The influence of nonlocal interactions on valence transitions and formation of excitonic bound states in the generalized Falicov-Kimball model

TL;DR

This study uses DMRG to explore how nonlocal interactions influence valence transitions and excitonic bound state formation in a generalized Falicov-Kimball model, revealing contrasting effects of different interactions.

Contribution

It demonstrates the distinct impacts of nonlocal Coulomb interaction and correlated hopping on valence states and exciton condensation, highlighting their combined effects.

Findings

$U_{nn}$ suppresses zero-momentum exciton condensation.

$U_{ch}$ enhances exciton formation at zero momentum.

Combined interactions cause discontinuous changes in ground-state properties.

Abstract

We use the density-matrix-renormalization-group (DMRG) method to study the combined effects of nonlocal interactions on valence transitions and the formation of excitonic bound states in the generalized Falicov-Kimball model. In particular, we consider the nearest-neighbour Coulomb interaction $U_{nn}$ between two $d$, two $f$, $d$ and $f$ electrons as well as the so-called correlated hopping term $U_{ch}$ and examine their effects on the density of conduction $n_d$ (valence $n_f$) electrons and the excitonic momentum distribution $N(q)$. It is shown that $U_{nn}$ and $U_{ch}$ exhibit very strong and fully different effects on valence transitions and the formation (condensation) of excitonic bound states. While the nonlocal interaction $U_{nn}$ suppresses the formation of zero momentum condensate ($N(q$=$0)$) and stabilizes the intermediate valence phases with $n_d \sim 0.5, n_f \sim 0.5$, the correlated hopping term $U_{ch}$ significantly enhances the number of excitons in the zero-momentum condensate and suppresses the stability region of intermediate valence phases. The physically most interesting results are observed if both $U_{nn}$ and $U_{ch}$ are nonzero, when the combined effects of $U_{nn}$ and $U_{ch}$ are able to generate discontinuous changes in $n_f$, $N(q$=$0)$ and some other ground-state quantities.

Equations

$$H_0=\sum _{ij}{t}_{ij}{d}_i^{+}{d}_j+U\sum _i{f}_i^{+}{f}_i{d}_i^{+}{d}_i+E_f\sum _i{f}_i^{+}{f}_i,$$

$$H_{non}=U_{dd}\sum _{<ij>}{n}_i^d{n}_j^d+U_{df}\sum _{<ij>}{n}_i^d{n}_j^f+U_{ff}\sum _{<ij>}{n}_i^f{n}_j^f+U_{ch}\sum _{<ij>}{d}_i^{+}{d}_j(n_i^f+n_j^f),$$

Full text

1 Introduction

Since its introduction in 1969 the Falicov-Kimball model has become an important standard model for a description of correlated fermions on the lattice [1]. It has been used in the literature to study a great variety of many-body effects in metals, of which charge-density waves, metal-insulator transitions and mixed-valence phenomena are the most common examples [2]. In the past years the Falicov-Kimball model was extensively studied in connection with the exciting idea of electronic ferroelectricity [3, 4, 5, 6, 7, 8, 9, 10, 11, 12], which is directly related with the formation and condensation of excitonic bound states of conduction ($d$) and valence ($f$) electrons [13, 14, 15, 16, 17, 18, 19]. In its original form, the Falicov-Kimball model describes a two-band system of localized $f$ and itinerant $d$ electrons with short-ranged $f$-$d$ Coulomb interaction $U$:

[TABLE]

where $f^{+}_{i}$, $f_{i}$ are the creation and annihilation operators for an electron in the localized state at lattice site $i$ with binding energy $E_{f}$ and $d^{+}_{i}$, $d_{i}$ are the creation and annihilation operators of the itinerant spinless electrons in the $d$-band Wannier state at site $i$.

The first term of (1) is the kinetic energy corresponding to quantum-mechanical hopping of the itinerant $d$ electrons between sites $i$ and $j$. These intersite hopping transitions are described by the matrix elements $t_{ij}$, which are $-t_{d}$ if $i$ and $j$ are the nearest neighbors and zero otherwise (in the following all parameters are measured in units of $t_{d}$). The second term represents the on-site Coulomb interaction between the $d$-band electrons with density $n_{d}=\frac{1}{L}\sum_{i}d^{+}_{i}d_{i}$ and the localized $f$ electrons with density $n_{f}=\frac{1}{L}\sum_{i}f^{+}_{i}f_{i}$, where $L$ is the number of lattice sites. The third term stands for the localized $f$ electrons whose sharp energy level is $E_{f}$.

