# Scaling And Renormalization In Statistical Physics Cardy Pdf

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- Conformal Field Theory and Statistical Mechanics
- Scaling and renormalization in statistical physics
- Lecture Tuesday, Thursday 12:30 - 1:45 PM, Phelps 3523

*Quantum Field Theory is the tool as well as the language that has been developed to describe the physics of problems in such apparently dissimilar fields. Physics is the second half of a two-semester sequence of courses in Quantum Field Theory.*

He is best known for his work in theoretical condensed matter physics and statistical mechanics , and in particular for research on critical phenomena and two-dimensional conformal field theory. He was an undergraduate and postgraduate student at Downing College , University of Cambridge , before moving to the University of California, Santa Barbara , where he joined the faculty in He is most known for his contributions to conformal field theory. The Cardy formula for black hole entropy, the Cardy formula in percolation theory , [7] and the Cardy conditions in boundary conformal field theory are named after him. From Wikipedia, the free encyclopedia.

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Report this Document. Flag for inappropriate content. Download now. For Later. Related titles. Carousel Previous Carousel Next. Townsend: Solutions to selected problems. Problemsandsolutionsonelectromagnetism Conversion Gate Jump to Page. Search inside document. Whether these problems relate to the very large scale structure of the universe, to the complicated forms of everyday macroscopic objects, or to the behaviour of the interactions between the fundamental constituents of matter at very short distances, they all have the common feature of pos- sessing large numbers of degrees of freedom which interact with each other in a complicated and highly non-linear fashion, often according to laws which are only poorly understood.

Yet it is often possible to make progress in understanding such problems by iso- lating a few relevant variables which characterise the behaviour of these systems on a particular length or time scale, and postulating simple scaling relations between them. These may serve to unify sets of experimental and numerical data taken under widely dif- fering conditions, a phenomenon called universality. When there is only a single independent variable, these relations often take the form of power laws, with exponents which do not appear to be simple rational numbers yet are, once again, universal.

The existence of such scaling behaviour may often be explained through a framework of theoretical ideas loosely grouped under the term renormalization. Roughly speaking, this describes how the parameters specifying the system must be adjusted, under pu- tative changes of the underlying dynamics, in such a way as not to modify the measurable properties on the length or time scales of interest.

The simple postulate of the existence of a fixed point of these renormalization flows is then sufficient to explain quali- tatively the appearance of universal scaling laws. In trying to under- stand scaling arguments applied to such problems it is often diffi- cult, especially for newcomers, to understand why certain variables should be neglected while others are retained in such scaling de- scriptions, and why in some cases power law relations should hold while they fail in others.

Fortunately, there is a class of physical problems within which the concepts of scaling and renormalization may be derived sys- tematically, and which therefore have become a paradigm for the whole approach. These concern equilibrium critical behaviour. The systems which exhibit such behaviour are governed by the simple and well understood laws of statistical mechanics. Indeed, along with the high energy behaviour of quantum field theories, this was the area of physics in which the concepts of renormalization were first formulated.

Although the subject of equilibrium critical be- haviour is, apart from a few unsolved problems, no longer of the greatest topical theoretical or experimental interest, its study is nonetheless important in providing a solid grounding to anyone who wishes to go on to attempt to understand scaling and renor- malization in more esoteric systems.

Yet the typical student in condensed matter theory faces a problem in trying to accomplish this. Historically, the subjects of renormalization in quantum field theory as applied to particle physics and in equilibrium critical behaviour have developed in parallel. This is no coincidence — the two sets of problems have, mathematically, a great deal in com- mon, and, indeed, the most systematic formulation of the subject relies heavily on the property of renormalizability in quantum field theory.

However, much of the qualitative structure of renormaliza- tion may be introduced through the alternative real space meth- ods which are both simple and appealing.

But students who learn this approach, and then wish to go further in existing accounts of the subject, must make a complete change of gears to momentum space methods which require a great deal of investment of time and effort in digesting the whole formalism of Feynman diagrams and renormalization theory.

In my opinion, these field theoretic details are appropriate only for the relatively small number of students who wish to go on and apply these methods to particle physics, or for those who really need to compute critical exponents to O e?

For the majority, whose goal is to understand how scaling and renormal- ization ideas might be applied to the rich variety of complex phe- nomena apparent in many other branches of the physical sciences, the main object is to learn the concepts, and the best way to do this is by covering as many examples as possible. This small book was written with this goal in mind. It is, in fact, based on a set of lectures which were given, in various incarnations, to physics grad- uate students at Santa Barbara and Oxford.

A significant fraction of the audience consisted of students planning to do experimental rather than theoretical research. I have assumed that the reader has already had a basic course in statistical mechanics, and, indeed, has had some exposure to critical phenomena, a subject which is, nowadays, often discussed in such courses. However, for completeness, the basic phenomena and some simple models are recalled in the first chapter. As mentioned above, the simplest conceptual route to renormalization concepts is through real space methods, and T have chosen this approach.

