# Fractionalization of long-range ordered states in a Falicov-Kimball model

###### Abstract

A Falicov-Kimball model which thermodynamically reduces the local Coulomb interaction of particles to attraction or repulsion is studied within the dynamical mean-field theory. In the strong interaction regime a fractionalization of particles into charge and spin objects, the physical properties of which are different from the whole particles, is observed in both high- and low-temperature phases. At the high temperature and strong interactions the single-particle density of states opens an excitation gap, but the charge compressibility and the spin susceptibility exhibit the features of gapless excitations. The low-temperature phase has a long-range order, and the single-particle spectra always open a gap, while the charge and spin excitations are gapless in the strong interaction regime. In the fractionalized long-range ordered phase both the charge compressibility and the spin susceptibility are universal scaling functions of temperature.

###### pacs:

71.27.+a, 72.15.Qm, 75.20.Hr, 71.10.Fd## I Introduction

In the many-body physics the fractionalization is the phenomenon where the particles of the system can be constructed as combinations of objects with new quantum numbers. The physical properties of the system cannot be determined by combinations of its elementary constituents. One of the prominent examples is the one-dimensional system of interacting electrons Giamarchi (). In one dimension the electrons are fractionalized into charge and spin objects, and the low-energy properties of the system are determined by collective excitations of these charge and spin objects Giamarchi (). Other example may include the so-called orthogonal metal, which has recently attracted research attention Senthil (). It is a non-Fermi liquid, in which the transport and thermodynamics are like the Fermi-liquid ones, and the quasiparticles are absent. The fractionalization is an intriguing effect of strong electron correlations. It is not only fascinating in itself, but has also been suggested to be the key element in understanding the nature of different phenomena such as the Mott insulator, high-temperature superconductors. The Mott insulator can be interpreted as a quantum spin liquid, where its spin quasiparticles do not carry charge Ng1 (); Ng2 (). The normal state of high-temperature superconductors exhibits unusual metallic properties, which seem to be related to non-Fermi liquids Schofield ().

Recently, Hohenadler and Assaad have introduced a Falicov-Kimball model (FKM), which thermodynamically reduces the local Coulomb interaction to attraction or repulsion Assaad (). This FKM can be considered as a three-component generation of the standard spinless FKM Tran1 (); Tran2 (); Tran3 (). In contrast to the Hubbard model, the FKM explicitly contains localized fermions FK (); Gruber (); FZ (). The presence of localized fermions leads the metallic state, which occurs at weak correlations to be non-Fermi liquid FZ () or an Anderson localization Antipov (). At low temperature the FKM exhibits different exotic ordered states Freericks (); Lemanski (). The FKM has attracted research attention due to its rich physics and a simplification of the Hubbard model. Quantum Monte-Carlo simulations, which are performed for the FKM proposed by Hohenadler and Assaad on a two-dimensional square lattice, reveal an exotic metal in the strong correlation regime Assaad (). In the exotic metallic phase the single-particle spectra are gapped, but the charge and spin excitations are gapless Assaad (). This demonstrates while the charge and spin excitations are like the metallic ones, the quasiparticles are absent. The exotic metal is indeed a fractionalized state. The FKM proposed by Hohenadler and Assaad is rarely a minimal lattice model among more sophisticated ones, which can exhibit an electron fractionalization Kitaev (); Fu (). So far, the electron fractionalization is only realized in a metallic state without any long-range order.

