Lecture#1 Path integral and Mote Carlo simulations

Transcription

1 Lecture# Path ntegral and Mote Carlo smulatons Ken-Ich Ishkaa Hroshma Unv.

2 . Lattce Feld Theory? For eample: Standard Model QCD: Strong nteracton, Hadrons <= quarks, gluons Glasho-Wenberg-Salam: Electroeak QED: electromagnetc nteracton: charged partcles, photon Weak nteracton: Z,W bosons, leptons, Hggs mechansm These are based on Quantum feld theory QFT. Perturbatve analyss usng the couplng constant epanson. Rely on the smallness of the couplng. QCD: at lo-energy, couplng epanson fals. Non-preturbatve analyss s requred.

3 To understand the nature of the strong nteracton among Hadrons from the dynamcs of quarks and gluons, Quantum Chromodynamcs QCD has been ntroduced and nvestgated. QCD s ell under stood n the hgh-energy eperments here the asymptotc-free nature of the couplng constant of QCD enables us the perturbatve epanson analyss. Hoerver, at lo-energy, the perturbatve analyss fals due to the large couplng constant. 3

4 The lattce feld theory s one of the nonpreturbatve analyss method. Lattce QCD has been used and developed to understand the lo-energy nature of Hadrons. The varous technque for lattce feld theory s common and also has been used n LQCD. In ths lecture I ould lke to gve some lattce technque and numercal algorthms for LQCD as an eample of lattce feld theores. 4

5 . Path ntegral and lattce feld theory Feynman's path ntegral quantzaton s a fundamental bass for lattce feld theory. Eucldean feld s also requred to ntroduce ell defned numercally calculable path ntegral formulaton. Lattce QCD s based on SU3 gauge theory defned on a Eucldean 4Dm lattce unverse. 5

6 - Feynman s path ntegral quantzaton A quantum feld theory : S[] : Acton. : Feld to be quantzed real scalar for smplcty. : space-tme corrdnate. Feynman s path ntegral quantzaton. Generatng functonal for Green s functons correlaton func. N-pont Green s functon of the theory. T[ ˆ ˆ Z[ ] D ep S[ ] ˆ n ] n Z[0] Z[0] ep S[ ] We can etract varous nformaton from Green s functons bascally. n D n Z[ ] n n 0 6

7 Hoever, the analytc ntegraton of the path-ntegral s not alays avalable ecept for free feld theores. The ntegral also has a dffculty n Mnkosk metrc. The ntegral s a knd of Fresnel ntegrals and the ntegrant oscllates. Ths may prevents us to evaluate t numercally. In order to evaluate ths ntegral: Introduce Eucldean path ntegral Needs valdaton : Mnkosk Eucld relaton. Epermentally or constructve feld theory, Osteralder-Schrader aoms Dscretze Space-Tme => Lattce space-tme Needs valdaton: lattce spacng error Here e assume: there s a Eucldean feld theory for a target Mnkosk feld theory. 7

8 - Eucldean path ntegral S E [ E ] : Eucldean acton. E E : Eucldean feld. Real valued. E, y, z, : Eucldean 4D coordnate. They are usually obtaned from Mnkosk versons after Wck s rotaton. Generatng functonal for Eucldean Green s functons. Z E [ ] DE ep SE[ E ] E t If the Eucldean acton s real valued, the ntegral has a better property than the Mkosk verson. A chance to evaluate them by numercal ntegraton? The physcs nformaton can be obtaned from Eucldean Green s functons by nverse Wck s rotaton or nvestgatng the tau dependence., 0,0 E E Z [0] E ZE[ ], 0,0 0 Z [0] E D, 0,0ep p E p d E, E 0,0 e Ce E p :loest energy n ths channel ntermedate state. E E E S E [ ]/ E 8

9 -3 Eucldean path ntegral and lattce Path ntegral measure Integraton by feld shape confguraton D E D E Eucldean space tme, E, y, z, s contnuous. Dffcult to mantan DE for numercal evaluaton. Ths ll cause UV dvergences. The renormalzaton and regularzaton s requred. Introduce the lattce dscretzaton: As a regularzaton. As a ell defned ntegraton measure. Degree of Freedom DoF s stll fnte. IR regulator by lmtng system sze fnte volume. 9

10 Lattce a Lattce spacng an Z E a na Latt n Lattce regularzed path ntegral n [ ] DE ep SE[ E ] E D d n Latt n Latt Z Latt [ ] DLatt ep SLatt[ Latt] Multple ntegraton on the vector T Latt n, Latt n, Latt n3 n, n, n, n y, z Latt 4 0

