Statistical Field Theory - An Introduction to Exactly Solved Models in Statistical Physics PDF
Preview Statistical Field Theory - An Introduction to Exactly Solved Models in Statistical Physics
OUPCORRECTEDPROOF – FINAL,21/2/2020,SPi STATISTICAL FIELD THEORY OUPCORRECTEDPROOF – FINAL,21/2/2020,SPi OUPCORRECTEDPROOF – FINAL,21/2/2020,SPi Statistical Field Theory An Introduction to Exactly Solved Models in Statistical Physics Second Edition Giuseppe Mussardo SISSA (Scuola Internazionale Superiore di Studi Avanzati),Trieste 1 OUPCORRECTEDPROOF – FINAL,21/2/2020,SPi 3 GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries ©GiuseppeMussardo2020 Themoralrightsoftheauthorhavebeenasserted FirstEditionpublishedin2010 SecondEditionpublishedin2020 Impression:1 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressControlNumber:2019954655 ISBN978–0–19–878810–2 DOI:10.1093/oso/9780198788102.001.0001 Printedandboundby CPIGroup(UK)Ltd,Croydon,CR04YY OUPCORRECTEDPROOF – FINAL,21/2/2020,SPi Ulrichthoughtthatthegeneralandtheparticulararenothing buttwofacesofthesamecoin. RobertMusil,TheManWithoutQualities OUPCORRECTEDPROOF – FINAL,21/2/2020,SPi OUPCORRECTEDPROOF – FINAL,21/2/2020,SPi Preface to the first edition Thisbookisanintroductiontostatisticalfieldtheory,animportantsubjectoftheoretical physicsthathasundergoneformidableprogressinrecentyears.Mostoftheattractive- nessofthisfieldcomesfromitsprofoundinterdisciplinarynatureanditsmathematical elegance; it sets outstanding challenges in several scientific areas, such as statistical mechanics,quantumfieldtheory(QFT)andmathematicalphysics. Statistical field theory deals, in short, with the behaviour of classical or quantum systemsconsistingofanenormousnumberofdegreesoffreedom.Thosesystemshave different phases, and the rich spectrum of the phenomena they give rise introduces several questions: What is their ground state in each phase? What is the nature of the phase transitions? What is the spectrum of the excitations? Can we compute the correlationfunctionsoftheirorderparameters?Canweestimatetheirfinitesizeeffects? Anidealguidetothefascinatingareaofphasetransitionsisprovidedbytheremarkable Isingmodel. There are several reasons to choose the Ising model as a pathfinder in the field of critical phenomena.The first one is its simplicity—an essential quality to illustrate the key physical features of the phase transitions, without masking their derivation with worthlesstechnicaldetails.IntheIsingmodel,thedegreesoffreedomaresimpleboolean variablesσ(cid:2),whosevaluesareσ(cid:2)=±1,definedonthesitesi(cid:2)ofad-dimensionallattice.For i i theseessentialfeatures,theIsingmodelhasalwaysplayedanimportantroleinstatistical physics,bothatpedagogicalandmethodologicallevels. However,thisisnottheonlyreasonofourchoice.ThesimplicityoftheIsingmodel is, in fact, quite deceptive. Despite its apparent innocent look, the Ising model has shownanextraordinaryabilityindescribingseveralphysicalsituationsandaremarkable theoretical richness.For instance,the detailed analysis of its properties involves several branches of mathematics,quite distinguished for their elegance:here we mention only combinatoric analysis, functions of complex variables, elliptic functions, the theory of non-linear differential and integral equations,the theory of the Fredholm determinant, and,finally,the subject of infinite dimensional algebras.Although this is only a partial list,itissufficienttoprovethattheIsingmodelisanidealplaygroundforseveralareas ofpureandappliedmathematics. Equallyrichisitsrangeofphysicalaspects.Therefore,itsstudyoffersthepossibility to acquire a rather general comprehension of the phase transitions. Phase transitions areremarkablecollectivephenomena,characterizedbysharpanddiscontinouschanges of the physical properties of a statistical system. Such discontinuities typically occur at particular values of the external parameters (e.g. temperature or pressure); close to these critical values, there is a divergence of the mean values of many thermodynam- ical quantities, accompanied by anomalous fluctuations and power law behaviour of OUPCORRECTEDPROOF – FINAL,21/2/2020,SPi viii Prefacetothefirstedition correlation functions. From an experimental point of view, phase transitions have an extremelyrichphenomenology,rangingfromthesuperfluidityofcertainmaterialstothe superconductivity of others, from the mesomorphic transformations of liquid crystals to the magnetic properties of iron. For example, liquid helium He4 shows exceptional superfluidpropertiesattemperatureslowerthanT =2.19K,whileseveralalloysshow c phasetransitionsequallyremarkable,withanabruptvanishingoftheelectricalresistance forverylowvaluesofthetemperature. The aim of the theory of phase transitions is to reach a general understanding of all the phenomena mentioned here on the basis of a few physical principles. Such a theoretical synthesis is made possible by a fundamental aspect of critical phenomena: their universality. This is a crucial property that depends on two basic features: the internalsymmetryoftheorderparametersandthedimensionalityofthelattice.Inshort, this meansthat,despitethedifferencesthattwosystemsmayhaveattheirmicroscopic level, as far as they share the two features mentioned, their critical behaviours are surprisingly identical.1 It is for these universal aspects that the theory of the phase transitionsisoneofthepillarsofstatisticalmechanicsand,simultaneously,oftheoretical physics.Asamatteroffact,itembracesconceptsandideasthatprovedtobethebuilding blocks of the