Extraordinary new materials named quasicrystals and characterized by noncrystallographic rotational symmetry and quasiperiodic translational properties have attracted scrutiny. Study of quasicrystals may shed light on the most basic notions related to the quantum critical state observed in heavy-fermion metals. We show that the electronic system of some quasicrystals is located at the fermion condensation quantum phase transition without tuning. In that case the quasicrystals possess the quantum critical state with the non-Fermi-liquid behavior which in magnetic fields transforms into the Landau Fermi-liquid one. Remarkably, the quantum critical state is robust despite the strong disorder experienced by the electrons. We also demonstrate for the first time that quasicrystals exhibit the typical scaling behavior of their thermodynamic properties such as the magnetic susceptibility, and belong to the famous family of heavy-fermion metals. Our calculated thermodynamic properties are in good agreement with recent experimental observations.

Strongly correlated Fermi systems are among the most intriguing and fundamental systems in physics, whose realization in some compounds is still to be discovered. We show that ${\text{ZnCu}}_3{\left(\text{OH}\right)}_6{\text{Cl}}_2$ can be viewed as a strongly correlated Fermi system whose low-temperature thermodynamics in magnetic fields is defined by a Fermi quantum spin liquid. Our calculations of its thermodynamic properties are in good agreement with recent experimental facts and allow us to reveal their scaling behavior which strongly resembles that observed in heavy-fermion metals and two-dimensional ${}^3$He.