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Six lectures on lattice field theory Stone, Michael Nov 30, 1983

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TR IUMFS I X  LECTURES  ON L A T T I C E  F I E L D  THEORY  M i c h a e l  S t o n eL o o m i s  L a b o r a t o r y  o f  P h y s i c s ,  U n i v e r s i t y  o f  I l l i n o i sL e c t u r e s  g i v e n  a t  T R I U M F ,  May  2 4 - J u n e  6 ,  1 9 8 3MESON F A C I L I T Y  OF :U N I V E R S I T Y  OF ALBERTA  S IMON  FRASER U N I V E R S I T Y  U N I V E R S I T Y  OF V I C T O R I A  U N I V E R S I T Y  OF B R I T I S H  COLUMB I A T R I - 8 3 - 2TRIUMFTRI-83-2SIX LECTURES ON LATTICE FIELD THEORY Michael StoneLoomis Laboratory of Physics, University of IllinoisLectures given at TRIUMF, May 24-June 6, 1983Postal address:TRIUMF4004 Wesbrook Mall Vancouver, B.C.Canada V6T 2A3 November 1983CONTENTSIntroductionLecture 1Scalar fields - free and interacting Lecture 2Continuum limit, restoration of rotational symmetry and asymptotic freedomLecture 3Fractals, scaling and renormalization Lecture 4Gauge fields, Monte-Carlo and string tensionLecture 5 FermionsLecture 6Chiral symmetry-breakingPage121015202529iiiINTRODUCTIONThese lectures are intended as an elementary introduction to some of the ideas of lattice quantum field theory1 for an audience already familiar with field theory in the continuum.There are, I think, two not wholly unrelated reasons why one should know something about field theories on a lattice: Firstly for numericalresults in strongly interacting systems like QCD it seems to be the only procedure that works well enough to be implemented with some degree of control over the accuracy of the results. In the next few years practitioners of the art hope to be able to compute the hadron spectrum from essentially first principles (requiring only the quark masses and the QCD parameter as inputs). The results of this computation will either agree with what we see in nature or not. While I am sure that there will be much debate before a consensus is reached, first results in this direction seem promising.2The second reason is more philosophical: When one first learnsfield theory in a traditional course, from Bjorken and Drell for example, most people, myself included, feel ill at ease with the subject because of the infinities. In any perturbative calculation there is a well- defined procedure to follow which, provided one does the sums correctly, will give an answer that makes physical sense and will be the same no matter who does it. As soon as one moves away from perturbation calcu­lations towards general arguments, however, it is no longer clear what the rules of the game are and there exist situations in which wise men of good faith will disagree about the answer. With a lattice and view of the continuum limit being taken in the sense of critical phenomena (a view due essentially to K.G. Wilson), one has a well-defined game in which questions such as "is (X^)^ a free theory" can be posed precisely (if not yet answered decisively). This does away with the feeling of skating on thin ice and restores one's belief that we know what we are talking about even if we do not know all the answers.While the second reason adduced above is the aim of "constructive field theory" which is a very formal subject, I do not intend to be formal in the organization of this material. I do, however, hope that there is some logic in the order that topics are presented.2LECTURE 1. SCALAR FIELDS - FREE AND INTERACTINGThroughout these lectures I am going to work in Euclidean space time [signature (+,+,+,+)]• Most of the time I shall be using a discrete lattice in all four directions with hypercubic symmetry (i.e.2?1*). I shall occasionally allude to the Hamiltonian formalism which uses a continuous ”time! variable ( I R ® ^ 3) and of course mention the connection with the real world which has continuous time and space and has Minkowski signature For the moment just visualize a four-dimensionalsquare lattice. I am also going to assume that you are all familiar with relativistic quantum field theory in the continuum, preferably in a a functional integral approach.In this first lecture we will go back to basics and study the free scalar field theory on with the particular intention of exploring the particle/field duality which is a fruitful way to think in lattice theories.We put a real at each pointscalar variable $(n) S = (n^.-.n^) on2.d.We look at the integrald/ * j m  i V "  d l" 1 ( M n + i )  -  ^ ( n ) -) 2 m2= / d[<t>] exp]- > a  + -2- ,+,(£)J  I L.2 a2 2( 1 . 1 )In this expressiond[<)>] =  I I  d f  <J>(n))na = lattice spacing i = (0,... 1,0...)it*1 spaceOne should think of Z as the partition function of some statistical system of springs and masses. The exponent is clearly to be thought of as a lattice approximation toL  = / ^  ( 2  ( H ) 2  +  f ~  < t > 2 )  '  ( 1 > 2 )3We are really interested in the Green's functions which can be thought of as the correlation functions of the <f>(n) with respect to the probability distribution in (1.1)<tf)(n1) ... (n±)> = —  f d[(j>] c()(n1) ... (Kn^.) expj- ^  (...)(. (1.3)Z J  I n,i=l >We can evaluate these Green's functions by the simple but relatively unilluminating procedure of Fourier transforming <|>(n). Let„"Hr♦ (ft)- /-TTddk(2 i r )d$(*) . (1.4)Because of the discrete cubic lattice the t are restricted to lie in a Brillouin zoneThe converse of this formula issince5>(k) = X )  e in * ^ <f)^ne 2 d/“ ITddk(2TT)dpit* (n-n ) - x , x , x ,e ' ' °n n' °n n' ••* “njnl1 1  2 2 a  an meIn terms of $(k) (and setting a = 1),- / d [i] exP ~ J 2 + 2 i  (1-coski) |Kk)(1*5)y i  eiS*(t £’) = ^2 (2 it)d 6d( t - t '  + 2irm) . (1 .6 )(1.7)4and we can read off<i(tx) $(t2)> = 6d (61_^2) ------------  (1.8)m2 + 2 2 (1-cosk-.)0 j 1or, in configuration space,<4>(\) 4>(n2 )> = / -----    -£------- , (1.9)J  (2tt) + 2 2 (1-cosk^)1This is clearly the analog of the continuum propagator since 2(l-cosk) = k2 + OCk4).To get more insight into this expression let us reorganize the exponent in (1.1):Z =Rescale Jd[<j>] exp | + ^  <f>(n+i)(J>(n) ~  ^(2d+m2) | • (1-10)<J> -> V 2d+m2 <J> (1 .1 1 )and absorb the Jacobean into the measurez 'fi/t]exp!