CSRC Short Course on Renormalization Group Methods and Applications 21-25 March, 2016 Conference Room II, Level 3, CSRC Mon, 21/3 Tues, 22/3 Wed, 23/3 Thur, 24/3 Fri, 25/3 8:30 - Registration 9:00 – CSRC Orland Chen Leiming Chen Leiming Conference on 10:15 introduction (phi-4 (dynamic RG) (phase recent (CSRC Central, theory, transition in developments in Complex Sys) epsilon incompressible nonequilibrium expansion) flocks) StatMech 10:15 – tea break 10:45 Morning a) Quantum quench 10:45 – Tang (review of Orland Delamotte Delamotte and 12:00 equilibrium (phi-4 (NPRG) (NPRG) (pre)thermalization StatMech, theory, Legendre epsilon transforms, expansion) b) FDT violation in Ising model) timescale separated systems 12:00 – lunch break 14:00 Afternoon 14:00 – Lan Yueheng Delamotte Delamotte Delamotte 15:15 (nonlinear (NPRG) (NPRG) (NPRG) a) p-spin models dynamics, (paper by Berthier phase space and Kurchan, trajectory, fixed Nature Phys 2013) point analysis) b) Hard sphere 15:15 – tea break, discussions glass, colloids 15:45 under shear, and active colloids, 15:45 – Special seminar Delamotte Special Group similarities and 17:00 by Guo Wenan (NPRG) seminar by presentations differences, phase (quantum Xing Xiangjun diagram, scaling criticality with (Singular properties two length perturbations scales) and the RG) 18:30 – Tutorials on generating fns, functional 21:00 derivatives, gaussian integrals, etc., led by Lan Yueheng, Tang Leihan and Xing Xiangjun, project discussions 14/3/16 For updated information, please visit http://www.csrc.ac.cn/en/event/schools/2016-02-25/16.html CSRC Short Course on Renormalization Group Methods and Applications 21-25 March, 2016 List of lecture material (received 18 March 2016) Lecturer Topic File Lectures/tutorials General background reading ME_Fisher_98.pdf Kadanoff_Essay.pdf fourier.pdf Leihan Tang Phase transitions and critical phenomena, Ising model LHT_scaling.pdf Delamotte_tutorial_material.pdf Yueheng Lan Nonlinear dynamics, phase space trajectory, fixed point analysis Yueheng_Lan_renormimpv.pdf Henri Orland φ 4 theory, epsilon expansion LHT_RG_notes.pdf Bertrand Delamotte Nonperturbative RG Delamotte_cours2014-6lecture.pdf Delamotte_tutorial_material.pdf Leiming Chen Dynamic RG: Forster, Stephen and Nelson, PRA 16, 732 (1977) FSN_PhysRevA.16.732.pdf Phase transition in incompressible flocks Leiming_Chen_NJP.pdf Leiming_Chen_SM_NJP.pdf Special guest seminars Wenan Guo Quantum criticality with two length scales Wenan_Guo_Science_paper.pdf Xiangjun Xing Singular perturbations and the RG Xiangjun_Xing_notes-ODE-RG.pdf First course on StatMech: http://pages.physics.cornell.edu/~sethna/StatMech/ Renormalization group theory: Its basis and formulation in statistical physics* Michael E. Fisher Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742 The nature and origins of renormalization group ideas in statistical physics and condensed matter theory are recounted informally, emphasizing those features of prime importance in these areas of science in contradistinction to quantum field theory, in particular: critical exponents and scaling, relevance, irrelevance and marginality, universality, and Wilson’s crucial concept of flows and fixed points in a large space of Hamiltonians. CONTENTS I. INTRODUCTION It is held by some that the ‘‘Renormalization Foreword 653 I. Introduction 653 Group’’—or, better, renormalization groups or, let us II. Whence Came Renormalization Group Theory? 654 say, Renormalization Group Theory (or RGT) is ‘‘one of III. Where Stands the Renormalization Group? 655 the underlying ideas in the theoretical structure of IV. Exponents, Anomalous Dimensions, Scale Quantum Field Theory.’’ That belief suggests the poten- Invariance and Scale Dependence 657 tial value of a historical and conceptual account of RG V. The Challenges Posed by Critical Phenomena 658 theory and the ideas and sources from which it grew, as VI. Exponent Relations, Scaling and Irrelevance 661 viewed from the perspective of statistical mechanics and VII. Relevance, Crossover, and Marginality 663 VIII. The Task for Renormalization Group Theory 664 condensed matter physics. Especially pertinent are the IX. Kadanoff’s Scaling Picture 666 roots in the theory of critical phenomena. X. Wilson’s Quest 669 The proposition just stated regarding the significance XI. The Construction of Renormalization Group of RG theory for Quantum Field Theory (or QFT, for Transformations: The Epsilon Expansion 671 short) is open to debate even though experts in QFT XII. Flows, Fixed Points, Universality and Scaling 675 have certainly invoked RG ideas. Indeed, one may ask: XIII. Conclusions 676 How far is some concept only instrumental? How far is Acknowledgments 678 Appendix. Asymptotic behavior 678 it crucial? It is surely true in physics that when we have Selected Bibliography 678 ideas and pictures that are extremely useful, they ac- quire elements of reality in and of themselves. But, FOREWORD philosophically, it is instructive to look at the degree to which such objects are purely instrumental—merely use- ‘‘In March 1996 the Departments of Philosophy and of ful tools—and the extent to which physicists seriously Physics at Boston University cosponsored a Colloquium suppose they embody an essence of reality. Certainly, ‘On the Foundations of Quantum Field Theory.’ But in many parts of physics are well established and long pre- the full title, this was preceded by the phrase ‘A Historical cede RG ideas. Among these is statistical mechanics it- Examination and Philosophical Reflections,’ which set self, a theory not reduced and, in a deep sense, not di- the aims of the meeting. The participants were mainly rectly reducible to lower, more fundamental levels high-energy physicists, experts in field theories, and inter- without the introduction of specific, new postulates. 1 ested philosophers of science. I was called on to speak, Furthermore, statistical mechanics has reached a stage essentially in a service role, presumably because I had where it is well posed mathematically; many of the basic witnessed and had some hand in the development of theorems (although by no means all) have been proved renormalization group concepts and because I have with full rigor. In that context, I believe it is possible to played a role in applications where these ideas really mat- view the renormalization group as merely an instrument tered. It is hoped that this article, based on the talk I or a computational device. On the other hand, at one presented in Boston, may prove of interest to a wider extreme, one might say: ‘‘Well, the partition function audience.’’ itself is really just a combinatorial device.’’ But most practitioners tend to think of it (and especially its loga- rithm, the free energy) as rather more basic! *Based on a lecture presented on 2 March 1996 at the Boston Now my aim here is not to instruct those field theo- Colloquium for the Philosophy of Science: ‘‘A Historical Ex- rists who understand these matters well.2 Rather, I hope amination and Philosophical Reflections on the Foundations of to convey to nonexperts and, in particular, to any with a Quantum Field Theory,’’ held at Boston University 1–3 March 1996. philosophical interest, a little more about what Renor- 1The proceedings of the conference are to be published under the title Conceptual Foundations of Quantum Field Theory (Cao, 1998): for details see the references collected in the Se- 2Such as D. Gross and R. Shankar (see Cao, 1998, and Shan- lected Bibliography. kar, 1994). Note also Bagnuls and Bervillier (1997). Reviews of Modern Physics, Vol. 70, No. 2, April 1998 0034-6861/98/70(2)/653(29)/$20.80 © 1998 The American Physical Society 653 654 Michael E. Fisher: Renormalization group theory malization Group Theory is3—at least in the eyes of associated with Leo Kadanoff at Brown University: their some of those who have earned a living by using it! One focus on relevant, irrelevant and marginal ‘operators’ (or hopes such information may be useful to those who perturbations) has played a central role.10 might want to discuss its implications and significance or But, if one takes a step back, two earlier, fundamental assess how it fits into physics more broadly or into QFT theoretical achievements must be recognized: the first is in particular. the work of Lev D. Landau, who in reality, is the founder of systematic effective field theories, even though he might not have put it that way. It is Landau’s II. WHENCE CAME RENORMALIZATION invention—as it may, I feel, be fairly called—of the order GROUP THEORY? parameter that is so important but often underappreciated.11 To assert that there exists an order This is a good question to start with: I will try to re- parameter in essence says: ‘‘I may not understand the spond, sketching the foundations of RG theory in the microscopic phenomena at all’’ (as was historically, the 4 critical exponent relations and crucial scaling concepts of case for superfluid helium), ‘‘but I recognize that there is Leo P. Kadanoff, Benjamin Widom, and myself devel- a microscopic level and I believe it should have certain 5 oped in 1963–66 —among, of course, other important general, overall properties especially as regards locality 6 workers, particularly Cyril Domb and his group at and symmetry: those then serve to govern the most char- King’s College London, of which, originally, I was a acteristic behavior on scales greater than atomic.’’ Lan- member, George A. Baker, Jr., whose introduction of dau and Ginzburg (a major