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Fractionally Charged Polarons and Phase Separation in an Extended Falicov-Kimball Model

We present the first numerical study of the Falicov-Kimball model extended by an on-site hybridization away from the particle-hole symmetry point. Stable polaronic distortions of the charge-density wave (CDW) phase are observed when doping with a single hole. For moderate hole-doping, we find phase separation between a hole-rich homogeneous state and the usual CDW state. Excitonic order is enhanced around the hole-rich region, locally manifesting the global competition between this and the CDW phase. Associated with the hole-doping is a fractionalization of the electron density between the itinerant and localized electrons. The calculated local density of states at the polaron centre reveals this fractionalization to be the result of charge fluctuations induced by the hybridization. We propose a scanning tunneling microscope experiment to test our results.

We present the first numerical study of the Falicov-Kimball model extended by an on-site hybridization away from the particle-hole symmetry point. Stable polaronic distortions of the chargedensity wave (CDW) phase are observed when doping with a single hole. For moderate hole-doping, we find phase separation between a hole-rich homogeneous state and the usual CDW state. Excitonic order is enhanced around the hole-rich region, locally manifesting the global competition between this and the CDW phase. Associated with the hole-doping is a fractionalization of the electron density between the itinerant and localized electrons. The calculated local density of states at the polaron centre reveals this fractionalization to be the result of charge fluctuations induced by the hybridization. We propose a scanning tunneling microscope experiment to test our results. The Falicov-Kimball model (FKM) has found use far beyond its original role as a minimal model for valence transitions [1]. The FKM describes itinerant d-electrons interacting via a repulsive contact potential U with localized f -electrons of energy ǫ f : this simple physical picture is a useful starting point for more complex models of mixed-valence (MV) phenomenon [2], heavy fermions [3], electronic ferroelectricity [4,5,6] and unconventional superconductivity [7]. The "bare" form of the FKM has also attracted much interest: it exhibits a charge density wave (CDW) instability at half-filling in any dimension D and is considered a basic model of binary alloys [8].
Apart from the extreme limits D = 1 [9] and D → ∞ [10], very little is known about the FKM away from half-filling. The numerical results of Lemański et al. suggest a rich 2D phase diagram, with phase separated coexistence of many different orderings [11]; it would clearly be very interesting to assess the stability of these states within the context of a more realistic model. The effect of dilute hole doping on the CDW is also of interest: in related models displaying density-wave instabilities (e.g. the Hubbard model) hole-doping away from perfect nesting produces polarons [12]. Yin et al. have studied polarons in the strong-coupling FKM with a weak f -electron hopping term, obtaining non-trivial modifications of the usual t-J model results [6]. Of greater relevance, Liu and Ho have proposed that polarons form in a FKM with hybridization, as a precursor to a valence transition between states with integral and fractional occupation of the f -orbitals [3].
In this Letter we extend our previous study of the FKM with d-f hybridization away from the particlehole symmetry point [13]. We perform a Hartree-Fock (HF) decoupling of the interaction and exactly diagonalize the resulting real-space Bogoliubov-de Gennes (BdG) eigenequations. We work exclusively in the limit of zero temperature where the HF approach is known to be most accurate. Doping with a single hole produces a local pola-ronic distortion of the CDW phase. Unique to our model, we find a fractional partitioning of the electron density between the two orbitals localized at the doped hole; this is revealed by the local density of states (LDOS) spectra at the polaron centre. We discuss the origin of the fractionalization and a proposed experimental approach to observing this novel electronic state. For a moderate hole-doping, we find that phase separation between a hole-rich homogeneous phase and the usual half-filled CDW phase is the ground state. The global competition between these phases is manifested as a local enhancement of excitonic ordering.
We consider an extended FKM for spinless fermions: In addition to the usual FKM terms, we include a hopping t f ij for the f -electrons and an on-site hybridization V between the d-and f -orbitals [14]. Here we concentrate on the case V = 0 and t f ij = 0; we consider also a finite t f ij with V = 0 for comparison with previous work [5,6]. The hopping integrals t ij and t f ij are, respectively, t and t f for nearest neighbours, vanishing otherwise.