+^  £ s<*+t>+<*> - e • (i-i2>n,i hIf m2 is large it makes sense to expand out the first term in the exponent and obtain a typical "strong coupling" or "high temperature"^ expansion:Z = dn,iIf we use the formulae<b2 / 2 = irJ  /2ttf  _ * L _  i j ) 2 n + l  e -<}>2 / 2  =  o J /2ttf  -A t- 4,2 e -ct,2 / 2  = lJ /2tt/* d<^ - (j)14 e-^2/2 = 3 etc. , (1.14)J /2 t t5we can organize the expansion into diagrams which form closed loopswith a weight l/(m2+2d) for each link used. The factor of 3 in the ((A integral (1.14) means that a diagram such as thishas a weight of +3 which can be regarded as the sum of three diagramsAll the other n!, etc. in the expansion of the exponent in (1.14) lead to the plausible, and true, result that the expansion for Z is a gas (or solution) of closed, non-interacting, loops. If we were calculating Green's functions the <(>(n) act as sources where the lines can start or stop, so we have a Feynman diagram-like expansion for them together with vacuum loops. We will interpret these lines and loops as the world lines of the particles created and destroyed by the <j>(n) fields.We will now sum these diagrams and recover the results we obtained from the Fourier transform method. [This is easy here but these methods can be used to solve non-trivial problems such as the d=2 Ising model.]1*Random WalksLet us temporarily forget about our field theory and just look at the theory of random walks on a hypercubic lattice. Letr(n1,n2,t) = r(0,n2-n1,t)= number of walks from n^ to n2which are t steps long (1.15)6Then it only takes a moment to see thatT(0,n,t) = coeff. of x^l x^2 ... xnd in(X1 + ^~ + X2 + ^  + ••• + xd + ^ ) t * d * 16)If we put x^ = e^ki we can extract this coefficient as a closed formular(<If we then define G(0,n,p) by/ddk   e lk*n (2cosk + 2 cosk + ...)t . (1.17)(2 i r ) aG(0,n,p) = ^  e-ty r(0,n,t) (1.18)t=0we see that G is essentially the propagator (1.7)ddk 1G ( 0 ,h ,p) = I ------ e_lk*n-----------------l n u - f(2ir)d 1 - e~h 2 2 cosk.1 1>/;= (m2+2d) I  - 2 ^ -  e-1^  ------- ^ --------- (1.19)(2ir)d m2 + 2 2 (l_cosk.)o l  iprovided e“h = (m2 + 2d)-1. The factor in front of the integral reflects the scaling made in Eq. (1.11). We have thus proved that the sum over paths from n1 to n2 with the weight (m2 + 2d)-1 per link does indeed produce the propagator.If we want to sum up the closed vacuum loops we have to be a little bit more careful to avoid overcounting. If we fix a point on a loop and sum over all loops through that point of length t we find:e-Ut r(n,n,t) = J e-^  £  cosk^ (1.20)but if we want to sum on n to get all loops of length t we must divide by 2t. The factor t arises because all t vertices on the loop are equivalent and the factor of 2 occurs because the loop occurs twice with opposite orientation. (This factor would be absent if we were using complex <f> fields as there would be arrows on the loop just as for charged continuum fields.)The total sum over all configurations of one loop is therefore7N £ (J2coski e~“ )= - —  / * ddk *n(l - e"W 5 ^  2008^  2j(2Tr)d \ ^= c onst / -- ---- Jtnlm2 + 2 > _ (1-cosk.)2 J  (27v)d \ 0 1(N = # of sites in the lattice). (1.21)If we sum over a gas of n loops one must divide by n! to avoid over­counting, so finally:Z = exp - | ^ * n ( m 2  + £  (1-cosk.))= det_1/2 fA2 + m2) . (1.22)We could have obtained this directly from (1.7) of course, but I think it is rather nice to see the determinant expression for the vacuum loops come directly from the grand canonical ensemble of a "loop gas" with fugacity e-lJ for each "monomer" or link.I hope everyone is familiar with the continuum expressionsz  =  < ° o u t l ° i n >  =  e x p j -  J -  J £ n ( k 2+ m 2 )  j= exp(- E0T) (1.23)dEo _ V j _ d _k £n(k2+m2 ) = —  j  ^ ^ V^2 + m2 (+ m indep. term)2 J  (2ir)^ 2 J  (2x)3= j  hv/degree of freedom) x number of states Putting in Interactions (1*24)We have seen that a free scalar field theory - i.e. one whose functional integral is purely Gaussian - could be interpreted as a gas of non-inPeracting world lines (a better physical model would be solution of polymers). If we introduce a non-Gaussian X<jA term (X > 0) into the action= j ' d [ 4>] e x p j -n, l<|>(n+i) - <|)(n) 2 mj^Z^) X<Jil+(n)'   + —   +A!, (1-25)the Xf))1* term can be used in the large m expansion when lines cross. It8has the effect (because of the minus sign) of reducing the contribution to the sum over loops of configurations where lines cross. In other words it leads to a short-range repulsion between loops and between one part of a single loop and another. This type of interaction is actually used to model the effects of short-range repulsion in polymer solutions.If one were to take X < 0 then the integral would diverge at large <j>(n). This is reflected in the force between polymers being attrac­tive. The force does not saturate and if one has n walks close together the cost in action per unit length is proportional toS = an - gXn2for some a,8 (the factor n2 is because there are n(n-l)/2 pairs of lines to interact) . This is essentially the same as in the continuum where an n-particle bound state has mass:mn - KXn2and will become tachyonic if n is large enough. When this happens the the vacuum is filled with a tangled mess of world lines. A polymer precipitates out of solution if the interstrand forces become attractive.If