Springer Theses Recognizing Outstanding Ph.D. Research Adam Nahum Critical Phenomena in Loop Models Springer Theses Recognizing Outstanding Ph.D. Research Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected foritsscientificexcellenceandthehighimpactofitscontentsforthepertinentfield of research. For greater accessibility to non-specialists, the published versions includeanextendedintroduction,aswellasaforewordbythestudent’ssupervisor explainingthespecialrelevanceoftheworkforthefield.Asawhole,theserieswill provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special ques- tions.Finally,itprovidesanaccrediteddocumentationofthevaluablecontributions made by today’s younger generation of scientists. 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More information about this series at http://www.springer.com/series/8790 Adam Nahum Critical Phenomena in Loop Models Doctoral Thesis accepted by the University of Oxford, UK 123 Author Supervisor Dr. AdamNahum Prof. JohnChalker Department of Physics Theoretical Physics Massachusetts Instituteof Technology Universityof Oxford Cambridge,MA Oxford USA UK ISSN 2190-5053 ISSN 2190-5061 (electronic) ISBN 978-3-319-06406-2 ISBN 978-3-319-06407-9 (eBook) DOI 10.1007/978-3-319-06407-9 LibraryofCongressControlNumber:2014949370 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodology now known or hereafter developed. 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While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Publications Related to this Work (cid:129) A. Nahum, J. T. Chalker, P. Serna, M. Ortuño and A. M. Somoza, 3D Loop Models and the CPn(cid:1)1 Sigma Model, Phys. Rev. Lett. 107, 110601 (2011). (cid:129) A. Nahum and J. T. Chalker, Universal Statistics of Vortex Lines, Phys. Rev. E 85, 031141 (2012). (cid:129) A. Nahum, P. Serna, A. M. Somoza, and M. Ortuño, Loop Models with Crossings, Phys. Rev. B 87, 184204 (2013). (cid:129) A. Nahum, J. T. Chalker, P. Serna, M. Ortuño and A. M. Somoza, Phase Transitions in Three-Dimensional Loop Models and the CPn(cid:1)1 Sigma Model, Phys. Rev. B 88, 134411 (2013). (cid:129) A. Nahum, J. T. Chalker, P. Serna, M. Ortuño and A. M. Somoza, Length Distributions in Loop Soups, Phys. Rev. Lett. 111, 100601 (2013). (cid:129) Inpreparation:A.Nahum,J.T.Chalker,P.Serna,M.OrtuñoandA.M.Somoza, Deconfined Criticality in a Classical Loop Model; P. Serna, M. Ortuño, A. M. Somoza,A.NahumandJ.T.Chalker,CriticalBehaviourin3DUnorientedLoop Models;A.Nahum,RGFlowsfor Θ-PointPolymers;A.Nahum,FieldTheories forFully-PackedLoopModelsandRVBStates. ’ Supervisor s Foreword ItisapleasuretowritethisforewordforAdamNahum’sthesis.Whenhestartedthe research,Ihadlittleideahowfaritwouldgoorhowwide-rangingitwouldturnout to be. Problems involving loops or random curves occupy an important place in theoretical physics. A good example is provided by the connection that P.-G. de Gennesestablished, betweenself-avoiding walksinpolymerphysicsandthehigh- temperature expansion for the OðnÞ model in the n!0 limit. Nevertheless, one mighteasilyhavethought5yearsagothatmostoftheinterestingpointshadalready been made. The thesis shows in multiple ways that this was not the case. One of the appeals of the results described in the following pages comes from seeingcurrenttopicsinanewlight.Aninstanceistheformulationofaparticularly subtlephasetransitioninquantummagnetsasaloopprobleminclassicalstatistical physics, with all the simplifying features of real, positive configurational weights replacing the phases and interference effects of quantum systems. Another is the connection between certain exotic Anderson localisation problems and the phase transitions discovered here in two-dimensional loop models. A second attraction of this work comes from the understanding it gives of old results. For example, numerical studies of random curves in a variety of settings have found they have a fractal dimension close to two. The field-theoretic descriptiondevelopedhereshowshowthisstemsnaturallyfromstandardproperties of Goldstone modes, with two as the exact value. Researchshouldsometimesleadtosurprises.Formeinthefollowingitistofind anexactstatementthatcanbemadeaboutastronglycorrelatedstatisticalsystem— involving the probability distribution of loop lengths, which turns out to be a non- trivial yet calculable function of an infinite number of degrees offreedom. Computational work is frequently crucial in theoretical physics and collabora- tionscanbe keyeveninsmall-scale science.Adamwas fortunate inhavingexpert co-workers from the University of Murcia, whose simulations have provided tests and substantiation of many of the ideas here. vii viii Supervisor’sForeword Lookingahead,classicalloopmodelsseemsettoplayanimportantpartoverthe next few years in thinking about strongly correlated quantum phases. I hope some of the results in this thesis will be useful in that endeavour. Oxford, April 2014 Prof. John Chalker Abstract Many critical systems are best discussed in the language of random geometry, in particular of random curves—which might be worldlines, line defects, Feynman graphs,polymers,etc.Thisthesisexploressuchformulations,aimingtoextendthe geometric language in places where it is underdeveloped: three-dimensional (3D) loop ensembles in the first half of the thesis and 2D loop models lying outside the standard paradigms in the second. Many of the problems studied here relate to sigma models on complex or real projective space (CPn(cid:1)1 or RPn(cid:1)1), and this is also an investigation of critical behaviour in these theories. We begin with a family of classical 3D loop models that show transitions between phases with and without infinite loops. We map them to the above field theories,withnfixedbythefugacityforloops.Treatingnascontinuouslyvarying, we characterise both renormalisation group flows in the CPn(cid:1)1 model and the universal statistical geometry of the loops (and generalise the latter conclusions to someother3Densembles).Wearguethatcertainofthemodelsexhibitthephysics of deconfinement and emergent gauge fields, and connect them to 2D quantum antiferromagnets. Next we tackle the fractal geometry of vortex lines in disordered media. These show geometrical phase transitions analogous to percolation transitions, but, we argue, in distinct universality classes. Vortex geometry has been studied numeri- cally in many contexts, but field theory descriptions have been lacking. Via map- pings to lattice gauge theory, we argue that replica limits or supersymmetric versionsoftheabovetheoriesaretherequireddescriptions,andusethemtoclassify critical properties. 2D loop models are well studied—but there is more to discover if we look beyond the familiar ones in which loop crossings are forbidden. We reveal new universality classes of continuous phase transitions, driven by the unbinding of Z 2 pointdefectsintheRPn(cid:1)1 sigmamodel(withn\2).Wealsocharacteriseindetail the‘Goldstone’phasewhichappearsverygenericallyforloopswithcrossings.One of our models provides a close analogue for Anderson metal-insulator transitions, ix x Abstract andasimplercontextinwhichtostudysuchpoorlyunderstoodcentral-charge-zero critical points. In the final chapter we address some long-standing questions concerning the polymer collapse transition (Θ point). We give a field theory treatment for the standard lattice model of a polymer with crossings in 2D, explaining previous numericalresults.Surprisingly,thecollapsetransition inthismodelturnsouttobe governed by an infinite-order multicritical point, indicating that more generic models will show new universal behaviour.