Condensed Matter Physis as a Laboratory
for Gravitation and Cosmology
Fernando Moraes
LaboratoriodeFsiaTeoriae Computaional
Departamentode Fsia, UniversidadeFederalde Pernambuo
50670-901 Reife,PE,Brazil
E-mail: moraesdf.ufpe.br
Reeived7January,2000
Thegeometri languageofGeneralRelativity isnotnormallyrelated toCondensedMatter(CM)
Physis sine it is the eletromagneti and not the gravitational interation that dominates the
physisofCMsystems. WhatpointsinommonwouldthenCMPhave withCosmologyand the
dynamisofobjetsinagravitationaleld?Thereisatleastonethatisveryimportant: topologial
defetsformedinsymmetrybreakingphasetransitions. Toexplorethesimilaritiesanddierenes
herehasbeenaveryfruitfulexperiene for bothsides. Ononehand,topologialdefetsinsolids
startedtobedesribedbyagravity-liketheoryinludingtorsionand,ontheotherhand,experiments
havebeenproposedandperformedinCMsystemswiththepurposeoftestingosmologialtheories.
Someexamplesare: 1)LandaulevelsandtheAharonov-Bohmeetofeletronsmovinginarystal
ontainingasrewdisloationanbedesribedinasimplewayinageometriformalism;2)losed
timelike urves have been proposed in the viinity of vorties in superuid Helium; 3) Kibble
mehanism, for the generation of topologial defets, has been experimentally veried in liquid
rystals. Insummary,CondensedMatterPhysiswithitsrihdiversityofsystemsandphenomena
and of relatively easy aess to experiments, appears as a laboratory for testing hypotheses of
gravitationaltheoryandosmologyinvolvingtopologial defets. InthisworkIsummarizereent
resultsinthisinterfaeareafousingmainlyintheresultsobtainedbyourresearhgroup.
I Introdution
Topologialdefets,bothinCondensedMatterPhysis
andinCosmology,areformedduringsymmetry
break-ingphasetransitions[1℄. Theanalogyissostrongthat,
inthepastfewyears,some\osmologialexperiments"
have been arried out in liquid rystals[2℄ and
super-uid Helium[3℄, shading a new light on the dynamis
ofthedefetformationproess. Althoughquite
onve-nienttostudy thedynamis ofdefetformation,those
systemsdonotprovideaneasyexperimentalsettingfor
thestudyofstatipropertiesofthedefetbakground,
due to the high mobilityof the defets. Onthe other
hand,topologialdefetsinrystallinesolids,withtheir
verylowmobility,are justidealforthis purpose. But,
howsimilar arethese defetsto spae-timedefets? It
is the intent of this work to briey review these
sim-ilarities and to present a summary of results on this
researhareaobtainedbyourresearhgroup.
There is a formal equivalene between
three-dimensionalgravitywith torsionandthetheoryof
de-fetsin solids[4℄. Defetsin rystalsare formed when
theontinuoustranslationalandrotationalsymmetries
of thelattieduring thefreezing transition. Theyan
beoneptuallygeneratedbya\utandglue"proess,
knownintheliteratureastheVolterraproess[5℄. This
proessgivesaunifyingviewofthetopologialline
de-fets. That is, take a ylinder of a ontinuous elasti
materialandmakearadialutin it,fromitsaxisout.
Displaementofthesurfaesoftheutwithrespetto
eahother andsubsequentglueingwill generatealine
defetwhoseoreoinideswiththeaxis. Considering
ylindrialoordinates,ifthedisplaementis:
(i) along the z-diretion a srew disloation is
formed.
(ii) along the r-diretion, an edge disloation is
formed.
(iii)alongthediretion,whihimpliestheaddition
orremoval ofawedgeof material, leadsto a
dislina-tion.
(iv)aombinationof both(i)and(iii) itprodues
adispiration.
It is lear then, that the ore of suh line defets
areassoiatedtogeometrisingularities.Hereweome
lose to gravity theory, where geometri singularities
from theatMinkowskygeometry.
II Geometri model for line
de-fets in elasti solids
It iswellknownthatelastisolidswithtopologial
de-fetsanbedesribedbyRiemann-Cartangeometry[6℄.
Reently, Katanaev and Volovih[4℄ have shown the
equivalene between three-dimensional gravity with
torsion andthe theoryof defets in solids. Thedefet
ats as a soure of a \gravitational" distortion eld.
Themetridesribingthemedium surroundingthe
de-fetisthenasolutiontothethree-dimensional
Einstein-Cartanequation.
