Falicov-Kimball Model: Correlated Electrons

• The Falicov-Kimball model is a lattice framework distinguishing mobile d‐electrons from localized f‐electrons, enabling controlled study of electron correlations.
• It utilizes techniques like DMFT, Monte Carlo, and exact diagonalization to analyze charge order, metal-insulator transitions, and disorder-free localization.
• Extended variants incorporate excitonic, orbital, and topological phenomena, providing insights into fractionalized metals and pressure-induced transitions.

The Falicov-Kimball model (FKM) is a class of lattice models for correlated electrons in which one sector is itinerant and another is localized, so that the localized occupations are conserved variables while the mobile sector propagates in a background generated by them. In a widely used spinless form at half-filling, the model contains mobile $d$-electrons, immobile $f$-electrons, and an onsite $d$-$f$ Coulomb interaction; for any fixed $f$-configuration the remaining problem is quadratic, which is why the FKM has become a standard setting for charge order, metal-insulator physics, localization, and controlled extensions to multiorbital, multicomponent, and even Majorana formulations (Pradhan, 2016, Li et al., 2019).

1. Canonical formulation and solvable structure

A standard spinless FKM on a lattice is written as

$$H_0 = - \sum_{\langle ij \rangle} t\,d_i^\dagger d_j + U \sum_i d_i^\dagger d_i f_i^\dagger f_i + E_f\sum_i f_i^\dagger f_i ,$$

where $t$ is the hopping amplitude of the itinerant $d$-electrons, $U$ is the local Coulomb repulsion between $d$- and $f$0-electrons, and $f$1 is the localized $f$2-level energy. In this formulation the $f$3-electron occupations are classical annealed variables $f$4, while the $f$5-electrons remain quantum mechanical (Pradhan, 2016). An equivalent viewpoint maps the spinless FKM to a binary-alloy disorder problem with site potentials taking two values; in cluster formulations this mapping is used to construct alloy-analogy approaches to localization and interaction-driven band splitting (Haldar et al., 2016).

The central structural feature is the separation between mobile and immobile sectors. Because localized occupations are good quantum numbers, the many-body problem reduces to a weighted sum over static backgrounds, and this in turn enables exact diagonalization combined with Monte Carlo annealing, dynamical mean-field theory (DMFT), projector-based renormalization, density matrix renormalization group (DMRG), and sign-problem-free Monte Carlo in different regimes and variants of the model (Pradhan, 2016, Ribic et al., 2016, Ejima et al., 2013, Hohenadler et al., 2018).

The same logic extends beyond the canonical spinless form. Generalized FKMs with spin-dependent hopping can be mapped to an FKM after a suitable spin rotation if only one effective flavor remains mobile. In the Majorana-Falicov-Kimball construction, exact solvability is recovered when the kinetic term acts only on a subset of Majorana flavors, leaving the others as static $f$6 variables. This places the FKM in a broader class of exactly solvable interacting models that includes superconducting, topological, and spin representations (Li et al., 2019).

2. Charge order, spectral gaps, and localization

At half-filling in finite dimensions, the spinless FKM exhibits a checkerboard charge-density-wave (CDW) ground state. The associated single-particle density of states (DOS) has a gap of width $f$7 centered at the Fermi energy, and the checkerboard ordering of localized $f$8-particles constitutes the ordered phase (Žonda et al., 2019). In two dimensions, nonlocal correlations originating from CDW fluctuations further enhance insulating behavior: diagrammatic extensions of DMFT based on the local two-particle vertex produce a larger low-frequency self-energy and a more insulating spectrum than single-site DMFT (Ribic et al., 2016).

Finite temperature does not simply close the CDW gap. Instead, sharp subgap states appear inside the gap and fill it in as temperature increases. In finite dimensions these sharp resonances are traced to single $f$9-electron excitations from the perfect checkerboard state: adding or removing a single $d$0-electron yields a bound state inside the gap, while a displacement defect yields two symmetrically located in-gap bound states. Near a critical interaction strength $d$1, the subgap peaks merge at $d$2, creating a regime with finite DOS at the Fermi level even though the system remains CDW-ordered; in the two-dimensional case quoted in the study, $d$3 (Žonda et al., 2019).