In this work, we show the electron fractionalization can also coexist with a long-range order. In this electron fractionalization the single-particle spectra still open a gap. However, the gap opening is due to a long-range ordering. Despite the gap opening, the charge compressibility and the spin susceptibility exhibit the gapless excitation features. The opposite behaviours of electrons and their charge and spin counterparts lead the long-range ordered phase to be fractionalized. We will show this realization of the electron fractionalization by studying the low-temperature phases in the FKM proposed by Hohenadler and Assaad Assaad (). Actually, the FKM proposed by Hohenadler and Assaad is a special symmetric case of a generalized three-component FKM with a three-body interaction Tran1 (); Tran2 (); Tran3 (). The three-component FKM exhibits various Mott insulators with different natures Tran1 (); Tran2 (); Tran3 (). In contrast to the previous studies Assaad (); Tran1 (); Tran2 (); Tran3 (), in this work we focus on the low-temperature phases, where a long-range ordering may occur. We use the dynamical mean-field theory (DMFT) to investigate a possibility of electron fractionalization. The DMFT is a widely and successfully used tool for treating strong electron correlations in a self-consistent non-perturbative mannerGKKR (). At low temperature we find a charge (or spin) long-range ordered state, in which the single-particle spectra are gapped, while the charge compressibility and the spin susceptibility are like metallic ones in the strong correlation regime. In the fractionalized state the charge compressibility and the spin susceptibility obey universal scaling laws. In addition, the DMFT also allows us to study both the high-temperature metal-insulator transition (MIT) and low-temperature ordering in detail. Within the DMFT we could calculate the single-site double and triple occupancies, which are accessible for site-resolved imaging techniques Sherson (); Bakr (); Greif (). This gives us a possibility of comparing the theoretical results with experiments. With advantages of ultracold techniques the proposed FKM can be realized in an optical lattice, and this could verify the electron fractionalization in the proposed model.

The present paper is organized as follows. In Sec. II we present the model and its DMFT. The numerical results are presented in Sec. III. Finally, the conclusion is presented in Sec. IV .

## Ii Model and its dynamical mean-field theory

We study the FKM proposed by Hohenadler and Assaad for a fractionalized metallic state Assaad (). The proposed FKM describes a lattice of two-component moveable and single-component localized particles Tran1 (); Tran2 (); Tran3 (). Its Hamiltonian reads

(1) |

where () is the creation (annihilation) operator of a conduction electron with spin at lattice site . is the number operator. is the hopping parameter between the nearest-neighbour sites. The localized fermions are present in the model through their Ising degree of freedom . is a three-body interaction, which is a combination of the Hubbard interaction of conduction electrons and the Ising variable. When the three-body interaction is reduced to the repulsive (attractive) Hubbard interaction of conduction electrons. Hamiltonian in Eq. (1) is a special case of the three-component FKM Tran1 (); Tran2 (); Tran3 () with a three-body interaction

(2) | |||||

where is the energy level of localized spinless fermions, is the Falicov-Kimball interaction between conduction electrons and localized fermions, and is their three-body interaction. The Ising variable is connected to the localized spinless fermions in the three-component FKM via the relation , where is the number operator of the localized spinless fermions Tran1 (); Tran2 (); Tran3 (). Note the three-body interaction in Eq. (2) already contains the local two-body interaction of conduction electrons in the three-component FKM. When and , Hamiltonians in Eqs. (1)-(2) are identical. Hamiltonian in Eq. (1) or in Eq. (2) can be realized by loading ultracold atoms in an optical lattice. Actually, Hamiltonian in Eq. (1) is the Hubbard model with randomly alternating local interactions. The standard Hubbard model has already been realized by quantum simulations of ultracold atoms Jordens (); Schneider (). A spatial modulation of the local interaction has also been achieved Yamazaki (). This leads to a possibility of realizing the Hubbard model with spatially alternating local interactions by quantum simulations Koga (). Hamiltonian in Eq. (2) can also be simulated by loading two-component light and single-component heavy fermionic atoms, for instance Li and Yb, into an optical lattice. In a sufficient deep lattice, the heavy atoms can be localized, and only the light atoms are moveable through the lattice. The three-body and few-body interactions have also been achieved in ultracold atoms Will (). With a symmetric tuning of the model parameters, Hamiltonian in Eq. (1) can also be realized through the three-component FKM.

We consider a bipartite lattice, which can be divided into two penetrating sublattices and . The single particle properties of conduction electrons can be determined by their Green function

(3) |

where , and , are the creation operators for conduction electrons in the sublattice and , respectively. We use the DMFT to solve the FKM described in Eq. (2) with and at half filling. Within the DMFT, the self energy of conduction electrons is a local function of frequency. From the Dyson equation we obtain

(4) |

where is the self energy of conduction electrons in the sublattice (), and is the dispersion of conduction electrons, and is the chemical potential. The self energy is determined from a single correlated site embedded in an effective medium. The action of the embedded single site of the sublattice reads

(5) | |||||

where the Green function represents the effective mean-field medium. The effective mean-field medium represents the correlation effects of whole lattice except for the considered site in a mean-field manner. It can be described by a Green function that is determined by the Dyson equation

(6) |

Here, is the local Green function of conduction electrons in the sublattice . We consider the hypercubic lattice in infinite dimensions. The local Green function is calculated by

(7) |

where

is the density of states (DOS) of noninteracting conduction electrons in the infinite-dimensional hypercubic lattice GKKR (). is the rescaling hopping parameter in the infinite-dimensional limit GKKR (). We will use as the energy unit.

