# Phase Separation and Charge-Ordered Phases of the Falicov-Kimball Model at T0: Temperature-Density-Chemical Potential Global Phase Diagram from Renormalization-Group Theory

## Abstract

The global phase diagram of the spinless Falicov-Kimball model in spatial dimensions is obtained by renormalization-group theory. This global phase diagram exhibits five distinct phases. Four of these phases are charge-ordered (CO) phases, in which the system forms two sublattices with different electron densities. The CO phases occur at and near half filling of the conduction electrons for the entire range of localized electron densities. The phase boundaries are second order, except for the intermediate and large interaction regimes, where a first-order phase boundary occurs in the central region of the phase diagram, resulting in phase coexistence at and near half filling of both localized and conduction electrons. These two-phase or three-phase coexistence regions are between different charge-ordered phases, between charge-ordered and disordered phases, and between dense and dilute disordered phases. The second-order phase boundaries terminate on the first-order phase transitions via critical endpoints and double critical endpoints. The first-order phase boundary is delimited by critical points. The cross-sections of the global phase diagram with respect to the chemical potentials and densities of the localized and conduction electrons, at all representative interactions strengths, hopping strengths, and temperatures, are calculated and exhibit ten distinct topologies.

PACS numbers: 71.10.Hf, 05.30.Fk, 64.60.De, 71.10.Fd

## I Introduction

The Falicov-Kimball model (FKM) was first proposed by L. M. Falicov and Kimball Falicov () to analyze the thermodynamics of semiconductor-metal transitions in SmB and transition-metal oxides RamirezFalicov70 (); RamirezFalicov71 (); Plischke (); GoncalvesDaSilva (). The model incorporates two types of electrons: one type can undergo hopping between sites and the other type cannot hop, thereby being localized at the sites. Thus, in its introduction, FKM described the Coulomb interaction between mobile band electrons and localized band electrons. There have been a multitude of subsequent physical interpretations based on this interaction, including that of localized ions attractively interacting with mobile electrons, which yields crystalline formation Kennedy (); Lieb (). Another physical interpretation of the model is as a binary alloy, in which the localized degree of freedom reflects A or B atom occupation FreericksFalicov (); Freericks (). In this paper we employ the original language, with and electrons as conduction and localized electrons with a repulsive interaction between them.

Since there is no interacting spin degree of freedom in the Hamiltonian, the model is traditionally studied in the spinless case, commonly referred as the spinless FKM (SFKM) and which is in fact a special case of the Hubbard model in which one type of spin (e.g., spin-up) cannot hop Hubbard (). In spite of its simplicity, this model is able to describe many physical phenomena in rare-earth and transition metal compounds, such as metal transitions, charge ordering, etc.

Beyond the introduction of the spin degree of freedom for both electrons Brandt (); Farkasovsky99 (); Zlatic (); Minh-Tien (); Lemanski (); Farkasovsky05 (); BrydonGulacsi06a (); ZlaticFreericks01a (); ZlaticFreericks01b (); ZlaticFreericks03 (); MillerFreericks01 (); FreericksNikolic01 (); FreericksNikolic02 (); AllubAlascio96 (); AllubAlascio97 (); LetfulovFreericks01 (); Ramakrishnan03 (), there also exist many extensions of the original model. The most widely studied extensions include multiband hybridization Lawrence (); Leder (); Hanke (); Baeck (); LiuHo83 (); LiuHo84 (); Portengen96 (); BrydonGulacsi06b (), hopping Kotecky (); Batista (); Yin03 (); Batista04 (); Ueltschi (), correlated hopping Michielsen (); Tranh-Hai (); Gajek (); Wojtkiewicz01 (); CencarikovaFarkasovsky04 (), non-bipartite lattices CencarikovaFarkasovsky07 (); GruberMacris97 (), hard-core bosonic particles GruberMacris97 (), magnetic fields ZlaticFreericks01b (); AllubAlascio96 (); AllubAlascio97 (); LetfulovFreericks01 (); Ramakrishnan03 (); GruberMacris97 (); FreericksZlatic98 (), and next-nearest-neighbor hopping Wojtkiewicz06 (). Exhaustive reviews are available in Refs. FreericksZlatic03 (); Jedrzejewski (); GruberMacris96 (); GruberUeltschi (). The wider physical application of both the basic FKM and its extended versions have aimed at explaining valence transitions Farkasovsky99 (); ZlaticFreericks01a (); ZlaticFreericks01b (); ZlaticFreericks03 (), metal-insulator transitions Farkasovsky99 (); MillerFreericks01 (); FreericksNikolic01 (); FreericksNikolic02 (); FreericksNikolic03 (), mixed valence phenomena SubrahmanyamBarma88 (), Raman scattering FreericksDevereaux01 (), colossal magnetoresistance AllubAlascio96 (); AllubAlascio97 (); LetfulovFreericks01 (); Ramakrishnan03 (), electronic ferroelectricity Batista (); Batista04 (); Portengen96 (); Yin03 (), and phase separation Farkasovsky99 (); Ueltschi (); Michielsen (); Freericks96 (); Freericks02a (); Freericks02b ().

