CRM Series in Mathematical Physics

Springer-Science+Business Media, L L C

CRM Series in Mathematical Physics

Conte, The Painleve Property: One Century Later MacKenzie, Paranjape, and Zakrzewski, Soli tons: Properties,

Dynamics , Interact ions, Appl icat ions

Semenoff and Vinet, Particles and Fields

R. MacKenzie M.B. Paranjape WJ. Zakrzewski Editors

Solitons Properties, Dynamics, Interactions, Applications

With 55 Figures

Springer

R. MacKenzie Departement de physique Laboratoire Rene J. A . Levesque Universite de Montreal C P . 6128, succursale Centre-ville Montreal, Quebec H3C 3J7 Canada rbmack@lps.umontreal.ca

M . B . Paranjape Departement de physique Laboratoire Rene J. A . Levesque Universite de Montreal C P . 6128, succursale Centre-ville Montreal, Quebec H3C 3J7 Canada paranj@lps.umontreal.ca

W.J. Zakrzewski Department of

Mathematical Sciences Durham University South Road Durham DH1 3LE U K w ,j .zakrzewski @ durham. ac

Editorial Board

Joel S. Feldman Department of Mathematics University of British Columbia Vancouver, British Columbia V6T 1Z2 Canada feldman@math.ubc.ca

Duong H . Phong Department of Mathematics Columbia University New York, N Y 10027-0029 U S A phong@math.columbia.edu

Yvan Saint-Aubin Departement de mathematiques Universite de Montreal C P . 6128, succursale Centre-ville Montreal, Quebec H3C 3J7 Canada saint@dms.umontreal.ca

Luc Vinet Department of Physics M c G i l l University Rutherford Building Montreal, Quebec H3A 2T8 Canada vinet@vpa.mcgill.ca

Library of Congress Cataloging-in-Publication Data Solitons : properties, dynamics, interactions, applications

R. MacKenzie, M . B . Paranjape, W.J. Zakrzewski, [editors], p. cm. - (The C R M series in mathematical physics)

Includes bibliographical references. I S B N 978-1-4612-7063-8 I S B N 978-1-4612-1254-6 (eBook) DOI 10.1007/978-1-4612-1254-6 1. Solitons Congresses. I. MacKenzie, R. (Richard)

II. Paranjape, M . B . III. Zakrzewski, W.J. IV. Series: C R M series in mathematical physics. QC174.26.W28S645 1999

530.12'4-dc21 99-16040

Printed on acid-free paper. © 2000 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 2000 Softcover r epr in t o f the ha rdcover 1st e d i t i o n 2000

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ISBN 978-1-4612-7063-8

Series Preface

The Centre de recherches mathematiques (CRM) was created in 1968 bythe Universite de Montreal to promote research in the mathematical sci­ences. It is now a national institute that hosts several groups, holds specialtheme years, summer schools, workshops, and a postdoctoral program. Thefocus of its scientific activities ranges from pure to applied mathematics,and includes statistics, theoretical computer science, mathematical meth­ods in biology and life sciences, and mathematical and theoretical physics.The CRM also promotes collaboration between mathematicians and in­dustry. It is subsidized by the Natural Sciences and Engineering ResearchCouncil of Canada, the Fonds FCAR of the Province of Quebec, the Cana­dian Institute for Advanced Research and has private endowments. Currentactivities, fellowships, and annual reports can be found on the CRM webpage at http://www .CRM.UMontreal.CAl.The CRM Series in Mathematical Physics includes monographs, lecturenotes, and proceedings based on research pursued and events held at theCentre de recherches mathematiques.

Yvan Saint-AubinMontreal

Preface

Solitons were discovered by John Scott Russel in 1834 and have interestedscientists and mathematicians ever since. They have been the subject of alarge body of research in a wide variety of fields of physics and mathematics,not to mention engineering and other branches of science such as biology.The workshop, "Solitons: Properties, Dynamics, Interactions, Applica­

tions" was held at Queen's University, Kingston, Canada over the periodJuly 20-26, 1997. It was conceived as an interdisciplinary meeting whereintop researchers from several of these fields could meet, interact, and ex­change ideas across traditional boundaries of research. During the work­shop, research was presented on mathematical and numerical aspects of soli­tons, as well as on applications of solitons to nuclear and particle physics,cosmology, condensed matter physics, and to the recent developments instring theory.This volume consists of written versions of all talks presented at theworkshop. It is our hope that people with an interest in solitons in virtuallyany field of research will find the range of topics discussed both interestingand inspiring.