Since is a good quantum number, we can take the trace over it in calculating the partition function of the single site problem

(8) | |||||

where

(9) | |||||

(10) | |||||

in Eq. (10) is actually the action of an effective single site of the Hubbard model with the local interaction and the chemical potential shifted by . It gives the local Green function

(11) |

where is the self energy due to the Hubbard local interaction . The local Green function of the original effective single site described by the action in Eq. (5) reads

(12) |

where

(13) |

Equation (12) shows the local Green function contains electron correlations which are generated from both the repulsive () and attractive () interactions. represents the weight factor of the contributions of the repulsive () or attractive () interactions to the local Green function. Within the DMFT we are able to explicitly study the contributions of the repulsive or attractive interactions to the dynamics of the system. One can show that

(14) |

This shows the expectation value measures the difference of the weight factors of the repulsive and attractive interactions in the system. When both the repulsive and attractive interactions equally contribute to the Green function. indicates only the repulsive (or attractive) interaction plays the dominant role.

We calculate the self energy of the action in Eq. (10) by the exact diagonalization GKKR (); Krauth (). Within the exact diagonalization procedure the action is mapped into an Anderson impurity model

(15) | |||||

where the creation and annihilation operators , represent a finite set of orbitals which are a discrete mapping of the effective medium. The parameters , are determined by the minimization of the mapping difference of the effective medium Green function in the Matsubara frequency domain GKKR (); Krauth (). With the finite orbital set, Hamiltonian in Eq. (15) can exactly be diagonalized, and we are able to calculate the local Green function from the Lehmann spectral representation GKKR (); Krauth (). Following the iteration procedure of the DMFT GKKR (); Krauth (), we could obtain the local Green function and the self energy self consistently.

Once the self-consistent solution is achieved, we compute the charge compressibility and the spin susceptibility. The charge compressibility is defined as

(16) |

where , and is the number of lattice sites. In order to compute the spin susceptibility, we introduce an external magnetic field , which applies to conduction electrons

(17) |

We consider both uniform , and staggered magnetic fields. In the case of uniform magnetic field the spin susceptibility is a ferromagnetic (FM) one

(18) |

and in the case of staggered magnetic field, the spin susceptibility is an antiferromagnetic (AF) one

(19) |

We use the Ridder implementation of numerical derivatives to calculate the charge compressibility and the spin susceptibility in the numerical calculations NR ().

## Iii Numerical results

We numerically solve the set of the DMFT equations by iterations. The effective single impurity problem is solved by the exact diagonalization GKKR (); Krauth (). In numerical calculations we typically use orbitals. We have also checked the results with . We mainly focus the study on the half filling case, in which . At high temperature we obtain , while at low temperature . The high temperature solution leads to a homogeneous (HM) phase, where . At low temperature we obtain different long-range ordered solutions: charge ordered (CO) and AF phases. The solution is accompanied by the CO phase, while appears together with the AF phase. The CO phase is paramagnetic and it is characterized by staggered electron density , where is the order parameter of the charge ordering. In the AF phase the electron density obeys , where is the order parameter of the AF ordering. At half filling although the symmetry between the and sublattices is broken due to the long-range ordering, the sublattice symmetric solution is still maintained for the Ising variable. Away from half filling, a solution is obtained at low temperature. The low temperature solutions are obtained depending on the initial input self energy. An initial CO (AF) self energy leads to the CO (AF) solution. The CO and AF phases appear below the same critical temperature . In Fig. 1 we plot the critical temperature as a function of . The obtained qualitatively agrees with the Monte-Carlo simulation result Assaad (), despite it is the infinite-dimensional result. The critical temperature approaches to zero in both the limits of weak and strong interactions. The HM phase was previously studied by the Monte-Carlo simulation in Ref. Assaad, . However, there is a lack of studies on the long-range ordered phases. In addition, within the DMFT the explicit contributions of the repulsive and attractive interactions to the system dynamics are calculable in both the HM and the long-range ordered phases.