We perform the standard HF decomposition [2] of the Coulomb repulsion term: Here ∆ j = f † j d j is the excitonic average at site j. In the absence of a hybridization potential V , ∆ j = 0 indicates the excitonic insulator phase [15]; Portengen et al. interpreted such a "spontaneous" excitonic average as evidence of electronic ferroelectricity [4]. The resulting HF Hamiltonian is diagonalized by the canonical transform, γ n = j (u n j d j + v n j f j ). The quasiparticle wavefunction amplitudes, u n j and v n j , are derived by solving the associated BdG eigenequations: where the components of the Hamiltonian matrix are defined as H dd In terms of the diagonal basis the order parameters at site j are given by n d Our calculations are performed in the canonical ensemble at zero temperature and thus the sum over n extends over the first N tot occupied quasiparticle states. We solve the BdG eigenequations self-consistently using a numerical iteration scheme. Commencing with an initializing set of order parameters, we exactly diagonalize Eq. (2) and hence compute new order parameters using the obtained quasiparticle wavefunctions. These values are then used as an input for the next iteration; this procedure is repeated until a desired accuracy is reached. The calculations are performed for a N = 24 × 24 lattice with periodic boundary conditions. To calculate the quasiparticle LDOS we use the converged 24 × 24 lattice as a supercell in a 10 × 10 array. The d-electron hopping integral t defines our energy scale. The results presented below correspond to ǫ f = 0 and U = 2.0; for the case of a single hole we take Limit of a single hole. We dope a single hole into the CDW state by computing the order parameters for N tot = N − 1 (note N tot = N corresponds to half filling); the converged results are presented in Fig. 1. The hole appears as a localized vacancy in the occupied f -electron sub-lattice [see Fig. 1(a)]. This defect in the periodic HF potential U n f j experienced by the d-electrons produces a distortion of the d-electron CDW state which extends over several lattice constants [ Fig. 1(b,c)], defining the spatial extent of a polaron. Near the polaron, the excitonic average ∆ j [ Fig. 1(d)] shows a Friedel-like oscillation with its tails extending along the diagonals. The enhancement of |∆ j | about the f -hole indicates that the polaron in the FKM is a local manifestation of the competition between the global excitonic ordering and CDW states [13].
Away from the polaron δ d j is uniform to within the accuracy of the convergence, indicating the expected single d-electron per unit cell in the CDW bulk. Overall, however, we find a total d-electron population The hybridization allows electrons in the d-orbitals to tunnel into the f -orbitals and vice versa: this implies charge fluctuations between the two orbitals leading to the observed fractional populations. Although true also in the half-filled CDW state, it is only with the introduction of the inhomogeneities by hole-doping (i.e. the polarons) that the fractional charge can be directly observed. Fractionally charged polarons are therefore unique to multiorbital models such as the FKM considered here; there is no analogous effect in single-orbital systems.
We find that the fractional d-electron ∆N d (or felectron ∆N f ) is strongly dependent upon U and V : ∆N d increases with both these parameters, although it decreases as V is raised above the critical value that destabilizes the global CDW phase. According to our HF decomposition the excitonic average enters into the equations as an effective hybridization potential −U ∆ j . In the CDW state, however, this is only non-zero in the presence of a finite on-site hybridization [13]. As the magnitude of ∆ j is directly proportional to both U and V , the observed behaviour of ∆N d is easily understood. The fractionalization can be explicitly visualized by computing the LDOS both at the polaron site and in the CDW bulk. From our converged solutions for the order parameters, we calculate the LDOS of the d-and f -electrons at site j: Here δ α (ω) is the Lorentzian with phenomenological lifetime broadening characterized by the parameter α. In our plots of the LDOS spectra we always take α = 0.05. We plot the LDOS spectra in Fig. 2(a); in the accompanying Fig. 2(b) we plot the same spectra for the FKM extended by a finite f -hopping integral t f = −0.1.
The effect of the hybridization on the d-and fwavefunctions can be clearly appreciated in Fig. 2(a). The off-diagonal terms H df ij in Eq. (2) produce quasiparticles with mixed nature, as evidenced by the coincidence of the peaks in the d-and f -electron spectra. The situation is qualitatively different for the FKM extended by a finite f -hopping integral [ Fig. 2(b)]. Essentially a two-band Hubbard model, in this caseN d andN f separately commute with the Hamiltonian (i.e. H df ij = 0) and the quasiparticles retain purely d-or f -character. This can be seen in the polaron LDOS spectra as two distinct peaks for the d-and f -electrons within the CDW gap.
