Disorder and Correlations in Mott-Hubbard systemsqcmjc/talk_slides/QCMJC.2012.08.09... · Disorder...
Transcript of Disorder and Correlations in Mott-Hubbard systemsqcmjc/talk_slides/QCMJC.2012.08.09... · Disorder...
Disorder and Correlations in Mott-Hubbard systems
N. S. Vidhyadhiraja
Theoretical Sciences UnitJNCASR, BangaloreIndia
( Journal Club talk on the recent work by V.Dobrosavljevic's group)
Aug 2012
Background● Basic question – What is the effect of disorder on
quantum criticality?● Insulating quantum magnets – the answer is –
infinite randomness fixed point and the associated Griffiths phase
● Authors – What happens in itinerant systems?● Experimental motivation – existence of disorder
driven non-Fermi Liquid behaviour due to rare configurations
Griffiths singularity● Robert B. Griffiths – PRL 23 (1969) 17.● Random diluted Ising FMs● Occupancy – p● Magnetization M is a
non-analytic function of H atat H=0 for any temperature T < T
c(p=1).
● Landau theory is inapplicable between Tc(p) and
Tc(p=1).
R.B.Griffiths, PRL 23 (1969) 17.
Griffiths singularity - Continued● Rare configurations that are unfrustrated get locked ● Griffiths phase between the spin-glass phase and the
paramagnetic phase● Ordered/Glassy phase – Infinite (large) relaxation
times● Griffiths phase – non-exponential relaxation● Paramagnetic phase – exponential relaxation
Randheria, Sethna and Palmer, PRL 54 (1985) 1321.
Quantum criticality● T=0 continuous quantum phase transition● Example
Gegenwart et al, Nature Physics Vol.4, pp.186-197 (2008).http://www.scitopics.com/Quantum_Criticality_of_Heavy_Fermions.html
S. Sachdev http://arxiv.org/pdf/0907.0008NSV et al, Phys. Rev. Lett. Vol.102, pp.206407 (2009).
Previous work by the same group
● Miranda and Dobrosavljevic PRL 86 264 (2001).● Tanaskovic, Miranda and Dobrosavljevic, PRB 70, 205108 (2004).● Dobrosavljevic and Kotliar PRL 78 3943 (1997).
– Electronic Griffiths phase (EGP)– Bethe Lattice, Disorder driven MITs– Strong disorder– Generalized DMFT (Pre-Stat-DMFT)
Questions asked● Effect of weak to strong disorder on interaction
driven MIT?● Does an EGP emerge even for weak disorder in
contrast to the disorder-driven MIT?
What has been done – In brief● Andrade et al PRL 102, 206403 (2009)
Electronic Griffiths phase of the d=2 Mott transition– Paramagnetic Disordered Hubbard model on an LxL square
lattice with PBC– Site energies picked from a uniform distribution over
[-W/2,W/2]– Use the Kotliar-Ruckenstein (KR) slave bosons functional– Apply statistical DMFT with the slave boson mean field
impurity solver– Study the T=0 U-W phase diagram, the distribution of
quasiparticle weights, critical behaviour of the spatial inhomogeneity, similarity with the infinite randomness fixed point behaviour
What has been found – In brief● The MIT retains the second-order character● Existence of a Griffiths like phase – disorder induced spatial
inhomogeneities● Renormalized disorder only at low energies => Energy
resolved inhomogeneity of local spectral functions
DMFT
● Analogous to Curie-Weiss mean field theory
Trace out all Si
except S0
Ising model Mean field approximation
Effective field – self consistently determined
Magnetization
Georges et al, Rev. Mod. Phys. Vol.68, pp.13-125 (1996).
DMFT
● Now the fermionic case
Hubbard model
Effective single site action
Impurity Self energyLocal lattice Green's function
Dyson's equation
Stat-DMFT
● Maps a disordered lattice problem to L2 impurity problems.
● The hybridization is determined self-consistently for each site.
● Need a fast and accurate impurity solver● Andrade et al used slave-boson mean field theory
with the Kotliar-Ruckenstein functional.
Kotliar-Ruckenstein slave bosons● Model● Introduce 4 slave bosons
● Enlarged the Hilbert space – unphysical states● With local constraints, reduces to the original
problem Kotliar and Ruckenstein, PRL 57 (1986) 1362.Lavagna PRB 41 (1990) 142.Powell cond-mat/0906.1640v7.
KR slave bosons● KR functional
● Mean field (saddle point) approximation – Exact in d=∞ and reduces to Gutzwiller variational approach
● Provides a perturbative extension to the GVA● Contrast: Barnes-Coleman-Read slave bosons are
best for impurity models (N--> ∞). Kotliar and Ruckenstein, PRL 57 (1986) 1362.Lavagna PRB 41 (1990) 142.Powell cond-mat/0906.1640v7.
Uniform Mean fieldFree energy
Minimize the free energy function with respect to the seven parameters.
Result is the Brinkman-rice scenario, since theSaddle point approximation is equivalent to the Gutzwiller variational approximation.
Lavagna PRB 41 (1990) 142..
Andrade et al● Effective action within Stat-DMFT
● Self-consistency condition
Z=quasiparticle weightv=Effective energy level
Andrade et al● Free energy functional
● Approach to Mott transition – Vanishing of the typical quasiparticle weight
Evidence for Griffiths phase● Spatial inhomogeneities – Rare Mott droplets with
anomalously large susceptibility
(χi ~ 1/Z
i)