we return to the case X > 0 but now take m2 < 0 so that spontaneous symmetry-breaking occurs for small X, then a similar picture holds; having m2 negative means that:e“U > (2d)"1 . (1.26)At each step of a walk on a d dimensional hypercubic lattice one has a choice of 2d directions to go in. If (1.26) holds, the e-,J factor is not enough to discourage long walks and the vacuum fills with a spaghetti of vacuum lines now limited in the density they achieve by the interparticle repulsion. This is the world line picture of a Bose condensate.Gauge and Electromagnetic InteractionsIn order to discuss charged particles we must, just as in continuum physics, allow <f>(n) to be a complex field. This just has the effect of adding an arrow to the diagrams to distinguish the use of the now distinct hopping terms <J)*(ti+l)<|) (fi) and <j>(fi+t)(|)*(ft) .To introduce an electromagnetic interaction we modify the action by replacing the exponent withS = - e±eA(*+l>*)«),(£) 12 + m2<f,*j,} .This means that the field at n+1 is compared with the field at n after ithas been "parallelly transported" by multiplication with the phase factoron the link. For a non-Abelian gauge group the phase factor would bereplaced by a matrix living in the representation of the group to which <J)(n) belongs.9Since— ^  ^  ~ e^e^(n^ > n ) 4>(n) |2 + m^ij) =t,S- (2d + m2 )d>*4> + <()(n+1) eieA(fi+i^) «},(£)+ <()(n+x) e_^eA(^+^>^) <J>*(ri)the hopping terms are<f>*(n+l) eieA(n+l,n) ^(n), <j,(£+l) e~ieA(n+i,n) ,),*(£) ,which give rise to the diagrams:(j) (j>* 4>* 4> >—  —n+i n n+1 nwith weights for a world line of t steps of i  1 [ eieA(n+l,n) .<2<‘+»«> linksFor closed loops we find that each loop has a "Wilson loop” factoreieA(n+l,n)which is reponsible for the interactions. If we recollect that for a continuum<W> = / d [ A ]  expJu = e J " d t 64 (x-x(t))<W> = exp - j  J Jy (x) Gyv(x-y)Jv (y) d'+xwe see that after integrating over gauge fields the effect is, for Abelian fields, of providing a Biot-Savart-like force between the world lines which carry a current proportional to their charge. The factor of "i" in the phase factor has the important effect of changing the sign of the interaction so that parallel conductors repel one another and anti­parallel world lines attract.10LECTURE 2. CONTINUUM LIMIT, RESTORATION OF ROTATIONAL SYMMETRY AND ASYMPTOTIC FREEDOMTo discuss the topics in the title of this lecture I am going to use a simple solvable model which illustrates them all in a very transparent way. This is the two-dimensional 0(N) symmetric non-linear a model in the large N limit.The partition function is the same as the free scalar theory except for a constraint on <J>:z = / d m  601,2-1 ) expj - I  [<t»a (?i+l) - <(,“ (£)]! ; (2 .1 )* n,i ’here $ = (<pa ) is an N component vector in some internal space. We can solve this model in the N -v <» limit just as we can in the continuum. We introduce a "Lagrange" multiplier X(n) to enforce the constraintz = / d[X] f  <![<(>] exp J  [*(*+*) -4>(n)]2 + £  iX (*) [<|>2-1 ] J .4 ,1  ( 2 . 2 ) If we put g2N = ot = const, X = NX we can use the method of steepest descent. If we assume X is a constant we can perform the <(> integrals togetf dx exp - N [jr (2-2coski) - 2iXa) + iXT RIUU )d(2.3)so the X saddle point equation is,+ rr2ia / ddk/ + i = 0= 1 . (2.4)2 J  (2ir)2 2(2-2cosk-j) - 2iXa-TTor if we put -2iX = m2vHra / ddk _________1_________J  (2tt)2 2(2-2cosk.-) + m2-ITUsing this method to compute the correlation functions we find that^ “ (ftj) <|)B(?i2 )> = 6ag ^  A(f!x ,#i2 ,m2 ), A ,-> ■* o .  / ' " hTd d kwhere A(n ,n ,m2 ) = / ---- - ---------- ±— 3--- ^  (2 .5 )1 2 J  (2ir )d 2 2 (1-cosk-,-) + m2ITand (2.4) can be written as11N< £  <j>2 -  1> = 0 . (2.6)1In two dimensions the integral on the LHS of (2.4) is infrared divergent as m2 + 0, so for all a we can find a solution with m2 > 0. In more than two dimensions A(0,m2 ) is bounded above by the finite number A (0,0) so for a < ac = [A(0,0)]_1 there is no solution of (2.4) for real m and we must set m2 = 0. There is a phase transition at ac •In two dimensions, after some searching in Gradshteyn and Ryzhik, we discover thatA (0,m2 ) = —   J  k (  \ ^2ir (1-fm2 /2 ) \(l+m2/2 )/where K is a complete elliptic integral. As m + 0(2.7)1 32A(0,m2 ) = -—  £n —— + 0(m2 ) (2.8)MUU m zso we can solve for the connection between a and m2 when m2 is small:1 1  32-  = —  £n —  . (2.9)a  4 SI  mzThis means that m2 is small when g2 is small.When g2 and m2 are small most of the contribution to the propagator comes from small k where the integrand looks like (^-Hn2 )-1 which is rotationally invariant. Let us look in detail how this comes about:(x,m2 ) = J,+TT j. .d2k e ^ ' XA(x / -----  ---- - j ------------ . (2 .10)RIUUFI m2 (2-2cosk)Suppose 5 is becoming large in a particular direction specified by aunit vector e± - I-1* -2 = ! .x = | r | e eWe expect A to fall off exponentially:A ( re ,m2 ) . e_,c^ >  lr I . (2 *n )The problem is to compute <(e), the inverse correlation length in the direction e. We expect it to be anisotropic at large m2 but to become isotropic as m2 becomes small.Define „f ( 0  = f  dr e_1^r A(re,m2 ) . (2.12)oThis should be analytic in the lower half £ plane and the asymptotic behaviour at large |r| of A(re,m2 ) will be given by the nearest12singularity to the real £ axis (which will be on the positive imaginary axis if A does not oscillate)recall f  ^  - 2 —  S -  .  e"5orJ  2iriNow, d2k 6 (k‘ e - £ )f(S)■ * / ;/d2k S(k»e - £)(2tt)2 m2 + ]T (2-2coski) 1(2.13)(2ir )2 D(k)For small m the singularity in f(5) is expected to be caused by zeros of D(k) pinching the contour of integration. [Just as in the continuum:f ( i U  = Z*dk — -i— —  (2.14)J  k2+m2-?2where the integrand has poles at ± i/m2-^2 which pinch if £ = m. This makes the continuum propagator fall off as e-Itr as indeed we know it does. ]The condition for a pinch singularity at i£0 is, if \  = it,D(t) = 0 (2.15a)9 D—  = 0 on K.e = 5 . (2.15b)9ROne can impose the constraint in (2.15b) by means of a Lagrange multiplier2- |D(t) - At*!