R. MacKenzieM.B. ParanjapeW.J. Zakrzewski

Contents

Series Preface v

Preface vii

List of Contributors xvii

1 Berry Phase and Dissipation of Topological Singularities 1Ping Ao and Xiao-Mei Zhu1 Introduction.................... 12 Description of the Berry Phase and Dissipation 13 Effective Vortex Action 34 Discussion. 75 References........ 8

2 Normal Mode Spectra of Multi-Skyrmions 11W.K. Baskerville, C. Barnes, and N.G. Turok1 Introduction............... 112 Method . . . . . . . . . . . . . . . . . 123 Results and Interpretation of Spectra. 134 References................ 20

3 Standard-Model Dirac Particles Trapped in Flat(Noncompact) Higher Dimensions 21Ronald Bryan1 Introduction.............. 212 Dirac Particle in Eight Flat Dimensions 223 References................. 24

4 Planar QED in Magnetic or Electric Solitonic Backgrounds 27Gerald Dunne1 References........................... 30

5 Collective Coordinates and Inequivalent Coset SpaceQuantizations 33Juan Pedro Garrahan and Martin Kruczenski1 References............................. 36

x Contents

6 Spatial Solitons of the Nonlinear Schrodinger Equationof Arbitrary Nonlinearity with a Potential Hill 39Boris V. Gisin1 References.................... 0 • • 43

7 Hairs on the Unicorn: Fine Structure of Monopolesand Other Solitons 45Alfred S. Goldhaber1 Monopoles and Unicorns 0 451.1 Origin in medieval Europe . 461.2 Subject of a vast literature 461.3 Never confirmed or captured 461.4 Unique unity, not usual duplexity . 461.5 Illuminates much about the world 461.6 Beautiful . . . . 0 • • 461.7 Poor cousins exist . . 471.8 Instantly recognizable 471.9 Still hope of discovery 47

2 "Fundamental" and "Complementary" Solitons 473 Fractional and Peculiar Soliton Charges . . . . 494 Conditions for Integer F . . . . . 0 • • • • • • • 515 Questions About and Applications of the Jackiw-RebbiMonopole . 53

6 Conclusions 547 References. 54

8 A Model for Partially Polarized Quantum Hall States 57ToH. Hansson and U. Nilsson1 References ... 0 0 •••••• 0 •••••••••••• 0 • 61

9 Ordering Dynamics of Topological Defect Networks 63Mark Hindmarsh1 References........... 0 • 0 • • • • 0 • • • 0 0 • • 66

10 Gauge Theory Description of Spin Chains and Ladders 69Yutaka Hosotani1 References........ 0 • • • • • • 0 • • • • • • • 0 • • 72

11 Soliton Solutions of the Integrable Chiral Model in(2+1) Dimensions 75Theodora Ioannidou1 Introduction. 0 • 0 • • • • 0 0 • 0 752 Construction of Soliton Solutions 763 References.... 0 • • • 0 • • 0 • 79

Contents xi

12 String Winding Modes From Charge Nonconservationin Compact Chern-Simons Theory 81Ian 1. Kogan1 References 0 0 0 0 0 0 0 0 •• 0 • 0 0 0 0 •• 0 0 ••• 0 • 0 91

13 Holes in the Charge Density of Topological Solitons 93Mo Kugler1 Introduction 0 0 0 0 • • • 932 The Existence of Holes . 943 Discussion 0 964 References 0 0 0 0 0 0 • 0 96