11 Lattce regularzed path ntegral Z Latt [ ] DLatt ep SLatt[ Latt] Latt n, Ho to evaluate ths ntegral? Smlar to Canoncal partton functon n statstcal mechancs. 6666, dm , 536 Multple ntegraton on the vector T Dmenson of s very large. For real scalar on a 4D lattce th the sze If the eght ep s real and non-negatve, e can evaluate t usng Monte Carlo Methods. Note: Lattce acton should be desgned approprately. based on Symmetry, spectrum, relaton Mnkosk Eucld,.. When no real and non-negatve eght s derved, e encounter the sgn problem n the Monte Carlo method. E. System n fnte densty. Latt n n, n, n, n Z d ep S y Latt n, Latt n3 z 4,

12 -4 Integraton usng Monte Carlo Monte Carlo Methods. E. Integraton th a sngle varable. I f b f d a 0, and real valued. f Rectangular ntegraton a b I lm N f b, a / N a f b

13 Random samplng Pck up a number from the nterval [ a, b] randomly. Evaluate functon as f f. 3 Repeat - tmes, then e get samples f, f,, f N. We can estmate the ntegral as N I b a N The random number sequence,,, N has a unform dstrbuton n [a,b]. Ths means that the random varable has the follong probablty densty: Const [ a, b] P 0 [ a, b] Thus the statstcal averagng for f f means lm N N N f N f f P d / P d Denomnator s for the probablty normalzaton. 3

14 A defect and neffcent property of the smple rectangular and random samplng ntegraton. If the target functon f has a keen peak th narro dth W, The ntegraton may fal untl a b / N W s satsfed. One sample n the peak. Most of samples are unmportant. sample rato = /N. f a b W In mult dmensonal ntegratons, the stuaton becomes more orse. D-dmenson D I f d a b / N W for - th drecton TotalSample D Number N N D Only one sample s n the peak. Rato = /N D <<. 4

15 Importance Samplng Monte Carlo As seen before unform samplng s not effectve f the ntegrant has keen peaks. Eucldean Path-ntegral s a knd of huge-mult dmensonal ntegraton. The ntegral has narro peaks n general, and the hghest peak corresponds to the classcal soluton of the system. [0] Latt Latt Latt Latt y DLatt Latt Latt yep ZLatt In the classcal lmt h->0, the domnant contrbuton to the ntegral comes from: A soluton * Latt hch gves the hghest peak of Ths corresponds to the statonary or mnmum soluton of acton: SLatt [ * Latt ] 0 for any varaton. We kno that the classcal soluton gves a narro peak for ep-s * Latt S [ :classcal soluton. ] S [ ep Latt Latt * ]. Unform samplng for φ s not effectve to evaluate path ntegrals. 5

16 Importance Samplng Monte Carlo cont d To ntegrate ths functon f: If e can generate a sequence / ensemble {} so that the statstcal hstogram/dstrbuton of {} s. We have The error behaves as /SqrtN The error s mnmzed hen f=. The error behaves as /SqrtN even for the mult-dmensonal ntegratons. 6 valued. and real 0, f d f I b a f a b averagng. :statstcal lm }: { Dstrbuton of },,,, { 3 f f N d f d f I N N b a b a N / / / N O f N f f f d f I b a

17 Importance samplng for Eucldean pathntegrals. For the to-pont correlaton functon: Generate a sequence/ensemble: d ep S Z0,, So that the sample has the dstrbton : The to-pont correlaton functon can be estmated as: The error behaves as /SqrtN. Note: the dmenson of the ntegral/ s, 3, N C ep S N N k Lattce stes 6 Ho to generate such an ensemble? k 4 k for e. 7

18 Markov Chan Monte Carlo MCMC general descrpton A smple random samplng generaton s not effectve as seen before. A non-random generaton s requred. MCMC Set up. There est : random varable 3 t,,,, : t-stepsequence generated. MCMC adds a ne sample to the sequence as t Generate th a probablty dstrbuton P t t Then add the ne sample to the sequence. 3 t t,,,,, : t stepsequence generated. Where P : Transton probablty t t t t. n MCMC. 8

19 Markov Chan Monte Carlo MCMC cont d Ho to generate the desred dstrbuton from the transton P? Perron-Frobenus theorem. Pφ φ transton probablty can be treated as a matr element hch nde takes a value of state number. state : - th state, P ' P : : transton probablty from - th stateto - th state. The matr P satsfes P s called a postve matr. Perron-Frobenus theorem: P, Any postve real matr has a unque and largest egenvalue th =, and assocated egenvector th postve components. P 0, P, 0 Real postve Probablty. Probablty conservaton. 9