Since the local $f$-electron number $f^{+}_{i}f_{i}$ is strictly conserved quantity, the $d$-$f$ electron coherence cannot be established in this model. This shortcoming can be overcome by including an explicit local hybridization $H_{V}=V\sum_{i}d^{+}_{i}f_{i}+f^{+}_{i}d_{i}$ between the $d$ and $f$ orbitals. This model has been extensively studied in our previous work [20]. The numerical analysis of the excitonic momentum distribution $N(q)=\langle b^{+}_{q}b_{q}\rangle$ (with $b^{+}_{q}=(1/\sqrt{L})\sum_{k}d^{+}_{k+q}f_{k}$) showed that this quantity diverges for $q=0$, signalizing a Bose-Einstein condensation of preformed excitons. The stability of the zero-momentum ($q=0$) condensate against the $f$-electron hopping has been studied in our very recent paper [21]. It was found that the negative values of the $f$-electron hopping integrals $t_{f}$ support the formation of zero-momentum condensate, while the positive values of $t_{f}$ have the fully opposite effect. Moreover, we have found that the fully opposite effects on the formation of condensate exhibit also the local and nonlocal hybridization with an inversion symmetry. The first one strongly supports the formation of condensate, while the second one destroys it completely. In systems with equal parity orbitals, the local and nonlocal hybridization can coexist and for this case we have found [22] that the combined effect of both hybridizations strongly support the formation of zero-momentum condensate the existence of which can yield the reasonable explanation for the pressure-induced resistivity anomaly observed experimentally in $TmSe_{0.45}Te_{0.55}$ compound [23].

These results show, that the Falicov-Kimball model has a great potential to describe some of anomalous features of real complex materials like rare-earth compounds. On the other hand it should be noted that the original version of the model, as well as its extensions discussed above, represent too crude approximation of real rare-earth compounds, since neglect all nonlocal Coulomb interactions, that can change this picture. For the correct description of these materials one should take into account at least the following nonlocal Coulomb interaction terms:

[TABLE]

which represent the nearest-neighbour Coulomb interaction between two $d$ electrons (the first term), between one $d$ and one $f$ electron (the second term), between two $f$ electrons (the third term) and the so-called correlated hopping (the last term).

There is a number of papers, were the influence of individual interaction terms from (2) on the ground state properties of the Falicov-Kimball model has been studied (see Ref. [24] and references therein), however, there are only few, where the combined effects of two or three terms were considered [25, 26]. From this point of view the model Hamiltonian studied in this paper $H=H_{0}+H_{V}+H_{non}$, represents one of the most complex extensions of the Falicov-Kimball model used for a description of ground state properties of strongly correlated systems. Here we concern our attention exclusively on a description of two main problems, and namely, the process of formation and condensation of excitonic bound states and the problem of valence transitions in the generalized Falicov-Kimball model.

2 Results and discussion

To examine effects of the above mentioned factors on the ground states properties of the generalized Falicov-Kimball model we have performed exhaustive DMRG studies of the model Hamiltonian $H=H_{0}+H_{V}+H_{non}$ for a wide range of the model parameters at the half-filled band case $n=n_{d}+n_{f}=1$. To simplify numerical calculations we adopt here the following model $U_{dd}=U_{ff}=U_{df}=U_{nn}$, that allows us to reduce the number of model parameters and at the same time to keep all nonlocal interactions terms nonzero. In all our DMRG calculations we typically keep up to 500 states per block, although in the numerically more difficult cases,where the DMRG results converge slower, we keep up to 1000 states. Truncation errors [27], given by the sum of the density matrix eigenvalues of the discarded states, vary from $10^{-6}$ in the worse cases to zero in the best cases. As mentioned above the special attention in the current paper is devoted to the understanding of the process of formation and condensation of excitonic bound states. To describe this process in more detail (and at the same time to specify the role of different factors) we have calculated numerically the density of zero momentum excitons $n_{0}=\frac{1}{L}N(q=0)$, the total exciton density $n_{T}=\frac{1}{L}\sum_{q}N(q)$, the total $d$-electron density $n_{d}$ and the total density of unbond $d$ electrons $n^{un}_{d}=n_{d}-n_{T}$ as functions of the interaction parameters $U_{nn}$ and $U_{ch}$ for zero as well as finite values of $E_{f}$. Let us start our discussion with numerical results obtained for the density of zero momentum excitons $n_{0}$ as function of $U_{nn}$ and $U_{ch}$ at $E_{f}=0$. The numerical results for $n_{0}$ obtained from the DMRG calculations are summarized in Fig. 1 for $U=1,L=100$ and several different values of the local hybridization $V$.

