A local theory for Mott-Anderson localization

Disorder is ubiquitous in real materials, strongly influencing their properties mirlin_review_2008 ; mackinnon_1993 ; fifty_years . Another aspect of several condensed matter systems like the heavy fermions or transition metal oxides is the presence of strong electron-electron interactions Phys_today_DMFT ; DMFT_RMP . In particular, Coulomb correlations and disorder may individually drive a system towards a metal-insulator transition. While the Anderson metal-insulator transition Anderson is caused by quenched disorder, the Mott-Hubbard metal-insulator transition emerges from strong Coulomb repulsion DMFT_RMP . The simultaneous presence of disorder ($W$) and interaction ($U$) effects is known to influence material properties in subtle ways. Over the last few decades, several experimental works on a range of systems mirlin_review_2008 ; kravchenko_review_2004 ; Miranda1997_review ; perovskite_maiti ; perovskite_kim ; perovskite_athena ; cold_atom1 ; cold_atom2 ; cold_atom3 have highlighted the importance of the interplay of disorder and interactions. The early theoretical studies of such systems TVR_Lee_RMP mainly focused on the weak disorder limit, perturbing around Fermi liquid theory. It is now known that the subtle interplay of disorder and interactions may lead to non-Fermi liquid like responses in the thermodynamic quantities, as observed in several experiments Miranda1997_review ; therefore, one requires a non-perturbative framework that can deal with interactions and disorder on an equal footing.

One of the most intriguing aspects of strongly correlated electron systems is the appearance of low-energy scales Raja_dyn_sca_pam ; Si_Steglich_QCP_review ; Bulla2001 . For metals with strong electronic correlations a frequently observed scenario is the presence of long-lived quasi-particles representing a coherent Fermi liquid picture at the lowest temperatures ($T$) and energy scales. A universal low energy scale, $T^{*}$, lies at the heart of all strongly correlated electron systems that manifests in the universal transport properties of these materials Bulla2001 . Over the past few decades, the dynamical mean field theory (DMFT) has stood out as a very successful theoretical framework for understanding several aspects of the low energy physics of strongly correlated electron systems DMFT_RMP . Since, in DMFT, any lattice-fermion model reduces to a local quantum impurity model, the involvement of Kondo physics is inevitable. Thus, universal behavior due to the emergence of a universal low energy scale is generally attributed to the underlying Kondo effect. For example in the DMFT picture of the metallic phase of the Hubbard model DMFT_RMP ; Logan_mott_ins , the Kondo effect leads to full quenching of the electron spin degrees of freedom resulting in a non-degenerate Fermi liquid ground state characterized by a low energy Fermi liquid scale. In the vicinity of the Mott transition, the strongly correlated metal is therefore characterized by a low-energy scale corresponding to the coherence temperature of a Fermi liquid DMFT_RMP ; Logan_mott_ins .

The emergence of a single low energy/temperature scale may not be restricted to clean strongly-correlated electron systems, but has also been predicted in the context of diluted two-dimensional electron gases (2DEGs) 2DEGs_vlad . Phenomenological theories, based on experimental observations in 2DEGs kravchenko_review_2004 ; Kravchenko2 ; Kravchenko3 , established a similarity between the metal-insulator transitions in such disordered systems and the conventional Mott-Hubbard metal-insulator transition. Studies in these directions are important for understanding the true driving force behind metal-insulator transitions observed in disordered interacting systems. A finite temperature study of the effects of disorder on the non-zero temperature Mott transition disorder_Tneq0 also revealed the prevalence of a single parameter scaling of the distribution of quasi-particle weights in the vicinity of a disordered Mott transition.

A natural question that follows from these studies is whether such scaling with respect to a single low energy scale also manifests in the dynamics of the microscopic quantities like the single particle spectra or the disorder averaged scattering rate in a disordered interacting system at zero temperature. And if such universal dynamics exists, then how general is this observation across the $W-U$ phase diagram? The origin and evolution of such low energy scales with respect to $W$ and/or $U$ would then reflect upon the driving force behind the localization of the electrons.

