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Uncomputability of phase diagrams.

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(?), where [Formula: see text], for which this is the case. The H(?) 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.

SUBMITTER: Bausch J

PROVIDER: S-EPMC7815885 | biostudies-literature | 2021 Jan

REPOSITORIES: biostudies-literature

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Uncomputability of phase diagrams.

Bausch Johannes J Cubitt Toby S TS Watson James D JD

Nature communications 20210119 1

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 [Formula: see text], for which this is the case. The H(φ) are translationally-invariant, with nearest-neighbour couplings on a 2D spin lattice. As well as i ...[more]

PMID: 33469011

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