  As the temperature of a many-body system approaches absolute zero, thermal
fluctuations of observables cease and quantum fluctuations dominate.
Competition between different energies, such as kinetic energy, interactions or
thermodynamic potentials, can induce a quantum phase transition between
distinct ground states. Near a continuous quantum phase transition, the
many-body system is quantum critical, exhibiting scale invariant and universal
collective behavior \cite{Coleman05Nat, Sachdev99QPT}. Quantum criticality has
been actively pursued in the study of a broad range of novel materials
\cite{vdMarel03Nat, Lohneysen07rmp, G08NatPhys, Sachdev08NatPhys}, and can
invoke new insights beyond the Landau-Ginzburg-Wilson paradigm of critical
phenomena \cite{Senthil04prb}. It remains a challenging task, however, to
directly and quantitatively verify predictions of quantum criticality in a
clean and controlled system. Here we report the observation of quantum critical
behavior in a two-dimensional Bose gas in optical lattices near the
vacuum-to-superfluid quantum phase transition. Based on \textit{in situ}
density measurements, we observe universal scaling of the equation of state at
sufficiently low temperatures, locate the quantum critical point, and determine
the critical exponents. The universal scaling laws also allow determination of
thermodynamic observables. In particular, we observe a finite entropy per
particle in the critical regime, which only weakly depends on the atomic
interaction. Our experiment provides a prototypical method to study quantum
criticality with ultracold atoms, and prepares the essential tools for further
study on quantum critical dynamics.
