In 2015, it was shown that the problem of determining the existence of a spectral gap is undecidable in two or more dimensions. The authors used an aperiodic tiling of quantum Turing machines and showed that this hypothetical material becomes gapped if and only if the machine halts. The one-dimensional case was also proven undecidable in 2020 by constructing a chain of interacting qudits divided into blocks that gain energy if and only if they represent a full computation by a Turing machine, and showing that this system becomes gapped if and only if the machine does not halt.