Physical Review B

Universal power-law behavior of local electron correlation functions in the Falicov–Kimball lattice model

Yu-Liang Liu

Published Jul 15, 2005• 3 citations

Abstract

By studying the Falicov--Kimball lattice model, we demonstrate that the correlation functions of local electrons can be written directly in the form of the partition function of conduction electrons with a time dependent external potential. These general expressions of the correlation functions are equivalent to the exact ones obtained by the dynamical mean field theory in infinite dimension limit. Moreover, we also prove that the asymptotic power-law behavior of the local electron correlation functions is universal in any dimensions, as long as the conduction electron excitation spectrum is gapless, regardless whether they are Landau Fermi liquid or not. This conclusion is also valid for other quantum many-particle systems, if the coupling between the conduction and local electrons is on-site or short-range Coulomb interaction.

DOI: 10.1103/physrevb.72.045123

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Physical Review B

Universal power-law behavior of local electron correlation functions in the Falicov–Kimball lattice model

Yu-Liang Liu

Published Jul 15, 2005• 3 citations

Publisher

Abstract

DOI: 10.1103/physrevb.72.045123

Related papers

Exact dynamical mean-field theory of the Falicov-Kimball model

James K. Freericks, V. Zlatić

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 ${\mathrm{YbInCu}}_{4},$ ${\mathrm{EuNi}}_{2}({\mathrm{Si}}_{1\ensuremath{-}x}{\mathrm{Ge}}_{x}{)}_{2},$ ${\mathrm{NiI}}_{2},$ and ${\mathrm{Ta}}_{x}\mathrm{N}].$ 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 ${\mathrm{YbInCu}}_{4},$ ${\mathrm{EuNi}}_{2}({\mathrm{Si}}_{1\ensuremath{-}x}{\mathrm{Ge}}_{x}{)}_{2},$ ${\mathrm{NiI}}_{2},$ and ${\mathrm{Ta}}_{x}\mathrm{N}].$

arXiv:cond-mat/0301188Reviews of Modern PhysicsOct 30, 2003350 citations

Correlation effect of local electrons in a one-dimensional Falicov-Kimball model

Qing-Wei Wang, Yu-Liang Liu

With the eigenfunctional theory, we give a general exact expression of the local electron Green function in the Falicov-Kimball model, and apply it to study the correlation effects of the local electrons in one dimension. For the two local electrons case, the correlation exponent of the local electron Green function has weak even-odd oscillation with the distance between these two local electrons, and it approaches zero in the strong coupling limit. While, at half filling of the local electrons, the ground-state phase is complicated. When the conduction electron is near $0.5$ , the ground state configuration is the chessboard phase for a considerable range of $U∕{t}^{*}$ , and the correlation exponent increases from zero to a finite value as $U∕{t}^{*}$ increases. When the conduction electron is far away from $0.5$ , the ground state configuration is the segregated phase, and the correlation exponent first increases and then jumps to zero when $U∕{t}^{*}$ is larger than a finite value which is a function of ${n}_{c}$ . Our results are compared with previous numerical simulations.With the eigenfunctional theory, we give a general exact expression of the local electron Green function in the Falicov-Kimball model, and apply it to study the correlation effects of the local electrons in one dimension. For the two local electrons case, the correlation exponent of the local electron Green function has weak even-odd oscillation with the distance between these two local electrons, and it approaches zero in the strong coupling limit. While, at half filling of the local electrons, the ground-state phase is complicated. When the conduction electron is near $0.5$, the ground state configuration is the chessboard phase for a considerable range of $U∕{t}^{*}$, and the correlation exponent increases from zero to a finite value as $U∕{t}^{*}$ increases. When the conduction electron is far away from $0.5$, the ground state configuration is the segregated phase, and the correlation exponent first increases and then jumps to zero when $U∕{t}^{*}$ is larger than a finite value which is a function of ${n}_{c}$. Our results are compared with previous numerical simulations.

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Exact out-of-time-ordered correlation functions for an interacting lattice fermion model

Naoto Tsuji, Philipp Werner, Masahito Ueda

Exact solutions for local equilibrium and nonequilibrium out-of-time-ordered correlation (OTOC) functions are obtained for a lattice fermion model with on-site interactions, namely, the Falicov-Kimball (FK) model, in the large dimensional and thermodynamic limit. Our approach is based on the nonequilibrium dynamical mean-field theory generalized to an extended Kadanoff-Baym contour. We find that the density-density OTOC is most enhanced at intermediate coupling around the metal-insulator phase transition. In the high-temperature limit, the OTOC remains nontrivially finite and interaction dependent, even though dynamical charge correlations probed by an ordinary response function are completely suppressed. We propose an experiment to measure OTOCs of fermionic lattice systems including the FK and Hubbard models in ultracold atomic systems.Exact solutions for local equilibrium and nonequilibrium out-of-time-ordered correlation (OTOC) functions are obtained for a lattice fermion model with on-site interactions, namely, the Falicov-Kimball (FK) model, in the large dimensional and thermodynamic limit. Our approach is based on the nonequilibrium dynamical mean-field theory generalized to an extended Kadanoff-Baym contour. We find that the density-density OTOC is most enhanced at intermediate coupling around the metal-insulator phase transition. In the high-temperature limit, the OTOC remains nontrivially finite and interaction dependent, even though dynamical charge correlations probed by an ordinary response function are completely suppressed. We propose an experiment to measure OTOCs of fermionic lattice systems including the FK and Hubbard models in ultracold atomic systems.

