Table Of ContentA Primer to the Theory of Critical Phenomena
A Primer to the Theory
of Critical Phenomena
Jurgen Honig
PurdueUniversity
WestLafayette,IN,USA
Józef Spałek
JagiellonianUniversity
Kraków,Poland
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Preface
Thisbookisintendedtobeanintroductiontothebasictheoreticalbackground
forthegeneralareaofcriticalphenomenatakingplacenearacontinuousphase
transition.Thesephenomenaencompassasingularbehaviorofphysicalproper-
ties,thatis,thebehaviorofmeasurablephysicalquantitiesthatmaytakeinfinite
values at the critical point (e.g., at the critical temperature). Although many
books and review articles already deal with this topic, they are often written
atanadvancedlevelofmathematicalsophistication.Wethereforeattemptedto
provide the necessary background information readers will need to study the
advanced presentations. Particular emphasis is placed on developing the con-
ceptoftheorderparameterandonasystematicapproachstartingfromLandau
mean-fieldtheory.
We generally have tried to keep the discussion at a reasonably elementary
level.Somefamiliaritywiththeelementsofquantummechanics,Fourierseries
andtransforms,andcomplexvariablesisassumed.However,wehavenothes-
itatedtoincludesomespecialtytopicsrelevanttoouraim;thismoreadvanced
∗
material,markedwith( ),canbeomittedonafirstreading.Also,torenderthe
individual chapters more self-contained, we have repeated some presentations
indifferentchapters.Althoughwehavetriedtobecareful,nothingcaneverbe
expectedtobeerror-free.Wewillthereforeappreciatebeinginformedwherewe
havebeenremiss.
It remains to thank Elsevier personnel for their great patience in bringing
thisbooktoproduction.ParticularthanksareduetoDr.DanutaGoc-Jagłofor
her skills in reconfiguring part of the original Word text into the LATEX format
andeditingthefigures.
This project was partly supported by The National Science Centre (NCN)
throughGrantMAESTRO,No.DEC-2012/04/A/ST3/00342.
JurgenM.Honig
JózefSpałek
WestLafayette–Kraków,2016–2017
xi
Chapter 1
Introduction: Classical Phases
and Critical Points
Phasetransitions,andcriticalphenomenainparticular,remainamongthemost
spectacularpropertiesinphysics,materialsscience,andinotherdisciplines.At
thetransitionmaterialsinthiscategorydisplaymathematicalsingularities(in-
finities) in measurable physical quantities such as the specific heat, the bulk
compressibility,orthemagneticsusceptibility.Whatisequallyimportant,when
thecriticalpointisapproachedthesesystemsexhibitinfinitefluctuationsatall
scales. At the same time, a surprising universality of these phenomena is ob-
served,whichareassembledintoso-calleduniversalityclasses.Inthischapter,
elementary experimental observations are briefly summarized to exemplify the
principalcharacteristicsofdiscontinuousvs.continuousclassicalphasetransi-
tions,i.e.,thosedrivenbythermodynamicfluctuationsoftheorderingprocess,
asillustratedmorepreciselyinlaterchapters.
1.1 PHYSICAL EXAMPLES: MACROSCOPIC PROPERTIES
We start with a macroscopic characterization of phase transitions. Everyday
life is heavily dependent on the presence of water in its three manifestations
as vapor, liquid, and solid. Each of these is referred to as a phase, a state of
matterwithuniformpropertiesatequilibrium,thatextendoveralargeregionof
space.1Thephasesaredefinedbytheirbordersatwhichthemeasurablephysical
propertiesexhibiteithersingularitiesordiscontinuities.ThetemperatureT and
pressureP canbeindependentlyvariedoverawiderangeofconditionswithout
affectingtheoverallappearanceofeachphase.However,thereexistconditions
wheretwoofthesephasesdocoexist,undercircumstanceswhereT andP can
nolongerbeindependentlyaltered.Thissituationisdisplayedpictoriallyviaa
P–T phasediagramforwater,asshownonalog–logscaleinFig.1.1Aandona
semilogarithmicscaleinFig.1.1B.Thecurvesrepresentthephaseboundaries,
i.e.,theP–T dependencethatmustbemetforthephasecoexistence.Ofspecial
relevancetotheentiretopicofthisbookistheL-Gbranchthatterminatesatthe
so-calledcriticalpointCPwithT (cid:2)647KandP (cid:2)218atm,wherethemenis-
c c
1. Thedivisionintothreephasesisamatterofawell-testedconvention,inwhichsnowandclouds
areexcludedasseparatephases.