} = 0 . (2.15c)9KThese equations have a simple geometric interpretation. The set of points = E,0 is a straight line perpendicular to % and at a distancefrom the origin. Equations (2.15) say that if there is a pinch singularity at i£0 then this line must be tangent to the curve D(K) = 0.A more direct way to obtain this condition is to note that the points of intersection of e ?o with D(K) = 0 are the locations of the poles of the integrand in the complex 1c plane. Clearly they can pinch only if they are coincident and the line is tangent.In our case:D(K) = 4 + m2 - 2(cosh + cosh ) . (2.16)For large m2 , D(fe) = 0 is essentially a square with sides at±cosh-1 (2+m2 )/2.13A little geometry shows that£0 = (cosO + sine) cosh-1 (2-Hn2 )/2 « (cos9 + sin8) £n(2+m2 )1•• A(r£,m) ~ e~-o =(2 + m 2 )  l n l  l +  l n 2 l(2.17)(2.18)which means that at large m2 the propagator is dominated by the shortest route between 0 and n.<----------------- ii)-i'------------As soon as m2 is away from m2 = oo the corners of the curve round off and as m2 becomes small, the curve D(£) = 0 becomes circular and the correlation length becomes isotropic.It is easy to see that on axisS = cosh-1(l + m2/2)while at 45° to an axisS = /2 cosh-1(1 + m2/4) .Both expressions are equal to m when m2 is small.14We can now explain what is meant by the "continuum limit" of this lattice model: As we let the coupling constant g2 become smaller twothings happen(i) The correlation length measured in units of lattice spacing grows and approaches infinity as g2 approaches zero.(ii) The correlation functions become isotropic.We can regard (i) and (ii) as meaning that if we take g2 to zero and the lattice spacing a to zero in such a way that the correlation length measured in physical units (fermi) stays the same, then the anisotropy due to the lattice disappears and we have a continuum theory. The inverse of the correlation length is the mass f the particle created by the <J> field and is given byM2 = 4  = 32 exp - ( ^ - )  . (2.19)az az \gzN/The mass M2 satisfies a renormaliztion group equation' a -  + e(g2 ) - \ U 2(alg2) = 0  (2.20)3a 9g2 /wherep(g2 ) = + g4 *L . (2 .2 1)2nIt is common in lattice theories and statistical mechanics to define the 8 function this way (i.e. by varying a length instead of a mass) and we have a change of sign compared to the particle physics convention. Thus the plus sign in Eq. (2.21) means that this theory is asymptotically free.6Finally, a brief word on A parameters. (I will touch on them later.) Suppose, instead of a lattice cut-off, we solved a continuum version of this model using a Fauli-Villars cut-off. Equation (2.4) would become• 4«>RIUUF2 |_k2+m2 k2+y2J so  M2 =  p 2 e - ^  / g 2 ^g2N f  r _ A _  _ _ i_ _|  = !s4  (2 U ) Lk nThe quantities called the A parameters are defined byApV = P2 e-*"/g2N‘lattice ' a ' 2 e‘W 8 2 Nand since the physical quantity M2 must be independent of the regulariza­tion scheme we must have732 ‘lattice " »?v15LECTURE 3. FRACTALS, SCALING AND RENORMALIZATIONWe saw last time that to obtain a continuum limit of a lattice theory we need to find a "critical point" where the correlation length tends to infinity. I do not wish to discuss the formal theory of criti­cal phenomena and field theory here as there are excellent discussions in the literature. What I do want to do is to be more intuitive and try to bridge the generation and culture gap that has grown up between those who learned renormalization in the traditional way and those who grew up on the work of Wilson. We all do the same sums but the language is often very different. I can paraphrase (parody?) the essentials of the two world views as follows:1) "A field theory is a set of Feynman rules together with a pre­scription (called renormalization) for getting rid of unwanted ultraviolet divergences.”2) "A field theory is the large distance behaviour of a system near a critical point. Renormalization is the language in which to describe how infrared divergences miraculously make this behaviour independent of the details of the short distance interactions."Perhaps not coincidentally most people who prefer world view #1 spend their working day in the Fourier transform of Minkowski space while those who favour world view #2 live and work in Euclidean space. Let me begin the discussion by askingHow Long is the Coastline of England?8To be more mathematical let us ask how long is the curve generated recursively from a square by successively replacing each line element by another one as follows:So we eventually obtain a figure with a "self similarity" property:16The length of the limiting set is clearly infinite but if we measure with a fixed resolution we get a finite number that depends on the resolving power. If we increase the resolving power by a factor of four we find that the measured length has increased eight times. Such a set is called a fractal and the way in which the length scales with the resolving power is related to the Hausdorff dimension of the set. The Hausdorff dimension is defined by trying to cover the set with little discs of radius e. If the number of discs required to do this increases asNumber = (3.1)then a is the Hausdorff dimension.For a straight line a = 1, for a plane figure a = 2, etc. In our caseif we reduce e by 1/A we need eight times as many discs so a = 3/2. Thetotal length we estimate is:Length = (3.2)Notice the way that an additional length scale (renormalization point?) has crept into the length to soak up the extra "anomalous