modern understanding of the fundamental interactions in nature. The universalbehaviour,forinstance,hasitsnaturaldemonstrationwithinthegeneralideas of the renormalization group, while the existence itself of a phase transition can be interpretedasaspontaneouslysymmetrybreakingoftheHamiltonianofthesystem.As well known,both are common concepts in QFT,another important area of theoretical physics,i.e.thetheorythatdealswiththefundamentalinteractionsofthelastconstituents ofthematter—theelementaryparticles. Therelationshipbetweentwotheoriesthatdescribessuchdifferentphenomenamay appear, at first sight, quite surprising. However, it becomes more comprehensible if we take into account two aspects: first, that both theories deal with systems of infinite degrees of freedom, and second, that, close to the phase transitions, the excitations of thesystemshavethesamedispersionrelationsoftheelementaryparticles.2 Duetothe essential identity of the two theories, we should not be surprised to discover that the two-dimensional Ising model, at temperature T slightly away from T and in absence c of an external magnetic field,is equivalent to a fermionic neutral particle (a Majorana fermion),whichsatisfiesaDiracequation.Similarly,atT =T butinthepresenceofan c externalmagneticfieldB,thetwo-dimensionalIsingmodelmayberegardedasaQFT witheightscalarparticlesofdifferentmasses. The use of QFT, i.e. those formalisms and methods that led to brilliant results in thestudyofthefundamentalinteractionsofphotons,electronsandallotherelementary particles,hasproducedremarkableprogressbothintheunderstandingofphasetransi- tionsandinthecomputationoftheiruniversalquantities.Asexplainedinthisbook,our 1 Thisbecomesevidentbychoosinganappropriatecombinationofthethermodynamicalvariablesofthe twosystems. 2 Theexplicitidentificationbetweenthetwotheoriescanbeprovedbyadoptingforboththepathofintegral formalism. OUPCORRECTEDPROOF – FINAL,21/2/2020,SPi Prefacetothefirstedition ix studysignificantlybenefitsfromsuchapossibility:sincephasetransitionsarephenomena that involve the long distance scales of the systems—the infrared scales—the adoption of the continuum formalism of field theory is not only extremely advantageous from a mathematical point of view,but also is perfectly justified from a physical point of view. By adoptingthe QFT approach,the discretestructureof the original statisticalmodels showsitselfonlythroughanultravioletmicroscopicscale,relatedtothelatticespacing. However,itisworthpointingoutthatthisscaleisabsolutelynecessarytoregularizethe ultravioletdivergenciesofQFTandtoimplementitsrenormalization. The main advantage of QFT is that it embodies a strong set of constraints coming from the compatibility of quantum mechanics with special relativity. This turns into general relations, such as the completeness of the multi-particle states or the unitarity oftheirscatteringprocesses.Thankstothesegeneralproperties,QFTmakesitpossible to understand,in a very simple and direct way,underlying aspects of phase transitions that may appear mysterious, or at least not evident, in the discrete formulation of the correspondingstatisticalmodel. There is one subject that has particularly improved thanks to this continuum for- mulation: the set of two-dimensional statistical models, for which we can achieve a classification of the fixed points and a detailed characterization of their classes of universality.Letusbrieflydiscussthenatureofthetwo-dimensionalQFTs. Right at the critical points, the QFTs are massless. Such theories are invariant under the conformal group,i.e.the set of geometrical transformations that implement a scaling of the length of the vectors while preserving their relative angle. But, in two dimensionsconformaltransformationscoincidewithmappingsbyanalyticfunctionsof acomplexvariable,characterizedbyaninfinite-dimensionalalgebraknownasVirasoro algebra. This enables us to identify first the operator content of the models (in terms of the irreducible representations of the Virasoro algebra) and then to determine the exact expressions of the correlators (by solving certain linear differential equations). In recent years,thanks to the methods of conformal field theory CFT,physicists have reached the exact solutions of a huge number of interacting quantum theories, with thedeterminationofalltheirphysicalquantities,suchasanomalousdimensions,critical exponents,structureconstantsoftheoperatorproductexpansions,correlationfunctions, partitionfunctions,etc. Awayfromcriticality,QFTsare,instead,generallymassive.Theiranalysiscanbeoften carriedoutonlybyperturbativeapproaches.However,therearesomefavourablecases that give rise to integrable models of great physical relevance. The integrable models are characterized by the existence of an infinite number of conserved charges.In such fortunatecircumstances,theexactsolutionoftheoff-criticalmodelscanbeachievedby means of S-matrix theory.This approach allows us to compute the exact spectrum of the excitations and the matrix elements of the operators on the set of these asymptotic states. Both these data can be thus employed to compute the correlation functions by the spectral series. These expressions enjoy remarkable convergence properties that turn out to be particularly useful for the control of their behaviours both at large and short distances. Finally, in the integrable cases, it is also possible to study the exact thermodynamical properties and the finite size effects of the QFTs. Exact predictions
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