Introduction

The metal-insulator transition (MIT) is a fascinating problem in physics, which continues to attract significant attention from theoretical physicists [1]. Notably, the Mott insulating state cannot be described by the band theory of solids because of strong electron-electron interactions, which are neglected in such frameworks. A system becomes a Mott insulator when its conduction electrons become localized due to these strong repulsions [2]. In theoretical models studying the MIT, the local Coulomb interaction plays a decisive role in determining the physical properties of the system. The study of Mott transitions has been extended following the successful loading of ultracold fermionic atoms into optical lattices, which are created by the interference of laser beams [3].

While the MIT has been extensively studied in three-component Hubbard models with equal particle masses [4], [5], the realization of optical lattices has opened a new avenue for exploring multi-component correlated systems with mass imbalances. Motivated by these experimental advances, this paper investigates the three-component Falicov-Kimball model (FKM). This system can be viewed as an asymmetric limit of the three-component Hubbard model, where a single-component fermion species is completely localized due to its heavy mass, whereas the two-component fermions remain itinerant. The MIT in this framework is driven by the competition between the hopping energy of the light particles and the on-site Coulomb interaction. Ultimately, our primary focus is to establish the precise conditions under which the MIT emerges in the three-component FKM.

Since exact solutions for multi-component strongly correlated systems are generally unattainable, various approximations have been employed to study these systems. The MIT in the three-component FKM has been investigated using methods, including dynamical mean-field theory (DMFT) [6], the slave-boson (SB) approach [7] and the coherent potential approximation (CPA) [8]. However, previous studies employing the SB [7], and CPA [8] approximations have not yet derived a generalized condition for the MIT within the FKM framework. Specifically, although the previous CPA study [8] observed the MIT, it only focused on a few specific particle fillings and interaction strengths. It did not provide a general condition for the phase transition across all particle concentrations. The behavior of the system at arbitrary fractional fillings remained unclear. By investigating the entire parameter space, our present work fills this gap. In this paper, we employ the CPA to establish the generalized condition for the MIT within the three-component FKM. This approach has demonstrated significant efficacy in investigating strongly correlated electron systems, where Coulomb interactions play a critical role in driving the MIT [8], [9], [10], [11]. Compared with DMFT and SB, the CPA offers distinct analytical simplicity. This method yields valuable analytical results and does not require much computational demand, making it well-suited for implementation on personal workstations.

Model and Formalism

We consider a three-component FKM, the Hamiltonian of the model is given by [6] [7] [8]:

(1)

where is the creation (annihilation) operator for light fermion particles with spin at lattice site i , and is the creation (annihilation) operator for localized fermion particles at lattice site i . t is the hopping parameter between the nearest-neighbor light fermion particles. The energy level is the chemical potential of the localized fermion particles, which controls the localized fermion atom filling . is the chemical potential of the system, which controls the filling in the system , with . is the on-site Coulomb interaction of the light fermion particles, and is the on-site Coulomb interaction between the light and localized fermion particles.

We apply the alloy analogy approach (AAA) to the model. By viewing the system in terms of a disordered alloy where light fermion particles with spin up can hop in the potential of light fermion particles with spin down and localized fermion particles. The many-body Hamiltonian (1) may be approximated by a one-particle Hamiltonian of the form:

(2)

where energy levels and configuration probabilities are given in Table 1. Here is the filling of localized fermion particles, is the filling of light fermion particles with spin down in the lattice site that not occupied (occupied) by localized fermion particles.

In principle, the Green's function corresponding to Hamiltonian (2) has to be averaged over all possible disorder configurations. Since this averaging cannot be performed exactly, we employ the CPA to replace the original system with an effective Hamiltonian characterized by a periodic system . It is the CPA self-energy of the system. Generally, it complex and energy-dependent single-site and independent of vector wave . Within this CPA framework, the effective Hamiltonian is expressed as:

(3)

Table 1

Energy levels and configuration probabilities for light fermion particles with spin up hopping in the potential of light fermion particles with spin down and localized fermion particles [12]

The lattice Green function of light fermionic particles can be expressed as:

(4)

where is an infinitesimally small positive number, is the Matsubara frequency, and denotes the bare density of states (DOS), which is chosen here for convenience to have a semicircular profile corresponding to the infinite-dimensional Bethe lattice [11], [13]

(5)

where W is the half bandwidth and we will use it as the energy unit. The average Green's function is calculated via the configurational probabilities and the partial Green's functions

(6)

where represents the partial Green's functions with configuration , defined as [12]:

(7)

Within the CPA framework, the self — consistent condition dictates that the average Green's function in equation (6) must match the local Green's function in equation (4). Consequently,

(8)

After determining the Green's function, we can calculate the hopping particle filling as follows

(9)

Results and discussions

The self-consistent equation (8) is solved numerically to determine the self-energy and the Green's function via an iterative procedure [14]. Given an initial guess for the self-energy , the lattice Green's function and the average Green's function are calculated from equation (4) and (6), respectively. Subsequently, an updated self-energy is evaluated as

(10)

This procedure is iterated until convergence is achieved. During the numerical calculations, the value of in equation (4) is chosen within the range of 10 ^–3 to 10 ^–2 for iteration convergence to happen. Using a value smaller than 10 ^–3 significantly increases the computational cost and typically prevents the iterative process from converging.

