R. Häggkvist, A. Rosengren, P. Lundow, K. Markström, Daniel Andrén, P. Kundrotas
{"title":"关于简单立方晶格的Ising模型","authors":"R. Häggkvist, A. Rosengren, P. Lundow, K. Markström, Daniel Andrén, P. Kundrotas","doi":"10.1080/00018730701577548","DOIUrl":null,"url":null,"abstract":"The Ising model was introduced in 1920 to describe a uniaxial system of magnetic moments, localized on a lattice, interacting via nearest-neighbour exchange interaction. It is the generic model for a continuous phase transition and arguably the most studied model in theoretical physics. Since it was solved for a two-dimensional lattice by Onsager in 1944, thereby representing one of the very few exactly solvable models in dimensions higher than one, it has served as a testing ground for new developments in analytic treatment and numerical algorithms. Only series expansions and numerical approaches, such as Monte Carlo simulations, are available in three dimensions. This review focuses on Monte Carlo simulation. We build upon a data set of unprecedented size. A great number of quantities of the model are estimated near the critical coupling. We present both a conventional analysis and an analysis in terms of a Puiseux series for the critical exponents. The former gives distinct values of the high- and low-temperature exponents; by means of the latter we can get these exponents to be equal at the cost of having true asymptotic behaviour being found only extremely close to the critical point. The consequences of this for simulations of lattice systems are discussed at length.","PeriodicalId":7373,"journal":{"name":"Advances in Physics","volume":null,"pages":null},"PeriodicalIF":35.0000,"publicationDate":"2007-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00018730701577548","citationCount":"30","resultStr":"{\"title\":\"On the Ising model for the simple cubic lattice\",\"authors\":\"R. Häggkvist, A. Rosengren, P. Lundow, K. Markström, Daniel Andrén, P. Kundrotas\",\"doi\":\"10.1080/00018730701577548\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The Ising model was introduced in 1920 to describe a uniaxial system of magnetic moments, localized on a lattice, interacting via nearest-neighbour exchange interaction. It is the generic model for a continuous phase transition and arguably the most studied model in theoretical physics. Since it was solved for a two-dimensional lattice by Onsager in 1944, thereby representing one of the very few exactly solvable models in dimensions higher than one, it has served as a testing ground for new developments in analytic treatment and numerical algorithms. Only series expansions and numerical approaches, such as Monte Carlo simulations, are available in three dimensions. This review focuses on Monte Carlo simulation. We build upon a data set of unprecedented size. A great number of quantities of the model are estimated near the critical coupling. We present both a conventional analysis and an analysis in terms of a Puiseux series for the critical exponents. The former gives distinct values of the high- and low-temperature exponents; by means of the latter we can get these exponents to be equal at the cost of having true asymptotic behaviour being found only extremely close to the critical point. The consequences of this for simulations of lattice systems are discussed at length.\",\"PeriodicalId\":7373,\"journal\":{\"name\":\"Advances in Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":35.0000,\"publicationDate\":\"2007-09-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1080/00018730701577548\",\"citationCount\":\"30\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1080/00018730701577548\",\"RegionNum\":1,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"PHYSICS, CONDENSED MATTER\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1080/00018730701577548","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, CONDENSED MATTER","Score":null,"Total":0}
The Ising model was introduced in 1920 to describe a uniaxial system of magnetic moments, localized on a lattice, interacting via nearest-neighbour exchange interaction. It is the generic model for a continuous phase transition and arguably the most studied model in theoretical physics. Since it was solved for a two-dimensional lattice by Onsager in 1944, thereby representing one of the very few exactly solvable models in dimensions higher than one, it has served as a testing ground for new developments in analytic treatment and numerical algorithms. Only series expansions and numerical approaches, such as Monte Carlo simulations, are available in three dimensions. This review focuses on Monte Carlo simulation. We build upon a data set of unprecedented size. A great number of quantities of the model are estimated near the critical coupling. We present both a conventional analysis and an analysis in terms of a Puiseux series for the critical exponents. The former gives distinct values of the high- and low-temperature exponents; by means of the latter we can get these exponents to be equal at the cost of having true asymptotic behaviour being found only extremely close to the critical point. The consequences of this for simulations of lattice systems are discussed at length.
期刊介绍:
Advances in Physics publishes authoritative critical reviews by experts on topics of interest and importance to condensed matter physicists. It is intended for motivated readers with a basic knowledge of the journal’s field and aims to draw out the salient points of a reviewed subject from the perspective of the author. The journal''s scope includes condensed matter physics and statistical mechanics: broadly defined to include the overlap with quantum information, cold atoms, soft matter physics and biophysics. Readership: Physicists, materials scientists and physical chemists in universities, industry and research institutes.