After the initial works on the FKM Falicov (); RamirezFalicov70 (); RamirezFalicov71 (); Plischke (); GoncalvesDaSilva (), the literature had to wait 14 years for the celebrated first rigorous results. Two independent studies, by Kennedy and Lieb Kennedy (); Lieb () and by Brandt and Schmidt BrandtSchmidt86 (); BrandtSchmidt87 (), proved for dimensions that, at low temperatures, FKM has long-range charge order with the formation of two sublattices. Various methods have been used in the study of the FKM. In most of these studies, either the infinite-dimensional limit or low-dimensional cases have been investigated. Studies include limiting cases such as ground-state analysis or the large interaction limit. Renormalization-group theory Wilson () offers fully physical and fairly easy techniques to yield global phase diagrams and other physical phenomena.

This non-trivial nature of SFKM motivated us to determine the global phase diagram of the model, which resulted in a richly complex phase diagram involving charge ordering and phase coexistence, as exemplified in Fig. 1.

We use the general method for arbitrary dimensional quantum systems developed by A. Falicov and Berker AFalicov () to obtain the global phase diagram of the SFKM in , in terms of both the chemical potentials and the densities of the two types of electrons, for all temperatures. The outline of this paper is as follows: In Sec.II, we introduce the SFKM and, in Sec.III, we present the method AFalicov (). Calculated phase diagrams are presented in Sec.IV, for the non-hopping () classical submodel and for the hopping () quantum regimes of small, intermediate, and large . We conclude the paper in Sec.V.

## Ii Spinless Falicov-Kimball Model

The SFKM is defined by the Hamiltonian

(1) |

where and denotes that the sum runs over all nearest-neighbor pairs of sites. Note that, as in all renormalization-group studies, the Hamiltonian has absorbed the inverse temperature. The dimensionless hopping strength can therefore be used as the inverse temperature. Here and are respectively creation and annihilation operators for the conduction electrons at lattice site , obeying the anticommutation rules and , while and are electron number operators for conduction and localized electrons respectively. The operator takes the values or , for site being respectively occupied or unoccupied by a localized electron. The particles are fermions, so that the Pauli exclusion principle forbids the occupation of a given site by more than one localized electron or by more than one conduction electron.

The first term of the Hamiltonian is the kinetic energy term, responsible for the quantum nature of the model. The system being invariant under sign change of (via a phase change of the local basis states in one sublattice), only positive values are considered. The second term is the screened on-site Coulomb interaction between localized and conduction electrons, with positive and negative values corresponding to attractive and repulsive interactions. We consider only the repulsive case, since the attractive case can be connected to the repulsive one by the particle-hole symmetry possessed by either type of electrons. Particle-hole symmetries are achieved by the transformations of for the localized electrons and , for the conduction electrons, where, for a bipartite lattice, for one sublattice and for the other FreericksFalicov (); GruberUeltschi (). The last two terms of the Hamiltonian are the chemical potential terms with and being the chemical potential for a localized and conduction electron.

In order to carry out a renormalization-group transformation easily, we trivially rearrange the Hamiltonian given in Eq.(1) into the equivalent form of

(2) |

where, for a -dimensional hypercubic lattice, , , , and is the two-site Hamiltonian involving only nearest-neighbor sites and .

