周光辉-syk模型学习(20170318)

AReviewofSachdev-Ye-KitaevModelandAd𝑆2/CF𝑇113物理周光辉WhatisAdS?•Theanti-deSitterspaceofsignature(p,q)canbeisometricallyembeddedinthespace𝑅𝑝,𝑞+1with(x1,...,xp,t1,...,tq+1)andthemetric𝑑𝑠2=𝑑𝑥𝑖2𝑝𝑖=1−𝑑𝑡𝑗2𝑞+1𝑗=1asthequasi-sphere𝑥𝑖2𝑝𝑖=1−𝑡𝑗2𝑞+1𝑗=1=−𝑎2Whereaisanonzeroconstant.Symmetry:O(p,q+1)WhatisCFT?•Aconformalfieldtheory(CFT)isaquantumfieldtheorythatisinvariantunderconformaltransformations.•Conformalsymmetryisasymmetryunderscaleinvarianceandunderthespecialconformaltransformationshavingthefollowingrelations𝑃μ,𝑃𝜈=0[D,𝐾μ]=−𝐾μ[D,𝑃μ]=𝑃μ[𝐾𝜈,𝐾μ]=0[𝐾𝜈,𝑃μ]=𝑔μ𝜈𝐷−𝑖𝑀μ𝜈wheregeneratestranslations,generatesscalingtransformationsasascalarandgeneratesthespecialconformaltransformationsasacovariantvectorunderLorentztransformation.Modelsofholography•Matrixmodelsaretoodifficulttobeexplicitlysolvable,whilevectormodelsaretoosimpletohavethesamerichproperties.Weneedamodelliesinbetween-SYK.•AtlargeN,thedominantFeynmandiagramsformatrixmodelsareplanardiagrams,whereasthedominantdiagramsforvectormodelsarebubblediagrams.TheSYKmodelisdominatedbyanewclassofFeynmandiagrams,namedsunsetorwatermelondiagrams.SYKItcontainsNMajoranafermionswiththeanticommutationrelation𝜒𝑖,𝜒𝑗=δ𝑖𝑗Theactionis𝑆=𝑑τ(12𝜒𝑖𝑑𝑑τ𝑁𝑖=1𝜒𝑖+(𝑖)𝑞2𝑞!𝐽𝑖1,…𝑖𝑞𝜒𝑖1…𝜒𝑖𝑞𝑁𝑖1,…𝑖𝑞=1),wherethecoupling𝐽𝑖1,…𝑖𝑞istotallyantisymmetricand,foreach𝑖1,…𝑖𝑞ischosenfromaGaussianensemble.•Thetwopointfunctionofthe𝐽𝑖1,…𝑖𝑞istakentobe,1𝑞−1!𝐽𝑖1,…𝑖𝑞𝐽𝑖1,…𝑖𝑞𝑁𝑖2,…𝑖𝑞=1=𝐽2Atleadingorderin1/N,itisequivalenttothesimplernormalization,𝐽𝑖1,…𝑖𝑞𝐽𝑖1,…𝑖𝑞=𝑞−1!𝐽2𝑁𝑞−1.Atzerocoupling,theEuclideantwopointfunction𝑇𝜒𝑖(τ)𝜒𝑗(0)≡𝐺(τ)δ𝑖𝑗isgivenby,𝐺0τ=12𝑠𝑔𝑛(τ),𝐺0ω=𝑖ω•𝐺0τ=12𝑠𝑔𝑛(τ),𝐺0ω=𝑖ω𝐺0τ=𝑖ω𝑒−𝑖ωτ𝑑ω𝐼𝑛𝑓−𝐼𝑛𝑓.𝐷𝐹(𝑥−𝑦)≡𝑑4𝑝(2π)4𝑖𝑝2−𝑚2+𝑖ε𝑒−𝑖𝑝.