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Universality classes of matrix models in 4-έ dimensions Jaimungal, Sebastian 1996

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UNIVERSALITY CLASSES OF MATRIX MODELS IN 4-e DIMENSIONS by SEBASTIAN JAIMUNGAL B A . S c , The University of Toronto, 1994 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T O F T H E REQUIREMENTS F O R T H E D E G R E E O F M A S T E R O F SCIENCE in T H E F A C U L T Y O F G R A D U A T E STUDIES Department of Physics We accept this thesis as conforming to the required standard T H E UNIVERSITY O F BRITISH C OLU MBIA May 1996 © Sebastian Jaimungal, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada Date M q ^ 0 1 . m & DE-6 (2/88) Abstract The role that matr ix models, in (4 — e) dimensions, play in quantum crit ical phenomena is explored. We begin wi th a traceless Hermitean scalar matr ix model and add opera-tors that couple to fermions, and gauge fields. Through each stage of generalization the universality class of the resulting theory is explored. We also argue that chiral symme-try breaking in (2 + 1) dimensional Q C D can be identified wi th Neel ordering in two dimensional quantum antiferromagents. When operators that drive the phase transition are added to these theories, we postulate that the resulting quantum cri t ical behavior lies in the universality class of gauged Yukawa matr ix models. As a consequence of the phase structure of this matr ix model, the chiral transition is typically of first order wi th computable cri t ical exponents. 11 Table of Contents Abstract li List of Figures fv 0 Introduction 1 1 The Renormalization Group and Spontaneous Symmetry Breaking 4 1.1 Universality 4 1.2 Spontaneous Symmetry Breaking and the Coleman Wienberg Phenomena 7 1.3 Renormalization Group Flow and Firs t Order Phase Transitions 10 1.4 The Stabil i ty Wedge and Restrictions on Fixed Points 14 2 Matrix Models 16 2.1 A Traceless Hermitean Scalar M a t r i x Mode l 16 2.2 Yukawa Coupl ing to Fermions 20 2.3 The Introduction of a Color Gauge F ie ld 25 3 Four Fermi Models 31 3.1 The Gross-Neveu Model 31 3.2 A Four Fermi Theory with Gauged Flavor 34 4 Spin Systems and Lattice Gauge Theories 37 4.1 Spin systems 37 4.2 Hamil tonian lattice gauge theory 40 i i i 5 Nature of the Phase Transitions 45 5.1 The Yukawa Ma t r i x Model and Four Fermion Theories 45 5.2 The Chi ra l Phase Transition and The Gauged Yukawa Ma t r i x Model . . 50 Bibliography 58 i V List of Figures 1.1 Typica l diagrams occurring in the one-loop expansion of the effective po-tential 9 2.1 Typica l vertices for the Yukawa coupled model 22 2.2 Propagators in the theory. . 22 2.3 Yukawa correction to the g2 vertex 23 2.4 Fermion bubble correction to the scalar wavefunction 23 2.5 Corrections to the Yukawa interaction 23 2.6 Ma t r i x correction to the fermion wave function normalization 24 2.7 New diagrams appearing in the model due to the gauge field interactions. 27 2.8 Addi t ional corrections to the Yukawa vertex 27 2.9 New corrections to the Fermion propagator • . 27 2.10 Corrections to the U(l) gauge field propagator 28 2.11 Corrections to the abelian gauge field interaction 28 3.1 Bubble corrections to the scalar field propagator 32 3.2 Loop correction to the Fermion self energy 33 3.3 New corrections to the Fermion propagator. The lower graph is suppressed by a factor of —1/NF due to the SU(NF) group structure 35 5.1 Each line corresponds to a fixed choice of Nc, as one moves from the lower region to the top Np is decreased from infinity down to it's lowest allowable value, NFrtt(Nc) for a second order phase transition to occur. 48 v 5.2 These are the renormalization trajectories projected down to the coupling . constant space (g\, g2). The dotted line represents the boundary of the sta-bil i ty wedge. The stars, circles, and crosses indicates where that particular flow hit the stability surface wi th e = 1, 0.5, and 0.25 respectively. . . . 49 5.3 Each line corresponds to a fixed choice of NC, as one moves from the top left region to the bottom, NF is increased from \\/2Nc up to it's lowest allowable value. The dotted line indicates where g\ + g2 = 0, and also specifies the upper critical NFNT(NC) for a second order phase transition to occur 56 VI Chapter 0 Introduction This thesis is organized into five chapters. The first chapter is a review of universality and spontaneous symmetry breaking. We review how radiative corrections to the classical potential can introduce a global minima away from zero field, as first shown by Coleman and Weinberg [1]. The onset of spontaneous symmetry breaking can also be understood through the beta functions of the theory alone, this work was carried out by A m i t [2] and Yamagishi [3]. We apply his work to a matrix valued field theory, and derive conditions necessary for first order behavior. The second chapter introduces the matr ix field theories that are to be studied in chapter five, the beta functions and crit ical exponents are also calculated here. Chapters three and four discuss some other models of interest to us. In chapter three a four fermion theory wi th gauged color, is introduced, and it 's crit ical exponents calculated. Chapter four reviews the connection between two lattice models, a generalized quantum Heseinberg antiferromagnet and lattice quantum chromodynamics ( Q C D ) . In chapter five the phase structure of the matrix models are discussed, and the connections between these models and those found in chapter three and four is explored. A more detailed discussion of the connection between the various chapters is given below. In the second chapter we introduce a traceless Hermitean matr ix scalar theory wi th in-ternal SU(NF) symmetry. This theory contains two relevant couplings, tr^ and (trcf)2)2. A