“…In the case of a complex Hamiltonian, however, the associated eigenfunctions need not be analytic in the parameters of the Hamiltonian, and phase transitions can be seen in finite matrix Hamiltonians (see, e.g., [2]). This situation is reminiscent of the analysis proposed by Lee and Yang [3,4], where the breakdown of analyticity associated with the canonical density function in Equation (1) can be explained by extending the parameters into a complex domain (see, e.g., [5] for a heuristic but informative exposition of the Lee-Yang theory). In this case, the canonical density function can exhibit lack of analyticity even in a system with finitely many degrees of freedom, in a way that resembles the eigenstates of finite complex Hamiltonians (see also [6] for a related point of view on these issues).…”