arXiv:1510.05233v1 [cond-mat.str-el] 18 Oct 2015 Exciton condensation in an extended spinless Falicov Kimball model in the presence of orbital magnetic ﬁelds S. Pradhan 1* and A. Taraphder 1,2 1 Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur - 721302, India 2 Center for Theoretical Studies, Indian Institute of Technology Kharagpur, Kharagpur - 721302, India (Dated: May 10, 2018) An extended, spinless Falicov-Kimball model in the presence of perpendicular magnetic ﬁeld is investigated employing Hartree-Fock self-consistent mean-ﬁeld theory in two dimensions. In the presence of an orbital ﬁeld the hybridization-dependence of the excitonic average Δ =<di † fi > is modiﬁed. The exciton responses in subtle diﬀerent ways for diﬀerent chosen values of the magnetic ﬂux consistent with Hofstadter’s well-known spectrum. The excitonic average is suppressed by the application of magnetic ﬁeld. We further examine the eﬀect of Coulomb interaction and f -electron hopping on the condensation of exciton for some rational values of the applied magnetic ﬁeld. The interband Coulomb interaction enhances the Δ exponentially, while a non-zero f -electron hopping reduces it. A strong commensurability eﬀect of the magnetic ﬂux on the behaviour of the excitons is observed. PACS numbers: 71.35.-y, 71.45.Lr, 77.80.-e, 71.35.Ji, 71.10.Fd, 71.28.+d, 71.27.+a I. INTRODUCTION The Falicov-Kimball model is perhaps the simplest model to study correlation in fermionic systems on a lattice. It involves a conduction d-band, a localized f - electron state and an on-site Coulomb interaction U be- tween the d and f -electrons. Since its introduction 1 in 1959, to describe valence or metal-insulator transition in some transition metal oxides, the model has been used successfully in describing various many-body eﬀects 2 like metal-insulator transition, 3 mixed-valence phenomena, 4 the formation of ionic crystals 5,6 and orbital 7 and charge- density waves (CDW). 8 It was found 8–10 that on a bi- partite lattice at half-ﬁlling (n d = n f =0.5), f -electrons occupy sites of one sublattice only, the well-known checkerboard phase. For D ≥ 2 dimension, the chess- board charge pattern exists below a critical temperature, T CDW , above which a disordered phase is obtained. 2 Fur- ther, using dynamical mean ﬁeld theory (DMFT), 11 ex- act in dimension D = ∞, Brandt and Mielsch conﬁrmed the existence of inhomogeneous CDW phase. 12 An ex- tended version of Falicov-Kimball model is also in use to account for the homogeneous mixed valence prob- lem. It has been used since last two decades to account unconventional ferroelectricity 13–15 in mixed-valent com- pounds. Customarily, ferroelectricity appears due to the struc- tural phase transition. However, it is also possible that there is a nonvanishing d − f coherence in a system lead- ing to a hybridization term of purely electronic origin. This coherence could give rise to electric polarization due to Bose-Einstein condensation of d − f excitons when the two bands diﬀer by odd parity. Portengen and co- authors studied an extended FKM with a k-dependent hybridization term in Hartree-Fock approximation fol- lowing Leder’s work. 13 They found that the Coulomb in- teraction U between itinerant d-electron and localized f -electron gives rise to a nonvanishing d − f coherence <d i † f i > even in the limit of vanishing hybridization V → 0 in the presence of a putative homogeneous ground state solution of the HF approximation. In the weak- coupling mean-ﬁeld theory, the formation of an order pa- rameter and the condensation thereof are concomitant. Quite interestingly, this condensation of the excitonic order parameter, they pointed out, signiﬁes a “sponta- neous” polarization in the system when the parities of the electron-hole partners in the condensate diﬀer by one. Later, Czycholl 16 showed that an imhomogeneous ground state, as obtains in FKM on a square lattice at half-ﬁlling, leads to <d i † f i >→ 0 as V → 0. Therefore, the FKM in the half-ﬁlled limit does not admit a “spon- taneous” symmetry breaking, consistent (but not contra- dictory at T = 0, where Elitzur’s theorem does not forbid an order) with the local U (1) symmetry in the f -band at V = 0. For a small non-zero hybridization V , the inho- mogeneous (CDW) phase is stable, and the order param- eter is ﬁnite. This CDW phase, though, melts beyond a critical hybridization strength. Similar conclusions were reached in a triangular lattice as well 17 . For a one- dimensional extended FKM, using exact-diagonalization and DMRG techniques, Farka˘ sovsk´ y has ruled out the possibility of spontaneous excitonic averages at zero temperature 18 , which is expected in one dimension. Em- ploying the same numerical technique, Farka˘ sovsk´ y inves- tigated the eﬀects of local and non-local 19,20 hybridiza- tion on valence transitions. Zlati´ c et al. 21 conﬁrmed that the static excitonic susceptibility diverges at T = 0 in the ordinary FKM (V = 0), from an exact solution of the model in inﬁnite dimension. In dimensions D> 1, a ﬁnite f -electron bandwidth breaks the local U (1) sym- metry and induce a non-zero polarization even in the ab- sence of d − f hybridization as expected from symmetry grounds. This is easily shown 22 by mapping an extended FKM (V = 0 but t f ﬁnite) on to a Hubbard model with asymmetric hopping (t ↑ = t ↓ ) and thence to an eﬀec- tive anisotropic XXZ, s =1/2 spin model with a “ﬁeld”