generic feature of field theories wi th more than one coupling is that the phase transi-tions are typically of first order. The multidimensionality of the coupling constant space is responsible for this, simply because there are more directions for the fixed point to 1 Chapter 0. Introduction 2 be unstable in . The classic example of this is the massless scalar electrodynamics stud-ied by Coleman and Weinberg [1]. There, in the two dimensional plane of the Higgs self-coupling there are no infrared (IR) stable fixed points, and hence a l l phase tran-sitions are of first order. Another example occurs in the complex matr ix scalar field theory wi th SU(NL) x SU(NR) symmetry, which represents the universality class of the finite temperature chiral transition in Q C D [4]. There are two coupling constants for the renormalizable interactions tr(M^M)2 and (trM^M)2, and when the matrices are larger than 2 x 2 the phase transition is first order. When the matrices are 2 x 2 the model is equivalent to a vector theory wi th one coupling constant and the phase transition is second order. Similar reasoning has been used to argue that the (4 — e) dimensional traceless Hermitean matr ix scalar field theory has a second order phase transition only when Nf = 2 and has a fluctuation-induced first order phase transition when Nf > 2 [5]. Two generalizations of the traceless Hermitean matr ix scalar theory are also intro-duced in chapter two. The first stage is the introduction a Yukawa interaction to fermions. We calculate the beta functions of the theory, however, the analysis of the phase struc-ture is left unt i l chapter five, where we show that the existence of a non-trivial Yukawa fixed point tends to make the fixed points of the theory I R stable. This theory is shown to have the same symmetries as a four fermi theory with gauged flavor, and hints that they might lie in the same universality class. This suspicion is confirmed when one com-pares the crit ical exponents of the two theories. This work was originally carried out by [6]. One of the goals of that work was to develop a theory that lies in the universality class of (2 + 1) dimensional Q C D . Unfortunately this matrix model does not have the same symmetries, and hence can not describe it's crit ical behavior. Once a U(Nc) gauge field is introduced into this generalization of the traceless Hermitean matr ix theory, the symmetries are identical and the resulting theory can lie in the same universality class as Q C D . Chapter 0. Introduction 3 In chapter four we discuss an intriguing feature of quantum spin systems - their rela-tionship to gauge theories. This connection was originally used to study chiral symmetry breaking in Q C D , where the strong coupling l imi t resembles a spin system [7]. More recently, the analogy has been exploited to prove that certain gauge theories break chiral symmetry in the strong coupling l imit [8, 9, 10]. It has also been used to formulate mean field theories for magnetic systems [11]. For the most part, these works use the formal similarity between a gauge theory and a spin system at the lattice distance scale. Recently it has been suggested that the analogy is much broader in that it can account for the quasi-particle spectrum and other infrared features of the two systems [12]. In this thesis evidence for the latter wi l l be presented by discussing a common feature of the phase diagrams of 2-dimensional quantum antiferromagnets and 3-dimensional Q C D . The dependence of the chiral symmetry breaking pattern on the number of flavors and colors of quarks in Q C D is similar to that of the antiferromagnet where the rank of the spin algebra and the size of its representation play the same role as the number of flavors and colors, respectively. We shall also study the crit ical behavior associated wi th a chiral or Neel phase transition. Such a transition must be driven by operators which are added to the Q C D or antiferromagnet Hamil tonian and which have the appropriate symmetries. We argue that these transitions fall into a universality class which can be analyzed using the epsilon expansion. In particular we argue that the universality class is that of the gauged traceless Hermitean matrix scalar theory wi th Yukawa coupling to fermions. In chapter five we show that in many cases the phase transitions are fluctuation induced first order ones. Chapter 1 The Renormalization Group and Spontaneous Symmetry Breaking 1.1 Universality Phase transitions are abundant in physics. One example is the phase transition that occurs in hot quantum chromodynamics ( Q C D ) wi th more than two flavors of fermions [4]. Here chiral symmetry is broken at finite temperature as the fermions pick up a mass. This type of transition, where the free energy contains a discontinuity is known as a first order phase transition. If the free energy is continuous but has a singular derivative 1 the theory is said to undergo a second order phase transition at that point. A n example of a second order transition occurs in the 2 dimensional Ising model. The Onsager[13] solution shows that the phase transition associated wi th this model is of second order. The phase transitions in the above two examples occurred as the result of tunning thermodynamic parameters of the theory, namely the temperature. We are, however, interested in quantum crit ical phenomena where one deals wi th transitions, typically at zero temperature, which results from the tunning of mechanical parameters in the theory, such as coupling constants or particle masses. A n important difference between first and second phase transitions is that the lat-ter exhibit a property known as universality. This appears in second order transitions because near the crit ical point one can approximate the scaling behavior of the