## Iii Renormalization-Group Theory

### iii.1 Suzuki-Takano Method in

In the Hamiltonian in Eq.(2) is

(3) |

The renormalization-group procedure traces out half of the degrees of freedom in the partition function Suzuki (); Takano (),

(4) |

Here and throughout this paper primes are used for the renormalized system. Thus, as an approximation, the non-commutativity of the operators beyond three consecutive sites is ignored at each successive length scale, in the two steps indicated by in the above equation. Earlier studies AFalicov (); Suzuki (); Takano (); Tomczak (); TomczakRichter96 (); TomczakRichter03 (); Hinczewski05 (); Hinczewski06 (); Hinczewski08 (); Kaplan2 (); Kaplan (); Sariyer () have established the quantitative validity of this procedure.

The above transformation is algebraically summarized in

(5) |

where are three successive sites. The operator acts on two-site states, while the operator acts on three-site states. Thus we can rewrite Eq.(5) in matrix form as

(6) |

where state variables , , , , and can be one of the four possible single-site states at each site , namely one of , , , and . Eq.(6) indicates that the unrenormalized matrix on the right-hand side is contracted into the renormalized matrix on the left-hand side. We use two-site basis states, , and three-site basis states, , in order to block-diagonalize the matrices in Eq.(6). These basis states are the eigenstates of total localized and conduction electron numbers. The set of and are given in Tables 1 and 2 respectively. The corresponding block-diagonal Hamiltonian matrices are given in Appendices A and B.

Two-site basis states | |||
---|---|---|---|

Three-site basis states | |||
---|---|---|---|

Phase | The interaction constants at the phase sinks | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

sink | ||||||||||||

CO | ||||||||||||

CO | ||||||||||||

CO | ||||||||||||

CO |

Phase | The runaway coefficients at the phase sinks | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

sink | ||||||||||||

, | ||||||||||||

, | ||||||||||||

CO, CO | ||||||||||||

CO, CO |

Phase | The expectation values at the phase sinks | Character | |||||||||||

sink | |||||||||||||

dilute - dilute | |||||||||||||

dilute - dense | |||||||||||||

dense - dilute | |||||||||||||

dense - dense | |||||||||||||

CO | dilute - charge ord. dilute | ||||||||||||

CO | dilute - charge ord. dense | ||||||||||||

CO | dense - charge ord. dilute | ||||||||||||

CO | dense - charge ord. dense |

Phase | Boundary | Interaction constants at the boundary fixed points | Relevant | eigenvalue | |||||||||||

boundary | type | exponent | |||||||||||||

CO/CO | 1st | 3 | |||||||||||||

order | |||||||||||||||

CO/ | 2nd | 0.273873 | |||||||||||||

order | |||||||||||||||

CO/ | 2nd | 0.273873 | |||||||||||||

order | |||||||||||||||

CO/ | 2nd | 0.273873 | |||||||||||||

order | |||||||||||||||

CO/ | 2nd | 0.273873 | |||||||||||||

order | |||||||||||||||

CO/CO | 2nd | 1.420396 | |||||||||||||

order | |||||||||||||||

CO/CO | 2nd | 1.420396 | |||||||||||||

order | |||||||||||||||

With these basis states, Eq.(6) can be rewritten as

(7) |

Once written in the basis states , the block-diagonal renormalized matrix has 13 independent elements, which means that renormalization-group transformation of the Hamiltonian generates 9 more interaction constants apart from , , , and . In this 13-dimensional interaction space, the form of the Hamiltonian stays closed under renormalization-group transformations. This Hamiltonian is

(8) |

where is a local operator that switches the conduction electron states of sites and : with for and otherwise. When is applied, further below, to three consecutive sites , , , with for and otherwise.

To extract the renormalization-group recursion relations, we consider the matrix elements . With , , , and , 12 out of 16 diagonal elements are independent and, with , only one of the 4 off-diagonal elements is independent, summing up to 13 independent matrix elements. Thus we obtain the renormalized interaction constants in terms of , defining for the diagonal elements and for the only independent off-diagonal element:

(9) |

The matrix elements of the exponentiated renormalized Hamiltonian are connected, by Eq.(7), to the matrix elements, of the exponentiated unrenormalized Hamiltonian,