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A complexity classification of spin systems with an external field.

Leslie Ann Goldberg1, Mark Jerrum2

  • 1Department of Computer Science, University of Oxford, Oxford OX1 3QD, United Kingdom;

Proceedings of the National Academy of Sciences of the United States of America
|October 8, 2015
PubMed
Summary

This study reveals three computational complexity levels for approximating partition functions in q-state spin systems. All nontrivial systems are computationally equivalent to fundamental two-state spin models.

Keywords:
computational complexitypartition functionspin system

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Area of Science:

  • Computational Complexity Theory
  • Statistical Mechanics
  • Quantum Information Science

Background:

  • Approximating partition functions is crucial for understanding complex systems.
  • Spin systems with external fields present significant computational challenges.

Purpose of the Study:

  • To determine the computational complexity of approximating the partition function for q-state spin systems with an external field.
  • To classify these complexities based on interaction strengths.

Main Methods:

  • Analysis of computational complexity.
  • Mapping spin systems to Ising models.
  • Investigating different interaction strengths (ferromagnetic, antiferromagnetic).

Main Results:

  • Identified three distinct levels of computational difficulty.
  • Demonstrated that all nontrivial q-state spin systems are computationally equivalent to specific two-state spin systems.
  • Classified systems as efficiently computable, or equivalent to ferromagnetic or antiferromagnetic Ising models.

Conclusions:

  • The computational landscape of q-state spin systems is reducible to two fundamental two-state models.
  • Understanding these equivalences simplifies the analysis of complex spin systems.
  • Provides a framework for classifying the computational hardness of spin system models.