relevant functions as power laws. These powers depend upon the particular Hamil tonian under 1This need not be the first derivative, as long as some derivative contains a singularity the phase transition is known as a second order one. 4 Chapter 1. The Renormalization Group and Spontaneous Symmetry Breaking 5 study, yet i f two systems have identical cri t ical exponents (as these powers are called) then the theories exhibit the same cri t ical behavior. To see how universality arises consider a general Hamil tonian wi th a single field variable <j>, °° 1 r W l = 52 ~ / dDx1...dDxn <j>(Xl) x ... x 4>{xn) xHn(xu ...,xn) (1.1) n=0 n - J The n-point functions of this theory are defined as, W^(xu...,xn) = JD[</>] </>(Xl) x ... x <f,{xn) x e - M (1.2) The renormalization group method maps %[4>\ to a scale dependent Hamil tonian TL\\(j)\ such that the respective n-point functions are related as follows, win\Xl,...,xn) = Z-n/2(\)WW(\Xl,...,\xn) (1.3) where Z(X) is a renormalization factor. This mapping is interesting i f it has a fixed point /H\[<f>] —> %*[<j>] as A —i oo. Assuming such a map the n-point functions wi l l also have a fixed point, W^(Xxu...,Xxn) = l i m Z+n'2(\)WW(xu...,xn) (1.4) A—¥00 A curious relation between W* at two different scales can be seen by introducing another scale parameter, W^iXfiXu.^XfjLXn) = l im Z+n^(X)Win\fixly...,^xn) X—too = \imZ+n/2(Xfi)W^(xu...,xn) (1.5) A—¥00 which then urges one to write the following equality, W j B V i , . . . , M * „ ) = Ztn'2{n)W^\xl,...1xn) (1.6) Chapter 1. The Renormalization Group and Spontaneous Symmetry Breaking 6 where the fixed point renormalization constant has been introduced, = l im Z(\u.)/Z{\) (1.7) A—>oo Since the above equations are valid for all scales / i the Z*(u) must obey the dilatation operation, and hence form a representation of the dilatation group. Thus we can imme-diately infer, ZM = vr"* (1.8) for some positive constant d^. The correlation functions then have a very simple scaling behavior parametrized by d^, W^(fxxu...^xn) = fJ,-nd*w!;nXx1,...,xn) ix » 1 (1.9) This is a remarkable result since the R . H . S . depends only on the fixed point Hamil tonian. This is what is meant by universality, after renormalization flow to a fixed point the behavior of the system depends only on the fixed point. So i f two different theories flow to the same fixed point, then they are in the same universality class and hence have the same crit ical exponents (in this single variable example that exponent would be d$). Recal l that (1.3) leads to the Callan-Symanzik [14] equation, ("I+ Pi+ n^w) w"(xu =0 (L10) where, - ^ 9 i = A ^ i ( A ) , . . . , 5 n ( A ) ) (1.11) \~Ar\XnZ{X) = 7GH(A), . . . ,3„(A)) (1.12) These equation actually tells us under what conditions second order phase transitions can occur. Since the renormalization flow is governed by the beta functions of the theory Chapter 1. The Renormalization Group and Spontaneous Symmetry Breaking 7 a necessary condition is that the beta functions have an infra red (IR) fixed point. This for the case of one coupling constant. This tells us that if the scale parameter , A, is to diverge (so that a second order phase transition can occur) a necessary condition is that the beta function has a zero. The zero must be I R stable, otherwise the couplings would flow away from that point and never cause a divergence in A. This concludes our discussion of universality. 1.2 Spontaneous Symmetry Breaking and the Coleman Wienberg Phenom-Spontaneous symmetry breaking is a well known phenomena. In the classical world these processes are observed daily. For example, holding a pencil vertically wi th it's point on the table and then releasing it causes the pencil to fall . The direction in which the top points to as it falls is arbitrary, since a l l directions have the same free energy. However, we observe only one direction in reality. The universe has picked one state from a set of states that all correspond to the same free energy - this is what is known as spontaneous symmetry breaking. Even though the theory posses some symmetry, the ground state does not. The situation is analogous in the quantum realm. A system of spins on a lattice wi th Ising type interactions have a degeneracy in the magnetization direction. Bu t as the temperature is cooled below some cri t ical temperature T c the spins a l l align in a particular direction. Coleman and Weinberg found another mechanism for generating spontaneous symmetry breaking in field theories. They noticed that loop (or radiative) corrections to the classical potential can produce a min ima away from the origin, and can also be seen by integrating (1.11) to obtain, (1.13) ena Chapter 1. The Renormalization Group and Spontaneous Symmetry Breaking 8 hence introduce states that lie below the classical ground state. Even at the level of one-loop corrections the onset of symmetry breaking can be realized. Phase transitions that occur in this manner are known as fluctuation induced first order transitions. Such transitions occur i f the effective potential contains a global min ima at a non-zero value of the classical field. If the beta functions of the theory does not vanish then any phase transition that occurs must be of first order. This is a direct consequence of the argument in the last section, which showed that a necessary condition for second order behavior is that the beta functions contain a zero. One can see this explicitly by following the work of Coleman and Weinberg [1]. In this section we wi l l review how the (f> field in the massless scalar <p4 theory picks up a mass, as shown by [1]. Consider the (j)4 Euclidean action, S = JdDxS^(l + z^d^<j> + ^zmcl>2 + ^(l + z x ) ^ (1.14) where z$, zm and z\ are the wave-function, mass and coupling constant renormalization constants respectively. To obtain the effective potential we wi l l perform a loop-wise expansion of the functional integral. Classically the (j>A potential is given by, V = ^ c (1.15) This also corresponds to the tree level approximation to the effective potential, as it should. The next order in the semiclassical or loopwise expansion consists of polygons wi th arbitrary number of sides with zero momenta entering the ring, see fig.