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, • 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: [T]=x-Ta x2 , Exp: [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.

( 0, )vs k A k

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. Frustration.

k

k2 2

k

Ek=t(k)+Re ( , 0)

= Im ( , 0)

( , 0)Ek

k

k

A k

Evolution of the k resolved Spectral Function at zero frequency. ( O. Parcollet G. Biroli and

GKotliar PRL, 92, 226402. (2004))

Uc=2.35+-.05, Tc/D=1/44

U/D=2 U/D=2.25

( 0, )vs k A k

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

Configurational Coordinate in the space of Hamiltonians

T

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 S

U U

TLog[2J+1]

Uc

~1/(Uc-U)

S=0

???

• 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

(2004) ]

U/t=4.

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.