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Real-space cluster dynamical mean-field approach to the Falicov-Kimball model: An alloy-analogy approach

Prosenjit Haldar, M. S. Laad, Syed R. Hassan

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 $(1/d,\phantom{\rule{0.16em}{0ex}}d=$ 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.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 $(1/d,\phantom{\rule{0.16em}{0ex}}d=$ 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.

arXiv:1603.00301Physical review. B./Physical review. BMar 13, 201712 citations

Static and dynamic properties of the spinless Falicov-Kimball model

K. W. Becker, Steffen Sykora, V. Zlatić

The spinless Falicov-Kimball model is studied by the use of a recently developed projector-based renormalization method for many-particle Hamiltonians. The method is used to evaluate static and dynamic quantities of the one-dimensional model at half-filling. These include the quasiparticle excitation energy ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{k}$, the momentum distribution ${n}_{k}$ of the conduction electrons, and the spatial correlation functions of the localized electrons. One of the most remarkable results is the appearance of a gap in ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{k}$ at the Fermi level of the order of the Coulomb repulsion $U$ , which is accompanied by a smooth behavior for ${n}_{k}$ . The density of states for the conduction electrons and the one-particle spectral functions for the localized electrons are also discussed. In both quantities, a gap opens with increasing $U$ .The spinless Falicov-Kimball model is studied by the use of a recently developed projector-based renormalization method for many-particle Hamiltonians. The method is used to evaluate static and dynamic quantities of the one-dimensional model at half-filling. These include the quasiparticle excitation energy ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{k}$, the momentum distribution ${n}_{k}$ of the conduction electrons, and the spatial correlation functions of the localized electrons. One of the most remarkable results is the appearance of a gap in ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{k}$ at the Fermi level of the order of the Coulomb repulsion $U$, which is accompanied by a smooth behavior for ${n}_{k}$. The density of states for the conduction electrons and the one-particle spectral functions for the localized electrons are also discussed. In both quantities, a gap opens with increasing $U$.

arXiv:cond-mat/0701433Physical Review BFeb 5, 20076 citations

Densities of states of the Falicov–Kimball model off half filling in infinite dimensions

Ihor V. Stasyuk, O. B. Hera

An approximate analytical scheme of the dynamical mean field theory (DMFT) is developed for the description of the electron (ion) lattice systems with Hubbard correlations within the asymmetric Hubbard model where the chemical potentials and electron transfer parameters depend on an electron spin (a sort of ions). Considering a complexity of the problem we test the approximation in the limiting case of the infinite- $U$ spinless Falicov--Kimball model. Despite the fact that the Falicov--Kimball model can be solved exactly within DMFT, the densities of states of localized particles have not been completely investigated off half filling. We use the approximation to obtain the spectra of localized particles for various particle concentrations (chemical potentials) and temperatures. The effect of a phase separation phenomenon on the spectral function is considered.An approximate analytical scheme of the dynamical mean field theory (DMFT) is developed for the description of the electron (ion) lattice systems with Hubbard correlations within the asymmetric Hubbard model where the chemical potentials and electron transfer parameters depend on an electron spin (a sort of ions). Considering a complexity of the problem we test the approximation in the limiting case of the infinite-$U$ spinless Falicov--Kimball model. Despite the fact that the Falicov--Kimball model can be solved exactly within DMFT, the densities of states of localized particles have not been completely investigated off half filling. We use the approximation to obtain the spectra of localized particles for various particle concentrations (chemical potentials) and temperatures. The effect of a phase separation phenomenon on the spectral function is considered.

arXiv:cond-mat/0502669Physical Review BJul 20, 20054 citations

{\mathrm{YbInCu}}_{4},

${\mathrm{EuNi}}_{2}({\mathrm{Si}}_{1\ensuremath{-}x}{\mathrm{Ge}}_{x}{)}_{2},$

{\mathrm{NiI}}_{2},

and

{\mathrm{Ta}}_{x}\mathrm{N}].

0.5

, the ground state configuration is the chessboard phase for a considerable range of

U∕{t}^{*}

, and the correlation exponent increases from zero to a finite value as

U∕{t}^{*}

increases. When the conduction electron is far away from

0.5

, the ground state configuration is the segregated phase, and the correlation exponent first increases and then jumps to zero when

U∕{t}^{*}

is larger than a finite value which is a function of

{n}_{c}

. Our results are compared with previous numerical simulations.

(1/d,\phantom{\rule{0.16em}{0ex}}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.

{n}_{k}

of the conduction electrons, and the spatial correlation functions of the localized electrons. One of the most remarkable results is the appearance of a gap in ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{k}$ at the Fermi level of the order of the Coulomb repulsion

U

, which is accompanied by a smooth behavior for

{n}_{k}

. The density of states for the conduction electrons and the one-particle spectral functions for the localized electrons are also discussed. In both quantities, a gap opens with increasing

U

U

spinless Falicov--Kimball model. Despite the fact that the Falicov--Kimball model can be solved exactly within DMFT, the densities of states of localized particles have not been completely investigated off half filling. We use the approximation to obtain the spectra of localized particles for various particle concentrations (chemical potentials) and temperatures. The effect of a phase separation phenomenon on the spectral function is considered.