14 From Two-dimensional Black Holes to sine-GordonSolitons 99J 0 Gegenberg and Go Kunstatter1 Introduction 0 0 0 0 0 • 0 0 0 • • 0 0 0 0 0 0 0 0 992 Black Holes in Jackiw-Teitelboim Gravity 0 0 1003 From Black Holes to sine-Gordon Solitons 0 • 0 1034 . 0 0 and Back Again 1045 Speculations. 1056 References 0 • 0 • • 0 106

15 Solitons and Exciton Superfluidity1. Loutsenko and Do Roubtsov1 References. 0 0 0 0 • 0 0 • 0 • 0 0 • 0

107

113

16 Quantum Effects on Higgs Winding Configurations 115Arthur Lue1 Introduction 0 0 0 ••• 0 0 • 0 0 • 0 0 0 0 0 0 • 1152 Asymptotic Behavior of the Effective Action 0 1163 Discussion 0 1174 References 0 0 0 0 0 0 ••• 0 0 0 0 0 0 118

17 Solitons and Their Moduli SpacesNoS. Manton1 References.. 0 0 • 0 • 0 • • 0 0 0 0 0

18 Deformed SkyrmionsLo Marleau1 Introduction 0 0 0 0 0

2 The Static Oblate Soliton3 Collective Variables .4 Discussion 0

5 References. 0 0 0 0 0

119

129

131

131131132134134

xii Contents

19 The Large-Nc Renormalization GroupNicholas Dorey and Michael P. Mattis1 Introduction..............2 Large-Nc Hadron Models .3 Summing the Leading-Order Graphs4 Solving the Classical Field Equation5 Large-Nc Renormalization Group6 References...............

20 Instantons in Nonirreducible Representations of theLorentz GroupD.G.C. McKeon1 References.............................

21 Fermion Vacuum Effects on Soliton StabilityStephen G. Naculich1 Nontopological Solitons .....2 Kinks in the Linear Sigma Model3 References.............

22 Soliton Solutions of the u-Model and DisorientedChiral CondensatesPrasanta K. Panigrahi and C. Nagaraja Kumar1 References...................

23 Dynamics of Topological Magnetic SolitonsN. Papanicolaou1 Introduction............2 Vorticity and Conservation Laws3 Ferromagnets . .4 Antiferromagnets5 Superfluids6 References....

24 Fun with Baby-SkyrmionsT. Gisiger and M.B. Paranjape1 Introduction..............2 Symmetries . . . . . . . . . . . . . .3 Static and Spinning Baby-Skyrmions4 The Model on 8 3 or R3

5 References............

25 Skyrmions and Domain WallsB.M.A.G. Piette and W.J. Zakrzewski1 Introduction.............

137

137139140145146151

153

157

159

159160162

163

166

167

167168172175178180

183

183183184185186

187

187

Contents xiii

2 Domain Wall Solutions . 1883 References........ 190

26 Fun with Electroweak Solitons 191Edward Farhi, Jeffrey Goldstone, Arthur Lue,and Krishna Rajagopal1 Introduction............. 1911.1 The model 1921.2 The soliton and the sphaleron . 1941.3 Over the barrier ., . . . . . . 1951.4 Fermion production 1971.5 Relating the model to the real world 198

2 Classical Dynamics for eNear e . . . . 1993 Quantum Processes in the Fixed ~E Limit 2054 Concluding Remarks 2105 References................... 211

27 Neutral and Charged Spin Excitations in the QuantumHall Ferromagnet 213Rashmi Ray1 Introduction.......... 2132 Notation and Formulation . . 2143 Ferromagnetic Ground State. 2154 Effective Action for the Magnons 2165 Charged Spin Skyrmions . 2176 Conclusions 2177 References......... 217

28 Quantum Corrections to MonopolesG. Chalmers, M. Rocek, and R. von Unge1 References................

219

223

29 Nonabelian Dyons 225B.J. Schroers1 Outline of the Problem. . . . . . . . . . . . . . . . . . . . 2252 SU(3) Monopoles and Their Moduli Spaces " . . . . . . 2273 Dyonic Quantum States and the Emergence of U(2) I>< JR4 2304 Discussion and Outlook 2325 References......................... 233

