Download Correlated Electrons: A Dynamical Mean Field

Quantum Criticality and
Fractionalized Phases.
Discussion Leader :G. Kotliar
Grodon Research Conference on
Correlated Electrons 2004
• Steve Julian: overview of phase transitions
in heavy fermion systems.
• Z.X. Shen ARPES Investigations of
Cuprate Superconductors.
• Senthil Deconfined Criticality.
Heavy Fermions.
• Fermi Liquid . . Correspondence between a
system of non interacting particles and the
full Hamiltonian.
• High temperatures, system of moments
and light electrons.
• Low temperature, heavy fermion liquid or
ordered magnet .Doniach Criteria .
• In the early 80’s , poster boy for Fermi
Liquid Theory.
Late 80’s early 90’s Cuprate
Superconductivity
Broad region of parameters
where Fermi liquid theory
does not apply .
Search for new paradigms, need
a starting point to describe this
phenomena.
Heavy Fermions. 90’s.
No longer a poster boy
for Fermi liquid theory.
Several examples
exhibiting, magnetism,
superconductivity, and
their disapearence, and
regimes where Fermi
liquid theory failed to
give a proper
description.[Megan
Aronson this morning]
Steve Julian’s talk
Quantum Phase Transitions: Standard
approach [Hertz, Moriya, Millis ]
• Identify order parameter (s), e.g. m, D
• Write effective Lagrangian for the order
parameter.
• Imaginary time is like an additional dimension.
• Carry out standard R.G, make predictions which
can be compared with experiments.
• Caveat, fermions have been integrate out,
and these are low energy degrees of freedom.
Ignore Berry phases.
• In several materials,[e.g. YbRh2Si2]
experimental evidence has accumulated
showing that some AF to FL transitions ARE
NOT described by the standard approach.
• Senthil’s talk, a new class of quantum critical
point, where degrees of freedom which are not
manifestly present in a description based on an
order parameter play a fundamental role.
Quantum Criticality in
Cuprates[from Tallon and Loram ]
Use the existence of an hypothetical or physical quantum critical point, as a
tool to approach the physics of the cuprates . Theoretical ideas, S. Sachdev.
The QCP paradigm: the area of
influence of a quantum critical point
A different Approach
• The evolution of the electronic structure
away from the Mott insulating state, is key.
• Need to understand this problem and the
correponding phases, before the phase
transitions.
• Talk by Z.X. Shen . Photoemission studies
of high Tc.
• Recent progress in understanding this
problem using cellular DMFT. )
RVB phase diagram of the Cuprate
Superconductors
• P.W. Anderson.
Baskaran Zou and
Anderson.
Connection between
high Tc and Mott
physics.
• <b> coherence order
parameter.
• K, D singlet formation
order paramters.
G. Kotliar and J. Liu Phys.Rev. B
38,5412 (1988)
• High temperature superconductivity is an
unavoidable consequence of the need to
connect with Mott insulator that does not break
any symmetries to a metallic state.
• Tc decreases as the quasiparticle residue goes
to zero at half filling and as the Fermi liquid
theory is approached.
• Early on, accounted for the most salient features
of the phase diagram. [d-wave
superconductivity, anomalous metallic state,
pseudo-gap state ]
Problems with the approach.
• Numerous other competing states. Dimer phase, box
phase , staggered flux phase , Neel order,
• Stability of the pseudogap state at finite temperature.
• Missing finite temperature . [ fluctuations of slave bosons
,]
• Temperature dependence of the penetration depth [Wen
and Lee , Ioffe and Millis ] Theory:
 r[T]=x-Ta x2 , Exp: r[T]= x-T a.
• Theory has uniform Z on the Fermi surface, in
contradiction with ARPES. [see however Varma and
Abrahams ]
Evolution of the spectral
function at low frequency.
A(  0, k )vs k
Ek=t(k)+Re( k ,   0)  
 k = Im( k ,   0)
A( k ,   0) 
k
 k 2  Ek 2
If the k dependence of the self energy is
weak, we expect to see contour
lines corresponding to Ek = const
and a height increasing as we
approach the Fermi surface.
Study a model of kappa organics.
Evolution of the k resolved Spectral Function at
zero frequency. ( O. Parcollet G. Biroli and
GKotliar PRL, 92, 226402. (2004))
A(  0, k )vs k
U/D=2
Uc=2.35+-.05, Tc/D=1/44
U/D=2.25
Keeps all the goodies of the slave boson mean
field and make many of the results more solid
but also removes the main difficulties.
• Can treat coherent and incoherent spectra.
• Not only superconductivity, but also the
phenomena of momentum space differentiation
(formation of hot and cold regions on the Fermi
surface) are unavoidable consequence of the
approach to the Mott insulator.
• Can treat dynamical fluctuations between
different singlet order parameters.
• Surprising role of the off diagonal self energy
which renormalizes t’.
Lattice and cluster self energies
Mechanism for hot spot formation: nn
self energy ! General phenomena.
Mott transition in cluster (QMC)
• General result ? YES. Application to
model with isotropic t and t’ with possible
relevance cuprates: M. Capone, M.
Civelli, V. Kancharla, O. Parcollet, and
G.K. Switch to ED solver. [ See poster by
M. Civelli ].
• Switch of hot-cold regions in electron and
hole doped system.
Energy Landscape of a Correlated
Material and a finite temperature
approach to correlated materials.
Energy
T
Configurational Coordinate in the space of Hamiltonians
Am under pressure. Lindbaum
et.al. PRB 63,2141010(2001)
ITU [J.C. Griveaux J. Rebizant G.
Lander]
Overview of rho (p, T)
of Am
• Note strongly
increasing
resistivity as f(p)
at all T. Shows
that more
electrons are
entering the
conduction band
• Superconducting
at all pressure
• IVariation of rho
vs. T for
increasing p.
DMFT study in the fcc structure. S.
Murthy and G. Kotliar
fcc
LDA+DMFT spectra. Notice the
rapid occupation of the f7/2 band.
One electron spectra. Experiments (Negele) and LDA+DFT
theory (S. Murthy and GK )
Mott transition in open (right) and
closed (left) shell systems.
S
T
Log[2J+1]
S
???
Uc
U
S=0
 ~1/(Uc-U)
U
• Approach the Mott transition, if the
localized configuration has an OPEN shell
the mass increases as the transition is
approached.
Consistent theory, entropy increases
monotonically as U  Uc .
• Approach the Mott transition, if the
localized configuration has a CLOSED
shell. We have an apparent paradox. To
approach the Mott transitions the bands
have to narrow, but the insulator has not
entropy.. SOLUTION: superconductivity
intervenes.
Mott transition in systems
evolving towards a closed shell.
• Resolution: as the Mott transition is
approached from the metallic side,
eventually superconductivity intervenes to
for a continuous transition to the localized
side.
• DMFT study of a 2 band model for
Buckminster fullerines Capone et. al.
Science ( 2002).
• Mechanism is relevant to Americium.
One dimensional Hubbard model .
Compare 2 site cluster (in exact diag with Nb=8) vs exact Bethe Anzats,
[V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][ [M.
CaponeM.Civelli V Kancharla C.Castellani and GK Phys. Rev. B 69, 195105
U/t=4.
(2004) ]
What to do as a chair?
• Humour
• Make a connection ?
• Give a bit of orientation, for students
postdocs, historical backround.
• Point of view of the relevance of quantum
critical phenomena.
• Advertise a different philosophy, and
approach.