At this level, all of the qualitative properties of scaling and universality may then be discussed. In this book I describe an approach, which is certainly not new but deserves to become better known, by which at least the first or- der perturbative renormalization group equations may be derived from a simple continuum real space approach, thus linking up directly with the earlier more intuitive considerations.

It relies on the operator product expansion of field theory, an impressive sounding name for something which is basically very intuitive and simple, and easy to calculate with in lowest order.

The details of these are summarised in a brief Appendix, for readers unfamiliar with these simple formulae. After this, the book embarks on a tour of many of the impor- tant applications of renormalization group to critical phenomena. Then come accounts of the application of similar meth- ods to critical behaviour near surfaces, to systems with quenched random impurities, and to the configurational statistics of large polymers in solution.

These are all problems in equilibrium criti- cal behaviour, bet the next chapter brings in the dynamics. This is a tremendously rich subject, indeed one which deserves a whole book in itself at this level, and it is therefore impossible to do it justice in a single chapter.

However, I have tried to include a number of examples apart from the standard ones, including in particular directed percolation, an example from the rapidly ex- panding subject of dynamic critical behaviour in systems far from equilibrium. Finally, the tour ends with an elementary account of some of the recent developments in the application of the ideas of conformal symmetry to equilibrium critical behaviour.

The non- mathematical reader may find this section slightly harder going than the earlier chapters, although all that is in fact required is a basic knowledge of tensor calculus and complex analysis. Similarly, the modern ap- proach to localisation of waves and electrons in random systems is replete with scaling arguments, but it too requires too extensive an introduction. The dynamics of phase ordering following a rapid temperature quench is another fascinating related subject where scaling arguments play a central role, but for which, as yet, no systematic renormalization approach has been formulated.

A more profound apology is required for the lack of any detailed reference to comparison with experimental data. I hope that this does not create the wrong impression.

The subject of critical phe- nomena is one which is, ultimately, driven by observation and experiment, and it is important that all theorists continue to bear this in mind. However, the basic experiments which established the phenomena of scaling and universality in critical behaviour were performed some time ago, and their results are by now ade- quately summarised in a number of standard references.

It is not the purpose of this book to make detailed comparison with ex- perimental results on particular systems, but rather to emphasise the generality of the principles involved. In this sense the current status of the theory is akin to that of quantum mechanics, where, in similarly introductory texts, it is considered adequate nowa- days to illustrate the theoretical principles with applications to simple and rather idealised systems, rather than by comparison with detailed experimental data.

Since this is an introductory account, I have not included bib- liographic references in the text. Rather, I have provided a list of selected sources of further reading at the end. I have also included a number of exercises, the aim of which is to lead the inquisitive reader into further examples and extensions of the ideas discussed in the text.

I thank my graduate students and colleagues at Santa Barbara and Oxford who have helped me formulate the material of this book over the years.

I am particularly grateful to Benjamin Lee for a careful reading of the manuscript, and to Reinhard Noack for helping produce the Ising model pictures in Chapter 3.

Now divide it into two roughly equal halves, keeping the external variables like pressure and temperature the same. The macroscopic properties of each piece will then be the same as those of the whole. The same holds true if the process is re- peated. But eventually, after many iterations, something different must happen, because we know that matter is made up of atoms and molecules whose individual properties are quite different from those of the matter which they constitute.

The length scale at which the overall properties of the pieces begin to differ markedly from those of the original gives a measure of what is termed the correlation length of the material. It is the distance over which the fluctuations of the microscopic degrees of freedom the posi- tions of the atoms and suchlike are significantly correlated with cach other.

The fluctuations in two parts of the material much fur- ther apart than the correlation length are effectively disconnected from each other.

Therefore it makes no appreciable difference to the macroscopic properties if the connection is completely severed. Usually the correlation length is of the order of a few inter- atomic spacings. This means that we may consider really quite small collections of atoms to get a very good idea of the macro- scopic behaviour of the material.

This statement needs qualifi- cation. In reality, small clusters of atoms will exhibit very strong surface effects which may be quite different from, and dominate, the bulk behaviour. It is well known that systems may abruptly change their macroscopic behaviour as these quantities are smoothly varied.

The points at which this happens are called crit- ical points, and they usually mark a phase transition from one state of matter to another. There are basically two possible ways in which such a transition may occur. In the first scenario, the two or more states on either side of the critical point also co- exist exactly at the critical point. However, even then they are distinct from each other, in that they have different macroscopic properties. Slightly away from the critical point, however, there is generically a unique phase whose properties are continuously con- nected to one of the co-existent phases at the critical point.

In that case, we should expect to find discontinuous behaviour in various thermodynamic quantities as we pass through the critical point, and therefore from one stable phase to another. Such transitions are termed discontinuous or first-order. Well-known examples are the melting of a three-dimensional solid, or the condensation of a gas into a liquid. In fact, such transitions often exhibit hysteresis, or memory effects, since the continuation of a given state into the opposite phase may be metastable so that the system may take a macroscopically long time to readjust.

The correlation length at such a first-order transition is generally finite. However, the situation is quite different at a continuous tran- sition, where the correlation length becomes effectively infinite. The fluctuations are then correlated over all distance scales, which thereby forces the whole system to be in a unique, critical, phase.