That finite DOS does not imply an ordinary metal. The inverse participation ratio,

$d$4

remains finite and large in the gapless CDW regime, showing that the relevant subgap states are strongly localized and tied to single-site or few-site defects. This distinction matters conceptually: a high DOS at the Fermi level can coexist with localized single-particle states and vanishing macroscopic conduction in the thermodynamic limit (Žonda et al., 2019).

The localization aspect persists in other settings. In a one-dimensional long-range FK model with staggered power-law Ising interactions, a finite-temperature CDW transition is stabilized for $d$5, while the fermionic sector displays disorder-free localization generated by thermal fluctuations of the classical background. At any nonzero temperature all fermion states are localized in the thermodynamic limit, although very large system sizes are required when the defect density is low (Hodson et al., 2021). Real-space cluster DMFT reaches a related conclusion from a different angle: the irreducible two-particle vertex develops non-analyticities before the Hubbard-type band-splitting transition, which was interpreted as the onset of an unusual strong localization regime (Haldar et al., 2016).

3. Extended, multiorbital, and generalized variants

A major branch of the literature studies extended FKMs with additional density interactions, orbital structure, or correlated hopping. In the half-filled extended FKM on the Bethe lattice in large dimensions, onsite $d$6 and nearest-neighbor $d$7 density-density interactions generate two ordered insulating ground states separated by a discontinuous transition at $d$8. The ground state is insulating for any nonzero value of $d$9 or $f$0, and at $f$1 the DMFT solution coincides with the static broken-symmetry Hartree-Fock approximation (HFA) (Lemański et al., 2017). At finite temperature the comparison changes: HFA still captures the discontinuous transition between ordered phases at $f$2 for small temperatures and qualitatively reproduces the continuous order-disorder transition, but it fails to describe various metal-insulator transitions and the change in order of the continuous transition that DMFT finds for larger $f$3 (Kapcia et al., 2020).

Triangular-lattice variants make the role of frustration explicit. In a spinless extended FKM with correlated hopping $f$4, Monte Carlo searches identify regular, bounded, hexagonal, stripe, and segregated ground states, together with valence and metal-insulator transitions. Phase segregation is enhanced by correlated hopping, while the non-bipartite geometry supports many competing states close in energy (Yadav et al., 2010). In a different triangular-lattice extension with two localized orbitals $f$5 and $f$6 and one itinerant $f$7-band, local $f$8-occupations remain conserved and the model shows stripe, bi-stripe, and three-sublattice orbital order, mixed valence, and inversion-symmetry breaking interpreted as orbitally driven ferroelectricity. In that setting, homogeneous mean-field solutions can show spontaneous $f$9, but the true inhomogeneous ground states remove this spontaneous symmetry breaking as $f$0 (Yadav et al., 2010).

Multi-component versions extend the model’s metal-insulator phenomenology. In a three-component FKM relevant to mixtures of one-component and two-component fermions in optical lattices, DMFT with exact diagonalization finds commensurate and incommensurate Mott transitions between one third and two thirds filling, species-selective and inverse transitions, a reentrant insulator-metal-insulator sequence at half-filling for sufficiently strong $f$1, and phase coexistence or phase separation depending on filling and interaction regime (Nguyen et al., 2013).

4. Excitonic, hybridization-driven, and topological regimes

Once the localized sector is allowed to disperse, the FKM becomes a standard platform for excitonic-insulator physics. In the one-dimensional half-filled extended FKM,

$f$2

finite-size DMRG, exact diagonalization, and Green-function methods yield a ground-state phase diagram with a band insulator (BI), a staggered orbital ordered (SOO) phase, and an intermediate excitonic insulator (EI) with power-law correlations and central charge $f$3. The anomalous spectral function, condensation amplitude, coherence length, and binding energy show a Coulomb-interaction-driven crossover from BCS-like electron-hole pairing fluctuations to tightly bound excitons; a mass imbalance disfavors the EI in favor of SOO, but does not shift the BCS-BEC crossover regime (Ejima et al., 2013).

