Phase transitions are usually defined through emergence of singularities of free

energy of the system with changing some external parameter like temperature.

Familiar examples include liquid-gas transitions, or transitions between

paramagnetic and ferromagnetic phases. Quantum phase transitions are very similar

except that they occur at zero temperature and are driven by quantum fluctuations.

In this talk I will introduce a concept of dynamical quantum phase transitions,

where the role of parameter driving the system between different phases is played by

the evolution time following some dynamical process like a quench. Mathematically

these transitions can be understood through singularities of the Loschmidt type

return amplitude, which serves as a natural real time generalization of the

partition function. Physically these transitions are related to the singularities in

the work distribution function. These transitions are naturally explained though the

language of zeros of the return amplitude in the complex time plane, which are

similar to Fisher zeros of the partition function in the complex temperature plane

in equilibrium systems.