易辛模型與微擾易辛模型之探討

Introduction to the Ising Model and the Ising Model with Disorders

馬宗儀(Tsung-Yi Ma)

指導教授 ： 李志煌

國立臺灣大學/理學院/數學系/碩士(2023年)

https://doi.org/10.6342/NTU202301483

摘要

易辛模型是一種在統計物理學中用於研究磁鐵材料系統的數學模型。該模型由稱為”自旋”的離散元素組成，可以取兩個不同的值，通常表示為+1和-1。模型考慮了自旋之間的相互作用並考慮了溫度的影響，產生了各種豐富多樣的現象，如相變和臨界現象。
在現實世界的情況下，可能存在外部磁場和磁極之間的干擾。因此，數學家引入了隨機場易辛模型和隨機鍵易辛模型。這些模型分別在與外部磁場和磁極之間相關的項中引入了隨機係數。

關鍵字

易辛模型 ； 隨機場易辛模型 ； 隨機鍵易辛模型 ； 混沌多項式展開 ； 高溫展開

並列摘要

The Ising model is a mathematical model employed in statistical physics to investigate systems of ferromagnetic materials. The model consists of discrete elements known as ”spins,” which can assume two distinct values, often represented as +1 and -1.
The model incorporates the interaction between spins and considers the influence of temperature, resulting in a rich variety of phenomena such as phase transitions and critical phenomena.
In real-world scenarios, external magnetic fields and disturbances between magnetic poles may exist. Therefore, mathematicians have introduced the Random Field Ising Model (RFIM) and the Random Bond Ising Model (RBIM). These models incorporate random coefficients in the terms related to external magnetic fields and interactions between magnetic poles, respectively.

並列關鍵字

Ising model ； random field Ising model ； random bond Ising model ； Wiener chaos expansion ； high-temperature expansion

參考文獻

Michael Aizenman, Hugo Duminil-Copin, and Vladas Sidoravicius. Random currents and continuity of ising model’s spontaneous magnetization. Communications in Mathematical Physics, 334:719–742, 2015.

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Dmitry Chelkak, Cl ́ement Hongler, and Konstantin Izyurov. Conformal invariance of spin correlations in the planar ising model. Annals of mathematics, pages 1087 - 1138, 2015.

Hugo Duminil-Copin. Parafermionic observables and their applications to planar statistical physics models. Ensaios Matematicos, 25:1–371, 2013.