Emergence of Quantum Phases
in Novel Materials
Ramón Aguado
Mª José Calderón
Leni Bascones
Belén Valenzuela
Teoría y Simulación de Materiales
Instituto de Ciencia de Materiales de Madrid
E. Bascones
leni@icmm.csic.es
More is different
“The ability to reduce everything to simple fundamental laws
does not imply the ability to start from those laws and
reconstruct the universe.”
“The behavior of large and complex aggregates of elementary
particles, it turns out, is not to be understood in terms of a
simple extrapolation of the properties of a few particles.
Instead, at each level of complexity entirely new properties
appear.”
1972, Anderson
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Emergence
The arising of novel and coherent structures, patterns and
properties during the process of self-organization in
complex systems
Goldstein, Economist (1999)
Figs: wikipedia
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Emergence in condensed matter
Electrons interact with each other and with the environment
Collective many-body behavior
(strongly correlated materials & mesoscopic systems)
To understand the complex behavior
Isolate the fundamental processes
More than 30 years of intensive research effort and
continuously new surprises appear
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Emergence in condensed matter
SCES
Strongly
correlated
electron
systems
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Serious
Challenge
of
Established
Standards
Emergence in condensed matter
Band theory: independent particle picture
Fermi liquid theory
Perturbative versus non-perturbative effect of interactions
Phase transition & symmetry breaking
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Emergence in condensed matter
Quantum states: Magnetism: Ferromagnetic, Antiferromagnetic
Superconductivity, many more
Phase transition: Classical or quantum
Fig: physics.bu.edu
Dimension: Influence in phase transitions but also in “normal
state”
Electrons:
itinerant or localized
Origin of magnetism
Scattering processes
Metallicity
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Interactions: localization and superconductivity
Mott transition:
a material predicted by band theory to be metallic becomes
an insulator due to electronic interactions
(not to be confused with Anderson localization due to disorder)
Superconductivity:
Below a critical temperature resistivity goes to zero.
(due to phonons we know, but probably also
due to magnetic fluctuations or Mott physics)
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Interactions: localization and superconductivity
Cuprates: high temperature superconductors
From an antiferromagnetic Mott insulator to a superconductor and then a metal
Pseudogap: Is it a
new state of matter or
can it be explained with
conventional concepts?
Antiferromagnetic
Mott insulator
Metal
High-temperature
superconductor
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Magnetism and superconductivity often appear close to each other
High temperature superconductors
Iron
superconductors
Cuprates
Their critical temperatures do not look high but
…
Organic
salt
Heavy
fermion
Fig: Faltelmeier,
PRB (2007)
Fig:www.supraconductivite.fr
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Emergence in condensed matter
Strongly correlated electron systems:
very sensitive to changes(Pressure, Magnetic Field, doping…)
Organic material
 Iron ferromagnetism, not completely understood yet
Fig: www.physics.berkeley.edu
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Emergence in condensed matter
Kondo effect:
A minimum in the resistivity of
metals with magnetic impurities
with a lot of physics behind
One dimension:
Dramatic effect of interactions:
Luttinger liquids
Fig: Kasai et al, (2001)
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Fig: Jonpol et al, Science (2009)
Emergence in mesoscopic systems
Kondo effect in
Quantum Dots
Fig:www.purdue.edu
Fig: www.gaia3d.co.uk
Luttinger liquid behavior
in carbon nanotubes
Fig:iopscience.iop.org
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Different phases put in contact
(superconducting & ferromagnetic )
Emergence of Quantum Phases in Novel States: Outline
 Introduction: Emergence and Basic Concepts (L. Bascones)
 Fermi liquid theory (L. Bascones)
 Mott Physics (L. Bascones)
Single band systems (Mott transition)
Multi band systems (Hund metal, OSMT, heavy fermions)
 Luttinger liquids (L. Bascones)
 Kondo effect (R. Aguado)
 Anderson localization due to disorder (R. Aguado)
 Phase transitions: classical, quantum & topological (B. Valenzuela)
 Magnetism (M.J. Calderón)
 Superconductivity (M.J. Calderón)
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Basic concepts: the Fermi gas
• Focus on electrons. Work at fixed
• Electrons are fermions: Fermi-Dirac Statistics
energy
f( ) =
e-(
1
)/KBT+1
chemical
potential
Zero temperature
High temperatures
(classical gas)
Step function
KBT 
States filled up to
the Fermi Energy
which defines a
Fermi surface
f( )  e-
F
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B
MaxwellBoltzmann
distribution
function
Fig: Wikipedia
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/K T
Basic concepts: the Fermi gas
• Low temperatures
• High temperatures
(quantum gas)
(classical gas)
KBT 
Free energy
Cv=
o Specific heat
Cv= T
F
T
KBT 
o Specific heat
Linear in
temperature
Cv= independent of T
Magnetization
o Spin susceptibility
M
=
s
H
Magnetic field
s=
Bohr
magneton
