites
Uncorrelated
Correlated
The Brinkman-Rice transition from the metallic state.

25. The correlated metallic state: Gutzwiller wave function
Correlated
Uncorrelated
The Brinkman-Rice transition from the metallic state.

26. The correlated metallic state: Gutzwiller wave function
<K>uncorrelated
<K>correlated
<U>correlated
<U>uncorrelated
Kinetic Energy
is reduced
Average potential energy
reduced due to reduced
double occupancy
The Brinkman-Rice transition from the metallic state.

27. The Brinkman-Rice transition
W
Heavy quasiparticle
(reduced Kinetic Energy)
Quasiparticle disappears
Correlated metallic state
U

28. The Brinkman-Rice transition
W
Heavy quasiparticle
(reduced Kinetic Energy)
Quasiparticle disappears
Correlated metallic state. Fermi liquid like aproach
Reduced
quasiparticle residue
Quasiparticle disappears
at the Mott transition

29. Mott-Hubbard vs Brinkman-Rice transition
UW
W
W
Gap
U-W
The Mott-Hubbard transition (insulator) Uc=W
The Brinkman-Rice transition (metallic) Uc=2W
W
Heavy quasiparticle
(reduced K.E.)
Reduced quasiparticle residue
Quasiparticle disappears
F* ~Z F

30. The correlated metallic state: Gutzwiller wave function
<K>uncorrelated
<K>correlated
<U>correlated
<U>uncorrelated
Transition happens when
double occupancy
dissapears
The Brinkman-Rice transition from the metallic state.
Energy of
independent
localized electrons

31. Large U limit. The Insulator. Magnetic exchange
Antiferromagnetic interactions
between the localized spins
(not always ordering)
J ~t2/U
Effective exchange interactions
Antiferromagnetic correlations/ordering can reduce the energy
of the localized spins
Double occupancy is not zero

32. The correlated metallic state: Gutzwiller wave function
Correlated
Metal
Uncorrelated
Metal Correlated
Insulator
Uncorrelated
insulator
Transition between correlated metal and insulator
t2/U
Transition happens
with non vanishing
double occupancy

33. Mott-Hubbard vs Brinkman-Rice transition
UW
W
W
Gap
U-W
The Mott-Hubbard transition (insulator)
The Brinkman-Rice transition (metallic)
W
Heavy quasiparticle
(reduced K.E.)
Reduced quasiparticle residue
Quasiparticle disappears
F* ~Z F

34. U
Gap U- W
between the
Hubbard bands
opens at
Uc1=W=2D
F* ~Z F
Heavy quasiparticle which disappears when
F* vanishes at Uc2 > Uc1
Mott-Hubbard + Brinkman-Rice transition
- Density of States: Quasiparticle and Hubbard
Bands three peak structure.
- Two energy scales: F* and the gap between
the Hubbard bands

35. Hubbard bands
(incoherent)
Heavy quasiparticles
(coherent)
Georges et al , RMP 68, 13 (1996)Infinite dimensions
U/D=1
U/D=2
U/D=2.5
U/D=3
U/D=4
Three peak structure
Two energy scales: F* and the gap between the Hubbard bands
F*
Mott transition. Paramagnetic state. DMFT picture
F* Fermi liquid, F* Non-Fermi liquid

36. Mott transition. Paramagnetic state. DMFT picture
Georges et al , RMP 68, 13 (1996)
Infinite dimensions
U/D=1
U/D=2
U/D=2.5
U/D=3
U/D=4
Transfer of spectral weight
from the quasiparticle peak
to the Hubbard bands
Quasiparticles disappear at the Mott transition
The gap between the
Hubbard bands
opens in the metallic state

37. 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
At the Mott transition
the Fermi surface disappears
Localization
In real space
Delocalization
in momentum space
Luttinger theorem
(original version):
Fermi surface volume
proportional
to carrier density

38. The Mott transition. Paramagnetic state. DMFT picture
Georges et al , RMP 68, 13 (1996)
DMFT numerical results can depend on the a
approximation used to solve the impurity problem
Quasiparticle weight vanishes at the Mott transition
but double occupancy does not

39. 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)

40. 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 Uc1
The quasiparticle peak
disappears at Uc2
Georges et al , RMP 68, 13 (1996)
Mott transition
At zero temperature the Mott transition happens at Uc2
when the quasiparticle peak disappears

41. The Mott transition. Finite temperatures
First order transition
The system becomes
insulating with
increasing temperature
Georges et al , RMP 68, 13 (1996)

42. The Mott transition. Finite temperatures
First order transition
The system becomes
insulating with
increasing temperature
Georges et al , RMP 68, 13 (1996) McWhan et al, PRB 7, 1920 (1973)

43. The Mott transition. Finite temperatures
Critical point:
No distinction of
what it is a metal
and what an insulator
at higher temperatures
Also in liquid gas transition

44. The Mott transition. Finite temperatures
Histeresis
First order
Critical point
Limelette et al, Science 302, 89 (2003)

45. The Mott transition. Finite temperatures
T=0.03 D
T=0.05 D
T=0.08 D
T=0.10 D
The quasiparticle weight Z decreases with increasing temperature
U/D=2.5

46. The Mott transition. Finite temperatures
U/D=2.4
Change from metallic to insulating
like behavior at a given temperature
Resistivity increases
with temperature
(metal)
Resistivity decreases
with temperature
(insulator)
Georges et al, J. de Physique IV 114, 165 (2004), arXiv:0311520

47. Not so clear distinction between a metal and an insulator at finite temperatures
The Mott transition. Finite temperatures
Georges et al, J. de Physique IV 114, 165 (2004), arXiv:0311520

48. The Mott transition. Finite temperatures
The slope of the linearT
dependence increases
with interactions
C ~ T ~ m*
Fermi liquid: Specific heat
linear with temperature
Mass enhanced
with interactions
3.1
3
2.85
2.65 2.45
2.25
2
DMFT Georges et al , RMP 68, 13 (1996)

49. The Mott transition. Finite temperatures
The slope of the linearT
dependence increases
with interactions
C ~ T ~ m*
Fermi liquid: Specific heat
linear with temperature
Mass enhanced
with interactions
Linearity is lost at a temperature which
decreases with increasing interactions
U/D=12
2.25
2.452.65
2.85
3
3.1
DMFT

50. The Mott transition. Finite temperatures
U/D=4
U/D=2
Activated behavior at low temperatures
(Insulating)
T-linear dependence
at low temperatures
(Metallic)
Change to insulating
Like behavior at high
temperatures
DMFT Georges et al , RMP 68, 13 (1996)

51. 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
 Charge transfer insulators: Lowest excitation with different orbital
character than the one which opens the gap
UW
W
W
Gap
U-W
U=0 Degenerate
Non-degenerate

52. 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
 Z as an order parameter for the transition
W
Heavy quasiparticle
(reduced K.E.)
Reduced quasiparticle residue
Quasiparticle disappears
F* ~Z F

53. Summary II-c: The Mott transition.
Half-filling. Zero T . Paramagnetic state
U/D=1
U/D=2
U/D=2.5
U/D=3
U/D=4
DMFT:
3-peak spectral function Hubbard
bands+ quasiparticle peak
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

54. Summary II-d: The Mott transition. Finite temperatures
First order transition & critical point
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
Incoherence increases with increasing
temperature & quasiparticles can
disappear
T=0.03 D
T=0.05 D
T=0.08 D
T=0.10 D
For intermediate U/t
U/D=4
U/D=2