论三维伊辛模型的不可控性

IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL
Wojciech Niedziółka, Jacek Wojtkiewicz
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引用次数: 0

摘要

众所周知,二维伊辛模型的分区函数可以用格拉斯曼积分来表示格拉斯曼变量中的双线性作用。证明这一等价性的关键在于证明格拉斯曼积分中出现的所有多边形都有固定的符号。对于三维模型,分割函数也可以用格拉斯曼积分来表示。然而,低温(L-T)展开产生的作用包含四次项,无法明确计算积分。我们想检查--显然还没有探索过--使用高温(H-T)展开会导致作用只包含双线性项的可能性(在二维中,L-T 和 H-T 展开是等价的,但在三维中,它们彼此不同)。然而,通过格拉斯曼积分得到的多边形对于格拉斯曼变量在位点上的任何排序都没有固定的符号。这样,三维伊辛模型的分割函数就无法用双线性作用的格拉斯曼积分来表示。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On nonintegrability of three-dimensional Ising model

It is well known that the partition function of two-dimensional Ising model can be expressed as a Grassmann integral over the action bilinear in Grassmann variables. The key aspect of the proof of this equivalence is to show that all polygons, appearing in Grassmann integration, enter with fixed sign. For three-dimensional model, the partition function can also be expressed by Grassmann integral. However, the action resulting from low-temperature (L-T) expansion contains quartic terms, which do not allow explicit computation of the integral. We wanted to check — apparently not explored — the possibility that using the high-temperature (H-T) expansion would result in action with only bilinear terms (in two dimensions, L-T and H-T expansions are equivalent, but in three dimensions, they differ from each other). It turned out, however, that polygons obtained by Grassmann integration are not of fixed sign for any ordering of Grassmann variables on sites. This way, it is not possible to express the partition function of three-dimensional Ising model as a Grassmann integral over bilinear action.

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来源期刊
Reports on Mathematical Physics
Reports on Mathematical Physics 物理-物理:数学物理
CiteScore
1.80
自引率
0.00%
发文量
40
审稿时长
6 months
期刊介绍: Reports on Mathematical Physics publish papers in theoretical physics which present a rigorous mathematical approach to problems of quantum and classical mechanics and field theories, relativity and gravitation, statistical physics, thermodynamics, mathematical foundations of physical theories, etc. Preferred are papers using modern methods of functional analysis, probability theory, differential geometry, algebra and mathematical logic. Papers without direct connection with physics will not be accepted. Manuscripts should be concise, but possibly complete in presentation and discussion, to be comprehensible not only for mathematicians, but also for mathematically oriented theoretical physicists. All papers should describe original work and be written in English.
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