In two dimensions, projector-based renormalization of the spinless extended FKM with dispersive $f$4-electrons confirms both the excitonic insulator and an exciton environment above the transition on the semiconductor side of the semimetal-semiconductor transition. The imaginary part of the dynamical electron-hole pair susceptibility displays sharp excitonic bound states below the continuum threshold. Above $f$5, a zero-momentum excitonic bound state exists in the semiconducting regime, whereas in the semimetallic regime only finite-momentum excitonic resonances persist (Phan et al., 2011).

Hybridization terms produce another line of development. In a spinless extended FKM with nonlocal nearest-neighbor $f$6-$f$7 hybridization $f$8, pressure is modeled by the empirical parametrization $f$9 rather than $$H_0 = - \sum_{\langle ij \rangle} t\,d_i^\dagger d_j + U \sum_i d_i^\dagger d_i f_i^\dagger f_i + E_f\sum_i f_i^\dagger f_i ,$$0. DMRG then yields a critical hybridization $$H_0 = - \sum_{\langle ij \rangle} t\,d_i^\dagger d_j + U \sum_i d_i^\dagger d_i f_i^\dagger f_i + E_f\sum_i f_i^\dagger f_i ,$$1 at which pressure-induced metal-insulator and valence transitions occur; the resulting gap and valence trends qualitatively reproduce experiments on SmB$$H_0 = - \sum_{\langle ij \rangle} t\,d_i^\dagger d_j + U \sum_i d_i^\dagger d_i f_i^\dagger f_i + E_f\sum_i f_i^\dagger f_i ,$$2 (Farkasovsky, 2018).

Magnetic field effects in the excitonic sector are by now studied directly through Peierls substitution. In a spinless two-orbital extended FKM on the square lattice, Hartree-Fock calculations find a nonmonotonic response of the excitonic order parameter to increasing magnetic field: Landau-level crossings can enhance condensation at small fields, but sufficiently strong fields suppress excitonic order and may drive a disordered insulating state with partially occupied $$H_0 = - \sum_{\langle ij \rangle} t\,d_i^\dagger d_j + U \sum_i d_i^\dagger d_i f_i^\dagger f_i + E_f\sum_i f_i^\dagger f_i ,$$3 and $$H_0 = - \sum_{\langle ij \rangle} t\,d_i^\dagger d_j + U \sum_i d_i^\dagger d_i f_i^\dagger f_i + E_f\sum_i f_i^\dagger f_i ,$$4 orbitals carrying opposite nonzero Chern numbers. In the excitonic supersolid phase, orbital order remains robust under field while excitonic condensation is suppressed (Ohta et al., 17 Jan 2025).

5. Fractionalization, transport, and metal-insulator phenomenology

The FKM also supports metallic regimes that are not Fermi liquids in the conventional sense. In a two-dimensional SU(2) FKM,

$$H_0 = - \sum_{\langle ij \rangle} t\,d_i^\dagger d_j + U \sum_i d_i^\dagger d_i f_i^\dagger f_i + E_f\sum_i f_i^\dagger f_i ,$$5

auxiliary-field quantum Monte Carlo reveals a phase with a single-particle gap, gapless spin and charge excitations, finite and nonsaturating conductivity, and no coherent quasiparticle peak. An exact duality to an unconstrained $$H_0 = - \sum_{\langle ij \rangle} t\,d_i^\dagger d_j + U \sum_i d_i^\dagger d_i f_i^\dagger f_i + E_f\sum_i f_i^\dagger f_i ,$$6 slave-spin theory identifies this phase as a fractionalized or orthogonal metal: the physical electron fractionalizes as $$H_0 = - \sum_{\langle ij \rangle} t\,d_i^\dagger d_j + U \sum_i d_i^\dagger d_i f_i^\dagger f_i + E_f\sum_i f_i^\dagger f_i ,$$7, the slave spins are disordered, and the $$H_0 = - \sum_{\langle ij \rangle} t\,d_i^\dagger d_j + U \sum_i d_i^\dagger d_i f_i^\dagger f_i + E_f\sum_i f_i^\dagger f_i ,$$8-fermions remain metallic even though electron insertion is gapped (Hohenadler et al., 2018).