20 Markov Chan Monte Carlo MCMC cont d Usng Perron-Frobenus theorem, e have For a gven ntal state: v 0 : f ntal dstrbuton. thesystems n - th state, v k-step MCMC corresponds to v v 0 k 0 The Perron-Frobenus theorem says that lm v k k Pv P k lm k v, v 0 P k v 0 Pv,, and other componentsare zero. k The convergence to the fed dstrbuton s usually eponental. After many MCMC step the dstrbuton s almost dentcal to the mamum egen vector. If s has the desred dstrbuton e can generate the desred sequence.,, v k here P Pv. 0

21 Markov Chan Monte Carlo MCMC cont d Generate ntal state. If the system s n the -th state, generate -th state th the probablty P 3 Add the ne state to the ensemble. 4 Goto Where e assumed that the state s dscrete and countable, P s a postve matr. Etenson to Non-negatve matres, and contnuum state s also possble. The property that the estence of a unque real mamam egenvalue and postve egenvector of the theorem stll holds, but some specal propertes are requred on P. Here I omt the detals of the etenson. rreducble, No the problem to the path-ntegral s Ho to construct P ' so that the mamum egen vector s C ep S?

22 -5 Detaled Balance Condton Ho to construct Transton probablty P ' to make a desred dstrbuton C ep S? One suffcent condton s the so called detaled balance condton. Recallng that the fed pont dstrbuton s a egenvector of the transton probablty th real unt egenvalue. Egen equaton for transton probablty matr P P or P d ' P ' ' The detaled balance condton requres P P or P ' ' for dscrete statespace for contnuous statespace P ' Problem- Ths s a suffcent condton for egenvector th unt evenvalue. In ths case, P s a reversble Markov chan s a smultaneous left and rght evec.

23 Some MCMC eamples that satsfes the detaled balance condton. Metropols-Hastngs algorthm Metropols et al. 953, Hastngs 970 step 0 Gven ntal setp Generate a step 3 t t, t state step Take net state candtate state t t goto step. 0 as, t 0 th probablty th probablty th probablty t t q t. Where mn, and q q 3

24 Ths algorthm s equvalent to the follong transton probablty P q r mn, Ths transton probablty matr satsfes the detaled balance condton. A More concrete eample for Metropols algorthm. Isng model th spns.,, r kq k k Problem- Z S ep S, 4

25 Isng model th spns. S Z To compute the spn average and spn correlaton Z0 Z0,, We generate the ensemble 0, N,,, th the dstrbuton ep ep Then e can estmate the squared spn average by, ep S ep S Z0 S 0 S tanh N N, N N 5

26 Dstrbuton/ Populaton/ Weght= We have only 4 states. State #, State # 3, State #, State # 4, The eghtprobablty s calculable C s the normalzaton const = Z0 # C ep #3 C ep, # C ep, #4 C ep # # #3 #4 State space We can generate ths dstrbuton th the Metropols Algorthm 6

27 Metropols algorthm for the Isng model th -spns. t step 0 Randomly choose ntal state, t 0 setp Generate a candtate state s th probablty q t / 4 t step Computethe eght mn, ep S s S Step 3 Generate a random real number U from [0,. t step 4 Take net state as t s hen U Accept t otherse Reect step 5 t t, goto step for desred sample numbers. Then e obtan ensemble:,,, 0, Correspondng Fortran Program: [ N Metropols test Metropols accept/reect step 7

28 Correspondng Fortran Program Metropols test Canddate generaton 8

29 Results Weght hstogram Beta dependence of Weght Z ep # # ep ep ep ep / 4 / Cosh ep / 4 / Cosh 4Cosh 9

30 Results Spn average ensemble hstory Frst 500 samples are plotted. Random alkng n the state space 4states Spn average can take one of the values -,0, spn average=0 can occur for state # and #4. spn average=+ occurs for state #. spn average=- occurs for state #3. As ncreasng beta, the state stays at state # or #4. spn average = 0 states. 30

31 Results Spn correlaton ensemble hstory Frst 500 samples are plotted. Random alkng n the state space 4states Spn corr. can take + or -. spn corr.=+ can occur for state # and #3. spn corr= - occurs for state # and #4. At small beta populaton of + and - s almost same. As ncreasng beta, state th spn corr.=- domnates. state # and #4 3

32 Results Spn average epectaton value Averagng the hstory data e obtan zero. Ths s consstent th the theoretcal one. N N 0 3

33 Results Spn correraton epectaton value Averagng the hstory data e obtan TanhBeta. Ths s consstent th the theoretcal one. e e e e 4Cosh Tanh N N 33