The understanding of the behavior of the low energy scales in a strongly correlated system thus stand out as a key prerequisite irrespective of the presence or absence of disorder. Several theoretical frameworks have attempted to understand the interplay of disorder and strong correlations AA ; Punnoose2005 ; finkelstein . However, the study of emergent low energy scales in the simultaneous presence of disorder and electron-electron interactions require non-perturbative frameworks. Studies using the framework of the DMFT have provided several insights in these directions. A computationally inexpensive approach involves the framework of DMFT and utilization of the ‘typical’ density of states (TDoS, $\rho_{typ}$) vlad_nikolic ; schubert for self-consistently obtaining the effectively local hybridizing medium, $\Gamma(\omega)$Byczuk2005 ; Aguiar_2006 ; Aguiar2009 ; 50_years_vollhardt ; aguiar_2013 , in which the single impurities are embedded. This construction of the DMFT bath utilizing the $\rho_{typ}(\omega)$ is known as the TMT-DMFT framework 50_years_vollhardt . The TDoS is most appropriately approximated as the geometric average of the local density of states (LDoS), $\langle\rho(\omega)\rangle_{geom}=\rho_{typ}(\omega)=\exp\langle\ln\rho_{i}(\omega)\rangle$ with $\rho_{i}(\omega)$ being the local density of states. Another way of representing this is $\rho_{typ}=\exp\int dV_{i}P(V_{i})\ln\rho_{i}(\omega)$, where $V_{i}$ represents the bare random potential and $P(V_{i})$, the probability distribution followed by these bare site energies. While, $\rho_{typ}(0)$ is critical at the Anderson transition Vollhardt_review ; first_TMT_Vlad2003 , the average density of states (ADoS) given by $\rho_{arith}(\omega)=\int dV_{i}P(V_{i})\rho_{i}(\omega)$ is not critical. The TDoS behaves like an order parameter for the metal-insulator transition originating in the metallic phase; in the insulating phase it is trivially zero at all frequencies. Thus, in principle the TMT framework is designed to capture the physics of the metal-insulator transition approaching from the metallic phase.

The TMT-DMFT method was first applied to the Anderson-Hubbard model by Byczuk et al. Byczuk2005 who explored the $W-U$ paramagnetic phase diagram using numerical renormalization group (NRG) as the impurity solver. Three distinct phases were identified namely, a correlated metallic phase, a Mott insulating phase and an Anderson insulating phase. Additionally, a coexistent regime of the metal-Mott insulating phase was reported. The Mott and Anderson insulator phases were found to be continuously connected. The characterization of these phases were based on the behavior of the band center of the TDoS ($\rho_{typ}(0)$) and the ADoS ($\rho_{arith}(0)$). The metallic phase featured a non-zero $\rho_{typ}(0)$ and $\rho_{arith}(0)$. For weak to moderate $W$, a sharp transition from the metallic to a gapped insulating phase was observed where both $\rho_{typ}(0)=0$ and $\rho_{arith}(0)=0$. This metal-insulator transition was similar in characteristics with the conventional single band Hubbard model and hence this insulating phase was termed as the Mott insulator. Moreover, the density of states in this phase featured prominent Hubbard subbands. Additionally a metal-Mott insulator coexistence regime similar to the p-h symmetric single-band Hubbard model was identified in the $W$-$U$ plane that terminated at a single $W$. The Anderson insulator was characterized as a phase that featured $\rho_{typ}(0)=0$ and $\rho_{arith}(0)\neq 0$. Additionally, the Hubbard bands were broad and diffused.