In the following, we consider , where a, b are integers (0 < a < b ). Fig. 1 indicates the filling of hopping particles as a function of the chemical potential µ with b = 5 and a = 1 , 2 , …, b — 1 for large U _cc and U _cf > W . It can be seen that there are three plateaus for large U _cc . Therefore, insulating states exist at three different fillings at , and which correspond to the total filling , , and , respectively. The total filling can be arbitrary in the range 1 < n < 2. Fig. 2 describes the two-component particle filling as a function of the chemical potential for different values of U _cc at with U _cf > W . When U _cc has small values, the system is insulating at . For general filling , the system is insulating at , which is equivalent to . This result can be understood through its equivalence to the spinless FKM when U _cc = 0. This indicates that weak correlations of hopping particles do not affect the insulating phase at .

Fig. 1. (Color online) The filling of hopping particles as a function of the chemical potential at with b = 5, 0 < a < b for U _cc =3.0 and U _cf = 2.0. The horizontal dotted lines show , where i = 1, 2,…, 2b — 1.

When U _cc increases, it drives the system into a metallic phase. For large values of U _cc , the insulating state occurs at , and . Therefore, the reentrance of the insulating phase can happen at , which is equivalent to . In general, for the case U _cf > W , in the domain of parameters ( n, n _f ) and for large values of U _cc , the system is insulating at total filling n = 1 and n = 2 with arbitrary 0 < n _f < 1 and at arbitrary total filling 1 < n < 2 with . In addition, for small values of U _cc , the insulating phase occurs at , which is equivalent to with 0 < n _f < 1. The MIT at can be considered an inverse MIT.

Turning to the case U _cf < W , Fig. 3 describes the filling of two-component atoms as a function of the chemical potential µ at with b = 5, 0 < a < b for U _cc = 3 . 0 and U _cf = 0 . 6. The plot of has only one plateau at for large U _cc . This indicates that the insulating phase appears only at total filling . Therefore, in the domain of parameters ( n, n _f ) for U _cf < W , the insulating phase occurs with at arbitrary 1 < n < 2. This is also shown in Fig. 4, which plots the filling of two-component atoms as a function of the chemical potential µ for different values of U _cc and fixed U _cf = 0 . 6 at . We can see that for U _cf < W and small values of U _cc , the system is in the metallic phase. When U _cc increases, it drives the system into an insulating phase, and the MIT can be observed at arbitrary 1 < n < 2 with , which is equivalent to . .

Fig. 2. (Color online) The filling of hopping particles as a function of the chemical potential for different values of U _cc and fixed U _cf = 2.0 at . The horizontal dotted lines show

Fig. 3. (Color online) The filling of hopping particles as a function of the chemical potential at with b = 5, 0 < a < b for U _cc =3.0 and U _cf = 0.6. The horizontal dotted lines show

Generally, the parameter regimes allowing for the MIT in the three-component FKM are summarized in Fig. 5. As illustrated, the transition behavior differs significantly depending on the interaction strength of U _cf relative to W . The black dot at , indicates the reentrant effect.

Fig. 4. (Color online) The filling of hopping particles as a function of the chemical potential for different values of U _cc and fixed U _cf = 0.6 at . The horizontal dotted lines show

Fig. 5. Diagram showing the conditions for the MIT in the (n, n _f ) plane for two regimes: (a) U _cf > W and (b) U _cf < W. Solid lines show the conditions under which the MIT can occur, and the dashed line shows the inverse MIT. The black dot at , indicates the reentrant effect

Conclusion

We have applied the CPA to study the condition under which the MIT can occur in the three-component FKM. This condition of phase transition is found based on the particle fillings. As a result, the MIT can only be observed in the region where the total filling is 1 ≤ n ≤ 2; outside this region, the MIT does not occur. In particular, when 1 < n < 2 and , the MIT can be observed at regardless of whether U _cf > W or U _cf < W . Therefore, under these conditions, the MIT is influenced by the local Coulomb interaction U _cc between the hopping particles. This means that in this case, each hopping particle having a light mass occupies a lattice site and Coulomb interaction U _cc prevents the double occupancy of the two-component particles.

In the case of n = 1 or n = 2, the MIT only occurs for U _cf > W with arbitrary 0 < n _f < 1. In this case, the roles for MIT of the two-component particles are the same as those of the single-component particles. Therefore, the same MIT can be witnessed in the three-component Hubbard model in which all components of the model have the same masses [4] [5]. In addition, the inverse MIT can be observed when and U _cf > W with small values of U _cc . Under this condition, each lattice site captures a single-component particle or a two-component particle, and Coulomb interaction between the single-component and two-component particles prevents them from occupying the same lattice site. Moreover, at half filling, the system can exhibit the reentrant effect. Comparing these results with the DMFT in the three-component FKM [6], there is obviously a good agreement between the two methods although the CPA that was applied is more analytically simple. The results of this study contribute to the completion of the MIT problem in the three-component FKM within the CPA [8].

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