dimension".This behaviour is very similar to the behaviour of Green's functions in a massless field theory. A propagator canonically varies as p-2 but when interactions are present it may vary as p-2-r' (e.g. the massless Thirring model).Let us see if we can relate such an anomalous dimension to some sortof self similarity in the field theory. Consider an Ising model nearTc . We define a "renormalization group transformation" by letting blocks of nine spins vote9 :In an Ising model the original spins S have a probability distribution given by the usual Ising Hamiltonian17P(Si) = Z-1 exp SiSj . (3.3)From this we can find the distribution P(S') for the new spins. If weimagine this used to describe spins on a lattice 1/3 the size we cancompare it with the original P:P(S) (vote, reduce by 1/3) p '(s ') * (3 *4)In general P'(S') is not exactly of Ising form; it will containinteractions and all sorts of junk. A very important feature of the critical point is, however, that if we keep iteratingP(S) P'(S') > P"(S") + ...eventually we reach a fixed distribution which does not change under this process. If we are not at a critical point this does not happen. (This is very similar to the central limit theorem of probability - keep adding random variables from the same distribution and eventually the distribu­tion of the sum is Gaussian.) This stability of the distribution is the analog of self similarity for the fractal. It essentially says that inside islands of up-spin there are smaller islands of down-spin and so on ad infinitum. Let us use this to obtain a power law for the spin-spin correlation functions:LI R .  G-. T18A formal statement of this stability is as follows:Let P(S' = y', S' = y ’; L) be the probability for two blocks of ninespins on the*original lattice, separated by a distance L, to vote for thevalues y ’ y '. Let P(S, = y' S = y', L/3) be the probability of indi­vidual spins at distance L/3 to have the values y^ , y ^ • Then at the fixed (stable point)P (S 1 = yi ’ S2 = W2 ; L) = P (S 1 = Wl’ S2 = W2 ’ U 3 )  * (3*5)The spin-spin correlation function is<S1S2>L = E  PiV12 P (S1 = V  S2 = V  L) * (3*6)P 1 »lJ2~±1Now P(S 1 = V  S2 = W2 ; L) = E  P(S1 = yJ S ' = yP P(S2 = *^2 IS2 =Uiy 2x PCSJ = yj, S?' = y^; L) (3.7)= E  P(Sj = u J S *  = y ’)P(S2 = y2 |S2 = y ‘)y iy 2x P(S1 = y*, S2 = y2 ; L/3) . (3.8)If the interaction is short range then P(S = y ^ | = y^) only depends onthe value of the Block spin. So"£2 y1P(Sl = y J S J  = yj) = f(y') (3.9)ylis a function of y' only. From symmetry considerationsP(Si = wJSi = Wj) = Z U 2 V{ • (3* 10 )ylThus we find from (3.10) and (3.6), (3.8) that<8 ^ 2  > = Z <S1 S2>L/3 <SlS2>L03n = zn<SlS2>L0 or <S S> oc *IL |71n = -Jin z/tn(3) .(Recall that anomalous dimensions in field theory are defined as=  z )  \Y d(Jln y)We can rewrite this by introducing a lattice spacing a and a renormaliza­tion point R019<S S> =LaL Rexp p £naL,'oR R0<S S> = <S S>renorma^£ze(} x Z(a,RQ )As a 0 we have logarithmic divergences in the wave function renormali­zation factor.This is at Tc and corresponds to a massless theory. If we want a massive theory we would work close to, but not quite at, Tc and let T approach Tc as we take a to zero.The idea that there is a fixed point in the probability distributionhas a number of consequences for continuum field theories which are worth appreciating:1) There are many different lattice approximations to any given continuum theory. To give rise to the same theory they just have to be in the domain of attraction of that fixed point (so-called universality class).2) Theories that are in the same universality class differ by what are called "irrelevant" interactions. These irrelevant interactions cannot affect the continuum theory. In the conventional language of field theory these irrelevant interactions were called "non-renormalizable” .3) Operator ordering: We all know that in quantum mechanics it mattersin which order quantum operators appear in the Hamiltonian - but we never seem to worry about this in continuum field theory. This is essentially due to point 1). The operator ordering ambiguities come about in the functional integral formalism because of different ways of discretizing the functional integral. As the dimension of space time increases more interactions become non-renormalizable or irrelevant and the size of the universality classes increases. This means that most operator-ordering problems disappear as we take the continuum limit.20LECTURE 4. GAUGE FIELDS, MONTE-CARLO AND STRING TENSIONWe briefly mentioned in the first lecture how matter fields interact with background gauge fields via "parallel transport". We now need to discuss the form of the gauge field action.The gauge field degrees of freedom live on the links of the hyper- cubic lattice and are elements of the gauge group. For an SU(2) group, for example,It is less obvious than in the scalar field case what motivates this action. The basic idea is to parallel transport around the four sides of a plaquette and to look how far the resulting group element is away from the identity (measured in the intrinsic Riemannian geometry on the group manifold). For slowly varying fields we have(4.1)V * )  = \  as T± A*(£) = ~  ag t* A^(n) . (4.2)When we need links in a backward direction we associate them with U^Cfo):U_y (n-Hi) = U“l(fc) • (4.3)The most commonly used action is the Wilson action1 which is made up out of a sum over plaquettes (y,v):S (4.4)Tr {uw (fc) Uy ffe+u ) Uyfctfrtf) U_y (n+v )^„ e* "^*"a^ V * ^  e-:^ v= ei(Bll+Bv+aailBv )-l/2[By ,Bv ] e-i(B|1+Bv+a3vBjJ )-l/2 [By ,BV ](using ex e^ = ex+y-1/2[x,y]+... ^* eiaO p Bv-3vBp-[BpBv])e (4.5)where F = 3 A - 3 A -g[A ,A ] . yv y v v y  61 y ’ v J (4.6)Now(4.7)21So 5252 — - Tr(u u U U) * - / d'+x -  F F , (4.8)-  tT? 2g2 J  4 yv yv ’ vn yvwhich is the usual continuum action and is proportional to the square ofthe distance of the group elements from the identity.The simplest quantity to study with some physical significance in apure gauge theory is the vacuum expectation value of a Wilson loop orproduct of the U ’s around a closed curve. If we take a rectangular loop of sides T,R then the expectation value will give the vacuum-vacuum amplitude in the presence of a quark-antiquark pair from which we canread off the extra energy the pair have above the vacuum:<nu> = e-TV(R> (4.9)V(R) = potential energy of the quark-antiquark pair at distance R. If there is a linear confining potential between the particles thenV(R) = oR + const , (4.10)where a is the "string tension", i.e.