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Kosterlitz–Thouless Physics: a Review of Key Issues Accepted for Publication 7 October 2015 Published 28 January 2016 Reports on Progress in Physics REVIEW Related content - Berezinskii–Kosterlitz–Thouless transition Kosterlitz–Thouless physics: a review of key and two-dimensional melting V N Ryzhov, E E Tareyeva, Yu D Fomin et issues al. - Depinning and nonequilibrium dynamic phases of particle assemblies driven over To cite this article: J Michael Kosterlitz 2016 Rep. Prog. Phys. 79 026001 random and ordered substrates: a review C Reichhardt and C J Olson Reichhardt - Quantum phase transitions Matthias Vojta View the article online for updates and enhancements. Recent citations - Isolating long-wavelength fluctuation from structural relaxation in two-dimensional glass: cage-relative displacement Hayato Shiba et al - Disappearance of the Hexatic Phase in a Binary Mixture of Hard Disks John Russo and Nigel B. Wilding - Berezinskii—Kosterlitz—Thouless transition and two-dimensional melting Valentin N. Ryzhov et al This content was downloaded from IP address 128.227.24.141 on 25/02/2018 at 11:47 IOP Reports on Progress in Physics Reports on Progress in Physics Rep. Prog. Phys. Rep. Prog. Phys. 79 (2016) 026001 (59pp) doi:10.1088/0034-4885/79/2/026001 79 Review 2016 Kosterlitz–Thouless physics: a review of key © 2016 IOP Publishing Ltd issues RPPHAG J Michael Kosterlitz 026001 Department of Physics, Brown University, Providence, RI 02912, USA J M Kosterlitz E-mail: [email protected] Received 1 June 2015, revised 5 October 2015 Kosterlitz–Thouless physics: a review of key issues Accepted for publication 7 October 2015 Published 28 January 2016 Printed in the UK Abstract This article reviews, from a very personal point of view, the origins and the early work on ROP transitions driven by topological defects such as vortices in the two dimensional planar rotor model and in 4Helium films and dislocations and disclinations in 2D crystals. I cover the early papers with David Thouless and describe the important insights but also 10.1088/0034-4885/79/2/026001 the errors and oversights since corrected by other workers. I then describe some of the experimental verifications of the theory and some numerical simulations. Finally applications to superconducting arrays of Josephson junctions and to recent cold atom experiments are 0034-4885 described. 2 Keywords: two dimensions, topological defects, renormalization group, superfluid Helium, superconducting films, two dimensional crystals, cold atom arrays (Some figures may appear in colour only in the online journal) 1. Introduction films and domain walls in the Ising model of magnetism. This opened my mind to the world of condensed matter physics From my point of view, the discovery of defect mediated where, to my uneducated mind, there seemed to be a plethora phase transitions started in 1970 when I was a postdoc in of unsolved physical problems. This was a big contrast to high high energy physics at the Istituto di Fisica Teorica of Torino energy physics where there seemed to be very few interesting University in Italy. I was, and still am, a disorganized person problems with several very smart people working on every with a tendency to wait until the last possible moment or even problem. However, there was another difficulty which was later to do something important like submitting a job applica- my almost total ignorance of statistical mechanics which I tion. Consequently, in September 1970, I found myself in the had more or less ignored as a graduate student in high energy Department of Mathematical Physics, Birmingham University, physics. England instead of at CERN in Geneva, Switzerland where I had been warned by my colleagues that David could be dif- I had intended to go. During my first year at Birmingham, ficult because he was reputed to not suffer fools gladly. There I did a set of complicated calculations on a model which was a was little question that I was an ignorant fool as far as phase precursor to string theory, but I was always beaten by a group transitions were concerned so I was quite nervous listening in USA. At least twice, I was in the process of writing up my to David’s thoughts about some low dimensional problems in calculations when a preprint by my competitors appeared in phase transitions. He explained that he had been discussing the department. with Phil Anderson the problem of a phase transitions in the Somewhat discouraged, I began to look at other branches 1D Ising chain with long range interactions between spins fall- of physics in the hope of finding an interesting but tractable ing off as 1/r2. He had recently shown [1] that, in this model, problem and found myself talking with David Thouless about there was a phase transition with some peculiar properties. strange excitations such as vortices in superfluid 4Helium It was known that an Ising chain with shorter ranged