# Recent Developments in the Field of the Metal-Insulator Transition in Two Dimensions
## 5 Band Flattening: Possible Condensation of Fermions  
The creation and investigation of flat-band materials is a forefront area of modern condensed matter physics [36; 37; 38; 39]. In particular, the interest is ignited by the fact that, due to the anomalous density of states, the flattening of the band may be important for the construction of room temperature superconductivity. The appearance of a flat band is theoretically predicted [40; 41; 42] in a number of systems including high-temperature superconductors, heavy fermion systems, \({}^{3}\)He, and 2D electron systems. As the strength of fermion-fermion interaction is increased, the single-particle spectrum becomes progressively flatter in the vicinity of the Fermi energy, eventually forming a plateau. The flattening of the spectrum is related to the increase of the effective fermion mass \(m_{\rm F}\) at the Fermi level and the corresponding peak in the density of states. The experimental data obtained in strongly-correlated 2D electron systems can be divided into two groups:  
1. data describing the electron system as a whole, like thermodynamic density of states, magnetization of the electron system, or the magnetic field required to polarize electron spins fully and
2. data related only to the electrons at the Fermi level, like the amplitude of the Shubnikov-de Haas oscillations. This yields the effective mass \(m_{\rm F}\) and Lande \(g\)-factor \(g_{\rm F}\) at the Fermi level. Usually, the data in the first group are interpreted using the language of quasiparticles, in which the energy-averaged effective mass, \(m\), and Lande \(g\)-factor, \(g\), are used. To determine the values, the formulas that hold for the case of non-interacting electrons are used. Although this approach is ideologically not correct, the results for \(m\) and \(g\) are often the same as, or close to, the results for \(m_{\rm F}\) and \(g_{\rm F}\). For example, the simultaneous increase of the energy-averaged effective mass and that at the Fermi level in Si MOSFETs was reported in publications described in the previous section. In [43], the magnetic field required to polarize the electrons' spins fully in the ultra-high mobility 2D electron system in SiGe/Si/SiGe quantum wells of unprecedented quality [44] has been investigated. The product \(g_{\rm F}m\) that characterizes the whole 2D electron system can be determined in the clean limit from the equality of the Zeeman splitting and the Fermi energy of the spin-polarized electron system:  
\[g_{\rm F}\mu_{\rm B}B^{*}=\frac{2\pi\hbar^{2}n_{\rm s}}{mg_{\rm v}}, \tag{1}\]  
where \(\mu_{\rm B}\) is the Bohr magneton. Simultaneously, the authors studied Shubnikov-de Haas oscillations, yielding the product \(g_{\rm F}m_{\rm F}\) at the Fermi level. It turned out (see Figure 5) that with decreasing electron density (or increasing interaction strength), the product \(g_{\rm F}m_{\rm F}\) at the Fermi level monotonically increases in the entire range of electron densities, while the energy-averaged product \(g_{\rm F}m\) saturates at low densities. Since the exchange effects in the 2D electron system in silicon are negligible, this difference can only be attributed to the different behaviors of the two effective masses. Their qualitatively different behavior reveals a precursor to the interaction-induced single-particle spectrum flattening at the Fermi level in this electron system. It seems natural to interpret these experimental results within the concept of the fermion condensation [46; 47; 48]. The fermion condensation occurs at the Fermi level in a range of momenta, unlike the condensation of bosons. As the strength of electron-electron interactions increases, the single-particle spectrum flattens in a region \(\Delta p\) near the Fermi momentum \(p_{F}\) (top inset to Figure 5).