  At low temperature a thermodynamic system undergoes a phase transition when a
physical parameter passes through a singularity point of the free energy,
corresponding to formation of a new order. At high temperature the thermal
fluctuations destroy the orders; correspondingly the free energy is a smooth
function of the physical parameter and the singularity would occur only at
complex values of the parameter. Since a complex parameter is unphysical, at
high temperature no phase transition is expected with varying the physical
parameter. Here we show that the quantum evolution, driven by a designed
interaction, of a system initialized in thermal equilibrium is equivalent to
the partition function of a complex parameter. Therefore we can access the
complex singularity points of thermodynamic functions and observe phase
transitions even at high temperatures. This discovery makes it possible to
study thermodynamics in the complex plane of physical parameters.
