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).

Disorder induced non-Fermi liquid behaviour

Stewart, Rev. Mod. Phys., 73 (2001) 797.

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

Details of solution● 2-D Hubbard Model (LxL, L=30,40,50) with PBC

● Statistical DMFT

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

Findings in detail● Phase diagram

Evidence for Griffiths phase● Probability distribution of quasiparticle weights

For small z

Evidence for Griffiths phase● Exponent

Evidence for Griffiths phase● Spatial inhomogeneities – Rare Mott droplets with

anomalously large susceptibility

(χi ~ 1/Z

i)

Evidence for Griffiths phase● Spatial distribution of local DOS

ω=0.1

ω=0.0

Thank You