V. A. Khodel
Toward a self-consistent theory of Fermi systems with flat bands

A model of fermion condensation, advanced more than 25 years ago, still remains the subject of hot debates, due to the fact that within its frameworks, non-Fermi-liquid (NFL) behavior, ubiquitously exhibited by strongly correlated Fermi systems, including electron systems of solids, is properly elucidated. The model is derived with the aid of the same Landau postulate that the ground state energy E is a functional of its quasiparticle momentum distribution n, giving rise to the conventional Landau state. However, the model discussed deals with completely different solutions, emergent beyond a critical point, at which the topological stability of the Landau state breaks down, and therefore relevant solutions of the problem are found from the well-known variational condition of mathematical physics δE(n)/δn(p) = µ where µ is the chemical potential. Since the left side of this condition is nothing but the quasiparticle energy ϵ(p), the variational condition does imply formation of the flat band or, in different words, a fermion condensate (FC). In fact, variational condition furnishes an opportunity to find solely the FC quasiparticle momentum distribution n_∗(p ∈ Ω).

A missing point is concerned with the single-particle spectrum of quasiparticles in the complementary domain p ∉ Ω. Originally, at variance with recent experimental data, the model spectrum ϵ(p ∉ Ω) was assumed to be gapless. To clarify the situation with the true character of this spectrum we employ a microscopic approach to theory of Bose liquid created by S. Belyaev, where the interaction between the condensate and non-condensate particles, giving rise to the emergence of a singular part of the self-energy, is treated properly.

Unfortunately, in systems having a FC, evaluation of any FC propagator in closed form is impossible, since in contrast to the BC, the FC occupies a finite domain of momentum space. As a result, methods appropriate to evaluation of the multi-particle BC propagators, fail to deal with corresponding FC propagators. In this situation, the only practical approach to solution of the problem involves the implementation of an iterative procedure, appealing to the smallness of the ratio η = ρ_c/ρ where the numerator is the FC density and the denominator, total density. In the article, leading, in the limit η → 0, contributions to the singular part Σ_𝐬 of the self- energy are calculated along the Belyaev’s theory lines. By virtue of the finite range of the FC domain, the struc- ture of Σ_𝐬 turns out to be different, compared with that in the boson case, treated by Belyaev. As a result, the evaluated spectrum of single-particle excitations of Fermi systems, hosting flat bands, acquires a gap, so that the FC itself becomes a midgap state.

In obtaining the gap solution we assumed the effective interaction between particles in the particle-particle channel to have sign, which prevents Cooper pairing, implying that the ground state constructed is not super- fluid. In this situation, the gap in the single-particle spectrum is a Mott-like gap.

The theory constructed is applied to the explanation of the metal-insulator transition in low-density homogeneous 2D electron systems, like those, which reside in silicon field-effect transistors. These systems are known to become insulators at T → 0 provided electron density declines below a critical value n_c≃ 0.8×10¹¹cm⁻² [1–4], its value being substantially larger than that dictated by a standard Wigner crystallization scenario. Importantly, the electron effective mass M^∗(n), extracted from corresponding measurements of the thermopower diverges at almost the same value n_t = 0.78×10¹¹cm⁻² [5], in agreement with the proposed scenario for the metal-insulator transition in MOSFETs, triggered by the onset of fermion condensation and subsequent opening the Mott-like gap in the electron single-particle spectrum.

This scenario is also applied to the elucidation of a challenging phenomenon uncovered in the analysis of ARPES data on two-dimensional anisotropic electron systems of cuprates. It consists in breaking down of the Fermi line into several disconnected segments located in the antinodal region, (the Fermi arc structure), and usually attributed to superconducting fluctuations [6]. In the context of this article, the existence or disappear- ance of the Fermi line is related to the FC arrangement. In cuprates, the FC occupies four different spots, every of which is associated with its own saddle point. Such a configuration of the FC spots promotes the occurrence of a well pronounced gap in the spectrum of single-particle states, located in the antinodal region. However, for single-particle states, located in the nodal region, the situation with the gap is opposite and therefore in this domain of momentum space, the spectrum ϵ(p) remains gapless, in agreement with the experiment.