  Determining the phase diagram of interacting quantum many-body systems is an
important task for a wide range of problems such as the understanding and
design of quantum materials. For classical equilibrium systems, the Lee-Yang
formalism provides a rigorous foundation of phase transitions, and these ideas
have also been extended to the quantum realm. Here, we develop a Lee-Yang
theory of quantum phase transitions that can include thermal fluctuations
caused by a finite temperature, and it thereby provides a link between the
classical Lee-Yang formalism and recent theories of phase transitions at zero
temperature. Our methodology exploits analytic properties of the moment
generating function of the order parameter in systems of finite size, and it
can be implemented in combination with tensor-network calculations.
Specifically, the onset of a symmetry-broken phase is signaled by the zeros of
the moment generating function approaching the origin in the complex plane of a
counting field that couples to the order parameter. Moreover, the zeros can be
obtained by measuring or calculating the high cumulants of the order parameter.
We determine the phase diagram of the two-dimensional quantum Ising model and
thereby demonstrate the potential of our method to predict the critical
behavior of two-dimensional quantum systems at finite temperatures.