In spite of the orbital-mixing, the LDOS spectra of the two species is still an experimentally accessible quantity; for example, we can imagine a scanning tunneling microscope (STM) experiment where the tunneling matrix only couples the tip to the d-orbitals. The polaron could then be identified by examining the spatial distribution of the d-electron LDOS at the energies corresponding to the two intra-gap peaks in Fig. 2(a). This will distinguish between the fractionally charged polaron and the usual Hubbard-like polaron: in our system a peak in the d-electron LDOS will be observed at both energies, as opposed to only a single peak at the higher energy when V = 0. Using Eq. (3) we calculate the variation of ρ d j (ω) across the lattice for ω = −0.131 and ω = 0.168. The results are presented in Fig. 3(a) and (b), respectively; the corresponding LDOS for the f -electrons are given in (c) and (d). As expected, for both species we find distinct peaks in the LDOS at the polaron site for the two intra-gap energies. The d-electron LDOS Fig. 3(a) for the (primarily f -character) ω = −0.131 peak displays the diagonal star-shape seen before in the plot of ∆ j [ Fig. 1(d)]. Since the enhancement of |∆ j | near the polaron indicates strong d-f mixing, this structure in the LDOS is a defining signature of the fractionally charged polaron. A much weaker diagonal structure is present in the felectron LDOS Fig. 3(c). The same effect occurs in the ω = 0.168 d-electron plot [ Fig. 3(b)], but it is somewhat obscured by the higher d-electron peak; no such structure is visible in the associated f -electron image.
Finite hole doping. To study the effects of moderate hole-doping, we solve the BdG eigenequations for N tot = N − 10. The results for V = 0 suggest a large variety of possible charge-orderings [11]; to find the ground state we therefore consider a number of different initial order parameter sets. These sets range from vertical and diagonal stripe orderings to a random distribution of quasiparticle weight across the lattice. Since we work in the zero temperature limit, we test the relative stability of the converged results by calculating the total energy E = n E n − U j ( n d j n f j − |∆ j | 2 ). We present the variations of δ d j for a representative sample of these initializing sets in Fig. 4(a-d); these figures are labelled with decreasing energy from E/N = −1.5554 for Fig. 4(a) to E/N = −1.5586 for Fig. 4(d). Fig. 4(e) gives the varia- tion of ∆ j for the lowest-energy solution. In Fig. 4(f) we present δ d j for finite t f . Our results clearly demonstrate the tendency for the f -holes to cluster. Such phase separation is well known in the V = 0 FKM, being reported in the 2D model at all coupling strengths [11,16]. Our observation of f -hole clustering again confirms the relevance of the V = 0 results as a guide to the physics of the extended model [13]. An attractive interaction between f -holes in a closelyrelated FKM has recently been cited as a possible mech-anism for the unconventional "valence fluctuation superconductivity" observed in several Ce compounds [7]. Although our results cannot confirm such a mechanism, they lend support to this scenario. With the inclusion of a finite f -hopping, we see from Fig. 4(f) that there appears to be some tendency for the phase separation to transform into a diagonal stripe phase. This tendency can be understood by the observation that the spinless FKM model with t f = t can be mapped onto the Hubbard model, where the stripe phase is always obtained upon finite hole doping [17].
The fractionalization effect is present in Fig. 4(a-d); in particular, for Fig. 4(d) we find ∆N d = 4.0038, indicating the transfer of four electrons from the f -orbitals. As before, we find no evidence of this transfer in Fig. 4(f) where N d = N d0 = 288. Also anticipated by our singlehole results, we find enhancement of the excitonic order parameter |∆ j |, mainly at the boundary of the f -hole cluster [ Fig. 4(e)]. This again locally evidences the competition between the excitonic and CDW orders.
In conclusion, we have investigated the FKM extended by the d-f hybridization away from the half-filling symmetry point. Throughout this report, we have compared and contrasted our results with the extended FKM proposed in [5]. We have studied the formation of polarons when a single hole is doped into the CDW state. Associated with the polaron, we have discovered a fractional partitioning of the electron density between the d-and f -orbitals. Such an effect is unique to our extended FKM: analysis of the LDOS spectra at the polaron centre reveals the origins of the fractionalization as the hybridization term. We propose a STM experiment to verify our results. For moderate hole-doping, we have demonstrated that phase separation into hole-rich homogeneous and CDW states is energetically favorable. In both doping limits, enhancement of the excitonic ordering is found around the hole-rich region, locally manifesting the global competition between the two phases [13].