(l.l). The renormalization terms also appear here, so that the effective potential becomes, (1.16) Performing the formal sum in the integral takes care of the apparently dangerous infrared divergent terms, (1.17) Chapter 1. The Renormalization Group and Spontaneous Symmetry Breaking 9 Figure 1.1: Typica l diagrams occurring in the one-loop expansion of the effective poten-t ia l . A cutoff at the high momentum scale, implemented at k2 = A 2 , is introduced so that the sti l l ultraviolet divergent integral can be performed. The resulting expression is, A 4 1 , 1 . A A 2 , 2 A V Veff = ^ + -2Zm*c + -ZX*e + ^ 1 + ^ 2 2At (lnW I ' (1.18) 2 A 2 2 J A s usual terms that vanish as A —>• oo are ignored. It now remains to find the normal-ization constants. The action (1.14) contains no mass term at the tree level potential, thus, i f the theory is to remain massless at the current level of the loop-wise expansion, the second derivative of the effective potential must vanish at zero field. This forces the mass normalization constant to be, A A 2 3 2 7 T 2 (1.19) exactly what one would calculate using diagrammatic methods. A t tree level the coupling constant is found by taking the fourth derivative of the potential at <f> = M, where M is Chapter 1. The Renormalization Group and Spontaneous Symmetry Breaking 10 some arbitrary mass scale. This does not change as the order of the loop-wise expansion increases, i.e. the coupling normalization constant is found by setting, d4V 3A 2 / A M 2 1 1 \ The effective potential with al l normalization constants inserted becomes, A 4 A 2 ^ / 4>l 2 5 \ Veff = ^ + 2E6^{]n^--e) ( L 2 1 ) A t tree level, the minima in the potential occurred at zero field, now there is a global minima located at a new non-zero field value < 4>> given by the expression, < <f> > 2 = - ^ T T 2 + O(A) (1.22) (j) =< (j) > is now the ground state of the system, and at this point the second derivative does not vanish. d2V/d(l)2\<(j>> is proportional to the mass of the (f) field thus a mass is spontaneously generated. Coleman and Weinberg argue that this minima is in fact outside the perturbative regime, hence there is no reason to believe that a first order phase transition actually occurs. However, this is attributed to the simplicity of the model. In fact they show, using the renormalization group method, that the phase transition is of second order. They also show that in a more realistic model - massless scalar electrodynamics, the new min ima developed there is indeed in the perturbative regime, and a first order phase transition occurs. 1.3 Renormalization Group Flow and First Order Phase Transitions In the previous section, first order phase transitions were shown to occur as the effective potential develops a minima away from zero field. It is, however, useful to find conditions for first order behavior which depends solely on the beta functions of the theory without Chapter 1. The Renormalization Group and Spontaneous Symmetry Breaking 11 directly obtaining the effective potential. Amit[2] and Yamagishi[3] noticed that a con-nection between the beta functions and first order behavior existed. This approach wi l l be applied to a matr ix valued field theory so that the results here can be quoted later on. We have already pointed out that if the beta functions support no IR stable fixed points, then a second order phase transition cannot take place. The st i l l open question is then: "Does the existence of IR stable fixed points guarantee second order behavior?" We wi l l see that there is no guarantee, instead the resulting phase transition depends upon the ini t ia l conditions of the coupling constants and where the fixed points lie in coupling constant space. In later chapters we wi l l be concerned wi th matrix valued field theories. In particular we w i l l like the scalar matr ix field, </>, to be traceless and Hermitean. Also the theories under consideration wi l l have the symmetry <f> -> W(j)U, wi th U G SU(N). In addition the possibility of interactions with other fields is left open. W i t h these considerations there are only two renormalizable self-interactions of the (j) field. Here is the part of the action that contains only <f> fields, where we have normalized the terms so that planar diagrams are suppressed in the large N l imi t . In determining whether spontaneous symmetry breaking occurs the symmetry breaking pattern must first be found. The first question to be answered is what class of traceless Hermitean matrices minimize the effective potential. To answer this, we must consider the form of the effective potential at the min imum point. Since flows generated by the renormalization group cannot alter the form of the self interactions, i.e. gx and g2 can only pick up some dependence on a dimensionless combination of <j> and the mass scale yu, say t = ln((j)/n~2~), the effective potential at some scale is given by, 8 T T V 4! { $ j ( W ) 8 + f T r * « } (1.23) (1.24) Chapter 1. The Renormalization Group and Spontaneous Symmetry Breaking 12 where n(t,gi) is some dimensionless function. Let </>a contain the eigenvalues of the 4> field, where 0 is some fixed value, and the TV-dimensional unit vector a is variable. The effective potential can then be written as, W , ^9i) = ^ { ^ + 9~f-j:<4) e"<*«"M> (1.25) The configuration which minimizes U depends on the sign of g2: i) if g2 > 0 then we must minimize