<riU> = e-aRT+c(R+T>+const (4.11)or the logarithm of the Wilson loop has an area term - the string tension, a perimeter term which is the quark self energy and possibly a constant term. One can determine the string tension by Monte-Carlo procedures which I will now attempt to describe. We want to evaluate the integralf  d[U] II U e"BS(U)= < n u >  (4 .1 2 )f d[U] e~$s(u )by a stochastic method. One might try choosing sets of U's at random and weighting the result by e-$s. This is very inefficient and a better procedure is to let the action term play a role in the choice of the U's.10 We will choose a sequence of configurations U by a Markov process governed by a master equation for the probabilities Pn (U) at step n:pn+l<u > = 5 2  P(U’+U)Pn (U') + P(U+U)Pn (U) . (4.13)U'*UWriteP(U*U) = 1 - / , P(U+U')U'*UPn+i(U) = Pn (U) + (p(U'*U)Pn (U') - P(U+U')Pn (U)) . (4.14)W U 'so thatWe want to choose the transition probabilities P(U-»-U') so that as n increases the Pn converge to a stationary distribution:P(U) = e_e S ( U ) / ^  e"SS(U) . (4.15)U22We can arrange this by making each term in the sum on U in (4.14) vanish (the principle of "detailed balance"), i.e.There are two popular methods for satisfying (4.16):1) The Metropolis algorithm11One starts by generating a table of about 5C matrices chosen randomly from the gauge group. No particular distribution for these matrices is needed but one hopes that between them they will generate the whole group. (Just in case they do not, one occasionally generates a new set.) Then one proceeds, one link at a time, to try replacing the U on that link by U multiplied by one of the fifty matrices. If this new U' leads to a configuration with lower action, then one goes ahead and replaces U by it. If this U' leads to a larger action S(U') one may still accept it but with a probabilityIf one does not accept the change you just leave the original U in place and move on to the next link.2) The heat bath12This is similar to the Metropolis algorithm in that one proceeds one link at a time but now one chooses a new U' with no reference to the old value on the link. One chooses U' with a probability proportional to:It is easy to check that both these procedures satisfy the detailed balance principle.Suppose now that we have iterated our Markov chain enough times that Pn has settled down to its asymptotic distribution. We use this to estimate the Wilson loops by evaluating them for a sequence of configura­tions and taking the time averageAs N °° this should become equal to the expectation value of W with respect to the distribution P (this__is the content of the ergodic theorem) . One can easily see that W is an unbiased estimator for <W>:P(U'*U) = P(U), „ e-e(S(U)-S(U')) (4.16)P(U+U') P(U')p = e-fl(s(U’)-S(U)) .e-BS(U') .(4.18)(4.19)since the distribution for all the Un is equal to P. Provided that the W(Un ) are only correlated over a finite number of Markov steps:23<W(Un )W(Un .> - <W>2 oc e ln~n ' XTU L (4.20)one can see that the variance of W tends to zero:<(W - <fi»2> = L- ^  i <W(Un )W(Un « )> - <W>2N n,n'“ T0/N . (4.21)In this case we see that, almost certainly, as N +  ® the time average of the W(Un ) is equal to the ensemble average (4.12). We also find that the errors decrease as N 172 provided we take independent samples, i.e. at intervals greater than t0 * Unfortunately near a critical point t0 becomes large and actually at a critical point the correlations decay algebraically making the errors decrease more slowly than N~1/2.To estimate the string tension it is convenient to take combinations of Wilson loops in which the constant term and the perimeter term cancel out. For example:provided W is of the form (4.11). Using this combination estimates of the string tension have been made for values of the couplings sufficiently small that o scales as any quantity of dimensions (mass)2 should:for SU(N). By fitting (4.23), (4.24) to the Monte-Carlo data one can estimate c and find13 :As this stands it is not much use but fortunately one can calculate the connection between the E C parameter and the A parameters for continuum schemes just as we did in Lecture 27 :o (4.22)a = c A2 (4.23)wheree'1 /Bog2 (4.24)(4.25)/o = (79 ± 12) EC ONRIF/a = (220 ± 66) AL SU(3) . (4.26)Amom/AL = 83.5 in SU(3) . (4.27)24Now one also believes that~ 450 MeV (4.28)from either potential models of quarkonia or from string models where there is a connection between a and the Regge slope a'S i  = a • (4.29)2 t tFrom (4.26), (4.27), (4.28), we find an estimate for AmoraAmom = 180 MeV (4.30)which is roughly consistent with what is seen in deep inelastic scaling violation.25„LECTURE 5. FERMIONSWe now come to the problem of incorporating fermions. Putting fermions on a lattice is a little tricky and there are a number of problems that have not yet been solved in a really satisfactory manner.We will begin by looking at the "naive" fermion action:s = ^ 2  |'Kn)'Kn) + K ’K n) Yu |'Kn+u) “ V ) ^ V • (5.1)n 1 p 'Comparing (5.1) with the usual continuum actiondl+x(^Y,J3p^ + nnp\p) (5.2)shows that we need to take’('continuum = '('lattice (5.3)® = —  • (5.4)2KI am using y matrices which are Hermitian and satisfy| Y u , Y v |  =  2 6 y v  .  ( 5 . 5 )We want to use this action in a functional integral. In order to get the correct Feynman rules we need to make the s into Grassman variablesand use the Berezin integral. So the i|>1 s and i^ 's anticommuteand{'K^)'p(n* ) | = { t(n) ,iKn ' ) } = {\p(n) ,ip(n' ) } = 0 (5.6)J "  d(i(>) = 0 / d( * »  = 1, etc. (5.7)With these definitions one can see why I chose the coefficient of the ipi/-' term to be one. We can either use the ipij; term at each site, to satisfy the requirements for a non-zero answer, or a pair of hopping terms. In this way, just as in lecture one, we quickly see, at least as long as we do not try to use the same link twice, that we get a path expansion for the propagator:<t(n')*(n)> = ^ 2  kIL I n • (5.8)pathThe yy here is shorthand for ±Yy depending on whether we traverse a link in the positive or negative direction. At first one may think, however, that the sum should be over self-avoiding paths or else we get in trouble with the i[j2 = ip = 0 conditions. The problem is actually illusory as there is a conspiracy between graphs contributing to the propagator and closed loop graphs contributing to the vacuum diagrams which allows us to sum over unconstrained paths in Eq. (5.8). To see how this comes aboutxlet us look at a simple example. Consider the path in the figure below:The part of the diagram where the two lines share the same link is for­bidden in the expansion, but consider also the diagram where the propa­gator goes straight through and there is a vacuum loop:Again we cannot put the loop in contact with the propagator line but the closed vacuum loop has a minus sign while the figure with the coil in the propagator has a plus sign. Thus adding both forbidden diagrams changes nothing:27That this conspiracy works in general is best seen from the identity for the diagonalized actionZ = J d[*]d[,[r] exp (  £  ^i'Piti = n  Xi) (5.9)<^k^k> = J  J dt^JdtipJtk'Pk exP ( £  xi^ ±'f'i)- ■ s i r  ( A * 1) " *  • ( 5 -10)One can see that there is co-operation between the vacuum terms and the other integral, which leads to the propagator being the inverse of the matrix in the exponent. We can now sum E q . (5.8) with a trick similar to the one used on the scalar fields in lecture one:■ /ddk - i t . t  eRIUU )d 1 - £ y y2i sinkpPddk -ik* n 1 + u Yy2i sinkp 1 4K^ sin^kpThis looks like the continuum propagator for small k:f  J.lv.„ _ — ----= / ---- - e ----  ^ —  • (5.11)J  (27T )d/ d<*k -ik.n (m + i/k) T  e lfc n — ------------------------ (5.12)RIUU )d m2 + k2but there is a problem. If we were to try to compute, say, \pip:r  ddk 1< M >  = /-----  r-------- (5.13)J  (2x)d 1 +  4K2 2  sin2 kjjBecause sinir = 0 we get similar contributions from many other places in the domain of integration. We actually get 2d times the correct answer (16 time in 4 dimensions). This is the notorious "fermion doubling problem" - we thought we had only one species of fermion in our sums but we turn out to have 16.There are several ways that have been proposed to circumvent this multiplication of degrees of freedom:1. Wilson Fermions1We replace the iy^ factors in the hopping terms by 1 ± y ^ . Then ■/ ddk -ik.n f 1 +  2ik £  Yu  slnkp + 2 k  £  cosk^)G = / -----  e  —  . (5.14)(2tr )2 (l - 2 £  coskp)2 + 4k2 ^  sinkp282. Staggered or "Kogut-Susskind" fermions11*These are a little more arcane. The best route to understandingthem is due to Kawamoto and Smit . 15 We replace ip by ip1 whereK n x • •• i\) = Y ^  Y 33 Y^2 Y^1 ^ ' ( ^  * * * \ ) *^ ( ^  ... n^) = ^ ' ( ^  ... n^JY^1 Y22 ^33 ^  so (5.1) becomesS = | * (n)ip * (n) +  ( - l ) ^  »n ) ip' (n) j ip(n+jj ) - ip(n-y)}| .n * ’The phases (-1 ^ » n ) arise from commuting the y ’s through one another, e.g.^(n)Yx |'KT1+1) _ 'P(n_1)}= ip (n) Yfl y " 2 Y 33 Y^4 (Yi ){y^ Y 33 Y^2 Y^1+1 4> (n+t) ~ Y^1* Y 33 y " 2 y " 1+1 ip'(n-l)}=  ( - l ) 1^ 4 ^  xfT' ( n )  j i p '  ( n + 1 )  -  ip '  ( n - 1 )  |  .Similarly<p(2 ,$) = n3 + n^<p(3 , ? i )  =<p(4,n) = 0The only remnant of the y algebra is now the fact thatn  ( - i ) Y  =  - 1plaquettewhich arises from yPyVyPyV = _1 if 11 ^ v. The action is now diagonal in the spin label and if we just keep one of the spins we reduce the number of fermi species by one quarter leaving, in four dimensions, four flavours of fermions. These are the Kogut-Susskind fermions.29LECTURE 6. CHIRAL SYMMETRY-BREAKINGThe Kogut-Susskind fermion actionS = |ip(n) M n) + i{;(n) ^  (_i)4>(n »u)  ^^  (n+p ) - ^(n-u)}| (6 .1 )n vhas a number of symmetries. Two of them are continuous:(a) "Vector" symmetryi|>(n) *  e i0 i|;(n)TfT(n) -► e-10  ^ (n) . (6 .2)(b) "Axial" symmetry (only good if m = 0)n odd (i.e. n^ = odd)n even . (6.3)The first symmetry simply leads to fermion number conservation (i.e. con­tinuity of world lines) while the second is only good when K -*• °° and the world-line picture is no use. This second symmetry can be broken spontaneously and leads to the existence of massless Goldstone bosons - "pions".To discuss these symmetries it is easiest to go back to the naive fermion language where the "axial" symmetry takes the form■ i0Yc , ip -*■ e 5 ipip -*■ ip . (6.4)(It should be said that this formula is a trifle deceptive because the actual symmetry, in terms of the flavours that are really present, is not the U(l) axial symmetry. It is a U(l) subgroup of the axial symmetry but in a combination with a flavour matrix which is anomaly free.) From this one can derive a Ward identity by the usual methods. I shall, following the general spirit of the lectures, prove it by a diagrammatic argument.The Ward identity