30 Electroweak Baryon Properties in Soliton Models 235Norberto N. Scoccola1 Introduction......... 2352 The Model 2353 Decuplet Radiative Decays. 236

xiv Contents

4 Hadronic Weak Decays of Octet Baryons. 2375 Conclusions 2386 References.................. 238

31 Solitons, Duality, and Supersymmetric Gauge Theories 241Alfred D. Shapere1 History 2412 Seiberg and Witten's Solution. 2443 Solitons and Singularities 2474 References...... . . . . . 250

32 Solitonic Strings and KnotsR.A. Battye and P.M. Sutcliffe1 References...........

253

260

33 Toward a String Formulation of Vortex Dynamics 263Elsebeth Schroder and ala Tornkvist1 Introduction........... 2632 String Formulation . . . . . . . 2643 The String Equation of Motion 2644 Conclusions and Outlook 2665 References............ 267

34 Domain Walls in a Chern-Simons Theory 269M. Torres1 The Model .. 2692 Domain Walls. 2703 References... 272

35 Microphysics of Gauge Vortices and Baryogenesis 273Mark Trodden1 Introduction....................... 2732 The Electroweak Theory and Sakharov . . . . . . . . 2743 Electroweak Symmetry Restoration around Vortices 2744 Defect-mediated Electroweak Baryogenesis . 2755 Conclusions 2766 References.......... 276

36 On a Dual Standard Model 279Tanmay Vachaspati1 Motivation 2792 Construction of the Dual Model . 2803 Confinement. 2814 Families . 2815 Fermions... 282

Contents xv

6 Conclusions 2847 References. 285

37 From Skyrmions to the Nucleon-Nucleon Potential 287Jochen Wambach and Thomas Waindwch1 Introduction........................ 2872 The Skyrme Model . . . . . . . . . . . . . . . . . . . . 2883 Interacting Skyrmions and the Gradient Flow Method 2894 The Nucleon-Nucleon Potential. 2915 References........................ 293

38 Two-dimensional Solitons at Finite Temperature 295M. Kacir and 1. Zahed1 Introduction......... 2952 Model Field Theory .... 2963 High Temperature Behavior 2984 Soliton Rest Mass .... 2995 Propagating Soliton Mass 3006 Energy Shift . 3047 Conclusions 3058 References.. 306

39 Nontopological Structures in the Baby-Skyrme Model 309B.M.A.G. Piette and W.J. Zakrzewski1 Introduction... 3092 Pseudobreathers 3103 Conclusions 3124 References.... 312

Contributors

Ping Ao, Department of Theoretical Physics, Umea University, 901 87Umea, Swedenao<otp.umu.se

C. Barnes, Stanford Linear Accelerator Center, P.O. Box 4349, Stanford,CA 94309, USAcbarnes<Oslac. stanford. edu

WK. Baskerville, Center for Particle Theory, Department ofMathematical Sciences, Science Laboratories, South Road, DurhamDH13LE, UKw.k.baskerville<Odurham.ac.uk

R.A. Battye, Cambridge University, Cambridge CB3 9EW, UKr.a.battye<Odamtp.cam.ac.uk

Ronald Bryan, Department of Physics, Texas A&M University, CollegeStation, TX 77843, USAbryan<Ophys.tamu.edu

G. Chalmers, Institute for Theoretical Physics, State University of NewYork, Stony Brook, NY 11794, USAchalmers<Oinsti.physics.sunysb.edu

N. Dorey, Department of Physics, University of Wales Swansea, SingletonPark, Swansea SA2 8PP, UKn.dorey<Ophython.swan.ac.uk

Gerald Dunne, Department of Physics, University of Connecticut, Storrs,CT 06269, USAdunne<Ohep.phys.uconn.edu

E. Farhi, Center for Theoretical Physics 6-410, Massachusetts Institute ofTechnology, 77 Massachusetts Ave., Cambridge, MA 02139, USAfarhiCOmitlns.mit.edu

Juan Pedro Garrahan, Departamento de Fisica, Facultad de CienciasExactas y Naturales, Universidad de Buenos Aires Pabellon I,Ciudad Universitaria (1428) Buenos Aires, Argentinagarrahan<Odf .uba. ar

xviii Contributors

J. Gegenberg, Department of Mathematics and Statistics, University ofNew Brunswick, Fredericton, New Brunswick E3B 5A3, Canadalenin~math.unb.ca