These results show that the nonlocal interactions $U_{nn}$ and $U_{ch}$ exhibit fully different effects on the condensation of preformed excitons to the zero-momentum state. The nonlocal interaction $U_{nn}$ strongly suppresses the condensation of preformed excitons to the ground state and already relatively small values of $U_{nn}$ ($U_{nn}\sim 0.4$) practically fully destroy the zero-momentum condensate. On the other hand, the correlated hopping represented by the $U_{ch}$ interaction term exhibits fully opposite effect on $n_{0}$. In this case the number of zero momentum excitons is strongly enhanced with increasing $U_{ch}$ and reaches its maximum at $U^{max}_{ch}\sim 0.6$. The Fig. 1d shows that the enhancement $\Delta=n_{0}(U_{ch})/n_{0}(U_{ch}$=$0)$ of the zero momentum excitons due to the $U_{ch}$ term is indeed considerable and at the point of the global maximum it reaches values $\Delta\sim 5$, for $V=0.6$ and $\Delta\sim 10$, for $V=0.1$, pointing on the crucial role of the correlated hopping in the process of formation and condensation of excitonic bound states. Regarding the influence of the local hybridization on the formation of zero-momentum condensate we have found that in all examined cases the nonlocal hybridization does not change qualitatively the behavior of $n_{0}$ and therefore all calculations in the rest of the paper are done for the representative value of $V=0.1$.

Till now we have presented results exclusively for $E_{f}=0$. Let as now discuss briefly the effect of change of the $f$-level position. This study is interesting also from this point of view that taking into account the parametrization between the external pressure $p$ and the position of the $f$ level ($E_{f}\sim p$), one can deduce from the $E_{f}$ dependences of the ground state characteristics also their $p$ dependences, at least qualitatively [28]. The resultant $E_{f}$ dependences of the density of zero momentum excitons $n_{0}$, the total exciton density $n_{T}$, the total $d$-electron density $n_{d}$ and the total density of unbond $d$ electrons $n^{un}_{d}$ obtained by DMRG method are shown in Figs. 2-4 for several values of $U_{nn}$ and $U_{ch}$ at $U=1$ and $L=100$.

Let us first discuss the effect of the $U_{nn}$ term (see Fig. 2a). One can see that the density of zero momentum excitons is nonzero over the whole interval of $E_{f}$ values and is considerably enhanced in some regions, due to the significant enhancement of the $d$ electron population in the $d$ band (see panel b). Generally the increasing nonlocal interaction $U_{nn}$ destroys the zero-momentum condensate and this effect is most pronounced near the half-filled band point $n_{d}=n_{f}=1/2$, obviously due to the formation of the charge-density-wave phase that is the ground state of the model at this point for $U_{nn}$ and $U_{ch}$ equals zero. Another general trend recognized from our numerical calculations concerns the valence transition behaviour of the model, and particularly, the stabilization of intermediate valence plateau at $n_{d}=n_{f}=1/2$ with increasing $U_{nn}$. This effect is very strong and already very small values of the nonlocal interaction $U_{nn}$ ($U_{nn}=0.4$) are able to stabilized the intermediate valence phase with $n_{f}\sim 1/2$ for a wide range of $E_{f}$ values. The total exciton density $n_{T}$ exhibits qualitatively the same behaviour as the total $d$-electron density $n_{d}$, however their difference $n_{d}-n_{T}$, which represents the total density of unbond $d$ electrons, exhibits strong dependence on both $U_{nn}$ as well as $E_{f}$. With increasing $U_{nn}$, the total density of unbond $d$ electrons $n^{un}_{d}$ is generally enhanced, while with increasing $E_{f}$, the total density of unbond $d$ electrons increases for $E_{f}<0$ and decreases for $E_{f}>0$, with obvious tendency to form the plateau around $E_{f}=0$ for $U_{nn}$ sufficiently large (small deviations from this behaviour are observed only near the point $E_{f}=0$ for $U_{nn}=0$).

In comparison to $U_{nn}$ effects, the correlated hopping term exhibits the fully opposite effects on both the formation and condensation of excitons as well as valence transitions. Indeed, Fig. 3a shows that the density of zero momentum excitons $n_{0}$ is now generally enhanced with increasing $U_{ch}$ (this effect is especially strong near $E_{f}=0$) and contrary to the previous case, where the intermediate phase with $n_{d}\sim 0.5$ has been stabilized, it is now suppressed, and due to, the valence transitions become more steeper.

The fully different behaviour (in comparison to the $U_{nn}>0$ case) exhibits also the total density of unbond $d$ electrons $n^{un}_{d}$, which is generally suppressed with increasing $U_{ch}$.

The physically most interesting case corresponds, however, to the situation when both ($U_{nn}$ as well as $U_{ch}$) interactions are switched on simultaneously. The results of our DMRG calculations obtained for the case $U_{nn}=U_{ch}$ are displayed in Fig. 4.

