Disorder in strongly correlated metals

  Quantum Ripples in Strongly Correlated Metals: Phys. Rev. Lett. 104, 236401 (2010)
Energy-Resolved Spatial Inhomogeneity of Disordered Mott Systems: Physica B 404, 3167 (2009)
Electronic Griffiths Phase of the d = 2 Mott Transition: Phys. Rev. Lett. 102, 206403 (2009)

 Recommended and commented on in the Journal Club for Condensed Matter Physics, Jan/2012

   Introduction

    In recent years, fascinating new evidence is emerging revealing that genuinely new phenomena (Miranda and Dobrosavljevic, 2005) arise in presence of disorder and impurities. These phenomena often dominates the observable properties in many systems, ranging from colossal magnetoresistance (CMR) manganites and cuprates, to diluted magnetic semiconductors, Kondo alloys, and even spin liquids.
    Disorder is also useful as a probe to reveal the underlying texture of several system. A primary example of disorder as a probe comes from the scanning tunneling microscopy (STM) in regular metals, cuprates, and more recently heavy-fermion systems. Therefore, understanding the combined effect of disorder and electron–electron interactions is not only of immense experimental relevance, as it also constitutes one of the most challenging problems in theoretical condensed matter physics.

   Electronic Griffiths phase near Mott metal-insulator transitions

The metal-insulator transition (MIT) represents an important class of quantum criticality, one that often cannot be reduced to breaking any static symmetry. Conventional theories of the MIT in disordered systems, based on the diffusion mode picture, strongly resemble standard critical phenomena and thus do not easily allow for rare event physics or ‘‘infinite randomness fixed point (IRFP) behavior. There currently exists, however, a large body of experimental work, documenting disorder-induced non-Fermi-liquid behavior due to rare disorder configurations, even in systems far from any spin or charge ordering.

In this study, we presented the first detailed model calculation investigating the effects of moderate disorder on the Mott metal insulator in two dimensions. As the simplest description of the effects of disorder on the Mott transition, we work within a Brinkman-Rice scenario, where a Gutzwiller variational approximation is applied to a disordered two dimensional Hubbard model (in its standard notation, but with random on-site energies εi):

hubbard


While restricted to paramagnetic phases and to low-temperatures, this methodology captures interaction-induced local physics (i.e. formation and screening of local moments) and disorder induced spatial variations in the single-particle sector and thus is well suited to deal with both Mott and Anderson localization. Our results demonstrate that:


(i) for sufficiently weak disorder the transition retains the second order Mott character, where electrons gradually turn into localized magnetic moments;

(ii) disorder induced spatial inhomogeneities (rare regions) give rise to an intermediate electronic Griffiths phase that displays IRFP character at the transition, even when we approach the Mott insulator at weak disorder. This Griffiths phase is characterized by a singular power-law distribtuion of the quasiparticle weight P(Z) ~ Zα-1 at low-Z, which imediately drives a singular thermodynamic response: χ ~ Tα-1 and C/T ~ Tα-1;
(iii) the renormalized disorder seen by quasiparticles is strongly screened only at low energies, resulting in pronounced energy-resolved inhomogeneity of local spectral functions.
phase diagram
Fig. 1: T = 0 phase diagram of the disordered half-filled paramagnetic Hubbard model in d = 2, as a function of the interaction U at weak to moderate disorder strength W < U. An intermediate electronic Griffiths phase emerges separating the disordered Fermi liquid metal and the Mott insulator. Inset: typical (Ztyp) and average (Zav) values of the local quasiparticle weight Zi as a function of U. The Mott transition is identified by the (linear) vanishing of Ztyp. Note that Zav is finite at UC, indicating that a fraction of the sites remain s nearly empty or doubly occupied.
rare event
Fig. 2: (a) Spatial distribution of the (normalized) local spin susceptibilities χi ~ Zi-1, illustrating a typical disorder realization containing a rare event with χtyp « χi. (b) Disorder fluctuations are eliminated outside a box of size l = 9, without appreciably affecting the rare event. (c) When the box is further reduced (here l=3), the rare event is rapidly (exponentially) suppressed, establishing the nonlocal nature of the rare event, in strong support of the "infinite randomness fixed point" picture.





ldos


Fig. 3: Spatial distribution of the local density of states ρi normalized by its clean and non-interacting value for one given realization of disorder and two distinct values of interaction: U/UC=0.87 (a) and U/UC=0.96 (b). We consider two different energies E=0 (bottom) and E>0 (top) in  both (a) and (b). Because of strong disorder screening, the renornmalized site energies |vi| « W close to UC. Thus, at the Fermi energy, the local density of states distribution becomes homogeneous as we approach the Mott transition. Conversely, if we move even slightly away from the Fermi energy, the fluctuations in Ztyp/Zi come into play and  the system becomes in fact more inhomogeneous close to Mott metal-insulator transition


  Quantum Ripples in Strongly Correlated Metals






    • We focus on single nonmagnetic impurity scattering in an otherwise uniform, strongly interacting, paramagnetic metal, where the analysis is most straightforward and transparent, but this general issue is of key relevance also for the diffusive regime.
    Once more, a Gutzwiller variational approximation is applied to a disordered two dimensional Hubbard model and we investigate how well-known effects of the long-ranged Friedel oscillations are affected by strong electronic correlations. Our mostly analytical results demonstrate that:


    (i) for sufficiently weak correlations we recover the results of the Hartree-Fock approximation, in which the effective scattering potential generated by the impurity is set by the long-ranged Friedel oscillations;


    (ii) as we approach the Mott transition, however, these oscillations are strongly suppressed as the charge screening becomes more and more local, corresponding to a shorter ‘‘healing length’’;


    (iii) a combination of ‘‘healing’’ and inelastic scattering strongly suppresses the Friedel oscillation effects even for moderate correlations.
    friedel
    Fig. 4: Electronic density deviations δni = ni - 1 (compared to the homogeneous system at half-filling n0=1) displaying characteristic Friedel oscillations. From top to bottom we have m/m* = 1.00 and 0.30. The Friedel oscillations appear here as crosses because of the underlying Fermi surface anisotropy. As we enter the strongly correlated regime, these oscillations are suppressed. The color scale encodes only the positive values of δni.


     
    tau elastic
    		 tau inelastic
    Fig. 5:  Amplitude of the the energy dependent transport scattering rate, describing the (elastic) scattering processes of quasiparticles off the screened disorder potential, as a function of the interaction strength U. We compare the Hartree-Fock and Gutzwiller (slave-boson) results. Inset: Leading elastic temperature correction to τtr -1(T) as a function of the interaction U. Here we considered ρ(0)UC = 1.
    		 Fig. 6: Limiting temperatures to the linear in T regime of the resistivity of a electron liquid in d = 2 on a log-log scale, calculated under two different assumptions. T*bulk is the dominating cutoff above which the linear behavior is lost. Thus, the linear in T non-Fermi-liquid region is limited to very low temperatures. Ultimately, as this linear in T regime is also bounded from below by a crossover to the diffusive regime, the ballistic T interval in which these elastic corrections dominate may not be present at all.