(𝑥−𝑦).Toleadingorderin1/N,G(ω)−1=𝐺0(ω)−1−𝛴ω=−𝑖ω−𝛴ω,𝛴τ=𝐽2𝐺(τ)𝑞−1Atstrongcoupling,𝐽τ≫1,wecandroptheterm𝑖ω,toget,Gω𝛴ω=−1,𝛴τ=𝐽2𝐺(τ)𝑞−1.Wecanverifythat,𝐺τ=𝑏𝑠𝑔𝑛(τ)|𝐽τ|2△isasolutiononetakes△=1𝑞,𝑏𝑞=12π1−2△tan(π△).Resf(a)=1𝑛−1!𝑑𝑛−1𝑑𝑧𝑛−1𝑧−𝑎𝑛𝑓𝑧|𝑧⟶𝑎𝑓(𝑥)𝑒𝑖𝑚𝑥𝑑𝑥𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑦−𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑦=2π𝑖𝛴Res(𝑓(𝑥)𝑒𝑖𝑚𝑥,x=a),when𝑧⟶𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑦,𝑓𝑧⟶0,𝑚0.Forq=4,△=14,𝑏4=14π,𝐺τ=(14π)14𝑠𝑔𝑛(τ)|𝐽τ|12,𝛴τ=(14π)34𝐽2𝑠𝑔𝑛(τ)|𝐽τ|32.Forconvenience,wecanignorethoseconstantfactors,andonlyneedtoverifythatGω𝛴ωisindependentofω.Gω=𝑠𝑔𝑛(τ)|τ|𝐼𝑛𝑓−𝐼𝑛𝑓𝑒𝑖ωτ𝑑τ=1τ𝑒𝑖ωτ𝑑τ+−1τ𝑒−𝑖ωτ𝑑τ𝐼𝑛𝑓0𝐼𝑛𝑓0=2𝑖τsin(ωτ)𝑑τ𝐼𝑛𝑓0Similarly,𝛴ω=2𝑖τ32sin(ωτ)𝑑τ𝐼𝑛𝑓0.Usingmathematica,TheFouriertransformof𝐺τ,is𝐺ω=𝑏𝜓△𝐽−2△𝜔2△−1𝑠𝑔𝑛ωWhere𝜓△≡2𝑖π2−2△𝛤(1−△)𝛤(12+△).EquationGω𝛴ω=−1,𝛴τ=𝐽2𝐺(τ)𝑞−1,isinvariantunderreparameterizationoftime,τ⟶𝑓τ,𝐺(τ)⟶𝑓`𝜏△𝐺(𝑓(τ)),𝛴(τ)⟶𝑓`𝜏△(𝑞−1)𝛴(𝑓(τ))Twopointfunctions•𝛺|𝑇(𝜓𝑖(𝑡)𝜓𝑖(0))|)|𝛺=0|𝑇(𝜓𝑖(𝑡)𝜓𝑖(0)){1−𝑖𝐻𝑡1𝑑𝑡1+−𝑖−𝑖2!𝐻𝑡2𝐻𝑡3𝑑𝑡2𝑑𝑡3+⋯}|0.因为全反对称𝐽𝑖1,…𝑖𝑞的限制使得只有展开式的偶次项可以存在。Feynmanrule类似于𝜙4，传播子𝐷𝑡=12𝑠𝑔𝑛(𝑡).1PI•One-particleirreduciblediagramisanydiagramthatcan`tbesplitintwobyremovingasingleline.•Thefouriertransformofthetwopointfunctioncannowbewrittenas𝑑4𝑥𝛺|𝑇𝜓(𝑥)𝜓(𝑥)|𝛺𝑒𝑖𝑝𝑥=𝐺ω=𝑖ω1−𝑖ω𝛴(ω),G(ω)−1=𝐺0(ω)−1−𝛴ω=−𝑖ω−𝛴ω(2.6)•同理我们可以得到G（t）与𝛴（t）的关系𝛴τ=𝐽2𝐺(τ)𝑞−1（2.6）2.3Theconformallimit2.4Largeqlimit2.5q=22.6Computingtheentropy2.7Correctiontotheconformalpropagator3Fourpointfunctions3.1Theladderdiagrams3.2Usingconformalsymmetry就是把自由两点函数变成有作用量下的conformal极限下的两点函数3.2.1Thefourpointfunctionasafunctionofthecrossratio3.2.2Eigenfunctionsofthecasimir由Casimir算符列出的微分方程是二阶线性常微分方程，若在（0，2）内求解，可发现其在0和1处有奇点。使用正则奇点领域的级数解法，以1为奇点，1为半径进行计算。可得到以下递推关系式，3.2.3Theeigenvaluesofthekernelkc(h)3.2.5Thesumofallladders3.2.6Operatorsofthemodel3.3Propertreatmentoftheh=2subspace3.3.1Theh=2eigenfunctionsandreparameterizations