Critical Phenomena in Loop Models

150 Pages·2014·3.029 MB·English

by  Adam Nahum

Springer Theses
Recognizing Outstanding Ph.D. Research
Adam Nahum
Critical
Phenomena
in Loop Models
Springer Theses
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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
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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.