Undecidability in quantum thermalization

The investigation of thermalization in isolated quantum many-body systems has a long history, dating back to the time of developing statistical mechanics. Most quantum many-body systems in nature are considered to thermalize, while some never achieve thermal equilibrium. The central problem is to clarify whether a given system thermalizes, which has been addressed previously, but not resolved. Here, we show that this problem is undecidable. The resulting undecidability even applies when the system is restricted to one-dimensional shift-invariant systems with nearest-neighbour interaction, and the initial state is a fixed product state. We construct a family of Hamiltonians encoding dynamics of a reversible universal Turing machine, where the fate of a relaxation process changes considerably depending on whether the Turing machine halts. Our result indicates that there is no general theorem, algorithm, or systematic procedure determining the presence or absence of thermalization in any given Hamiltonian. The question whether a given isolated quantum many-body system would thermalize has currently no general answer. Here, Shiraishi and Matsumoto demonstrate the computational universality of thermalization phenomena already for simplified 1D systems, thus proving that the thermalization problem is undecidable.

Automated summary

The study investigates the undecidability of thermalization phenomena in isolated quantum many-body systems, even in one-dimensional systems. By constructing a family of Hamiltonians encoding dynamics of a reversible universal Turing machine, it is shown that the fate of a relaxation process can dramatically change depending on whether the Turing machine halts. The result indicates the lack of a general theorem, algorithm, or systematic procedure for determining the presence of thermalization in any given Hamiltonian.