APrimertotheTheoryofCriticalPhenomena.http://dx.doi.org/10.1016/B978-0-12-804685-2.00001-2
Copyright©2018ElsevierInc.Allrightsreserved. 1
2 APrimertotheTheoryofCriticalPhenomena
FIGURE1.1 (A)Schematicphasediagramforwater,depictingtherangeofstabilityofsolid(S),
liquid(L),andvapor(G).CPdenotesthecriticalpoint;thetriplepoint(TP)wherethethreephases
coexistisalsomarked.Thissketchservesasatemplateforphasediagramsforahugevarietyof
materialspresentundernormalconditions.(B)Samephasediagramforwateronasemilogarithmic
scale.
cusseparatingliquidfrom vapordisappearsentirely, indicativeof themerging
ofliquidandvaporintoasingleentitytermedacriticalfluid.Justaheadofthis
event the system changes from a translucent to a cloudy (misty) appearance;
thisisduetoalmostuninhibitedfluctuationsindensitythatgeneratecondensed
clusters of size comparable to or larger than the wavelength of visible light,
whichscatterincidentradiation.Forcompleteness,wementionthetriplepoint
in the phase diagram (T =273.16 K, P =6.03·10−3 atm), at which all the
threephasescoexist.
Anotherindicatoroftheapproachofwatertocriticalconditionsisthechange
initslatentheatofvaporizationLasthetemperatureisraised.Theenthalpyof
vaporization L=2.25 kJ/g at 273 K reflects the discontinuity in all physical
propertiesofwaterastheliquidisconvertedtovapor.Thisprocessisgenerally
referredtoasafirst-order(discontinuous)phasetransition.Fig.1.2displaysthe
progressive decrease of L with rising T that terminates close to T =647 K,
c
where the curve plunges to zero with infinite slope. Concomitantly, with ris-
ing temperature, the density of water diminishes, whereas that of steam rises
untilthetwomatchatT .Oncrossingthattemperature,thedensityofthesys-
c
Introduction:ClassicalPhasesandCriticalPoints Chapter | 1 3
FIGURE1.2 LatentheatofvaporizationLofwaterasafunctionoftemperatureT atthecritical
pressurePc.Theendsolidpointmarksthecriticalpointatwhichthespecificheatissingular,not
justdiscontinuousascharacterizedbyalatentheatL.
tem changes continuously, but other properties exhibit singularities, a process
referredtoasacontinuousphasetransition.2Theprecisemannerofthiscoales-
cenceisoneinstanceofthemoregeneralstudyofcontinuousphasetransitions
inwhichweshallbeinterestedatlength.
Adifferenthallmarkofcriticalityistheexistenceoflargespatialscalefluc-
tuations.Thisreflectstheinherentinstabilityofthesystemclosetocriticality,to
be documented in later chapters. Under these conditions, water and steam can
beinterconvertedwithnegligiblechangesintheenthalpy.Theprocessresultsin
the presence of islands of water of widely variable size coexisting with steam
asthecriticalpointisapproached.Theconstantbombardmentofthecondensed
islands by gaseous water molecules gives rise to wild alterations in the local
densityofthesystem.
The above discussion obviously applies with appropriate changes to other
systems.Fig.1.1Athenbecomesasortofgenericphasediagramthatdisplays
the P–T dependence for a huge variety of materials, for which T /P takes
c c
valuesthatrangefrom5.2K/2.3atmforHeto647K/218atmforwater.
Itmaycomeasasurprisethatcompletelydifferentphysicalsystemsexhibit
analogous if not exactly the same critical phenomena. Prominent among these
areferromagneticmaterials.Fig.1.3displaysthespontaneousmagnetizationM
of metallic iron and nickel as a function of temperature T relative to that for
2. Forhistoricreasons,thisisalsoreferredtoasasecond-orderphasetransition,butthisdesignation
hasnowfallenintodisuse,aswellastheterm“first-orderphasetransition.”Bothofthesetermswere
introducedbyPaulEhrenfest(1935)onphenomenologicalgrounds.Theyhavebeenreplacedbythe
notionofdiscontinuousvs.continuousphasetransitionsintroducedbyLandau(1936),basedonthe
conceptoftheorderparameteranditschangesatphasetransition;theselatterconceptswilllaterbe
elaboratedonindetail.
4 APrimertotheTheoryofCriticalPhenomena
FIGURE1.3 SchematicrepresentationoftherelativespontaneousmagnetizationofimpureFeand
Niasfunctionsoftemperature.NotethesteepdeclineclosetotheirrespectiveCurietemperatures
Tcwithaninfiniteslopeatthatpoint.
T →0K.Ignoringcomplicationsarisingfrommagneticdomainstructures,we
note with rising T a continuous decrease in M, followed by a steep decrease
withinfiniteslopeatM=0,i.e.,attheCurietemperatureT =T (1043Kand
c
631 K for iron and nickel, respectively). Above this temperature, the material
is in the paramagnetic, nanopolarized regime. The above reflects the gradual
change with T in atomic spin alignments. Within a single domain, all equili-
brated atomic moments are aligned at T =0 K; with rising T, due to thermal
fluctuations of the magnetic moment, dynamic islands of misaligned clusters
maketheir appearance;as T →T these growin size andnumber, therebydi-
c
minishing the overall magnetization.3 Near T , these islands compete in size
c
andprevalencewiththeoriginalspinconfiguration;infact,closetoT ,ween-
c
counter an enormous (infinite) range of size distributions of the continuously
fluctuatingalignedandrandomspinconfigurations,whichthencloseinonthe
paramagneticregimeabove T .Itturnsoutthatthetheoreticalanalysisforthe
c
critical characteristics of both classes of materials is the same. This certainly
shouldpiqueourinterest.Laterweintroducetheso-calleduniversalityclasses
ofcriticalbehavior.