Sineweare disussinglinedefets itis onvenient
to imagine at three-dimensional spae as a pile of
\panakes". This waywe animagineaVolterra
pro-ess for a single \panake", and then, extend the
de-formationtotherest ofthepile. Obviouslythemetri
desribingthespaeis
ds 2
=dz 2
+dr 2
+r 2
d 2
; (1)
in ylindrialoordinates.
A srew disloation orresponds to a pile of ut
\panakes" like the onein Fig. 1. The displaement
between the edges of the ut is the so-alled Burgers
vetorofmodulusbanddiretedalongthez-axis. The
orrespondingmetriis[7℄
ds 2
=(dz+ b
2 d)
2
+dr 2
+r 2
d 2
: (2)
Figure1. SrewDisloation
Figure2. Dislination
Fig. 2 orresponds to the dislination. The
met-riofadislinatedmedium, whihis in fat thespae
setionoftheosmistringmetri[8℄,isgivenby
ds 2
=dz 2
+dr 2
+ 2
r 2
d 2
; (3)
where givesameasure of the angular deit. Sine
aslieofangle2(1 )hasbeentakenoutof spae,
thetotalalglearoundthez-axisisnow2.
The above metris are in fat partiular ases of
theonedesribingthespinninghiralstringspae-time
onsideredbyGal'tsovandLetelier[9℄andTod[10℄
ds 2
=(dt+Jd) 2
dr 2
2
r 2
d 2
(dz+d) 2
; (4)
Figure3.Edgedisloationviewedasadislinationdipole.
An edge disloation may be formed [4℄ by plaing
a pair of opposing parallel dislinations next to eah
other asa dipole (Fig. 3). Eah dislination is made
byeitherremovingorinsertingawedgeofmaterial. In
theontinuumlimit,atlargedistanesfromthedefet,
themedium isdesribedbythemetri[4℄
ds 2
=dz 2
+
1+
m 2hrsin h 2
2
(dr 2
+r 2
d 2
in ylindrialoordinates. Thewedge anglesaregiven
bymandthedislinationsareatadistane hapart.
The Burgers vetor ~
b = mh ^x is therefore along the
x-axisandthedefetitselfisalongthez-axis.
Naturally,therearemoredefetsand,onsequently,
moremetris. Theyarealllisted in[7℄. Wewill be
in-terestedinjusttheones desribedabove.
III Some results
We havebeen mainly interested in the physial
prop-erties of rystals with defets where the defets have
the passive role of providing the bakground for
par-tiles (eletrons orholes) or elds (eletromagneti or
salar) to be around. We have also looked at
simi-lar defets in spae-time. The defets impose
bound-ary onditions on those objetsleading to bound and
sattering states for the partiles: binding of
ele-trons and holes to defets[12, 13℄, Landau levels in
thepreseneofthedefets[14,15℄, loalizedstates[16℄,
geodesi motion[11, 17℄, transport phenomena[18℄,
Aharonov-Bohmeet[19, 20, 21℄,quantum sattering
by defets[22℄, the aquisition of a quantum (Berry's)
phase[23℄. Fortheelds, theboundaryonditionsdue
tothedefetsleadtolassialandquantumeetslike
self-fore[24, 25, 26℄,Casimir eet[27, 28℄ and
orre-tionsto physialonstants[29℄. Amongthese results I
hoosetoommentbrieyon: (a)thegeodesisaround
anedge disloation[11℄, (b)theself-foreonaharged
partile also near an edge disloation[26℄, () the
or-retion to the magneti moment of the eletron near
adislination[29℄and, nally, (d)theAharonov-Bohm
eetaroundasrewdisloation[20,21℄.
(a)AsshowninFig. 3,anedgedisloationmaybe
formedbyremovingaretangularslieofmaterialand
glueingthelooseedges. Themediumwiththisdefetis
assoiatedtothemetri(5)whihdesribesessentially
aatthree-dimensionalspaewith singulartorsionon
thez-axis. Notietheassymmetryofthemetriasone
goaroundthez-axis. Thisbeomequiteevidentwhen
oneobservesthegeodesisinsuhmedium[11℄(Figures
4-6). Eahgureshowsabundleof geodesis,parallel
to eah other at innity, approahing and being
de-eted by the disloation loated at the origin of the
plots.
(b)Theboundaryonditionsimposedbya
topolog-ialdefeton theeletrield of apoint hargein its
neighborhood provokes a distortion of the eld lines.
