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arXiv:2103.11735 · Physical review. B./Physical review. B

One-dimensional long-range Falikov-Kimball model: Thermal phase transition and disorder-free localization

Thomas Hodson, Josef Willsher, Johannes Knolle

Published Jul 9, 2021• 7 citations
arXivHTMLPDF

Abstract

Disorder or interactions can turn metals into insulators. One of the simplest settings in which to study this physics is given by the Falikov-Kimball (FK) model, which describes itinerant fermions interacting with a classical Ising background field. Despite the translational invariance of the model, inhomogeneous configurations of the background field give rise to effective disorder physics which lead to a rich phase diagram in two (or more) dimensions with finite-temperature charge-density wave (CDW) transitions and interaction-tuned Anderson versus Mott localized phases. Here, we propose a generalized FK model in one dimension with long-range interactions which shows a similarly rich phase diagram. We use an exact Markov chain Monte Carlo method to map the phase diagram and compute the energy-resolved localization properties of the fermions. We compare the behavior of this transitionally invariant model to an Anderson model of uncorrelated binary disorder about a background CDW field which confirms that the fermionic sector only fully localizes for very large system sizes.

DOI
10.1103/physrevb.104.045116

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AcPPB•Oct 1, 2001•2 citations
Disorder or interactions can turn metals into insulators. One of the simplest settings in which to study this physics is given by the Falikov-Kimball (FK) model, which describes itinerant fermions interacting with a classical Ising background field. Despite the translational invariance of the model, inhomogeneous configurations of the background field give rise to effective disorder physics which lead to a rich phase diagram in two (or more) dimensions with finite-temperature charge-density wave (CDW) transitions and interaction-tuned Anderson versus Mott localized phases. Here, we propose a generalized FK model in one dimension with long-range interactions which shows a similarly rich phase diagram. We use an exact Markov chain Monte Carlo method to map the phase diagram and compute the energy-resolved localization properties of the fermions. We compare the behavior of this transitionally invariant model to an Anderson model of uncorrelated binary disorder about a background CDW field which confirms that the fermionic sector only fully localizes for very large system sizes.
The two-dimensional Falicov-Kimball (FK) model is analyzed using the Monte Carlo method. In the case of concentrations of both itinerant and localized particles equal to 0.5 we determine temperature dependence of specific heat, charge density wave susceptibility, and density-density correlation function. In the weak interaction regime we find a first order transition to the ordered state and anomalous temperature dependence of the correlation function. We construct the phase diagram of half-filled FK model. Also, the role of next-nearest-neighbor hopping on the phase diagram is analyzed. Lastly, we discuss the density of states and the spectral functions for the mobile particles in weak and strong interaction regime.
The Falicov-Kimball model was introduced in 1969 as a statistical model for metal-insulator transitions; it includes itinerant and localized electrons that mutually interact with a local Coulomb interaction and is the simplest model of electron correlations. It can be solved exactly with dynamical mean-field theory in the limit of large spatial dimensions, which provides an interesting benchmark for the physics of locally correlated systems. In this review, the authors develop the formalism for solving the Falicov-Kimball model from a path-integral perspective and provide a number of expressions for single- and two-particle properties. Many important theoretical results are examined that show the absence of Fermi-liquid features and provide a detailed description of the static and dynamic correlation functions and of transport properties. The parameter space is rich and one finds a variety of many-body features like metal-insulator transitions, classical valence fluctuating transitions, metamagnetic transitions, charge-density-wave order-disorder transitions, and phase separation. At the same time, a number of experimental systems have been discovered that show anomalies related to Falicov-Kimball physics [including YbInCu4,{\mathrm{YbInCu}}_{4},YbInCu4​, ${\mathrm{EuNi}}_{2}({\mathrm{Si}}_{1\ensuremath{-}x}{\mathrm{Ge}}_{x}{)}_{2},$ NiI2,{\mathrm{NiI}}_{2},NiI2​, and TaxN].{\mathrm{Ta}}_{x}\mathrm{N}].Tax​N].
The two-dimensional Falicov-Kimball (FK) model is analyzed using Monte Carlo method. In the case of concentrations of both itinerant and localized particles equal to 0.5 we determine temperature dependence of specific heat, charge density wave susceptibility and density-density correlation function. In the weak interaction regime we find a first order transition to the ordered state and anomalous temperature dependence of the correlation function. We construct the phase diagram of half-filled FK model. Also, the role of next-nearest-neighbor hopping on the phase diagram is analyzed. Lastly, we discuss the density of states and the spectral functions for the mobile particles in weak and strong interaction regime.
Disorder or sufficiently strong interactions can render a metallic state unstable, causing it to turn into an insulating one. Despite the fact that the interplay of these two routes to a vanishing conductivity has been a central research topic, a unifying picture has not emerged so far. Here, we establish that the two-dimensional Falicov-Kimball model, one of the simplest lattice models of strong electron correlation, does allow for the study of this interplay. In particular, we show that this model at particle-hole symmetry possesses three distinct thermodynamic insulating phases and exhibits Anderson localization. The previously reported metallic phase is identified as a finite-size feature due to the presence of weak localization. We characterize these phases by their electronic density of states, staggered occupation, conductivity, and the generalized inverse participation ratio. The implications of our findings for other strongly correlated systems are discussed.
It is long known that the best single-site coherent potential approximation falls short of describing Anderson localization. Here, we study a binary alloy disorder [or equivalently, a spinless Falicov-Kimball (FK)] model and construct a dominantly analytic cluster extension that treats intracluster (1/d,d=(1/d,\phantom{\rule{0.16em}{0ex}}d=(1/d,d= spatial dimension) correlations exactly. We find that, in general, the irreducible two-particle vertex exhibits clear nonanalyticities before the band splitting transition of the Hubbard type occurs, signaling onset of an unusual type of localization at strong coupling. Using time-dependent response to a sudden local quench as a diagnostic, we find that the long-time wave-function overlap changes from a power-law to an anomalous form at strong coupling, lending additional support to this idea. Our results also imply such ``strong'' localization in the equivalent FK model, the simplest interacting fermion system.
Both canonical and grand canonical phase diagrams of the one dimensional Falicov-Kimball model (FKM) with correlated hopping are presented for several values of the model parameters. Using the method of restricted phase diagrams the system has been studied in the thermodynamic limit. As compare to the conventional FKM, the diagrams exhibit a loss of the particle-hole symmetry. PACS D5.30.Fk, 71.30+h As the simplest, still non-trivial model of highly correlated electron systems, the Falicov-Kimball model (FKM) attracts growing attention among solid state physicists. It can describe variety of the most intriguing cooperative phenomena, as metal-insulator transition, mixed-valence phenomenon etc. (see e.g. the review [1]). The model deals with itinerant particles (electrons) that can hope between nearest-neighbor sites. Some of lattice sites are occupied by non-movable particles, playing a role of ions or localized electrons; we call them ”ions”. The only interaction in the system is the on-site, Coulomb-type interaction between the electrons and the ions. The interaction generates long-range correlations between the ions. The model has been investigated thoroughly in nineties. Numerous approximate results supplemented with some exact and rigorous statements reported up to now provide a good basis for further extensions of the model towards more realistic physical situations. These include the discussed here model with correlated hopping, according to which the electron hopping rate depends on occupations of relevant sites. The Hamiltonian of the FKM with correlated hopping reads: