E. Bascones Instituto de Ciencia de Materiales de Madrid (ICMM-CSIC)

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

2. Mott physics. Course Outline
 Metals and Insulators. Basic concepts: Fermi liquids, Mott
insulators, Slater insulators, nature of magnetism
 The Mott transition: Mott-Hubbard vs Brinkmann-Rice
transition, DMFT description. Charge-transfer vs Mott insulators.
Finite temperatures.
Doping a Mott insulator. The case of cuprates.
Single-orbital systems
Multi-orbital systems
 Mott physics in Multi-orbital systems (at & away half filling)
- Degenerate bands. Effect of Hund’s coupling. Hund’s metals
- Non degenerate bands:Orbital selective Mott transition. Hund
- Spin-orbital Mott insulators (iridates)
 Mott physics in iron superconductors

3. 1st Talk: Basic concepts
 Independent electron & Fermi liquid descriptions
 Mott transition: Breakdown of independent electron picture.
 Itinerant versus atomic description
 Magnetic exchange. Slater versus Mott insulators

4. Bloch theory for Fermi gas:
A(k, )  ( - (k))
A(k, ):
Band states are eigenstates,
i.e. infinite lifetime
Electron spectral function
Probability that an electron has
momentum k and energy
Band energy
States filled up to the Fermi level
Fermi surface in metals
Metals and Insulators. Independent electrons

5. Metals and Insulators. Independent electrons
Metallicity
in clean systems
Bands crossing
the Fermi level
(finite DOS)
Fig: Calderón et al, PRB, 80, 094531 (2009)
Insulating behaviour
in clean systems
Bands below
Fermi level filled
Fig: Hess & Serene, PRB 59, 15167 (1999)

6. Metals and Insulators. Independent electrons
Spin degeneracy:
Each band can hold 2 electrons per unit cell
Even number
of electrons
per unit cell
Insulating
Metallic (in case
of band overlap)
Odd number
of electrons
per unit cell
Metallic

7. Weakly correlated metals: Fermi liquid description
Band theory based on kinetic energy of electrons in presence of a lattice
but electrons interact!
Why does an independent electron theory works at all?

8. Weakly correlated metals: Fermi liquid description
Band theory based on kinetic energy of electrons in presence of a lattice
but electrons interact
Why does an independent electron theory works at all?
Fermi liquid theory (effective theory to describe small energy excited states):
- Elementary excitations: quasiparticles with charge e and spin ½
-The quasiparticles are not electrons but there is a one-to-one correspondence
with an electron
Mattuck

9. Bloch theory for Fermi gas:
A(k, )  ( - (k))
A(k, ):
Band states are eigenstates,
i.e. infinite lifetime
Electron spectral function
Probability that an electron has
momentum k and energy
Band energy
States filled up to the Fermi level
Fermi surface in metals
Weakly correlated metals: Fermi liquid description
Fermi liquid:
There is a Fermi surface.
Close to the Fermi surface the
elementary excitations are
quasiparticles with renormalized
energy *(k) and finite lifetime 1/
Spectral function is broadened
and peaks at *(k)

10. Weakly correlated metals: Fermi liquid description
A(k, ): Electron spectral function
Probability that an electron has momentum k and energy
Fig: Damascelli, Hussain, Shen, RMP 75, 473 (2003)

11. Fermi liquid description
Fig: Lu et al, Nature 455, 81 (2008)
Angle Resolved Photoemission
Experiments (ARPES) would
show energy bands

12. Weakly correlated metals: Fermi liquid description
There is a Fermi surface.
Quasiparticles with renormalized
energy *(k) and finite lifetime 1/
Spectral function is broadened
and peaks at *(k)
A quasiparticle is well defined if
F
ZeroT: quasiparticles at the Fermi Surface have infinite lifetime
~A ( - *F)2 + BT2
Temperature
In Fermi liquid (phase space arguments)
Close to the Fermi surface
quasiparticles are well defined
2

13. Weakly correlated metals: Fermi liquid description
Renormalized mass m*=m/Z
electrons become heavier
Renormalized band energy (k)
Z: quasiparticle weight 0  Z 1
smaller Z : larger effect of interactions
Z=0 there are no quasiparticles
Z also gives the quasiparticle
peak height in the spectral function

14. Fermi liquid description
Fig: Lu et al, Nature 455, 81 (2008)
Angle Resolved Photoemission
Experiments (ARPES) would
show energy bands but with a
renormalized bandwidth
How well defined it is the band and how much reduced is the bandwidth
give an idea of the value of Z.
If Z vanishes the band is not well defined. Smaller Z: narrower band

15. Fermi liquid behaviour
Metal (Fermi liquid)
Resistivity increases with temperature
~ 0 + A T2
A ~ m*2
Specific heat linear with temperature
C ~ T ~ m*
Magnetic susceptibility
does not depend on temperature
~ ~ m*
Experimental measurements
help to identify the strength
of interactions in metals
Not always easy to probe (phonons , …)

16. Metals and Insulators. Mott insulators
Fig: Pickett, RMP 61, 433 (1989)
Electron counting
La2CuO4: 2 La (57x2)+Cu (29) + 4 O (4x8)=175 electrons
Metallic behavior
expected