A related DMFT study of a symmetric FKM with a locally conserved Ising variable reaches the same broad conclusion in both disordered and ordered phases. For strong interaction, the single-particle DOS is gapped, while the charge compressibility and spin susceptibility remain finite. In the long-range ordered phase the DOS remains gapped, yet the low-energy charge and spin responses are gapless and obey universal scaling with $$H_0 = - \sum_{\langle ij \rangle} t\,d_i^\dagger d_j + U \sum_i d_i^\dagger d_i f_i^\dagger f_i + E_f\sum_i f_i^\dagger f_i ,$$9 in the strong-coupling regime (Tran, 2018). These results constrain a common misconception: a single-particle gap does not by itself determine whether collective charge or spin modes are gapless.

Transport in finite geometries shows a complementary finite-size aspect. In heterostructures where an FKM region is sandwiched between metallic leads, the transmission function

$t$0

becomes finite at the Fermi level in the gapless CDW regime of the finite-dimensional spinless model. The effect is substantial only for small systems, however, because generalized inverse participation ratios show that the relevant states remain localized even after hybridization with the leads; transmission then decreases rapidly with system size (Žonda et al., 2019).

6. External fields, dynamics, spectroscopy, and computational analysis

Magnetic fields reorganize the FKM spectrum even without invoking excitonic order. On finite square and triangular lattices, the Peierls-substituted spinless FKM produces Hofstadter-butterfly spectra whose gaps depend jointly on magnetic flux, Coulomb repulsion, and lattice geometry. At half-filling on the square lattice, the main gap grows linearly with $t$1, while repulsive interaction suppresses extra finite-size states inside the Hofstadter gaps. Orbital and edge currents are also geometry dependent: for the square lattice the edge current oscillates with flux and retains a similar pattern in the interacting case, whereas on the triangular lattice interactions redirect currents into narrow channels and destroy the regular oscillatory edge-current pattern (Pradhan, 2016).

Real-time dynamics provides another probe of the mobile sector moving in a correlated static background. Lattice Monte Carlo calculations in one and two dimensions show that the light-cone velocity of spreading charge correlations decreases with interaction strength at low temperature, while the phase velocity increases. At higher temperature the initial spreading is governed by the noninteracting Fermi velocity, but the maximum range of correlations shrinks as interaction increases. Charge-order correlations in the effective disorder potential enhance the range of propagation, and direct lattice simulations were found to be more accurate and more efficient for these observables than equilibrium or nonequilibrium dynamical cluster approximations (Herrmann et al., 2017).

Core-level spectroscopy directly resolves local occupancy sectors. Finite-temperature X-ray photoemission spectroscopy in the FKM, computed with a Wiener-Hopf sum equation approach within DMFT, shows that the core-hole spectral function contains peaks associated with creating a core hole on empty, singly occupied, or doubly occupied sites. In metallic states the spectrum has two side peaks and two nearly degenerate central peaks that merge at higher temperature; in insulating states it shows two peaks plus a strongly temperature-dependent low-energy peak associated with a thermally excited empty site (Pakhira et al., 2018).

Recent work also uses the FKM as a benchmark for data-driven phase classification. Unsupervised machine-learning analyses of two-dimensional Monte Carlo occupation snapshots, especially local principal component analysis (PCA), successfully identify the ordered-disordered phase boundary and distinguish weakly localized from Anderson-localized regimes in the disordered phase. In that study, PCA based analysis outperformed more complex predictors and autoencoders, indicating that the occupation patterns of the FKM already contain linearly accessible information about both symmetry breaking and finite-size localization crossovers (Frk et al., 2024).

Across these developments, the FKM functions less as a single narrowly defined Hamiltonian than as a solvable organizing framework for correlated lattice electrons. Its enduring role comes from a precise balance: the immobile sector is simple enough to preserve analytical and numerical control, yet rich enough to generate CDW order, disorder-free localization, excitonic pairing, fractionalized metals, field-induced topological structure, and sharp tests of many-body methods across equilibrium and nonequilibrium regimes (Žonda et al., 2019, Li et al., 2019).