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Canadian Journal of Physics

Nonexistence of excitonic insulators in one and two dimensions

M. B. Walker

Published Apr 1, 1968• 8 citations
Publisher

Abstract

The Bogoliubov inequality is used to prove the nonexistence of the excitonic insulating phase in one and two dimensions for systems of electrons described by three different many-band models with interactions. The proof is carried through (1) for arbitrary two-particle interactions with an interaction potential which goes to zero faster than [Formula: see text] (n being the number of dimensions) at large [Formula: see text], (2) for two-particle interactions of arbitrary range, provided only the so-called Coulomb and exchange parts of the interaction are retained and the remainder is neglected, and (3) for the simplified isotropic two-band model suggested for the excitonic insulator by Des Cloizeaux.

DOI
10.1139/p68-103

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The Bogoliubov inequality is used to prove the nonexistence of the excitonic insulating phase in one and two dimensions for systems of electrons described by three different many-band models with interactions. The proof is carried through (1) for arbitrary two-particle interactions with an interaction potential which goes to zero faster than [Formula: see text] (n being the number of dimensions) at large [Formula: see text], (2) for two-particle interactions of arbitrary range, provided only the so-called Coulomb and exchange parts of the interaction are retained and the remainder is neglected, and (3) for the simplified isotropic two-band model suggested for the excitonic insulator by Des Cloizeaux.
The variational cluster approximation is used to study the spontaneous symmetry breaking of the excitonic insulator (EI) state in the two-orbital Hubbard model defined on the two-dimensional square lattice. Evaluating the order parameter of the EI state, staggered magnetization, and number of particles on each orbital, we obtain the ground-state phase diagram of the model in a wide parameter space of the intra and interorbital Coulomb interactions. We also calculate the single-particle and anomalous Green's functions. We show that the normal metallic phase is unstable to the formation of the EI state and the EI phase appears in a wide parameter region between the band-insulator and Mott-insulator phases.
Using the projector-based renormalization method we investigate the formation of the excitonic insulator phase in the two-dimensional (2D) spinless Falicov-Kimball model with dispersive f electrons and address the existence of excitonic bound states at high temperatures on the semiconductor side of the semimetal-semiconductor transition. To this end we calculate the imaginary part of the dynamical electron-hole pair susceptibility and analyze the wave vector and energy dependence of excitonic resonances emerging in the band gap. We thereby confirm the existence of the exciton insulator and its exciton environment within a generic two-band lattice model with local Coulomb attraction.
We study the optical properties of an electron–hole two-band Hubbard model with interactions of electron–electron (hole–hole) repulsion U and electron–hole attraction U '. In the case of two types of correlated insulators at half filling, viz. , the Mott–Hubbard type and the biexciton type, interband linear absorption/gain spectra including excitonic effects are calculated analytically using two-site dynamical mean-field theory and ladder approximation. We show that an exciton-like peak structure emerges because of an “exciton” in the sea of correlated fermions when U ' is larger than a certain critical value. The binding energy of the photoinduced excitonic bound state and the spectral intensity are evaluated as a function of U '. The emergence and absence of the exciton-like peak suggests that there are two different insulating mechanisms in the Mott–Hubbard insulating state of the electron–hole system: one is attributable to only U and the other is attributable to both U and U '; the latter is called the excitonic Mott–Hubbard insulating state. The intrinsic difference of the mechanisms cannot be captured from only single-particle properties such as the density of states, but it appears clearly in optical properties reflecting two-particle correlations.
We study the excitonic phase transition in a system of the conduction band electrons and valence band holes described by the three-dimensional (3D) extended Falicov–Kimball (EFKM) model with the tunable Coulomb interaction UUU between both species. By lowering the temperature, the electron–hole system may become unstable with respect to the formation of the excitons, i.e, electron–hole pairs at temperature T=TΔT=T_{\Delta }T=TΔ​ , exhibiting a gap Δ\Delta Δ in the particle excitation spectrum. To this end we implement the functional integral formulation of the EFKM, where the Coulomb interaction term is expressed in terms of U(1) phase variables conjugate to the local particle number, providing a useful representation of strongly correlated system. The effective action formalism allows us to formulate a problem in the phase-only action in the form of the quantum rotor model and to obtain analytical formulas for the critical lines and other quantities of physical interest like charge gap, chemical potential and the correlation length.