interactions 0034-4885/16/026001+59$33.00 1 © 2016 IOP Publishing Ltd Printed in the UK Rep. Prog. Phys. 79 (2016) 026001 Review had no transition to an ordered state at any finite temperature for the domain wall at 0 xL well away from the ends [2] while the system with interactions decaying more slowly of the chain. Thus, the free energy ∆F of an isolated domain than 1/r2 had an ordered state at sufficiently low temperatures wall is [3] and, thus, a transition at a finite T > 0. Our early discussions of this and related problems intro- ∆=FT(),lLJ()−+kTB n/()La O()1. (3) duced me to more new concepts like topological defects The probability of a single domain wall in a system of length such as vortices in superfluid 4He and their role in disorder- La is ing a system. As I was operating from a position of extreme ignorance, these did not seem any stranger than other ideas e−∆FL(),/Tk()BT Pt(), L ∼ about phase transitions. In fact, they seemed more hopeful 1e+ −∆FL(),/Tk()BT than many others all of which had failed to make much of an 1 ⎧0, kTB < J, impression on this class of intractable problems. In an Ising ∼ Jk/1T − → ⎨ (4) 1/+ La ()B ⎩ 1, kTB > J. model, a topological defect is an easily visualized excitation () as it is just a domain wall between regions of opposite spin This implies that one can identify the critical temperature as orientation. It is not difficult to rewrite the Ising spin Hamilton kBcTJ= since there will be no isolated domain walls in the in terms of interacting domain walls which live on the lat- equilibrium system when TT< c so that the system will be tice dual to the original spin lattice. During these discussions, ordered. There will be finite domains of reversed spins below David gave me a set of papers by Anderson and coworkers Tc but still a finite magnetization. On the other hand, forTT > c, [4–8] and suggested I look at them as they may be useful. there is a finite probability that there will be some isolated free These were a major surprise because, as far as I was concerned, domain walls implying disorder and vanishing magnetization. their importance was that they solved the 1D Ising model with The original argument [1] went further to show that the mag- 1/r2 interactions and demonstrated that it is ordered at low netization has a discontinuity at Tc although the transition is temper atures and has a phase transition to a disordered state at continuous. This was later confirmed by more rigorous work a finiteT . I spent most of six months doing nothing but read- c [15–17]. ing and rereading these papers until I understood what they The free energy ∆F of equation (3) for an isolated domain were doing, especially the last in the series [6] which solved wall in the 1/r2 ferromagnetic Ising chain has important impli- the problem by a very strange method. I later realized that this cations which were to become relevant to later developments paper contained a very early version of the renormalization in the defect theory of phase transitions. Domain walls can group popularized by Wilson [9, 10] to formalize the scaling be regarded as topological defects in an Ising ferromagnet theory of Kadanoff [11] but, at the time, it just appeared to be and it is obvious that a finite concentration of these in thermal a very clever and unusual method of solving one particular equilibrium tend to disorder the system. Equation (4) shows example of a phase transition. that there will be no isolated domain walls in the system for kBTJ< and there will be long range ferromagnetic order. On 1.1. One dimensional Ising model the other hand, when kBTJ> , a finite density of these topolog- ical defects is present in thermal equilibrium thus disordering In 1969, Thouless was first exposed to the essential ideas the system. This argument is made more exact by Anderson, vital to theories of transitions driven by topological defects Yuval and Hamann [6, 8] who treated the 1/r2 Ising chain by during a visit to Bell Labs, one of the most important cen- a renormalization group method. Because this very early ver- ters of theoretical physics in the twentieth century. There he sion of a renormalization group treatment of a defect driven learned of the work by Anderson and colleagues on the trans- phase transition was so important to later developments by formation of the Kondo problem into a 1D Ising model with Thouless and myself [18 20], I review it here. 2 – 1/r interactions [4–8] and was asked if he knew anything The first step is to rewrite the partition function Z(K, y, h) about this intermediate case. It was known that the model of the 1D Ising model with Hamiltonian [6, 8], assuming peri- with interactions falling off more slowly than this have long odic boundary conditions range order [3], or finite magnetization, at low temperatures 2 and that systems with shorter ranged interactions are dis- H 1 ssij− 1 KT () Assh2 s ordered [2] at all T > 0.
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