Although, the NRG is highly efficient in capturing the Kondo effect, the distribution of Kondo scales, a natural occurrence in interacting disordered systems, was not explored in Ref. Byczuk2005, . Thus the role of the local Kondo scales could not be deduced from the above calculation. Such a direction was however explored using slave-boson mean-field theory calculations Aguiar2009 , highlighting the role of the local quasi-particle weights, $Z_{i}$, that may also serve as an order parameter for the localization physics in the Anderson-Hubbard model. Close to the disorder driven metal-insulator transition at $U/W<1$ a two fluid picture was proposed. Through TMT-DMFT calculations they proposed a spatially inhomogeneous picture where in certain regions there existed Mott fluid droplets with $Z_{i}\to 0$ and at other regions $Z_{i}\to 1$ representing Anderson localized particles. Irrespective of the spatially inhomogeneous picture, one would expect a metal-insulator transition to occur at a critical disorder strength, $W_{c}$, when the $U$ is fixed, and this would coincide with the vanishing of the impurity hybridization obtained from the TDoS. A conventional Mott-like picture was proposed to prevail for the $U$ driven metal-insulator transition at sufficiently small disorder strengths. A similar line of reasoning based on the behavior of the impurity hybridization would lead us to expect that the Mott upper critical interaction, $U_{c2}$, would coincide with the vanishing of the impurity hybridization obtained from the TDoS.

However, it is also well known that slave-boson based solvers fail to account for inelastic scattering and thus fail to predict the correct lineshape of spectral functions and scattering rates senechal2006theoretical ; SB_artefact1 . Moreover, the physics at low energies, may be highly affected by the physics at higher energy scales. Thus, in order to have a precise understanding of the spectral/dynamical properties in a correlated system, we require all energy scales and interaction strengths, from weak to strong coupling, to be handled within a unified theoretical framework. In this work we revisit the metallic phase of the Anderson-Hubbard model using the local moment approach (LMA) Eastwood1998 as an impurity solver within the TMT-DMFT. The LMA has been successfully applied for several impurity Eastwood1998 ; LMA_pseudogap_AIM1 ; LMA_NRG_soft_gap_AIM and lattice models Eastwood1997 ; LMA_KIs ; Raja_dyn_sca_pam (within DMFT). The LMA is known to capture the Kondo effect correctly while also capturing the correct lineshape of the spectral functions. With this set up we look into the evolution of the distribution of Kondo scales as a function of $W$ and $U$. Additionally, we explore the scattering dynamics within the current non-perturbative local framework, and identify universal dynamics and scaling similar to the clean interacting scenario. It should be noted that all the calculations presented in this work pertain to the metallic phase and an exploration of the insulating phases is beyond the scope of the current work. These results are therefore relevant in the context of the breakdown of the metallic phase towards Mott or Anderson localization.

The Anderson-Hubbard model is considered as a paradigmatic model for looking into the interplay of strong electron interactions and disorder. It is given by,

where, $c_{i\sigma}^{\dagger}$ ($c_{i\sigma}$) is the fermionic creation (annihilation) operator for an electron with spin $\sigma$ at site $i$, and $\hat{n}_{i\sigma}=c_{i\sigma}^{\dagger}c_{i\sigma}$, $t_{ij}$ is the nearest neighbour site to site hopping amplitude considered to be constant in this work, $U$ is the onsite Coulomb interaction energy. The lattice is represented by a three-dimensional cubic DoS with full bandwidth, $D=3$. The random local potential $V_{i}$ follows a ”box” distribution $P(V_{i})$ such that $P(V_{i})=\frac{1}{2W}\Theta(W-|V_{i}|)$, where $\Theta(x)$ is the Heaviside step function. A global particle-hole symmetry is imposed by $\mu=U/2$. At $W=0$, this model reduces to the particle-hole (p-h) symmetric single-band Hubbard model, which displays a first order Mott transition at zero temperature, $T=0$, as a function of $U$. On approaching this transition from the Fermi liquid (FL) side, the Kondo scale, $T_{K}^{0}$ vanishes at a critical point, $U_{c2}$, marking the transition to the Mott insulating state. On approaching from the Mott insulating side, the Mott gap vanishes at a critical point, $U_{c1}$, where, $U_{c1}<U_{c2}$. This scenario in the $W=0$ case motivates us to look at the regimes, $U<U_{c2}$ and $U>U_{c2}$ distinctly. For a three-dimensional simple cubic DoS, within the LMA, we have found out that the Mott MIT occurs at $U_{c2}/D\sim 0.8$ which corresponds to $U_{c2}\approx 2.3$, the bandwidth ($D$) being equal to 3. This result compares well with the value predicted by NRG calculations ($\sim 1.1D$) bonvca2009van .

