Uncomputability of phase diagrams

Johannes Bausch (), Toby S. Cubitt () and James D. Watson ()
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Johannes Bausch: University of Cambridge
Toby S. Cubitt: University College London
James D. Watson: University College London

Nature Communications, 2021, vol. 12, issue 1, 1-8

Abstract: Abstract The phase diagram of a material is of central importance in describing the properties and behaviour of a condensed matter system. In this work, we prove that the task of determining the phase diagram of a many-body Hamiltonian is in general uncomputable, by explicitly constructing a continuous one-parameter family of Hamiltonians H(φ), where $$\varphi \in {\mathbb{R}}$$ φ ∈ R , for which this is the case. The H(φ) are translationally-invariant, with nearest-neighbour couplings on a 2D spin lattice. As well as implying uncomputablity of phase diagrams, our result also proves that undecidability can hold for a set of positive measure of a Hamiltonian’s parameter space, whereas previous results only implied undecidability on a zero measure set. This brings the spectral gap undecidability results a step closer to standard condensed matter problems, where one typically studies phase diagrams of many-body models as a function of one or more continuously varying real parameters, such as magnetic field strength or pressure.

Date: 2021
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DOI: 10.1038/s41467-020-20504-6

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