  We report on a new state of matter manifested by strongly correlated Fermi
systems including various heavy-fermion (HF) metals, two-dimensional quantum
liquids such as $\rm ^3He$ films, certain quasicrystals, and systems behaving
as quantum spin liquids. Generically, these systems can be viewed as HF systems
or HF compounds, in that they exhibit typical behavior of HF metals. At zero
temperature, such systems can experience a so-called fermion-condensation
quantum phase transition (FCQPT). Combining analytical considerations with
arguments based entirely on experimental grounds we argue and demonstrate that
the class of HF systems is characterized by universal scaling behavior of their
thermodynamic, transport, and relaxation properties. That is, the quantum
physics of different HF compounds is found to be universal, emerging
irrespective of the individual details of their symmetries, interactions, and
microscopic structure. This observed universal behavior reveals the existence
of a new state of matter manifest in HF compounds. We propose a simple,
realistic model to study the appearance of flat bands in two-dimensional
ensembles of ultracold fermionic atoms, interacting with coherent resonant
light. It is shown that signatures of these flat bands may be found in
peculiarities in their thermodynamic and spectroscopic properties. We also show
that the FCQPT, in generating flat bands and altering Fermi surface topology,
is an essential progenitor of the exotic behavior of the overdoped
high-temperature superconductors represented by $\rm La_{2-x}Sr_xCuO_4$, whose
superconductivity differs from that predicted by the classical
Bardeen-Cooper-Schrieffer theory. The theoretical results presented are in good
agreement with recent experimental observations, closing the colossal gap
between these empirical findings and Bardeen-Cooper-Schrieffer-like theories.

Strongly Correlated Electrons;Superconductivity