z3£Li at • ll) i f 9i < 0 then J^iLi at must be maximized. Notice that N / N \ 2 JV-1 JV E A * = ( £ « ? ) - 2 E E i=l \i=l / i=l j=i+l N - 1 N = 1 - 2 E E i=l j=t+l = l - F ( a ) (1.26) Case i) then corresponds to maximizing F(a), the configuration with minimum energy points in the "diagonal" direction, f T7lfe(+1> " I , +1, -1, •••> 0, +1, -1), N e odd ; 3 = , (1.27) ( ^(+1, -1, +1, -1, +1, -1), N € even. When #2 < 0 the symmetry breaking configuration corresponds to minimizing F(a) and occurs along the "flat" direction, a = ^ ( + l , 0 , 0 , 0 , . . . , 0 , - l ) , V / V > 1 (1.28) When looking for spontaneously broken symmetry the (j) field is chosen to be in one of the above configurations even before flowing through the R G equations. The reason is simple: any other configuration would end up in a higher energy state, even if such a configuration develops a non-zero minima. The renormalization group equation for the effective potential of the theory can be writ ten in terms of the beta functions in the following manner, ( ^ + | > | ; + i * £ W m . * M ) = <> (1.29) Chapter 1. The Renormalization Group and Spontaneous Symmetry Breaking 13 A dimensionless form of this equation can be formed from knowing that the mass dimen-sion of the <j> field is [<fr] = (D - 2)/2, so that the effective potential has form, 2D U(d>(u.),= <j>v-»V(t,gi(t)) (1.30) where the dimensionless function V(t,gi(t)) is unknown. Substituting (1.30) into (1.29) gives the desired result, ( 2 D 7 + £ % ^ - | W ^ ) ) = 0 (1.31) \ D - 2 " & " d g i at where, The general solution to (1.31) is well known and is given by, f 2D rl \ V{t,9i{t)) = M(tigi))&ip^D—^Jo y{^(x,gi))dxj (1.33) where g'i(t,gi) are the solutions to the set of coupled differential equations, d g > i wi th i.e. g'i(0,9i) = 9i (1-34) dt The function /(<4(t,&)) can be fixed wi th the knowledge of what the symmetry breaking pattern is. The boundary t = 0 is simply 1, g2 > 0, N e even; 87T 2 f(&(0,9i)) = •^^(gi + a(g2,N)g2) where, a = { N N 2 ' 92 >0, AT G odd; (1.35) N 92<0,VN> 1. which is seen by comparing (1.33) wi th (1.25). The function /(&•(£, #i)) is then the natural extension from the t = 0 case, namely replace & with <7,-(£,<&) in (1.35). The effective potential in terms of the renormalized couplings is then, UfanMt)) = J ^ ^ f o i W + a(6,N)g'2{t))e&S:m*))* (1.36) Chapter 1. The Renormalization Group and Spontaneous Symmetry Breaking 14 The dependence of g\(t) on ^ is to be understood from here on. In order for a first order phase transition to occur, a global minima must occur at some non-zero 4>. Since we have tuned the effective potential to be exactly zero at <f> = 0, as long as there is only one other min ima it w i l l be a global one i f U\<<p> < 0. This implies that g[ + a(g'2jN)g2<0 (1.37) The first derivative vanishes at the min ima so another condition is, and finally the second derivative must be positive, 0>o. so that the resulting extrema is a local min ima rather than a maxima. The result is that if the flow crosses the surface (this surface wi l l be called the stability surface), V = 0 where V = D(g[ + a(g'2, N)g'2) + 8X + a(g'2, N)02 (1.40) in the region, g[+a(g'2,N)g'2 < 0 D(p1 + a(g'2,N)p2) + £ P i ^ ( h + a(92,N)p2) > 0 (1.41) i=l " l the theory w i l l undergo a first order phase transition, as the <f> field wi l l pick up a mass at this new non-zero global minima. These results wi l l prove to l imit the region of second order behavior even further than requiring the existence of I R stable fixed points. 1.4 The Stability Wedge and Restrictions on Fixed Points We have found the criteria under which a matrix model displays first order behavior as the couplings are flowed from some ini t ia l values along their renormalization trajectories. Chapter 1. The Renormalization Group and Spontaneous Symmetry Breaking 15 Thus far our analysis leave the ini t ia l choices of the couplings arbitrary. This is in error -the effective potential, U given by (1.24), must be bounded from below in the U V l imit , and hence the ini t ia l couplings are restrained. Obviously to bound the potential 1.36 from below the couplings must lie wi th in the region, 9x + a(g2,N)g2>0 (1.42) notice that the primes on the g's are removed since this is an ini t ia l restraint. As long as the couplings start in this regime, called the stability wedge, there w i l l be some flows that reach a fixed point i f one exists. Even more can be said along these lines. If a fixed point lies outside of the stability wedge then it is possible to show that al l flows must cross the stability surface. To see this notice that the stability curves are al l of the form V = D(gi +ag2) + fi\+af32 = 0 and $ « — tgi for small couplings. Then in the U V l imi t V ~ (D — t)(gi + ag2) > 0 since the couplings must begin wi thin the stability wedge. However at the fixed point the beta functions vanish so that V = D(gi + ag2) and i f this is negative the flow must have hit the stability surface, but this is negative i f and i f only the fixed point lies outside of the stability wedge. This restriction of the fixed points conspire to reduce the size of the conformal window 2 in the (NF, Nc) plane further than requiring the existence of I R stable fixed points. 