T. Gisiger, Laboratoire Rene J.-A. Levesque, Universite de Montreal,C.P. 6128, succ. Centre-ville, Montreal, Qc, H3C 3J7, Canadagisiger~lps.umontreal.ca

Boris V. Gisin, Department of Electrical Engineering-PhysicalElectronics, Tel-Aviv University, Tel-Aviv 69978, Israelgisin~eng.tau.ac.il

Alfred S. Goldhaber, Institute for Theoretical Physics, State University ofNew York, Stony Brook, NY 11794, USAgoldhab~insti.physics.sunysb.edu

J. Goldstone, Center for Theoretical Physics 6-410, MassachusettsInstitute of Technology, 77 Massachusetts Ave., Cambridge, MA02139, USAgoldstone~itlns.mit.edu

T.H. Hansson, Stockholm University, Fysikum, Box 6730, 11385Stockholm, Swedenhansson~physto.se

Mark Hindmarsh, Center for Theoretical Physics, Sussex University,Brighton BN1 9QH, UKmarkh~pcss.maps.susx.ac.uk

Yutaka Hosotani, School of Physics and Astronomy, University ofMinnesota, Minneapolis, MN 55455, USAyutaka~mnhepw.hep.umn.edu

Theodora Ioannidou, Department of Mathematical Sciences, DurhamUniversity, South Rd., Durham DH1 3LE, UKtheodora.ioannidou~durham.ac.uk

M. Kacir, Service de physique theorique, CEA-Saclay, F-91191,Gif-sur-Yvette, Francezahed~nuclear.physics.sunysb.edu

Ian 1. Kogan, Theoretical Physics, 1 Keble Rd., Oxford OX1 3NP, UKi.koganl~physics.ox.ac.uk

Marlin Kruczenski, Departamento de Fisica, TANDAR, ComisionNacional de Energia Atomica, Av. Libertador 8250, 1429 BuenosAires, Argentinakruczenz~tandar.cnea.edu.ar

Contributors xix

M. Kugler, Weizmann Institute, Rehovot, 76100 Israelfhkugler~wicc.weizmann.ac.ilk

C. Nagaraja Kumar, School of Physics, University of Hyderabad,Hyderabad 500 046, Indiapani-sp~ohyd.ernet.in

G. Kunstatter, Department of Physics, University of Winnipeg,Winnipeg, Manitoba, R3B 2E9, Canadagabor~theory.uwinnipeg.ca

1. Loutsenko, Centre de recherches matMmatiques, Universite deMontreal, C.P. 6128, suce. Centre-ville, Montreal, Qc, H3C 3J7,Canadaloutseni~crm.umontreal.ca

A. Lue, Physics Department, Pupin Laboratories #904, ColumbiaUniversity, 538 W 120th Street, NY, NY 10027, USAlue~cuphyb.phys.columbia.edu

N.S. Manton, Department of Applied Mathematics and TheoreticalPhysics, University of Cambridge, Silver Street, Cambridge CB39EW, UKn.s.manton~damtp.cam.ac.uk

L. Marleau, Departement de physique, University Laval, Ste-Foy, Qc,G1K 7P4, CanadaImarleau~phy.ulaval.ca

M.P. Mattis, Theoretical Division T-8, Los Alamos National Laboratory,Los Alamos, NM 87545, USAmattis~pion.lanl.gov

D. G. C. McKeon, Department of Applied Mathematics, University ofWestern Ontario, London, qntario, N6A 5B7, Canadatmleafs~apmaths.uwo.ca

Stephen G. Naculich, Department of Physics, Bowdoin College,Brunswick, ME 04011, USAnaculich~bowdoin.edu