Another feature is worth noting: Fig. 1.4 shows a set of liquid–gas coexis-
tencecurvesforeightdistinctfluidsneartheirrespectivecriticalpointsasplots
of the scaled temperature T/T vs. the scaled density ρ/ρ The collapse of
c c
thesedataformaterialswithsuchwidelyvaryingpropertiesontoasinglecurve
isimpressive,alertingusalsotosomekindofuniversalitythatcallsforanex-
planation.Furthermore,closetoT ,thedensitiesfortheliquidandgasconverge
c
on each other according to the relation |ρ−ρ |∼|T −T|β with β ≈0.326,
c c
ratherthanwithβ=0.5,asispredictedfromthevenerablevanderWaalsequa-
3. Strictlyspeaking,forT (cid:6)Tccollectivewaveexcitations–spinwaves(magnons)aretheelemen-
taryexcitations.Eachofthesecarriesaunitμofthereversedmagneticmoment.AsT approaches
Tcthesegraduallytransformintomisalignedclustersofvariablesize.AtTcitisdifficulttoidentify
suchclusters,astheycanbeofanysize.
Introduction:ClassicalPhasesandCriticalPoints Chapter | 1 5
TABLE 1.1 Values at critical L→G point
for selected systems. In the last row, the
characteristicsofliquid4Heatitsselected
boilingpointareprovidedforcomparison.
4Heisaquantumliquid
System Tc(K) Pc(atm) ρc(gcm3)
H2O 647.5 218.5 0.325
CO2 304.2 72.8 0.46
O2 154.6 49.7 0.41
Xe 189.8 57.6 1.105
Ar 150.8 48.3 0.53
4He 4.2 1 0.125
FIGURE1.4 PlotoftherelativetemperatureT/Tc vs.thescaleddensityρ/ρc ofeightdifferent
materialsunderconditionsclosetotheircriticaltemperaturesfortheliquid–gastransition.Thegas
phaseisrepresentedbytheleftpartofthediagram.Notethecollapseofallthedataontoasingle
curve.[AdaptedfromE.A.Guggenheim,J.Chem.Phys.13,253(1945).]
tionofstate.Thisdiscrepancyinβ initiallydidnotelicitmuchinterest,butlater
becameoneofthecatalyststhatprecipitatedtheentirestudyofcriticalphenom-
ena. We should note that in Fig. 1.4 the relative pressure P/P is determined
c
fromtheequationofstate.Theselectedvalues(T ,P ,ρ )atthecriticalpoint
c c c
are listed in Table 1.1. The universal plot obtained in ρ/ρ , T/T , and P/P
c c c
iscalledtheGuggenheimphasediagram,illustratingtheso-calledprincipleof
correspondingstates(Guggenheim,1945).
6 APrimertotheTheoryofCriticalPhenomena
1
FIGURE1.5 Experimentalscaledspontaneousmagnetizationdata(MdividedbyBδ)vs.scaled
temperatureacrossthecriticalpointforCrBr3,EuO,Ni,YIG,andPd3Fe.Thecollapseofthedata
intoasinglecurveisnoteworthy[afterH.E.Stanley,Rev.Mod.Phys.71,S358(1999)].
An analogous situation obtains for magnetic materials. In Fig. 1.5 we
show curves depicting the variation of the scaled magnetization M/B1/δ vs.
(T −T )/T forfivedifferentmaterialsofwidelydifferentmagneticcharacter-
c c
istics. The data collapse effect is apparent; note the asymptotic descent to the
paramagneticregime(M→0)fortemperatureswellaboveT .
c
1.2 SINGULARITY OF THE SPECIFIC HEAT
To demonstrate explicitly the divergence of the physical quantities at a con-
tinuous phase transition, we have plotted in Fig. 1.6A the overall temperature
dependence of the specific heat for a normal-liquid to superfluid transition in
condensed 4He. The singularity appears for T =T (cid:2)2.17 K. This transition
λ
iscalledtheλtransitioninliquid4Hesincethedataresembletheshapeofthe
Greekletterλ.Toshowexplicitlythatthisdependenceexhibitsamathematical
singularity,wehavereplottedinFigs.1.6B–Dthesamedependenceasafunc-
tionofT −T forgraduallynarrowerrangesintemperatureclosetoT .Wecan
λ λ
clearly see that, even at the microKelvin scale, the data exhibit the same type
ofsingularity,namelyaln|T −T |dependence4 withanadjustableconstantof
c
proportionality.
4. Strictlyspeaking,thefunctionisA±−Bln|T −Tc|,whereA+=−0.65forT >Tλ,A−=4.55
forT <Tλ,andB=−3.00inunitsofJ/molK(fromM.I.BuckinghamandW.M.Fairbank,Prog.
LowTemp.Phys.3,80(1961)).
