Ising models at complex temperature: partition function zeroes, helimagnetism, and complex thermodynamics

While the idea of studying the partition function of classical statistical mechanical models for complex temperature or magnetic field is several decades old, much less attention has been paid to the behaviour of the correlation functions in this case. In this talk, I shall derive the zero contour and correlation functions for the simplest case (the d=1 Ising chain) analytically, and then discuss the general features of complex-temperature correlation functions, which include an unusual type of helical magnetism. I will also argue that, contrary to the common interpretation, long-range order is possible only on the contour of partition function zeroes. This will lead to the conjecture that, for the d=2 square-lattice Ising model, there are extra partition function zeroes near the real-temperature line that are not captured in Fisher's original derivation.