Phenomena, Models and Understanding The Lenz-Ising Model and Critical Phenomena 1920–1971 Martin Niss 2005 Preface This dissertation has been prepared as the main component in the fulfilment of the re- quirements for the Ph.D. degree at the Department of Mathematics and Physics (IMFUFA) at the University of Roskilde, Denmark. The work has been carried out in the period from February 2001 to March 2005, interrupted by a leave of absence in the academic year 2002-2003 due to my reception of a scholarship to study mathematics abroad. Professor Jeppe C. Dyre was my supervisor. Acknowledgements I would very much like to thank my supervisor Jeppe. From the very beginning of my dissertation work he had enough confidence in me to allow me to go my own ways. Even though he is a professor of physics proper and not a professional historian of physics, he has been keenly interested in my project throughout its duration and his door has always been open when I needed any sort of help. I would also like to thank Dr. Tinne Hoff Kjeldsen who in her capacity as a historian of mathematics has helped me greatly on numerous scholarly issues. I have also benefited very much from her help and advice on non-scholarly problems which unavoidably appear during a Ph.D. study and need to be tackled. My father Mogens Niss has provided numerous supportive comments on my project on all scales of magnitude. The linguistic appearance of the dissertation has greatly benefited from his detailed comments. Jeppe, Tinne and Mogens have all discussed the project with me during the whole period on numerous occasions, and they have read and commented on a draft of the entire dissertation. I am very grateful for their advice and commentary, but needless to say all responsibility for this dissertation solely lies with me. At early stages of my project I discussed it with Evelyn Fox Keller, Helge Kragh, Andrea Loettgers and the Physics of Scales group at Dibner Institute comprising Babak Ashrafi, Karl Hall and Sam S. Schweber. I am grateful for all their advice. I would also like to thank Helge and Andrea as well as Anja Skaar Jacobsen and Stine Timmermann for their reading of parts of the dissertation. I am thankful for their commentary. I also wish to extend my thanks to Per Christian Hemmer and Cyril Domb, who most kindly answered some of my questions by mail at a time when the time was short and to Leo P. Kadanoff who most kindly sent me two of his books. I also want to thank Albena Nielsen for translating Frenkel (1946) from Russian into Danish. Furthermore, I would like to thank the people of IMFUFA where my studies were undertaken. They have provided a stimulating environment and I have benefited immensely from the interdisciplinary spirit and interest in mathematical modelling which are marked features of the department. Roskilde, March 2005 Martin Niss iv Preface to IMFUFA Text This IMFUFA text is identical to the dissertation I submitted for the fulfilment of the re- quirements for the Ph.D. degree, except that a few minor clarifications have been made and a number of typos have been corrected. Roskilde, August 2005 Martin Niss Summary Modelling is an essential part of the practice of modern physicists, but this activity has not received much attention from historians of physics. The present dissertation is a step towards remedying this situation by examining modelling in one area of physics in a histor- ical perspective, namely microscopic modelling of macroscopic phenomena in condensed matter physics from the 1920s onwards. More precisely, I focus on one single model, the Lenz-Ising model, an important model in modern condensed matter physics. However, the model is studied here because physicists’ perception of the model and the role it can or should play in the understanding of macroscopic models witnessed great changes. The dissertation traces these changes in the Lenz-Ising model in its capacity as a model, i.e., its ability to represent physical phenomena, since its proposal in the early 1920s until the early 1970s. The dissertation focuses on this aspect of the model’s history in contrast to the more technical-oriented accounts given in Brush (1967) and Hoddeson, Shubert, Heims and Baym (1992) and thus presents a new side and view on the model’s history. The dissertation falls in three parts. The first part examines the roots of the model in terms of three phases with respect to the physicists’ perception of the model and of the role it can play, in the epoch from 1920 to 1950. In the first phase the model was proposed by the German physicists Lenz and Ising in the early 1920s as a realistic representation of a ferromagnet. With the advent of quantum mechanics and the ensuring replacement of the foundation theory of the model, the view of the physical realism of the model changed. This in turn rendered the model irrelevant to physicists. In the third phase, which took place from the late 1930s to the 1950s, the model gained renewed interest, but now as a model of transition points in general rather than of ferromagnetism alone. The model was found to be relevant to the study of these phenomena because it could function as a proof by example that statistical mechanics can accommodate phase transitions. However, it was seen as irrelevant to a more detailed study of phase transitions because of its lack of physical realism. The driving forces behind these changes in perception of the model are analysed. The second part of the dissertation studies how the Lenz-Ising model came to play an important role in the theory of critical phenomena of the 1960s. This was not at all in the cards in the 1950s and it is shown in the dissertation that the model was found to be physically irrelevant within three of the four areas which became the central critical phenomena in the following decade. Moreover, in these three areas more realistic models were dominant; in the field of ferromagnetism and antiferromagnetism the Heisenberg model was the preferred model, while scientists interested in the condensation of gases were preoccupied with the Mayer model. I argue that the Lenz-Ising model eventually became seen as physically relevant in the 1960s, mainly because a surprising agreement was found between the model as a representative of gases and new experimental results on argon and oxygen by Voronel’ and his Russian group. This had a rub-off effect on the model as portraying magnetism. vi Next, it is documented that the Lenz-Ising model toppled the Mayer model in the area of gas condensation (but not the Heisenberg model within magnetism) in the 1960s. It is argued that this was mainly due to the realisation of the mathematical difficulties involved in handling the Mayer model rather than its deficiencies as a model. Having established that the Lenz-Ising model acquired a prominent position within the field of critical phenomena, the ways in which the model was used here are thoroughly examined. It is argued that the model was put to two different uses in the two halves of the 1960s. In the first half, the model was mainly employed to solve specific small-scale tasks, and it functioned to a large extent as a substitute for precise experimental results. It was first in the latter half of the decade that the model was put to a systematic use, namely in the fulfilment of what I term Fisher’s programme. This widespread programme attempted to elucidate the correspondence between features of model Hamiltonians and critical behaviour. For this purpose, simple and tractable models were used extensively, especially the Lenz-Ising model. The latter half of the 1960s also marked the emergence of the idea that a ‘universe’ constituted by the Lenz-Ising model could actually tell something about real systems; an effect found in this universe, it was assumed, reveals something about the real system portrayed by the model. This use of the Lenz-Ising model in the last half of the 1960s differs much from the previous use of models within the core areas of critical phenomena. I introduce the notion of modelling approach in order to describe these different modelling strategies. The notion of modelling approach is supposed to characterise different ways of using models to build phenomenological theories. The approach to modelling inherent in Fisher’s programme is contrasted with two previous approaches to the ‘core areas’ of critical phenomena – one advanced by J. E. Mayer and one by H. N. V.Temperley. In the final part of the dissertation, a hypothesis which I derive from a statement by Michael E. Fisher is examined, namely that the time around the late 1950s and the 1960s marks a watershed within condensed matter physics, with different approaches to modelling placed on each side of it. It is argued that this hypothesis is supported by the findings in the previous parts of the dissertation, which does indeed show such a change in the approach to critical phenomena. However, since the hypothesis concerns condensed matter physics more generally, I lay forth evidence from recollections both by protagonists of critical phenomena and by prominent solid state physicists such as P. W. Anderson, which further supports the hypothesis. However, it is beyond the scope of the dissertation to discuss the validity of the hypothesis in greater depth, so the hypothesis is laid out as a point of examination in a future research programme. On a more general note, the dissertation argues that historical studies of the perception of models and their roles are not only worthwhile but crucial if we are to understand large parts of physics, including solid state physics or its modern counterpart, condensed matter physics.