### iii.1 Homogeneous phase

The DMFT results in the HM phase support the reported results of the Monte-Carlo simulation Assaad (). Indeed, the HM phase is separated into two regimes. One is the weak correlation regime, where the DOS shows the metallic behavior. The other is the strong correlation regime, where the single-particle DOS opens a gap. In Fig. 2 we plot the DOS in both the weak and strong correlation regimes. We have used the Lorentzian broadening parameter for the delta functions in the DOS. The gap opening in the strong correlation regime indicates the single particle excitation is similar to the one in an insulator. Figure 2 also shows a MIT driven by electron correlations. In the intermediate correlation regime, a Kondo-like peak appears in the opening gap. However, it disappears at high temperature and before the long-range ordering when temperature decreases, as it is shown in Fig. 3. This is an insulator reentrance effect driven by temperature in the intermediate correlation regime. At high temperature the Kondo-like peak is absent, because the effective single site problem can induce a Kondo effect only at low temperature. The Kondo-like peak is present in the MIT of the repulsive Hubbard model Georges (); Rozenberg (). However, it is absent in the attractive Hubbard model Toschi (); Keller (). The disappearance of the Kondo-like peak when temperature decreases towards the critical temperature of the long-range ordering shows a competition between the repulsive and attractive interactions in the present model. It shows the dominance of the attractive interaction at temperature close to the long-range ordering. Due to the presence of both the repulsive and attractive interactions, the MIT in the proposed FKM is distinguishable from the one in the Hubbard and the spinless Falicov-Kimball models GKKR (); FZ ().

Within the DMFT we can calculate local quantities such as the double and triple occupancies from the effective single-site problem. The double occupancy is used to experimentally detect the MIT in optical lattices Jordens (); Schneider (). It can also be obtained from the site occupation, which is accessible by the site-resolved imaging experiments Sherson (); Bakr (); Greif (). From the action of the effective single site in Eq. (5), one can show

(20) | |||

(21) |

where is the double occupancy in the repulsive (attractive) Hubbard action described by Eq. (10). In the strong correlation regime the repulsive Hubbard interaction suppresses the double occupancy , while the attractive one binds the local pair of electrons with opposite spins, hence Keller (). Since in the HM phase , thus , and in the strong correlation limit. In Fig. 4 we plot the double and triple occupancies as functions of the interaction. One can imagine the HM state as a bonding (or antibonding) of the Mott insulator () and the electron pairing state (). Accidentally, the double occupancy in the HM phase at half filling is independent on the interaction strength as it is shown in Fig. 4. However, the double occupancy in the week and the strong correlation limits has different origins. In the limit , , while in the limit , is a strong correlation effect. The smooth dependencies of the double and triple occupancies on the interaction suggest that the correlation-driven MIT in the HM phase is just a continuous crossover from metal to insulator. With the advantages of the site resolved imaging technique, the double and triple occupancies would be measured as functions of the interaction, once the present model is simulated by ultracold atoms.

In the strong correlation regime, the single-particle DOS opens a gap, which is usually an insulator’s attribute. However, the charge and spin excitations show non-insulating behaviors. In Figs. 5 and 6 we plot the charge compressibility and the AF spin susceptibility. We obtain the FM spin susceptibility . The HM phase occurs at . In the HM phase the charge compressibility and the spin susceptibility are always finite for any finite interaction . This indicates the charge and spin excitations are gapless, and their behaviors are qualitatively the same for both the weak and strong correlation regimes. However, the single-particle excitation in the strong correlation regime are gapped. The opposite behaviors of the single-particle excitation and its charge and spin counterparts constitute the HM phase a fractionalized state in the strong correlation regime. In this fractionalized state the single-particle properties look like the insulating ones, but the charge and spin excitations exhibit the metallic feature. The finite value of the charge compressibility can also be seen from the dependence of the electron density on the chemical potential. In Fig. 7 we plot the electron density and as functions of the chemical potential in the strong correlation regime. The electron density monotonously increases with the chemical potential, hence the charge compressibility is finite. However, only at half filling . Away from half filling is finite, and the phase becomes ordered. The disordered state is just unique in the surrounding ordered phase. When the chemical potential is shifted from its half-filling value , the state nature is also changed. Therefore, the charge compressibility does not vanish despite opening a gap in the single-particle spectra at half filling.

