A Method for Investigating Thermodynamics of Inaccessible Objects by Determining the Co-Amoeba of their Partition Functions

Given an Ising-type system, the zeros of its complexified partition function, known as Lee--Yang zeros, form an algebraic variety that sheds light on the thermodynamics of that system. But in practice, we often confront with the inaccessibility of a system for direct experimentation and then a strategic approach is taken to investigate a probe that is interacting with the target system. Considering such an inaccessible system, reading the probe alone, we present a method to determine the co-amoeba, the images of the argument map of an algebraic variety corresponding to the Lee--Yang zeros of its partition function. In our method, starting from thermal equilibrium of target and probe, the target can be effectively initiated at any chosen value of its internal parameters by doing operations only on the probe. Thus, sampling the co-amoeba corresponding to a wide of range of physical parameters is possible which allows one to perceive the full algebraic variety and thereby determining the thermodynamics of the system. No prior knowledge about the internal parameters, neither any control over the target system under investigation is required for this protocol. Finally, we experimentally demonstrate this protocol via the nuclear magnetic resonance architecture using a system of three mutually interacting spin-1/2 nuclei.