Abstract
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.
Highlights
The investigation of thermalization in isolated quantum many-body systems has a long history, dating back to the time of developing statistical mechanics
The problem originated in the field of nonequilibrium statistical mechanics
A central problem in this field is whether a given system thermalizes[3,10]
Summary
The investigation of thermalization in isolated quantum many-body systems has a long history, dating back to the time of developing statistical mechanics. Almost all-natural quantum many-body systems are expected to thermalize, some systems, including integrable and localized systems, are known to never achieve thermalization[11,12,13,14,15] To resolve this problem, the eigenstate thermalization hypothesis (ETH) has been raised as a clue to understanding thermalization phenomena. We examine the difficulty of the problem from the viewpoint of theoretical computer science This type of approach is employed in some problems in physics, including prediction of dynamical systems[38], repeated quantum measurements[39], and the spectral gap problem[40]. The fate of thermalization in a general setup is independent of the basic axioms of mathematics, as implied in the Gödel’s incompleteness theorem[41] We prove this by demonstrating that the relaxation and thermalization phenomena in one-dimensional systems have the power of universal computation. Our result sets a limit on what we can know about quantum thermalization, and elucidates a rich variety of thermalization phenomena, which can implement any computational task
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