2 The conformal window refers to that region in which second order behavior is observed, as then the theory admits a massless and hence conformally invariant limit. Chapter 2 Matrix Models 2.1 A Traceless Hermitean Scalar Matrix Model The basic matr ix model in D = 4 — e dimensions is a simple generalization of the scalar 4>4— theory, S = Jd^x + | / ^ 4 } (2.1) The standard one-loop calculation of the </>4 beta function gives, ^ = - e 5 + ( J ? / + 0 ( 5 3 ) ( 2 - 2 ) It is easy to see that the zeros are given by, 9° = 0 9* = {-jf-e (2.3) Obviously g° corresponds to an ultraviolet ( U V ) fixed point and tells us nothing about the crit ical behavior of the theory. However, the slope of the beta function at the second fixed point, g*, is positive so that g* corresponds to an infra red (IR) stable fixed point. In the case of a single coupling constant the region of instabilities that lead to first order behavior is g < 0 (in analogy wi th (1.37)). However, since g* > 0 and the sign of the beta function does not change in the interval [0, g*] one can conclude that a second order transition wi l l occur. A massless </>4 theory is then allowed where the propagators become scale independent. When this situation is generalized to matr ix fields only a very l imited subset of the theories w i l l be seen to have a good conformal l imit . 16 Chapter 2. Matrix Models 17 To generalize the action (2.1), first notice that the scalar </>4 theory has the symmetry 4> —> U(j)U^ for U G U(l). A n obvious generalization of this symmetry that should accompany the introduction of a N x N matr ix field, <j>, is </> —>U4>U^  where U G U(N). Such a theory would have the action, where al l terms renormalizable in four dimensions that are consistent wi th the symmetry are included. Al though one can study this action, we do not need to consider such a complication to obtain interesting results. In addition, later on we wi l l see that if we restrict the matrix fields to be traceless and introduce couplings to fermions the theory lies in the universality class of certain four fermi theories. Such a restriction of (2.4) has Euclidean action, Again , a l l terms renormalizable in four dimensions that are in harmony wi th the pre-scribed symmetry are included. Notice that in contrast to (2.1) there are now two coupling constants. The introduction of just one more degree of freedom had a drastic affect on massless scalar electrodynamics model as shown by Coleman and Weinberg [1]. It was noted in their work, that due to the existence of two coupling constants all fixed points (within the perturbative regime) were IR unstable. Generically, when the coupling constant space is multidimensional, fixed points of the flow are not IR stable and the phase transitions , i f any, are first order. This applies to the present situation as well, as we w i l l see. As discussed in section 1.3, knowledge of the beta functions tells us about the crit ical behavior of the theory through the Yamagashi analysis. As such, we would like to obtain S = J (2.4) * = / (2.5) Chapter 2. Matrix Models 18 them here. However, using diagrammatics for the evaluation of the beta functions in this theory is quite cumbersome, as there are many contributions and it is difficult to know if a l l diagrams were included or not. Instead, we wi l l generate the terms that appear in the effective action which are infinite as e —> 0, and read off the normalization constants from there. Let (j) = (j)aTa where the Ta,s are generators of SU{N) normalized so that Tr TaTb = (l/2)Sab and the <j)a,s are scalar fields. B y perturbing around the constant classical solution 4> -> 4> + 4>c the action can be writ ten as, S = J d^Jy\n-5a%d, + M « V + ^ { ^ ( T r ( ^ ) ) 2 + | T r ( ^ ) } where, +0(J>4) 2-7T2 f 1 / fiab (2.6) c +g2 (Tr[^c2T^] + Tr[</. CT> CT 6]) j (2.7) and the notation T^aTb^ = TaTb + TbTa is used. Integrating out the scalar ^-fields from (2.6) leaves the effective action, Seff = A T T R l n p ^ (2.8) where T R means a trace in function space and indices. To one-loop the only contribution from the expansion of (2.8) is the second order term, -Mab • Mba f -^—^ — = - F ( £ / 2 ) MabMba (2 9) 2 J ( 2 T T ) 4 - ( P 2 ) 2 2(4TT)( 2-/ 2) To evaluate the contraction the Fierz identity is of use, Chapter 2. Matrix Models 19 A simple application of (2.10) produces the identities, Tr[ATa]Tr[BTa] = \ hx[AB] - h[ftA][TrB] Tr[ATaBTa] = i ^[TrA][TrS] — ~Tr[AB]j (2.11) for arbitrary matrices A and B. App ly ing these identities to the expression MabMba immediately produces the normalization constants of the theory, f N2 + 7 2N2-3 N2 + 3 g%\ 1 f 2 N2 — 9 ) 1 = 1 (2.12) The beta functions are then obtained by wri t ing the bare couplings in terms of the coupling at some scale fj, through the normalization constants, 9i = Vt9i% (2-13) Since a change of scale does not alter the bare coupling, applying the operator nd/djj, yields the beta functions in terms of the normalization constants: In calculating 3h Bj^ are set to their 0(e) term, i.e. = —egj. Then, keeping only terms up to 0(e) in expression (2.14) we obtain the beta functions, ^ + 7 , 2 i V 2 - 3 N2 + 3 o 2 N2 — 9 fa = -^92 + j^9i92 + -^-92 (2-15) The crit ical behavior of the action (2.5) can now be studied through the fixed points of these beta functions. A necessary condition for second order behavior is that the fixed Chapter 2. Matrix Models 20 point must be I R stable, this is achieved i f the stability matrix, (2.16) has a l l positive eigenvalues at the fixed points. Pisarski [15] noted that these matrix that satisfies this requirement , but a theory with such fields are known to be equivalent to a two-dimensional vector theory (since a 2 x 2 traceless matrix has only two degrees of freedom.) Thus any "real" traceless matr ix theory can have only first order phase transitions. We wi l l se that by adding a Yukawa coupling to fermions the theory is stabilized and a window of second order phase transitions is opened. 