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B
2N(
F)
independent of T
Density of states
at Fermi level
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Pauli susceptibility
o Spin susceptibility
s
1
T
Curie law
Basic concepts: the Fermi gas
Different temperature dependence of measurable quantities
in the degenerate (quantum) and classical limits
• Low temperatures
(quantum gas)
Linear in
temperature
Cv= T
o Specific heat
o Spin susceptibility
s=
KBT 
2N(
F)
B
independent of T
• High temperatures
(classical gas)
KBT 
Cv= independent of T
s
1
T
Curie law
Pauli susceptibility
Consequence of Pauli exclusion principle
Non-interacting fermions are correlated due to Fermi statistics & Pauli principle
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Basic concepts: the Fermi gas
Different temperature dependence
of measurable quantities
in the degenerate (quantum)
and classical limits
• Low temperatures
(quantum gas)
KBT 
• High temperatures
(classical gas)
KBT 
Fermi temperature TF= F/KB
Temperature below which quantum effects are important
In metals
F ~ 3 eV
Metals are to be treated
as degenerate gases
TF ~ 2400 K
Fig: Bruus’s and Flensberg’s book
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Basic concepts: lattice and energy bands vs the continuum
In free space: Parabolic band
2
k
Kinetic energy =
2m
• Isotropic
• No length scale
• Momentum conserved
In a solid:
• periodic potential of the ionic lattice
• The lattice constant a length scale
• Energy bands filled up to
F. Fermi
surface
• Momentum conserved modulus 2 /a
• In general, anisotropic potential
Fig: Calderón et al, PRB, 80, 094531 (2009)
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• Effective mass m-1= |2 /k2|
Basic concepts: lattice and energy bands vs the continuum
In free space: Parabolic band
2
k
Kinetic energy =
2m
Some phenomena can be described in
terms of parabolic bands which substitute
the energy bands close to the Fermi surface
Careful!
Fig: Calderón et al, PRB, 80, 094531 (2009)
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This is not always possible
Some effects are intrinsic to the lattice
Basic concepts: metals and insulators
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)
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Basic concepts: metals and insulators
Spin degeneracy:
Each band can hold 2 electrons per unit cell
Odd number
of electrons
per unit cell
Metallic
Insulating
Even number
of electrons
per unit cell
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Metallic (in case
of band overlap)
Basic concepts: strength of interactions
Strength of interactions: U
Kinetic energy: W bandwidth
U/W
Narrow bands
more sensitive to interactions
Interactions are not a small compared to kinetic energy
• Focus on electrons close to the Fermi surface.
• Core electrons + nucleus form the ion
Kinetic energy: tight binding model using atomic orbitals as a basis
Larger spread of wave function (less localized orbitals)
Larger overlap between orbitals
in neighbouring sites
(larger kinetic energy)
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The interaction between
two electrons on the same site
Is weaker
Basic concepts: strength of interactions
3d: competition between
kinetic energy & interaction
Interaction strength decreases in 4d & overall
in 5d due to larger extension of the orbital
s&p
electrons
kinetic energy
more important
4f overall and 5f orbitals are quite localized on a site.
Very small kinetic energy. Interactions win
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Basic concepts: strength of interactions
Assume 3D, free space (continuum), let n=Ne /V (n electronic density)
Ne: Number of electrons
Kinetic energy
-
Interaction energy
Coulomb interaction:
Interaction energy
Kinetic energy
F
V: Volume
d  n2/3
 <1/r>  n1/3
e2
r
 n2/3 =n-1/3
Careful !
n
0
Due to Pauli principle the importance of interactions decreases with increasing density
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Basic concepts: Screening and Hubbard model
Coulomb interaction:
Interaction is larger when
electrons are closer
e2
r
Long range character
Other electrons in the solid screen the Coulomb interaction. Interaction decays faster
Hubbard model : lattice model, 1 orbital per site
Interactions restricted to
electrons in the same site
ij
I,j lattice sites
spin
tij c c + h.c. + U
Kinetic Energy
(Tight binding)
jnjnj
Interaction Energy
Works better for
localized electrons
It looks simple but it is not well understood. Basic in the study of correlated electrons
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Representations in Quantum Mechanics
Table: Coleman’s book
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Basic concepts: Summary
 Electrons in a solid follow Fermi-Dirac statistic. In a typical metal the
degeneracy temperature is much higher than room temperature.

Cv= T
Linear in
temperature
=
s
B
2N(
F)
independent of T
 Parabolic bands (continuum) versus lattice models
 Band theory: metallic or insulating character on the basis of the number
of electrons per unit cell.
 U/W controls strength of interactions
 In 3 D in the continuum limit the importance of interactions decreases with
increasing density
 s and p orbitals less sensitive to interactions, f orbitals are the most
correlated. d-electrons interplay between kinetic energy & interactions
 Screening and Hubbard model
Representations in quantum mechanics
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