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arXiv:1702.04167 · Physical review. B./Physical review. B

Formation and condensation of excitonic bound states in the generalized Falicov-Kimball model

Pavol Farkašovský

Published Jan 3, 2017• 13 citations
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Abstract

The density-matrix-renormalization-group method and the Hartree-Fock approximation with the charge-density-wave instability are used to study a formation and condensation of excitonic bound states in the generalized Falicov-Kimball model. In particular, we examine effects of various factors, such as the $f\text{-electron}$ hopping, the local and nonlocal hybridizations, as well as the increasing dimension of the system on the excitonic momentum distribution $N(q)$ and especially on the number of zero-momentum excitons ${N}_{0}=N(q=0)$ in the condensate. It is found that the negative values of the $f\text{-electron}$ hopping integrals ${t}_{f}$ support the formation of a zero-momentum condensate, whereas the positive values of ${t}_{f}$ have the fully opposite effect. The opposite effects on the formation of the condensate exhibit also the local and nonlocal hybridizations. The first one strongly supports the formation of the condensate, whereas the second one destroys it completely. Moreover, it was shown that the zero-momentum condensate remains robust with increasing dimension of the system.

DOI
10.1103/physrevb.95.045101

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The density-matrix-renormalization-group method and the Hartree-Fock approximation with the charge-density-wave instability are used to study a formation and condensation of excitonic bound states in the generalized Falicov-Kimball model. In particular, we examine effects of various factors, such as the f-electronf\text{-electron}f-electron hopping, the local and nonlocal hybridizations, as well as the increasing dimension of the system on the excitonic momentum distribution N(q)N(q)N(q) and especially on the number of zero-momentum excitons N0=N(q=0){N}_{0}=N(q=0)N0​=N(q=0) in the condensate. It is found that the negative values of the f-electronf\text{-electron}f-electron hopping integrals tf{t}_{f}tf​ support the formation of a zero-momentum condensate, whereas the positive values of tf{t}_{f}tf​ have the fully opposite effect. The opposite effects on the formation of the condensate exhibit also the local and nonlocal hybridizations. The first one strongly supports the formation of the condensate, whereas the second one destroys it completely. Moreover, it was shown that the zero-momentum condensate remains robust with increasing dimension of the system.
The formation and condensation of excitonic bound states of conduction-band electrons and valence-band holes surely belongs to one of the most exciting ideas of contemporary solid state physics.In this short review we present the latest progress in this field reached by the density-matrix-renormalization-group (DMRG) calculations within various extensions of the Falicov-Kimball model.Particular attention is paid to a description of crucial mechanisms (interactions) that affect the stability of the excitonic phase, and namely: (i) the interband d-f Coulomb interaction, (ii) the f -electron hopping, (iii) the nonlocal hybridization with odd and even parity, (iv) combined effects of the local and nonlocal hybridization, (v) the nearest-neighbor Coulomb interaction between d and f electrons and (vi) the correlated hopping.The relevance of numerical results obtained within different extensions of the Falicov-Kimball model for a description of the real d-f materials is widely discussed.
We consider the excitonic insulator state (often associated with electronic ferroelectricity), which arises on the phase diagram of an extended spinless Falicov-Kimball model (FKM) at half-filling. Within the Hartree--Fock approach, we calculate the spectrum of low-energy collective excitations in this state up to second order in the narrow-band hopping and/or hybridization. This allows to probe the mean-field stability of the excitonic insulator. The latter is found to be unstable when the case of the pure FKM (no hybridization with a fully localized band) is approached. The instability is due to the presence of another, lower-lying ground state and not to the degeneracy of the excitonic phase in the pure FKM. The excitonic phase, however, may be stabilized further away from the pure FKM limit. In this case, the low-energy excitation spectrum contains important information about the properties of the excitonic condensate (including the strongly suppressed critical temperature).
The density matrix renormalization group (DMRG) method is used to examine the effects of the interband Coulomb interaction U between f and d electrons as well as the f -electron hopping on the stability of the excitonic phase in the one-dimensional Falicov-Kimball model (FKM) with local hybridization V . It is found that the interband Coulomb interaction significantly enhances the excitonic average and that this effect is especially strong in the limit of small hybridization, where the interacting excitonic value is enhanced several hundred times in comparison to its non-interacting value. The further increase in P df is observed due to the non-zero f -electron hopping, but these changes are considerable only if the d and f bands have opposite parity. In addition, the examination of the interplay between the excitonic effects and the charge density wave (CDW) instability showed that the CDW and excitonic phase coexist up to relatively large values of local hybridization . The ground-state phase diagram of the model in the plane is discussed.
We present a Hartree-Fock study of the Falicov-Kimball model extended by both on-site and nonlocal hybridization. We examine the interplay between excitonic effects and the charge-density wave (CDW) instability known to exist at zero hybridization. It is found that the CDW state remains stable in the presence of finite hybridization; for on-site hybridization the Coulomb interaction nevertheless strongly enhances the excitonic average above its value in the noninteracting system. In contrast, for nonlocal hybridization, we observe no such enhancement of the excitonic average or a spontaneous on-site hybridization potential. Instead, we find only a significant suppression of the excitonic correlations in the CDW state. A phenomenological Ginzburg-Landau analysis is also provided to understand the interplay.
We obtain exact ground states for the spinless Falicov-Kimball model with hybridization in one dimension (d=1).(d=1).(d=1). Ground-state energies and their corresponding wave functions are given for a restricted region of the parameter space in the limit of strong interactions for the half-filling condition. We find that the ground states possess a diagonal long range order and constitute a condensation of excitons in a zero momentum state. On the other hand, the correlation function $A=〈{d}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{f}_{i}〉$ is purely imaginary, indicating that the ground state does not exhibit spontaneous ferroelectricity.
We investigate the effects of magnetic fields on excitonic condensation in the extended Falicov-Kimball model, which is a spinless two-orbital Hubbard model with orbital splitting. In lattice systems under magnetic fields up to several tens of teslas, Zeeman effects on electron spins have been extensively studied, while the impact on orbital motion has often been considered negligible. However, the recent capability to generate ultra-high magnetic fields exceeding 1000 T has renewed interest in understanding their influence on ordered phases in correlated electron systems, beyond spin-related phenomena. To examine these effects, we incorporate a magnetic field into the extended Falicov-Kimball model by introducing the Peierls phase into the transfer integrals, enabling the study of orbital motion. Using the Hartree-Fock approximation, we reveal a nonmonotonic response of the excitonic order parameter to increasing magnetic fields. At sufficiently high fields, the excitonic order is suppressed, resulting in a disordered insulating state characterized by partial occupation of the two orbitals with nonzero Chern numbers. This state is distinct from a fully orbital-polarized configuration. Furthermore, our analysis of an excitonic supersolid phase, in which excitonic and orbital orders coexist, demonstrates that orbital order remains robust under magnetic fields, while excitonic condensation is suppressed. These findings provide insights into the interplay between orbital motion and magnetic fields in multi-orbital correlated electron systems.