U. Nilsson, Department of Physics, University of Stockholm, P.O. Box6730, S-11385 Stockholm, Swedenulfn~vanosf.physto.se

Prasanta K. Panigrahi, School of Physics, University of Hyderabad,Hyderabad 500 046, Indiapani-sp~ohyd.ernet.in

xx Contributors

N. Papanicolaou, Department of Physics, University of Crete, andResearch Center of Crete, Heraklion, Greecepapanico~pluto.physics.uch.gr

M.B. Paranjape, Laboratoire Rene J.-A. Levesque, Departement dePhysique, University de Montreal, C.P. 6128, succ. Centre-ville,Montreal, Qc, H3C 3J7, Canadaparanj@lpshpd.lps.umontreal.ca

B.M.A. G. Piette, Department of Mathematics, University Durham,South Rd., Durham DH1 3LE, UKb.m.a.g.piette~durham.ac.uk

K. Rajagopal, Center for Theoretical Physics, Massachusetts Institute ofTechnology, 77 Massachusetts Ave., Cambridge, MA 02139, USAkrishna@ctp.mit.edu

Rashmi Ray, Laboratoire Rene J.-A. Levesque, Universite de Montreal,c.P. 6128, succ. Centre-ville, Montreal, Qc, H3C 3J7, Canadarray~lps.umontreal.ca

D. Roubtsov, Departement de physique, Universite de Montreal,c.P. 6128, succ. Centre-ville, Montreal, Qc, H3C 3J7, Canadaroubtsod~physcn.umontreal.ca

M. Rocek, Institute for Theoretical Physics, State University of NewYork, Stony Brook, NY 11794, USArocek@insti.physics.sunysb.ed

Elsebeth Schroder, Materials and Surface Physics Group, Vasa 11,Department of Applied Physics, Chalmers University of Technology,S-412 96 G6teborg, Swedenschroder~fy.chalmers.se

N.N. Scoccola, Departamento de Fisica, TANDAR, Comision Nacional deEnergia Atomica, Av. Libertador 8250, 1429 Buenos Aires,Argentinascoccola~tandar.cnea.edu.ar

A.D. Shapere, Department of Physics, University of Kentucky, Lexington,NY 40506, USAshapere~pa.uky.edu

P.M. Sutcliffe, Institute of Mathematics, University of Kent, CanterburyCT27NF,UKp.m.sutcliffe~ukc.ac.uk

Contributors xxi

ala Tornkvist, Fermilab, Theoretical Astrophysics Group, MS 209, P.O.Box 500, Batavia, IL 60510, USAolat@fnal.gov

Manuel Torres, Institute de Ffsica, University Nacional Aut6noma deMexico, Apdo. Postal 20-364, 01000 Mexico, D.F., Mexicomanuel@teorical.ifisicau.unam.mx

Mark Trodden, Center for Theoretical Physics 6-410, MassachusettsInstitute of Technology, 77 Massachusetts Ave., Cambridge, MA02139, USAtrodden@ctpa04.mit.edu

N. G. Turok, DAMTP, Cambridge University, Silver Street, CambridgeCB3 9EW, UKn.g.turok@damtp.cam.ac.uk

R. von Unge, Joseph Henry Laboratories, Princeton University,Princeton, NJ 08544, USAunge@feynman.princeton.edu

Tanmay Vachaspati, Department of Physics, Case Western ReserveUniversity, Cleveland, OH 44106, USAtxv7@cwru.edu

Thomas Waindzoch, Institut fUr Theoretische Physik, Auf derMorgenstelle 14, Universitat Tiibingen, D-72076 Tiibingen,Germanythomas.waindzoch@uni-tuebingen.de

Jochen Wambach, SchoBgartenstr. 9, TU-Darmstadt, D-64289Darmstadt, Germanyjochen.wambach@physik.th-darmstadt.de

I. Zahed, Department of Physics, State University of New York, StonyBrook, NY 11794, USAzahed@nuclear.physics.sunysb.edu

w.J. Zakrzewski, Department of Mathematical Sciences, DurhamUniversity, South Rd., Durham DH1 3LE, UKw.j.zakrzewski@durham.ac.uk

Xiao-Mei Zhu, Department of Experimental Physics, Umea University,S-90187, Umea, Swedenxiaomei.zhu@tp.umu.se