This anbe interpreted as due to image harges, just
likein the aseof a point harge in the presene of a
ground onduting plane. Inthis asethe interation
energy between the harge and its image is the
self-energyoftheharge. Sineitdependsonthedistane
betweenthehargeandtheplane,thederivativeofthis
energywithrespettothisdistanegivesrisetoan
ef-samealsohappensforapointhargeandatopologial
defet[26℄. Fig. 7showsaplotoftheself-energyofa
lin-earhargedensityparalleltoanedgedisloation(and
'aretheoordinatesofthelineofhargewithrespet
tothedefet,whoseloationoinideswiththez-axis).
Thelakofaxialsymmetryofthedefetismanifestin
theplot. Notiethat,asthehargegoesaroundthe
de-fet,therearealternatingregionsofeetiveattration
andrepulsion, respetively.
() Analogousto the lassialeld phenomenon of
theself-foretherearealsoquantumeldeetsdueto
topologialdefets. TheCasimireet[27,28℄beingthe
bestknownexampleofsuhphenomena. Alessknown
eet,butveryimportantforondensedmatter
exper-imentsistheorretionofthemagnetimomentofthe
eletronduetoadefet[29℄. Quantumeletrodynamis
givesus = e~ 2m 1+ e 2 2~ +::: ; (6)
forthemagnetimomentoftheeletroninemptyspae.
TheorretiontotheDiravalue= e~
2m
isduetothe
ouplingoftheeletrontothequantizedmagnetield.
Whenthequantizedeletromagnetield issubmitted
to the boundary onditions imposed by a topologial
defettheorretionobviouslyhange. Foraneletron
boundto anegativedislinationitisfound[29℄
Æ = (1 p 2 )s 2 (p)e 6 288 4 ~ 3 3 2 ; (7)
where p=1=. Anegativedislinationorrespondsto
0<p<1, therefore Æ
>0,onrming experimental
evidene[30℄ that negative urvature dislinations
en-ouragelargerloalmoments.
(d) Here, just a brief omment on the
Aharonov-Bohmeetaroundasrewdisloation,whihhasbeen
studied in detail in[20, 21℄. The well-known original
Aharonov-Bohmeetinvolvesahargedpartile
mov-ing outsidea regionwith magneti ux. Ifthe ux is
onnedtoastring,itanbedesribed,inthe
Kaluza-Kleinapproah,astheve-dimensionalmetri
ds 2 =dt 2 dz 2 dr 2 r 2 d 2 (dx 5 + 2 d) 2 ; (8)
whereisthemagnetiuxandx 5
thefthdimension.
Now, ompare this metri with the four-dimensional
metriofasrewdisloation(seeEquation(2)
ds 2 =dt 2 (dz+ b 2 d) 2 dr 2 r 2 d 2 : (9)
NotiethattheBurgersvetorbplaysthesameroleas
themagnetiuxmakingitlearthatthe
Aharonov-Bohmeetshouldappearforapartilemovingin the
preseneofasrewdisloation. Indeed,theShrodinger
equation will look the same in either bakground (
Figure4. Geodesisaroundedgedisloation(1).
Figure5. Geodesisaroundedgedisloation(2).
Figure7. Self-energyofalinearhargedensityparallelto
anedgedisloation.
IV Other systems
Topologialdefetsappearinnatureasonsequeneof
symmetry-breaking phase transitions. An important
issue in osmology is whether the observed struture
of the universe ontains relis of topologial defets
formed as theearly universe ooled. What is the
de-fetdensityafteraphasetransition? In1976T.W.B.
Kibble[31℄ proposed a statistial proedure for
alu-lating theprobabilityof stringformationin thephase
transitionsin the earlyuniverse. In1991 Chuangand
oworkers[32℄ and in 1994 Bowiket al. made
experi-mentalveriationsoftheKibblemehanismforstring
formation in liquid rystals. Sine 1994 dierent
ex-perimental groups[3℄ have tested Kibble's mehanism
insuperuidHe 4
.
MoreinterestingthansuperuidHe 4
arethe
super-uidphasesofHe 3
whiharequantum liquidswith
in-terating fermioni and bosoni elds. Its rih
stru-turegivesrisetoanumberofanaloguesofosmologial
defets[33℄: 1) the dysgiration, that simulates the
ex-tremely massiveosmi string; 2) the singularvortex,
whihisanalogoustotherotatingosmistring,3)the
ontinous or ATC vortex, whose motion auses
\mo-mentogenesis",whihistheanalogueofbaryogenesisin
theearly universe; 4) planarsolitons, that haveevent
horizons similar to rotating blak and white holes; 5)
symmetri vorties (in a thin lm), whih admit the
existeneoflosed timelikeurvesthroughwhih only
superuidlustersofanti-He 3
atomsantraveland
gen-There are still thesuperondutorswith their
vor-ties; glasses with their urved spae desription that
requiredislinations andperhapsmonopoles; magnets
withtheirdislinationsanddomain walls...