34 Dffcultes n Nave Metropols Algorthm As seen before the ne canddate stateconfguraton s really added hen the Metropols test accept. In Statstcal Mechancs Canoncal ensemble, the eponent of the eght s the energy of the target system. The acceptance rato s governed by the Energy dfference S S[ Canddate State] S[Prevous State] mn, ep S[Canddate State] S[Prevous State] mn,ep S When s negatve, Metropols test alays accept the canddate. S S When s postve, the acceptance probablty decreases as ep S. When the target system has a huge number of d.o.f., the random samplng method to generate the canddate state almost alays large postve number for S. Ths s typcal n statstcal mechancs and huge multple dmenson ntegraton. Canddate generaton method th small energy dfference s mportant. See also D-Isng model. Heat-bath Gbbs sampler algorthm 34

35 Most of MCMC algorthms make use of the Metropols algorthm and ts etenson. For LQCD, the system has contnuous varables states. Naïve Metropols algorthm may fal due to the large energy dfferece. Hybrd Monte Carlo HMC algorthm Scalatar, Scalapno, Sugar, PRB34986; Duane, Kennedy Pendleton, Roeth, PL95B987 Ths algorthm s useful hen the varables are contnuous. Ths s an etenson of the Metropols algorthm th better canddate generaton. The HMC algorthm s a de fact standard algorthm for LQCD th dynamcal quarks. In the net lecture I ll descrbe the detals of the HMC algorthm. 35

36 Problems All programs are NO WARRANTY. Check that the detaled balance condton s a suffcent condton of the egenvector statonary dstrbuton of the transton matr. [page ] Check that the transton matr for the Metropols algorthm satsfes the detaled balance condton. [page 4]. 3 Complete the transton matr for the Isng model th -spns n a 44 matr form and Check the egenvector. [page 4-33] 4 Get and comple the Isng model th -spns program. Check the result numercally. [page 4-33] ths needs gfortran and gnuplot on Lnu 5 Evaluate the averaged acceptance rate of the Metropols test hen the energy dfference s a random varable from the Gaussan dstrbuton th mean= and varance=. [Hnt: Complementary error functon] S p S ep [Advanced] Consder a N-stes D Isng model th perodc boundary condton. P P P P P 3 4 P P P P 3 4 P P P P P P P P

37 Backup -ste Scalar model For a contnuous state model. I sho the -ste scalar model. a toy model for lattce scalar feld theores 37 s s s s s ds Z ep ep 0 0 ep 0 s s s s s s ds Z s s s s s s s s ds Z s s s s

38 Metropols algorthm t step 0 ntal state s, t 0 from Gaussan Dstrbuton N[0, var] t setp Generate a candtate state s from Gaussan Dstrbuton N[ s, var] t t s s Ths corresponds to q s s ep var var t step Computethe eght mn, ep S s S s Step 3 Generate a random real number U from [0,. t step 4 Take net state s as t s hen U Accept s t s otherse Reect step 5 t t, goto step for desred sample numbers.. We do not have a fnte dstrbuton at beta= th ths model. s We can not use unform samplng for canddate generaton because. 38

39 Fortran program: [ 0,000,000 samples are generated. But e save 0,000 samples th nterval 00. We use var= for canddate generaton. State eght/hstogram generated va Metropols algorthm Measured Theoretcal 39

40 State eght/hstogram generated va Metropols algorthm 40

41 State eght/hstogram generated va Metropols algorthm 4

42 State eght/hstogram generated va Metropols algorthm 4

43 State eght/hstogram generated va Metropols algorthm 43

44 Spn average and Spn correlaton hstory generated va Metropols algorthm Spn average Spn correlaton 44

45 Beta dependence of Spn average and Spn corr. Spn average Spn correlaton All programs are NO WARRANTY. 45

46 Metropols algorthm transton probablty for P s s' s, s' q s s' r s' s s' s, s' mn, ep ' r s ds s, s' q s s Delta functon S s S s', q s s' s' ep var s s ' s s' var 46

47 mn, and q q P P Problem ansers P f P log mn, f f because P f f f f f f f f f f P Smlarly P P 47

48 3 I sho beta>0 case only. Egenpars Thus MCMC converges to the desred dstrbuton for th e y y y y y y y P. th, y y y. th 0 0, 0 0 y 0 th, 3 3 Desred dstrbuton lm P N v N The convergence rate s governed by the dfference beteen and net largest egenvalue.

49 5 complementary error functons: erf erfc erf 0 e Averagng acceptance probablty: t dt e t dt error functon. complementary error functon. P acc S S 4 S mn, e e erfc 4 49

50 Hstory 03/0/3: Metropols test probablty s corrected. mn, and q q Wrong mn, and q q Rght 50