2.2 Yukawa Coupling to Fermions We now consider the first non-trivial matr ix valued field theory. As was shown above, the matr ix model by itself has only a very narrow range of values in which second order behavior is observed. In this section and beyond the dimension of the matr ix field w i l l be denoted by NF (rather than N as in the last section) indicating the number of quarks flavors in the theory. B y introducing a Yukawa coupling to an Nc x Np fermionic field, we wi l l show that due to a non-trivial fixed point in the Yukawa coupling constant, the range of second order behavior is extended tremendously (here Nc indicates the number of colors.) The most general renormalizable theory in harmony wi th the symmetries, <j> - » U(j)lJ\ V Uif> and $ ->• -0(7+ wi th U G SU(NF) has Euclidean action, The factors that appear along side the coupling constants are placed there in hind site so that planar graphs dominate and fermion loops are suppressed in the large NF l imi t . models have I R stable fixed points only if TV < 2 x 2 matrices are the only matrices S (2.17) Chapter 2. Matrix Models 21 Notice that the color symmetry has been kept intact ip —»• Vip and •0 — > ipV^. Determining the beta functions goes much the same as before. In fact there is no alteration in any of the matr ix self couplings, however some new diagrams which alter the normalization constants do appear. Previously the contribution to the normalization constants were read off through the effective potential using algebraic methods. Now, however, it is easier to obtain the new contributions directly from the diagrammatics. Since we had no need for diagrams previously, the Feynmann rules appear here for the first t ime. The vertices are shown in fig.(2.1), and the propagators in fig.(2.2). It is not difficult to convince oneself that the contribution to the matrix self coupling corrections alter only the g2 vertex, and is given diagrammatically in fig.(2.3). The other contribution to the old beta functions is introduced from the scalar matr ix wavefunction corrections. The relevant diagram is shown in fig.(2.4). The Yukawa coupling is also corrected by a one-loop diagram, see fig.(2.5). The fermion propagator, although not important for B\ and /32, is required for the calculation of By and also has a one-loop correction. F i g . (2.6) shows the relevant diagrams. Relegating the details of the loop integrals to the reader, one finds that the normal-ization factors for this model are: 7 - , , iN2 + 7 , 2 ^ 2 ~ 3 , N* + 3gj 3 y 4 ) 1 Z l ~ 1 + \ ~ ^ 9 l + ^ N ^ 9 2 + ^2J^71~WFJ^72j-e , f 2 N2 - 9 1 1 y2 i Zy = 1 U Za, = 1 SN2NC e y2 i 47V F e N l - 1 ,1 As before the matr ix self coupling beta functions are found by wri t ing the bare coupling Chapter 2. Matrix Models 22 a) b) c ) Figure 2.1: Typica l vertices for the Yukawa coupled model. Matrix x Fermion Figure 2.2: Propagators in the theory. Figure 2.5: Corrections to the Yukawa interaction. Chapter 2. Matrix Models 24 Figure 2.6: Ma t r i x correction to the fermion wave function normalization. in terms of the coupling at some scale / i as in (2.13), and applying the operator nd/d/j,, Z4 „ d ( Zi\ „ d ( Zi (2.19) Pi is then found by setting = -egj, we now have a new beta to set as well, j3y = -(e/2)y. Then al l goes as before keeping only up to 0(e) terms. The bare Yukawa coupling scales as, y° = »t/2y-and yields (3y by the above procedure. The results are, (2.20) Pi = NF + 7 2 2Nl 6N* fa = -eg2 + -^9x92 + ' F 3N2 Nl-9 3N2 -9\92 + NF + 3 2NF -92 + 2N, -y2g\ 9l 8NCN! 2NF y292 Py _e N2+2NFNc-3 3 2 V 16N2NC V (2.21) The second order behavior is again obtained through a study of the fixed points of the beta functions. As this analysis is much richer than that appearing in the basic matrix model a discussion wi l l be delayed unti l chapter 5 where we discuss the renormalization group flows, and stability in epsilon as well. Assuming that there is some regime of second order behavior it is possible to calculate the anomalous dimensions of the fields. Using (1.12) and wri t ing the wave function Chapter 2. Matrix Models 25 normalization constant as Z = 1 4- Y?k=i(ak/zk) we have, (2.22) I f d d\ (2.23) where = s 5 i ^e + . . . and By = syyt + .... W i t h this tool in hand the anomalous dimensions are, Notice there are no two pairs of (NF,NC) values wi th identical anomalous dimensions, hence every choice of Nc and Np corresponds to a different universality class. This theory has the possibility of being solved in the 1/Nc expansion, as can be seen from the fact the color indices are only contracted over the fermions. In chapter 5 we wi l l show that there exists a four fermi theory that has 1/NQ expansion wi th identical crit ical exponents as the one-loop analysis carried out here. In the next section a gauge field interaction is added to the present model in the name of generalization and in hopes of stabilizing the theory further. 2.3 The Introduction of a Color Gauge Field This wi l l be the last stage of generalization of the 4>A action (2.1). The action is once again the most general renormalizable action consistent wi th the same symmetries in the 7^  = •+ 14. = -I r2 _ i I2pNc [ V } (2.24) Chapter 2. Matrix Models 26 last section, but now a U(Nc) gauge field interaction to the fermions is included. Here is the relevant action, +ilie/2el%Afda0 + i^2e2dafi6ab^TrA^ ^ 1 ^ + 4 t r F ^ j (2.25) where A^ is the traceless part of A^ = Aai[Ta wi th T a ' s the generators of U(NC). The beta function for the non-abelian field is the just the standard Q C D result, as al l relevant normalization factors can be read off directly from ghost contributions and the gauge field normalization, neither of which detect the matrix field at the one-loop level. Thus (see e.g. [16]), To one loop the this gauge field does not affect the matrix self coupling normalization constants. However, the gauge field does have an affect on the Yukawa vertex through the new interactions shown in fig.(2.7). This introduces the contributions in fig.