For treating non-zero disorder in the presence of interactions (Eq. (1)) we employ the TMT-DMFT framework where we map the disordered lattice model on to an ensemble of single impurity Anderson models, each embedded in a self-consistently determined effective medium, $\Gamma(\omega)$, which is obtained from $\rho_{typ}(\omega)$, as described in Appendix B. The reader is also referred to several previous works Byczuk2005 ; Aguiar_2006 ; Aguiar2009 ; aguiar_2013 ; Vollhardt_review for the details of the formalism. In Appendix A and  B we also describe the implementation of the LMA within the TMT-DMFT framework. We typically solve for $\sim 10^{5}$ disorder realizations each of which involves the calculation of the local interaction self-energy, $\Sigma_{i}(\omega)$.

In the absence of interactions, Eq. (1) reduces to the Anderson model of non-interacting electrons Anderson . Here, the metal to insulator transition is not characterized by the vanishing of the DoS. Instead, the hybridization paths get canceled accompanied by weak localization of the wave functions due to coherent backscattering from impurities or exponential localization of the wave functions in deep-trapped states Bulka1985 ; Bulla2001 ; mackinnon_1993 . As a result, the electrons occupying such exponentially localized states are confined to limited regions in the space and cannot contribute to the transport. As the disorder potential, $W$, is increased, more and more regions in space become exponentially localized and the system undergoes a metal-insulator transition as a function of $W$. At the Anderson localization transition the average DoS given by, $\rho_{arith}(\omega)=\int dV_{i}P(V_{i})\rho_{i}(\omega)$, with $\rho_{i}(\omega)$ being the LDoS, is not critical. However, the geometrical mean of the LDoS, $\rho_{typ}=\exp\int dV_{i}P(V_{i})\ln\rho_{i}(\omega)$, better approximates the critical nature of the Anderson localization transition. The local TMT framework adopted here reproduces some of the expected features of the Anderson localization transition, but underestimates the critical disorder strength, $W_{c}$ first_TMT_Vlad2003 . Although by construction the local TMT framework is able to describe qualitatively the effects of strong localization due to disorder, all non-local coherent backscattering effects are lost. The localization mechanism explicitly contained within the TMT is essentially the physics due to deep-trapped states where the states initially above and below the bare band-edge become localized in deeply trapped states mackinnon_1993 ; Bulka1985 ; Bulka1987 . This effect is subsequently fed back into the hybridizing medium so that the band center also localizes. Within TMT, the band edge of $\rho_{typ}(\omega)$ then monotonically moves towards the band center such that at the critical disorder strength even states at the band center are exponentially localized.

Perturbative studies on the weakly interacting disordered electron gas dates back to the seminal work of Altshuler and Aronov AA . Later extensions include the two-loop large-$N$ reormalization group analysis of Punnoose and Finkelstein Punnoose2005 , that could describe a metal-insulator transition in a two-dimensional electron gas. However, in disordered interacting systems there also exists a number of relevant phenomena that are beyond the reach of perturbative methods. For example, the work by Milanović, Sachdev and Bhatt Milov1989 and later by Bhatt and Fisher Bhatt1992 showed the importance of disorder in describing the instability of a disordered, interacting Fermi liquid towards the formation of local moments. The treatment of interactions within the non-pertubative framework of DMFT DMFT_RMP can naturally incorporate the tendency towards the formation of local moments Aguiar2009 .

In this work, we revisit the paramagnetic phase of the AHM and try to elucidate the mechanism that could lead to the formation of such local moments in a disordered, interacting system. In particular, we look into the single particle quantities across a broad range of $U$ and $W$ parameters, putting particular emphasis on the scattering rate and the evolution of the distribution of Kondo scales with respect to $U$ and $W$.