is<ip(n)\p(n)> = <t(n ')Y5'Kn ') 'Kn)Y5'Kn)> • (6.5)n'To prove it consider a graph contributing to <\[nj/>:30Insert a y5 matrix at n and another at n' on the loop and slide n' round the loop. Since y 5 anticommutes with the Yp this gives rise to an alternation in sign. On adding all the terms they cancel except for one (where the two y ^ 's are coincident at 7i. In this way we get the LHS of(6.5).) The terms which are being added can also be regarded as contri­butions to the RHS of (6.5). By summing over all loops through n we get the whole of the (6.5).Rescaling the fields by /2K, as in the last lecture, to make contact with the continuum field normalization gives:< ^ >  = m<i{iY5\J>, ^y5ij)>p_0 • (6 .6)If <3M>> *  0 and we take m ♦ 0 we must have a divergence in the zero momentum Green's function which will be caused by a zero mass pion.There is a nice argument due to Brout and Englert^ which enables one to see how ipip becomes nonzero at strong coupling. When g2 * 0 the quarks and antiquarks have to pair up on each link so the diagrammatic expansion for is a sum of rooted trees:We can sura these recursively(NC) - 1 < M >  =  rr-   • (6.7)1 - x<H>(NC ) _1The factors of N, C are the number of fermi components and the number of colours, respectively. The x is the factor for each link of the tree. Since each link has a +yp and a -y^ and can point in any of 2d directions we see thatx = -2dK2 . (6 .8)31Thus:<n> " ' l} • (6.9)Again rescaling by / 2K and putting m = (2K) - 1 we can rewrite this as<ujn|j> = |/m2+2d - m j (6 .10)and as m 0 we find that (6 .10) goes to a finite limit of1.1<n>n=0 = NC /J ( 6 . 11 )One can also see the existence of the zero mass pion in this limit. Diagrammatically the pion looks likeWe can sum this by noticing that the effect of the y5,s at the ends cancels the (-1) for all the links in the "backbone" of the graph. Thus;= NC<yj>x2 NC^< W>X2NCp « h l/<jjnp>\2llL l NC /ddk(2ir)ik*n1 - K2 NC f  2 (2cosk1)( 6 . 1 2 )When m 0 (or equivalently K + 0) this goes on tof  ddk »n ) 11 I   -3J  (2tt )d3ik* n.d ~ ,   dwhich is the propagator for a massless particle.1 - l/2d ^ 2cosk^ 1(6.13)A number of groups17 have performed Monte-Carlo simulations of chiral symmetry-breaking in the "quenched approximation" where the fermion determinant is ignored. In this case the chiral symmetry-breaking order parameter is obtained from the inverse of the Dirac operator in thebackground gauge field:32< M >  = Tr G(x x) (6.14)= Tr <— — > • (6.15)y + mIn (6.15) the angular brackets can be read as averaging over gauge con­figurations produced with the Wilson action. If the eigenvalues of the Euclidean Dirac operator are iXn we can rewrite (6.15) as=  — N  --------------------  • ( 6 . 1 6 )V ^  iXn + mAs V becomes large the poles in (6.16) merge to form a branch cut and we can regard the large V limit of (6.16) as a dispersion relation<n> =  f  dx P-(-X-}- , (6.17)J  iX + mwhere p(X) is the density of states with eigenvalue X. The symmetry breaking now comes about because the integral is discontinuous across the branch cut (i.e. as m changes from positive to negative) and we find<i|>t>m=0 = irp(O) ♦ (6.18)This branch cut is in evidence in Eq. (6.10) because of the square root sign. We can interpret Eq. (6.10) in the light of these remarks as a calculation of the strong coupling density of eigenvalues of the Dirac operator. We findp(X) = / 2d-X2 . (6.19)dirThis can be compared with the free theory where X = |k| andp(X)dX = = |k|3 J l S -  (6.20)X LSS EC  X LSSE Tso p(X) « X3 for small X. In the quenched strong coupling calculation the symmetry-breaking comes about because the eigenvalues slump towards X = 0 to make up the semicircle distribution which seems characteristic of many random matrix problems.33References1. K.G. Wilson, Phys. Rev. D 14, 2455 (1974);A.M. Polyakov, Phys. Lett. 59B, 83 (1975);F. Wegner, J. Math. Phys. 12, 2259 (1971).For review articles, see:L.P. Kadanoff, Rev. Mod. Phys. 49, 267 (1977);J. Kogut, Rev. Mod. Phys. 51, 659 (1979);J. Kogut, Rev. Mod. Phys. 55, 775 (1983).2. H. Hamber and G. Parisi, Phys. Rev. Lett. 47, 1792 (1981);E. Marinari, G. Parisi and C. Rebbi, ibid. 1795 (1981);D. Weingarten, Phys. Lett. 109B, 57 (1982).3. M. VJortis, in Phase Transitions and Critical Phenomena, vol. 3, eds.C. Domb and M.S. Green (Academic, London, 1974);R. Balian, J.M. Drouffe and C. Itzykson, Phys. Rev. D 11, 2104 (1975);G. Munster, Nucl• Phys. B120 [FS2], 23 (1981).4. R.P. Feynman, Statistical Mechanics (Benjamin-Cummings, Menlo Park, 1972).5. I.S. Gradshteyn and I.M. Ryzhik, Table of Integral Series and Products (Academic, New York, 1980).6 . H.D. Politzer, Phys. Rev. Lett. 30, 1346 (1973);D.J. Gross and F. Wilczek, ibid., 1343;G. t'Hooft, unpublished remarks at Marseilles Conference of Gauge Theories 1972.7. A. Hasenfratz and P. Hasenfratz, Phys. Lett. 93B, 165 (1980);R. Dashen and D. Gross, Phys. Rev. D 23, 2340 (1981).8. B. Mandelbrot, Fractals, Form, Chance and Dimension (Freeman, San Francisco, 1977).9. K. Wilson, Sci. Am. 241, 158 (August, 1979).10. Monte Carlo Methods in Statistical Physics, ed. K. Binder (Springer- Verlag, New York, 1979).11. K.G. Wilson, in Recent Developments in Gauge Theories, ed. G. t'Hooft, Cargese 1979 (Plenum, New York, 1980).12. M. Creutz, Phys. Rev. D 21, 2308 (1980).13. M. Creutz, Phys. Rev. Lett. 45, 313 (1980).14. J. Kogut and L. Susskind, Phys. Rev. D 11, 359 (1975);L. Susskind, Phys. Rev. D 16, 303 (1977).15. N. Kawamoto and J. Smit, Nucl. Phys. B192, 100 (1981).16. J.M. Blairon, R. Erout, F. Englert and J. Greensite, University Libre de Bruxelles preprint (1980).17. J. Kogut, M. Stone, H.W. Wyld, J. Shigemitsu, S.H. Shenker and D.K. Sinclair, Phys. Rev. Lett. 48, 1140 (1982) (see also Ref. 2).

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