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<p>We discuss the relaxation time (inverse spectral gap) of the one dimensional <i>O</i>(<i>N</i>) model, for all <i>N</i> and with two types of boundary conditions. We see how its low temperature asymptotic behavior is affected by the topology. The combination of the space dimension, which here is always 1, the boundary condition (free or periodic), and the spin state <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3475_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {S}}^{N-1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>, determines the existence or absence of non-trivial homotopy classes in some discrete version. Such non-trivial topology reflects in bottlenecks of the dynamics, creating metastable states that the system exits at exponential times; while when only one homotopy class exists the relaxation time depends polynomially on the temperature. We prove in the one dimensional case that, indeed, the relaxation time is a proxy to the model’s topological properties via the exponential/polynomial dependence on the temperature.</p>

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Relaxation Time and Topology in 1D O(N) Models

  • 期刊论文

  • 发表日期：

    2025年7月11日

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  • Pietro Caputo,
  • Sébastien Ott,
  • Assaf Shapira

摘要

We discuss the relaxation time (inverse spectral gap) of the one dimensional O(N) model, for all N and with two types of boundary conditions. We see how its low temperature asymptotic behavior is affected by the topology. The combination of the space dimension, which here is always 1, the boundary condition (free or periodic), and the spin state \({\mathbb {S}}^{N-1}\) S N - 1 , determines the existence or absence of non-trivial homotopy classes in some discrete version. Such non-trivial topology reflects in bottlenecks of the dynamics, creating metastable states that the system exits at exponential times; while when only one homotopy class exists the relaxation time depends polynomially on the temperature. We prove in the one dimensional case that, indeed, the relaxation time is a proxy to the model’s topological properties via the exponential/polynomial dependence on the temperature.