We employed the dynamical mean field theory framework with a typical medium, to look into the interplay of disorder and strong correlations in the paramagnetic metallic phase of the particle-hole symmetric Anderson-Hubbard model using the local moment approach. Particularly, we explored the single particle dynamics by analyzing the disorder averaged self-energy and identified the existence of a universal ‘Kondo’ scale within such a local theoretical framework that considers the strong correlation physics in presence of disorder scattering only due to deep trapped states. Additionally, we showed that this scale could be represented by the peak ($T_{K}^{peak}$) in the distribution function of the Kondo scales. Moreover, the universal regime is shown to exist up to significantly high energies, although a strict Fermi liquid scattering dynamics holds true for $\omega\lesssim T_{K}^{peak}$. While such universal dynamics similar to that observed in the strong coupling limit of the conventional single-band Hubbard model Logan_mott_ins ; Zitko_Shastry is anticipated in the low disorder regime disorder_Tneq0 , the same is surprising in the proximity of a Anderson-Mott transition, where the disorder is much stronger in comparison to the interaction. But then, such an observation highlights the incipient disorder renormalised Kondo screening of the local moments to be the dominant mechanism determining the low energy physics of the system.

As mentioned before, within the local framework of the dynamical mean field theory in combination within the typical medium theory, the Anderson-Hubbard model is mapped onto an ensemble of impurity problems, where the host for the impurities is determined by the typical density of states. The tendency of an impurity site to form a local moment is governed by the impurity-host hybridization function that is determined by the typical impurity density of states. Thus the low energy physics will be determined by the peak of the distribution of the density of states and reinforced by this self consistency since all the sites see the same hybridization function. In this case, these sites are the ones with the lowest Kondo scale, which are at the peak of the distribution. They are the ones closest to Mott character. The inhibition of the low energy hybridization function would be felt by all the impurities leading to a pronounced tendency towards forming local moments.

Since the disorder, especially near an Anderson localization transition, strongly suppresses the hybridization to the impurity, our observations highlight that in a disordered interacting system, Anderson and Mott mechanism of localization may not be disentangled. It is worth noting that the behavior of the local Kondo scales and the density of states are in agreement with the previous works as in Refs. Byczuk2005, ; Aguiar2009, . In our work we perform a detailed investigation of the spectra, and find that the broad distribution of Kondo scales and the underlying universal scattering dynamics corroborate the physical picture of the emergence of the formation of local moments in the presence of metallic droplets, as proposed in Refs. Paalanen1988, ; Aguiar2009, . While the emergent local moments would tend towards a common Kondo scale, we speculate that the Kondo scales and hence the low energy physics associated with the metallic droplets could be inhomogeneous. These observations are particularly relevant for understanding the underlying mechanisms that lead to the breakdown of the metallic phase towards a Mott or Mott-Anderson localization transition. However, in order to assert the true nature of this spatial inhomogeneity we require to go beyond the local framework and incorporate non-local dynamical fluctuations.

The local moment approach is an inherently non-perturbative impurity solver neither confined to low energies like the slave-boson approach nor to weak coupling like the iterated perturbation theory or modified perturbation theory approaches. While for non-disordered correlated systems this has clearly been demonstrated LMA_SIAM1 ; LMA_NRG_benchmark1 , our present calculations show that it does capture the strong correlation physics in accordance with the numerically exact NRG calculations for disordered correlated systems Byczuk2005 . With this set up established, one then asks the question as to what happens if we include short-range dynamical fluctuations due to disorder. Such directions within the framework of the typical medium dynamical cluster approximation Chinedu_TMDCA2014 are currently under our consideration.

We would like to acknowledge fruitful discussions with Pinaki Majumdar and Subroto Mukerjee. S.S. acknowledges the financial support from CSIR, India and JNCASR, India. This material is based upon work supported by the National Science Foundation award DMR-1237565 and by the EPSCoR Cooperative Agreement EPS-1003897 with additional support from the Louisiana Board of Regents. Supercomputer support is provided by the Louisiana Optical Network Initiative (LONI) and HPC@LSU.