Uncomputability of phase diagrams
| Autor: | James D. Watson, Toby S. Cubitt, Johannes Bausch |
|---|---|
| Přispěvatelé: | Bausch, Johannes [0000-0003-3189-9162], Watson, James D. [0000-0002-6077-4898], Apollo - University of Cambridge Repository, Watson, James D [0000-0002-6077-4898] |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Work (thermodynamics)
639/705/1041 03D35 68Q17 81V70 82B26 Science math-ph FOS: Physical sciences General Physics and Astronomy 02 engineering and technology Parameter space 639/766/259 Computer Science::Digital Libraries 01 natural sciences Measure (mathematics) General Biochemistry Genetics and Molecular Biology Article symbols.namesake math.MP quant-ph 639/766/119/2795 0103 physical sciences Information theory and computation 129 010306 general physics Mathematical Physics Mathematical physics Phase diagram Physics Quantum Physics Multidisciplinary Zero (complex analysis) Mathematical Physics (math-ph) General Chemistry Function (mathematics) 021001 nanoscience & nanotechnology Applied mathematics Condensed Matter - Other Condensed Matter Phase transitions and critical phenomena cond-mat.other symbols Spectral gap Quantum Physics (quant-ph) 0210 nano-technology Hamiltonian (quantum mechanics) Other Condensed Matter (cond-mat.other) |
| Zdroj: | Nature Communications Nature Communications, Vol 12, Iss 1, Pp 1-8 (2021) |
| ISSN: | 2041-1723 |
| Popis: | 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in {\mathbb{R}}$$\end{document}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. Phase diagrams describe how a system changes phenomenologically as an external parameter, such as a magnetic field strength, is varied. Here, the authors prove that in general such a phase diagram is uncomputable, by explicitly constructing a one-parameter Hamiltonian for which this is the case. |
| Databáze: | OpenAIRE |
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