V Conlusion
Condensed Matter Physis has muh beneted from
tools and ideas from gravitation and osmology and
paysthatbakbyoeringalaboratoryfortestingsome
osmologialor gravitationalhypotheses.
Aknowledgments
My warmest thanks go to my numerous ollaborators
(see referenes [12-28℄) whomade this work ome out
of the vauum. This work was partially supported
byFACEPE,CNPqandFINEPthroughitsPRONEX
program.
Referenes
[1℄ TammayVahaspati,Phys.Rev.D44,3723(1991).
[2℄ MarkJ.Bowiketal.,Siene263,943(1994).
[3℄ W.H.Zurek,Phys.Rep.276,177(1996).
[4℄ M.O.KatanaevandI.V.Volovih,Ann.Phys.(NY)
216,1(1992).
[5℄ M. Kleman, Points, Lignes, Parois: dans les
u-ides anisotropes et les solides ristallins (Edition de
Physique,Frane1977).
[6℄ E. Kroner, Continuum theory of defets, in: Les
Houhes,SessionXXXV,1980-PhysisofDefets,eds.
R.Balianetal.(North-Holland,Amsterdam,1981)pp.
282-315.
[7℄ Roland A. Puntigam and Harald H. Soleng, Class.
QuantumGrav.14,1129(1997).
[8℄ A.Vilenkin,Phys.Rev.D23,852(1981).
[9℄ D. V. Gal'tsov and P. S. Letelier, Phys. Rev. D 47,
4273(1993).
[10℄ K.P.Tod,Class.QuantumGrav.11,1331(1994).
[11℄ FernandoMoraes,Phys.Lett.A214,189(1996).
[12℄ ClaudioFurtadoandFernandoMoraes,Phys.Lett. A
188,394(1994).
[13℄ ClaudioFurtadoandFernandoMoraes,Harmoni
Os-illatorInteratingwithConialSingularities,
submit-tedtopubliation.
[14℄ C.Furtadoetal.,Phys.Lett.A195,90(1994).
[15℄ Claudio Furtado and Fernando Moraes, Europhys.
Lett.45,279(1999).
[16℄ Sergio Azevedo, Claudio Furtado and Fernando
Moraes,PhysiaStatusSolidi(b),207,387(1998).
[17℄ A. de Padua, Fernando Parisio-Filho and Fernando
Moraes,Phys.Lett.A238,153(1998).
[18℄ A.P. Balahandran, Varghese John, Arshad Momen
and Fernando Moraes, Int. J. Mod. Phys.A 13, 841
(1998).
[19℄ Sergio Azevedoand Fernando Moraes, Phys. Lett. A
246,374(1998).
[20℄ ClaudioFurtado,FernandoMoraesandV.B.Bezerra,
Phys.Rev.D59,107504(1999).
[21℄ ClaudioFurtado,FernandoMoraesandV.B.Bezerra,
Loop Variables andAharonov-Bohm eet in
Kaluza-Kleintheory, Aeptedfor publiationinMod.Phys.
Lett.A(2000).
[22℄ Sergio Azevedo and Fernando
Moraes,Two-dimensionalsatteringbydislinationsin
monolayer graphite,submittedtopubliation.
[23℄ ClaudioFurtado,V.B.BezerraandFernandoMoraes,
Berry's Quantum Phase in DisloatedMedia,
submit-tedtopubliation.
[24℄ E.R. BezerradeMello, V.B.Bezerra,C.Furtadoand
F.Moraes,Phys.Rev.D51,7140(1995).
[25℄ Claudio Furtadoand Fernando Moraes, Class.
Quan-tumGrav.14,3425 (1997).
[26℄ A. M. de M. Carvalho, Claudio Furtado and
Fer-nandoMoraes,Self-foresoneletriandmagneti
lin-earsoures in the presene of a torsionaldefet,
sub-mittedtopubliation.
[27℄ FernandoMoraes,Phys.Lett.A204,399(1995).
[28℄ IvanPontualand FernandoMoraes,Phil. Mag.A 78,
1073(1998).
[29℄ FernandoMoraes,Mod.Phys.Lett.A10,2335(1995).
[30℄ L.H.BennetandR.E.Watson,Phys.Rev.B35,845
(1987).
[31℄ T.W.B.Kibble,J.Phys.AGen.Phys.9,1387(1976).
[32℄ I. Chuang, N. Turok and B.Yurke, Phys. Rev.Lett.
66,2472(1991).
[33℄ G.E.Volovik,P.Natl.Aad.Si.USA96,6042(1999)