17. Breakdown of independent electron picture
Fig: Pickett, RMP 61, 433 (1989)
Metallic behavior
expected
Insulating behavior is found
Breakdown of independent electron picture

18. Mott insulators
Fig: Pickett, RMP 61, 433 (1989)
Metallic behavior
expected
Insulating behavior is found
Mott insulator:
Insulating behavior due to electron-electron interactions
Do not be confused with Anderson localization which is due to disorder

19. Kinetic energy. Delocalizing effect
Fig: Calderón et al, PRB, 80, 094531 (2009)
atomic site (ij)Atomic
orbital
spin
Adding
electrons
Filling bands
(rigid band shift)
Kinetic energy
Going from one atom to another
Delocalizing effect

20. Interaction energy
1 Atomic level.
Tight-binding (hopping)
Intra-orbital
repulsion
E
Consider 1 atom with a single orbital
Two electrons in the same
atom repel each other
1 electron (two possible states)
E =0
2 electron (the energy changes)
To add a second electron
to single filled orbital
costs energy U
Energy states depend
on the occupancy
(non-rigid band shift)

21. Kinetic and 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)

22. Kinetic and 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

23. Mott insulators
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
For U >> t electrons localize: Mott insulator

24. The Mott transition
Atomic lattice with a single orbital per site and average occupancy 1
half filling
Hopping
saves energy t
Double occupancy
costs energy U
For U >> t electrons localize: Mott insulator
Small U/t
Metal
Large U/t
Insulator
Increasing U/t
Mott transition

25. The Bandwidth
Increasing coordination number increases kinetic energy gain and bandwidth
1 dimension: hops to two neighbors
2 dimensions square lattice:
hops to four neighbors
2 dimensions triangular lattice:
hops to six neighbors
Bandwidth: (half bandwidth) D, bandwidthW
Parameter controlling Mott transition U/D or U/W

26. Itinerant vs localized electrons
Fig: Calderón et al, PRB, 80, 094531 (2009)
Metal:
Electrons delocalized in real space,
localized in k-space.
Description in terms of electronic
bands
Mott Insulator:
Electrons localized in real space,
delocalized in k-space.
Spin models. Description as localized
spins is meaningful

27. Itinerant vs localized electrons
Metal (Fermi liquid) Mott insulator
Resistivity increases with temperature Resistivity decreases with temperature
~ 0 + A T2
A ~ m*2
Specific heat linear with temperature
C ~ T ~ m*
Magnetic susceptibility
does not depend on temperature
~ ~ m*
Specific heat activated like behavior
Magnetic susceptibility inversely
proportional to temperature
~ + C’/(T+ )

28. Itinerant vs localized electrons
s & p
electrons
generally
delocalized
3d: competition between
kinetic energy & interaction
Interaction strength decreases
in 4d & overall in 5d
4f electrons are localized, 5f are also expected to be quite localized

29. Metals and Insulators. Independent electrons
Spin degeneracy:
Each band can hold 2 electrons per unit cell
Even number
of electrons
per unit cell
Insulating
Metallic (in case
of band overlap)
Odd number
of electrons
per unit cell
Metallic

30. Slater vs Mott insulators
Antiferromagnetism doubles the unit cell
1 electron per site
2 electrons per unit cell
(even number of electrons/unit cell)
Slater insulators: Insulating behavior due to unit cell doubling
(Antiferromagnetism)
The shape of the Fermi can lead to an antiferromagnetic instability

31. Slater vs Mott insulators
Antiferromagnetism doubles the unit cell
1 electron per site
2 electrons per unit cell
(even number of electrons/unit cell)
Slater insulators: Insulating behavior due to unit cell doubling
(Antiferromagnetism)
Mott insulators: Insulating behavior does not require AF
The shape of the Fermi can lead to an antiferromagnetic instability

32. Slater vs Mott insulators
Paramagnetic
Mott
Insulator
Metal-Insulator
transition with
decreasing pressure
Increasing Pressure: decreasing U/W
Antiferromagnetism
McWhan et al, PRB 7, 1920 (1973)

33. Large U limit. The Insulator. Magnetic exchange
Mott insulator:
Avoid double occupancy
(no constraint on spin ordering)

34. Large U limit. The Insulator. Magnetic exchange
Virtual transition
t2/U
Mott insulator:
Avoid double occupancy
(no constraint on spin ordering)

35. 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

36. Nature of antiferromagnetism
Fermi surface instability Antiferromagnetic exchange
- Delocalized electrons. Energy
bands in k-space and Fermi surface
good starting point to describe
the system.
-The shape of the Fermi surface
presents a special feature (nesting)
-In the presence of small
interactions antiferromagnetic
ordering appears.
- Ordering can be incommensurate
Spin DensityWave
Magnetism driven by interactions
- Localized electrons. Spins localized in
real space
-Kinetic energy favors virtual hopping
of electrons (t2/ E ~ t2/ E ).
-Virtual hopping results in interactions
between the spins. Magnetic Exchange
Spin models
- Magnetic ordering appears if frustration
(lattice, hopping, …) does not avoid it.
- Commesurate ordering
Magnetism driven by kinetic energy

37. 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*

38. 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 to avoiding double occupancy not
with magnetism (Slater insulators)
 Magnetism:
Weakly correlated metals: Fermi surface instability
Mott insulators: Magnetic exchange (t2/U). Spin models