  We analyze measurements of the magnetization, differential susceptibility and
specific heat of quasi-one dimensional insulator
Cu(C$_4$H$_4$N$_2$)(NO$_3$)$_2$ (CuPzN) subjected to magnetic fields. We show
that the thermodynamic properties are defined by quantum spin liquid formed
with spinons, with the magnetic field tuning the insulator CuPzN towards
quantum critical point related to fermion condensation quantum phase transition
(FCQPT) at which the spinon effective mass diverges kinematically. We show that
the FCQPT concept permits to reveal and explain the scaling behavior of
thermodynamic characteristics. For the first time, we construct the schematic
$T-H$ (temperature---magnetic field) phase diagram of CuPzN, that contains
Landau-Fermi-liquid, crossover and non-Fermi liquid parts, thus resembling that
of heavy-fermion compounds.