One can see that there are some similarities, but also differences in comparison to the above discussed cases ($U_{nn}>0,U_{ch}=0$ and $U_{nn}=0,U_{ch}>0$). Let us first discuss the similarities. There are: (i) the strong suppression of the zero-momentum condensate in the region of $E_{f}$, where $n_{d}\sim 0.5$, (ii) the stabilization of the intermediate phase with $n_{d}\sim 0.5$ for increasing $U_{nn}=U_{ch}$ and (iii) the strong enhancement of the total density of unbond $d$ electrons $n^{un}_{d}$ with increasing of $U_{nn}=U_{ch}$. It should be noted, that these similarities with the case $U_{nn}>0,U_{ch}=0$ are quite surprising, since as discussed above $U_{nn}$ and $U_{ch}$ exhibit fully opposite effects on the formation of zero momentum condensate, as well as on the valence transitions.

More interesting are, however, differences which are produced by the combined effects of $U_{nn}$ and $U_{ch}$ interaction in comparison to the cases when $U_{nn}$ and $U_{ch}$ act individually. There are: the stabilization of zero momentum condensate for some values of the $f$-level energy $E_{f}$ in the weak coupling limit $U_{nn}=U_{ch}\sim 0.2$, (ii) the appearance of the discontinuous valence transitions for sufficiently large values of $U_{nn}=U_{ch}\sim 0.4$, (iii) the discontinuous disappearance of the density of zero momentum excitons, as well as the discontinuous changes in the total density of excitons $n_{T}$ and the total density of unbond $d$ electrons $n^{un}_{d}$ at the valence transition points.

The appearance of the discontinuous changes in some ground-state observables like the density of conduction $d$ (valence $f$) electrons, the density of zero-momentum condensate, the density of unbond electrons is very important result from the point of view of rare-earth compounds. In some of them, e.g., the mixed valence system SmS such discontinuous changes are observed experimentally in the density of valence electrons when the external hydrostatic pressure is applied [29], however, they were not satisfactorily described till now. Indeed, as mentioned above the SmS compound is the mixed valence system, with fluctuating valence and thus for its description one has to take into account the hybridization between the localized $f$ and conduction $d$ electron states. However, more reliable methods, like alloy-analog approximation [30], renormalization group method [31], exact diagonalization method [32] predict only the continuous valence transitions within the Falicov-Kimball model extended by the local hybridization. Here we show that considering the parametrization between the external pressure $p$ and the $f$-level position $E_{f}$, the pressure induced discontinuous valence transitions is possible to generate also in such a system under very realistic assumption, and namely, that nonlocal interactions are switched on. This opens new route to understanding various ground-state anomalies observed in the rare-earth compounds within the unified picture.

In summary, we have used the DMRG calculations to examine the combined effects of various nonlocal interactions on valence transitions and the formation of excitonic bound states in the generalized Falicov-Kimball model. In particular, we have considered the nearest-neighbour Coulomb interaction (i) between two $d$ electrons ($U_{dd}$), (ii) between one $d$ and one $f$ electron ($U_{df}$), (iii) between two $f$ electrons ($U_{ff}$) and (iv) the so-called correlated hopping ($U_{ch}$). To reduce the parametric space of the model we have restricted our study only to the case $U_{dd}=U_{df}=U_{ff}=U_{nn}$ and examined the individual and combined effects of $U_{nn}$ and $U_{ch}$ on valence transitions and the formation (condensation) of excitonic bound states of conduction $d$ and valence $f$ electrons. Analysing numerically the density of conduction $n_{d}$ (valence $n_{f}$) electrons and the excitonic momentum distribution $N(q)$, as functions of $U_{nn}$ and $U_{ch}$ it was found that these interactions exhibit fully different effects on the valence transitions as well as on the process of formation and condensation of excitonic bound states. While the nonlocal interaction $U_{nn}$ suppresses the formation of zero momentum condensate and stabilizes the intermediate valence phases with $n_{d}\sim 0.5,n_{f}\sim 0.5$, the correlated hopping significantly enhances the number of excitons in the zero-momentum condensate and suppresses the stability region of intermediate valence phases. However, the physically most interesting results are observed if both $U_{nn}$ and $U_{ch}$ are nonzero, when the combined effects of $U_{nn}$ and $U_{ch}$ are able to generate discontinuous changes in $n_{f}$, $N(q$=$0)$ and some other ground-state quantities of the model.

This work was supported by projects ITMS26220120047, VEGA 2-0112-18 and APVV-17-0020. Calculations were performed in the Computing Centre of the Slovak Academy of Sciences using the supercomputing infrastructure acquired in project ITMS 26230120002 and 26210120002 (Slovak infrastructure for high-performance computing) supported by the Research and Development Operational Programme funded by the ERDF.