### iii.2 Long-range ordered phase

Below the critical temperature a long-range ordering occurs. Both conduction electrons and the Ising variable are ordered. Although at low temperature we obtain two solutions, the CO phase with and the AF phase with , their charge compressibility and spin susceptibility are the same for both phases. Therefore, we focus the present study on the CO phase. The single-particle DOS in the CO phase is always gapped. In Fig. 8 we plot the DOS for both the weak and strong correlation regimes. The gap opening due to long-range ordering indicates the single-particle excitation is similar to the one in a Slater insulator. In Fig. 9 we plot the charge order parameter , , as well as the double and triple occupancies as functions of the interaction . In the strong correlation regime, one sublattice, for instance , is fully occupied, while the other () sublattice is empty. In the empty sublattice the double and triple occupancies vanish. In the occupied sublattice, the attractive interaction gives the dominant contributions to the double and triple occupancies, hence when . In addition, in the CO phase , and , which result in . The CO phase is also the pairing state, where pairs of electrons with opposite spins are bound at every sites of the occupied sublattice due to the attractive interaction.

Similar to the HM phase, the CO phase also consists of two distinct regimes, the week and strong correlation regimes. In Figs. 5 and 6 we have already plotted the charge compressibility and the AF susceptibility. The CO phase occurs in the region . One can see that both the charge compressibility and the spin susceptibility exhibit distinct behaviors in the week and strong correlation regimes. In the week correlation regime they are strongly suppressed like the ones in an insulator. However, in the strong correlation regime both the charge compressibility and the spin susceptibility are finite. This indicates the charge and spin excitations look like the ones in a metal. The finite charge compressibility in the strong correlation regime can also be seen from the dependence of the electron density on the chemical potential. In Fig. 10 we plot the dependencies of the total and the sublattice electron densities, as well as , as functions of the chemical potential in the strong correlation regime. It shows that the total electron density monotonously increases with the chemical potential. As a consequence, the charge compressibility is finite. One can also notice only at half filling . Away from half filling , i.e the Ising variable is antiferromagnetically ordered. Any small shift of the chemical potential from its value at half filling drives the ordering of the Ising variable from homogeneous to staggered ones. The opening gap in the single-particle DOS does not generate a plateau in the function around half filling . The opposite behaviors of the single-particle excitation and its charge and spin counterparts show the charge ordered phase is also fractionalized, like the ones in the HM phase. In the present model, the state fractionalization occurs both at high and low temperatures. It smoothly crosses from the weak to the strong correlation regimes. Figures 5 and 6 suggest in the strong correlation regime the charge compressibility and the spin suscepbility obtains universal scalings

(22) |

where , , and the scaling function is independent on . In the CO phase the scaling function can empirically be fitted with the following function

(23) |

where , , are the fitting parameters. In Figs. 5 and 6 we also plot the fitting function for a comparison. Although we cannot analytically derive the fitting function in Eq. (23), it fits well with the numerical results of the charge compressibility and the spin susceptibility in the CO phase.

## Iv Conclusion

We have showed the electron fractionalization in a symmetric three-component FKM. It is characterized by opposite behaviours of the single particles and their charge and spin counterparts. In the electron fractionalization the single particle spectra open a gap, while the charge and spin excitations are gapless. It occurs in both high- and low-temperature phases. When the interaction increases the ground state continuously changes from the normal state to the fractionalized one. At high temperature the phase is disordered, and strong electron correlations open a gap in the single-particle spectra, while the charge compressibility and the spin susceptibility remain finite like the ones in a metal. At low temperature the gap opening is due to a long-range ordering. In the strong correlation regime, despite the gap opening, the charge compressibility and the spin susceptibility are finite. They are universal functions of temperature in the fractionalized state.

So far we have only studied the special symmetric case of the three-component FKM. It seems that the three-component FKM contains very rich physics, which has not fully been explored yet. The three-component FKM can also been considered as an extreme of the mass imbalance in the three-component Hubbard model. The electron fractionalization in the three-component FKM suggests a possible fractionalization driven by the mass imbalance in the three-component Hubbard model. We leave this problem for further studies.

## Acknowledgement

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No 103.01-2017.13.

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