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<p>Rather than analyzing each individual spin interaction, mean-field theory (MFT) simplifies the intricate interactions found in a standard Ising model by substituting them with a single, effective magnetic field. We explore the critical transition temperature in the Ising Hamiltonian with four-spin interactions using MFT and reveal that the magnetization exhibits a discontinuous change, unlike the continuous phase transition seen in two-spin interactions. The interplay of both two-spin and four-spin interactions results in a tricritical phase transition, distinguished by varying critical exponent values derived from Landau’s theory of phase transitions. In this MFT framework, a fully-connected <i>p</i>-spin model demonstrates a first-order phase transition for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(p&gt;2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>&gt;</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, with zero-field susceptibility confirming Curie’s law rather than the Curie-Weiss law. We believe that this research will prove advantageous for both undergraduate and postgraduate students focusing on statistical mechanics.</p>

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Mean Field Analysis of Four Spin Interaction: A toy Model

  • 期刊论文

  • 发表日期：

    2026年4月22日

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  • Debnarayan Jana

摘要

Rather than analyzing each individual spin interaction, mean-field theory (MFT) simplifies the intricate interactions found in a standard Ising model by substituting them with a single, effective magnetic field. We explore the critical transition temperature in the Ising Hamiltonian with four-spin interactions using MFT and reveal that the magnetization exhibits a discontinuous change, unlike the continuous phase transition seen in two-spin interactions. The interplay of both two-spin and four-spin interactions results in a tricritical phase transition, distinguished by varying critical exponent values derived from Landau’s theory of phase transitions. In this MFT framework, a fully-connected p-spin model demonstrates a first-order phase transition for \(p>2\) p > 2 , with zero-field susceptibility confirming Curie’s law rather than the Curie-Weiss law. We believe that this research will prove advantageous for both undergraduate and postgraduate students focusing on statistical mechanics.