Boundary Conditions and the Two-Point Function Plateau for the Hierarchical \(|\varphi |^4\) Model in Dimensions 4 and Higher

IF 1.4 3区物理与天体物理 Q2 PHYSICS, MATHEMATICAL

Annales Henri Poincaré Pub Date : 2025-04-15 DOI:10.1007/s00023-025-01566-y

Jiwoon Park, Gordon Slade

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Abstract

We obtain precise plateau estimates for the two-point function of the finite-volume weakly coupled hierarchical \(|\varphi |^4\) model in dimensions \(d \ge 4\), for both free and periodic boundary conditions and for any number \(n \ge 1\) of components of the field \(\varphi \). We prove that, within a critical window around their respective effective critical points, the two-point functions for both free and periodic boundary conditions have a plateau, in the sense that they decay as \(|x|^{-(d-2)}\) until reaching a constant plateau value of order \(V^{-1/2}\) (with a logarithmic correction for \(d=4\)), where V is the size of the finite volume. The two critical windows for free and periodic boundary conditions do not overlap. The dependence of the plateau height on the location within the critical window is governed by an explicit n-dependent universal profile which is independent of the dimension. The proof is based on a rigorous renormalisation group method and extends the method used by Michta et al. (arXiv:2306.00896) to study the finite-volume susceptibility and related quantities. Our results lead to precise conjectures concerning Euclidean (non-hierarchical) models of spin systems and self-avoiding walk in dimensions \(d \ge 4\).

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4维及以上层次\(|\varphi |^4\)模型的边界条件和两点函数平台

我们获得了一维\(d \ge 4\)的有限体积弱耦合分层\(|\varphi |^4\)模型的两点函数的精确平台估计，包括自由边界条件和周期边界条件，以及场\(\varphi \)的任意数量\(n \ge 1\)分量。我们证明，在其各自有效临界点周围的临界窗口内，自由和周期边界条件的两点函数都具有平台，在某种意义上，它们衰减为\(|x|^{-(d-2)}\)，直到达到恒定的平台值\(V^{-1/2}\)级（对\(d=4\)进行对数修正），其中V是有限体积的大小。自由边界条件和周期边界条件的两个临界窗口不重叠。高原高度对临界窗口内位置的依赖是由一个明确的n依赖的通用剖面控制的，该剖面与尺寸无关。该证明基于严格的重整化群方法，并扩展了Michta等人（arXiv:2306.00896）研究有限体积磁化率及相关量的方法。我们的结果导致了关于自旋系统的欧几里得（非分层）模型和自避免在维度上行走\(d \ge 4\)的精确猜想。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Annales Henri Poincaré 物理-物理：粒子与场物理

CiteScore

3.00

自引率

6.70%

发文量

108

审稿时长

6-12 weeks

期刊介绍： The two journals Annales de l''Institut Henri Poincaré, physique théorique and Helvetica Physical Acta merged into a single new journal under the name Annales Henri Poincaré - A Journal of Theoretical and Mathematical Physics edited jointly by the Institut Henri Poincaré and by the Swiss Physical Society. The goal of the journal is to serve the international scientific community in theoretical and mathematical physics by collecting and publishing original research papers meeting the highest professional standards in the field. The emphasis will be on analytical theoretical and mathematical physics in a broad sense.