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Physical review. B./Physical review. B

Thermal fluctuations and mass imbalance in low-energy excitonic condensate excitations

Thi‐Hong‐Hai Do, Van‐Nham Phan

Published Dec 19, 2024• 3 citations
Publisher

Abstract

Low-energy excitations of the excitonic states in the semimetal and semiconducting materials described by the extended Falicov-Kimball model with phonon correlation are investigated. In the framework of the unrestricted Hartree-Fock approximation, we derive a set of self-consistent equations to find the quasiparticle energies and excitonic condensate order parameters. The real part of the optical conductivity and then the dynamical excitonic susceptibility function are analytically evaluated by means of the linear response theory. Numerical results reveal the significant expansion of the excitonic condensation window in the temperature--Coulomb interaction plane by lowering the mass imbalance. Under the influence of the mass imbalance and thermal fluctuations, the BCS-BEC crossover of the excitonic condensation states has been also addressed in the collective mode signatures of the optical conductivity. Out of the condensation state, the collective excitation of the excitonic bound state is examined by analyzing the imaginary part of the dynamical excitonic susceptibility function. At large mass imbalance and low thermal fluctuations, we find an extremely sharp peak with high intensity at low frequency in the imaginary part of the dynamical excitonic susceptibility spectrum, specifying the long-lifetime exciton existence before it becomes condensed.

DOI
10.1103/physrevb.110.235143

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Low-energy excitations of the excitonic states in the semimetal and semiconducting materials described by the extended Falicov-Kimball model with phonon correlation are investigated. In the framework of the unrestricted Hartree-Fock approximation, we derive a set of self-consistent equations to find the quasiparticle energies and excitonic condensate order parameters. The real part of the optical conductivity and then the dynamical excitonic susceptibility function are analytically evaluated by means of the linear response theory. Numerical results reveal the significant expansion of the excitonic condensation window in the temperature--Coulomb interaction plane by lowering the mass imbalance. Under the influence of the mass imbalance and thermal fluctuations, the BCS-BEC crossover of the excitonic condensation states has been also addressed in the collective mode signatures of the optical conductivity. Out of the condensation state, the collective excitation of the excitonic bound state is examined by analyzing the imaginary part of the dynamical excitonic susceptibility function. At large mass imbalance and low thermal fluctuations, we find an extremely sharp peak with high intensity at low frequency in the imaginary part of the dynamical excitonic susceptibility spectrum, specifying the long-lifetime exciton existence before it becomes condensed.
The ground‐state properties of the excitonic condensation state in the effects of the mass imbalance are discussed in the framework of the spinless extended Falicov–Kimball model, in which the hopping of the f electrons is involved. Using the unrestricted Hartree–Fock approximation, a set of self‐consistent equations are obtained so the excitonic condensate order parameter is determined. In this sense, the real part of the optical conductivity is analytically calculated based on the Kubo formula. Phase diagram releases a complex phase structure of the excitonic condensation state with a strong depression of its Bardeen–Cooper–Schrieffer (BCS)–Bose–Einstein condensation (BEC) crossover in the influence of the mass imbalance. The impression of the mass imbalance affecting the excitonic condensate is also addressed in the signature of the optical conductivity. The finding delivers the natural scenario of the excitonic condensation state applied for a wide range of materials, from transition‐metal dichalcogenides to double‐layer systems such as a semiconductor double quantum well, double‐bilayer graphene, or double‐layer graphene.
Using exact numerical techniques, we investigate the nature of excitonic (electron-hole) bound states and the development of exciton coherence in the one-dimensional half-filled extended Falicov-Kimball model. The ground-state phase diagram of the model exhibits, besides band-insulator and staggered orbital ordered phases, an excitonic insulator (EI) with power-law correlations. The criticality of the EI state shows up in the von Neumann entropy. The anomalous spectral function and condensation amplitude provide the binding energy and coherence length of the electron-hole pairs which, on their part, point towards a Coulomb interaction driven crossover from BCS-like electron-hole pairing fluctuations to tightly bound excitons. We show that while a mass imbalance between electrons and holes does not affect the location of the BCS-BEC crossover regime, it favors staggered orbital ordering to the disadvantage of the EI. Within the Bose-Einstein condensation (BEC) regime, the quasiparticle dispersion develops a flat valence-band top, in accord with the experimental finding for Ta2NiSe5.
We study the spontaneous symmetry breaking of the excitonic insulator state induced by the Coulomb interaction UUU in the two-dimensional extended Falicov-Kimball model. Using the variational cluster approximation (VCA) and Hartree-Fock approximation (HFA), we evaluate the order parameter, single-particle excitation gap, momentum distribution functions, coherence length of excitons, and single-particle and anomalous excitation spectra as functions of UUU at zero temperature. We find that in the weak-to-intermediate coupling regime, the Fermi surface plays an essential role and calculated results can be understood in close correspondence with the BCS theory, whereas in the strong-coupling regime, the Fermi surface plays no role and results are consistent with the picture of a Bose-Einstein condensate (BEC). Moreover, we find that HFA works well in both the weak- and strong-coupling regimes, and that the difference between the results of VCA and HFA mostly appears in the intermediate-coupling regime. The reason for this is discussed from a viewpoint of the self-energy. We thereby clarify the excitonic insulator state that typifies either a BCS condensate of electron-hole pairs (weak-coupling regime) or a BEC of preformed excitons (strong-coupling regime).
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).
Based on the SO(2)\mathrm{SO}(2)SO(2)-invariant slave-boson scheme, the static charge, orbital, and excitonic susceptibilities in the extended Falicov--Kimball model are calculated. Analyzing the phase without long-range order, we find instabilities toward charge order, orbital order, and the excitonic insulator (EI) phase. The instability toward the EI is in agreement with the saddle-point phase diagram. We also evaluate the dynamic excitonic susceptibility, which allows the investigation of uncondensed excitons. We find qualitatively different features of the exciton dispersion at the semimetal-EI and at the semiconductor-EI transition supporting a crossover scenario between a BCS-type electron-hole condensation and a Bose--Einstein condensation of preformed bound electron-hole pairs.
Motivated by recent experimental and theoretical studies of excitonic insulators, we use the variational cluster approximation to calculate the single-particle Green's function and anomalous Green's function of the Falicov-Kimball model extended by including a finite valence bandwidth. We thus determine the single-particle excitation gap due to the bound-state formation of an electron and a hole. We also evaluate the order parameter of the excitonic insulating state indicating quantum coherence of condensed excitons. We thereby discuss the excitonic insulator that typifies either a BCS condensate of electron-hole pairs (weak-coupling regime) or a Bose-Einstein condensate of preformed excitons (strong-coupling regime). A BCS-BEC crossover of the model thus manifests itself as a function of the Coulombic coupling strength.