At a continuous transition, therefore, the two or more phases on either side of the critical point must become identical as it is approached. Not only does the correlation length diverge in a continuous fashion as such a critical point is approached, but the differences in the various thermodynamic quantities between the competing phases, like the energy density and the magnetisation, go to zero smoothly.

It is the task of the theory to explain this behaviour in a quantitative manner.

## Conformal Field Theory and Statistical Mechanics

Quantum field theory is the mathematical framework of all of particle physics and the indispensable language of a large body of statistical mechanics and of quantum many body physics. It is an incredibly rich subject that you will keep learning for the rest of your scientific life. Learning QFT is qualitatively different from learning other advanced subjects, such as quantum mechanics and classical general relativity, which have a more established logical framework, and have been formalized even to the satisfaction of mathematicians. QFT is still an open subject, close to the conceptual frontier of theoretical physics. It is a work in progress, enriched by new viewpoints, simplified and extended in unexpected directions by each generation of theorists. We will review the renormalization of UV divergences in perturbation theory and then introduce the Wilsonian viewpoint on QFT - notably the connection between renormalization and critical phenomena. One-loop calculations in quantum electrodynamics and Yang-Mills theory.

Alday, D. Gaiotto, and Y. Phys , vol. DOI : Cardy , Scaling and renormalization in statistical physics , El-showk, M. Paulos, D.

Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. The lectures provide a pedagogical introduction to the methods of CFT as applied to two-dimensional critical behaviour. View PDF on arXiv. Save to Library. Create Alert. Launch Research Feed.

## Scaling and renormalization in statistical physics

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This course is the second quarter of statistical mechanics. It focuses on collective behavior of many-particle systems, hydrodynamics, and phase transitions. This course is going to be taught a little differently than usual. We will cover aspects of the subject through topical examples from recent research. Rather than proceed systematically through the subject, we will introduce a research example, and work through the necessary background in as efficient but not thorough!

Written in English. A gem. Very physical treatment of renormalization in the context of statistical mechanics. Other textbooks: To get oriented in a difficult subject that will likely challenge you both conceptually and technically, I very strongly recommend reading Zee's book: A. By explaining the fundamental principles of renormalization theory such as scale invariance and universality which lie behind all the technical variations, this book aims to guide the reader to a more unified understanding of today s physics.

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### Lecture Tuesday, Thursday 12:30 - 1:45 PM, Phelps 3523

Сьюзан ни слова не сказала об истинной причине своей беседы с Дэвидом Беккером - о том, что она собиралась предложить ему место в Отделе азиатской криптографии. Судя по той увлеченности, с которой молодой профессор говорил о преподавательской работе, из университета он не уйдет. Сьюзан решила не заводить деловых разговоров, чтобы не портить настроение ни ему ни. Она снова почувствовала себя школьницей.

- Парень снова сплюнул. - Поэтому все его последователи, достойные этого названия, соорудили себе точно такие. Беккер долго молчал. Медленно, словно после укола транквилизатора, он поднял голову и начал внимательно рассматривать пассажиров. Все до единого - панки. И все внимательно смотрели на .

#### Scaling and Renormalization in Statistical Physics. John Cardy

Неужели ему предстояло погибнуть по той же причине. Человек неумолимо приближался по крутой дорожке. Вокруг Беккера не было ничего, кроме стен. По сторонам, правда, находились железные ворота, но звать на помощь уже поздно. Беккер прижался к стене спиной, внезапно ощутив все камушки под подошвами, все бугорки штукатурки на стене, впившиеся в спину. Мысли его перенеслись назад, в детство. Родители… Сьюзан.

- Подумайте, - предложил. - Раз у человека в паспорте был наш номер, то скорее всего он наш клиент. Поэтому я мог бы избавить вас от хлопот с полицией. - Не знаю… - В голосе слышалась нерешительность. - Я бы только… - Не надо спешить, друг .

Повернувшись, она увидела, как за стеной, в шифровалке, Чатрукьян что-то говорит Хейлу. Понятно, домой он так и не ушел и теперь в панике пытается что-то внушить Хейлу. Она понимала, что это больше не имеет значения: Хейл и без того знал все, что можно было знать. Мне нужно доложить об этом Стратмору, - подумала она, - и как можно скорее.

Похоже, мне не уйти. Асфальт впереди становился светлее и ярче. Такси приближалось, и свет его фар бросал на дорогу таинственные тени. Раздался еще один выстрел.

Прочитав их, Беккер прокрутил в памяти все события последних двенадцати часов. Комната в отеле Альфонсо XIII. Тучный немец, помахавший у него под носом рукой и сказавший на ломаном английском: Проваливай и умри. - С вами все в порядке? - спросила девушка, заметив, что он переменился в лице.

Ключ к шифру-убийце - это число. - Но, сэр, тут висячие строки. Танкадо - мастер высокого класса, он никогда не оставил бы висячие строки, тем более в таком количестве.

* Soccoro! - Его голос звучал еле слышно. - Помогите. С обеих сторон на него надвигались стены извивающейся улочки.*