(2.8) to the normalization constant. The last beta function needed is the U(l) part of the gauge field. For this calculation the diagrams shown in figures (2.9), (2.10) and (2.11) are required. Leaving the details of the loop calculations to the reader, the set of normalization con-stants for the theory (excluding that for the non-abelian part of the gauge field, since we already have it's beta function) are displayed here: _ / N2 + 7 2N2-3 N2 + 3g2 3 y4 \ 1 Z l ~ 1 + \~JN^9l + ^ ^ 9 2 + ^N^7r4NFN^72f~e Chapter 2. Matrix Models 27 abelian non-abelian Figure 2.7: New diagrams appearing in the model due to the gauge field interactions. Figure 2.8: Addi t iona l corrections to the Yukawa vertex. Figure 2.9: New corrections to the Fermion propagator. Chapter 2. Matrix Models Figure 2.10: Corrections to the U(l) gauge field propagator. Chapter 2. Matrix Models 29 [ 2 T V " 2 - 9 ] 1 7 = i ( y2 , ^ c - 1 e i , e ' y y V 8 7 V 2 i V c 27VC 4TT 2 47r2y e Z e 2 " 1 ^ l e ^ ^ + l G T T 2 2 i V c e i + 3 2 i V F A r c y ) e y2 1 1 4 / V F e ( N F - 1 N c - 1 ej , e 2 7 — 1 _ b n,2 I ° 1 i Vl67v*|,/v"c y T 2NC 8TT 2 ^ 8 ^ ^ e z » = ^ w ^ i ( 2 - 2 7 > The beta functions are calculated by following the procedure in the previous section wi th the additional operator, p-i <2-28> added to (2.19) to accommodate the new degrees of freedom. This yields the beta func-tions for the gauged Yukawa matrix model, NJ. + 7 2 2NF-3 NF + 3 2 1 2 "ENT91 + ~WT9192 + ~2NV92 + 2N~FV 9 1 2 TV2 - 9 2 3 4 1 2 i v j 9 1 9 2 + -mr92" m*? + m;y 92 3 N c - 1 2 _ _3_ 2 ^YF + 2JV>iVc ~ 3 3 16TT2 NC 6 l V ~ 8 ^ e 2 y + WNFNC V UNC - 2NF 3 4 8 ^ 6 1 N C N F 3 1 2 7 T 2 6 2 A = m + 02 = Py = e 2 y Pei = e & 2 = - 5 * + (2.29) The analysis of the phase transitions of this model w i l l also be left unti l chapter 5 where a detailed discussion is given. As a final point, the anomalous dimensions of the theory can be calculated through Chapter 2. Matrix Models 30 a generalized form of (2.23) namely, 7 = 2 i s9i9i (2.30) where Be. = s^fiit + — Assuming that an I R fixed point of the coupled beta functions exists, the cri t ical exponents are, where the stared couplings are evaluated at the fixed points. Notice that the boson field exponents dependence on y* has not been altered by the introduction of the gauge field -this is a direct consequence of the lack of interaction between the matr ix and gauge fields. However, the numerical value of the crit ical exponents differ from the non-gauged model for any choice of Np and Nc showing that they lie in different universality classes. Also wi th in the present model alone, each choice of (NF, Nc) leads yields different exponents and hence lie in different universality classes. and, 14> = (y*)2 (2.31) 8NF Chapter 3 Four Fermi Models 3.1 The Gross-Neveu Model The Gross-Neveu model consists of Nc fermions, ipa ,a = 1,..., Nc, interacting through a four-fermion vertex which obeys a U(Nc) symmetry. The action is, s = jdDx {r^d.r - ^ - ( r r ) 2 } (3.1) and was first studied by Gross and Neveu [17]. It was introduced to study fermion mass generation, and also exhibits a number of other interesting phenomena such as dimensional transmutation, however we are only interested in the first. Notice that the model has C , P, T and discrete chiral symmetry in addition to the global symmetry ip —> Uip where U £ U(Nc)- These symmetries w i l l survive even after the generalization of the next section, and hints towards the type of matrix model to study later on. This model in solvable in the 1/Nc expansion. To see this firstly introduce an auxiliary field <p to "replace" the four fermi interaction, s = fd°x [ r ( 7„aM+</>) r +1^2} (3.2) This model is seen to be equivalent to the original one through the equations of motion of the field 0, or by carrying out the gaussian integration in the functional integral representation of the parti t ion function. In this form the fermions are easily integrated 31 Chapter 3. Four Fermi Models 32 out of the theory, leaving the effective action, Seff = -NCTR l n ( 7 / ^ + </>) + / d D x ^ \ (3.3) here T R means a trace in function space. In eq.(3.3) l / i V c takes over the roll of the Plank constant. A l l quantities in the theory are expandable in this parameter. In particular we are interested in scalar and fermionic propagators, as these objects lead directly to a computation of the crit ical exponents of the theory. To 0(1) the scalar field <j) is corrected by the fermionic bubble diagrams depicted in fig. 3.1. These diagrams contribute at this order since each bubble is a closed fermi loop and a factor of 7V C originates there, while each internal 4> propagator introduces a factor of 1/NC, so that the resulting contribution is 0 (1 ) . Summing up al l of these diagrams Figure 3.1: Bubble corrections to the scalar field propagator yields the dressed up propagator, A ^ { r dDk _ k(k-p) \ n f dDk J (2n)D _\_ Nc 1 Nc 1 - A dDk Tr_k(k-p) A D-2 (2n)D k2(k-p)2_ 'I _ _ 8 1 A ~ (47r)^/ 2 (D - 2 )T(D/2) 4 r ( i - / j /2) [ r (D/2)] 2 ( 4 T T ) D / 2 T(D-l) { P ' (3.4) where in the last line we have introduced a naive U V cutoff at momentum scale p2 = A 2 . Chapter 3. Four Fermi Models 33 From the above it is obvious that at the crit ical point, A — A c = ( 4 ? r ) P / 2 ( D - 2 ) r ( D / 2 ) A 2 - D (3.5) 8 the dressed propagator becomes scale independent, and takes on the form, ^ = ^ r ( " r % 2 T 2 ) ' 2 ( p 2 ) ' " s A { D ) i ^ ( 3 6 ) The first correction to the fermion propagator occurs at the next order in the 1 /Nc ex-Figure 3.2: Loop correction to the Fermion self energy pansion, and is shown diagrammatically in fig.(3.2). The correction consists of a dressed scalar connecting the external fermion lines. The dressed scalar appears since a l l subdia-grams in the dressed propagator are 0 (1) , so a l l of them must be included in the fermion self-energy to obtain the exact 0(1/Nc) contribution. Comput ing the loop integral gives the inverse propagator, dDk k+p S ~ 1 { P ) = l p - A k ) I ( 2 ^ (D)J (27r)D(p2)i-i(k+p)2 = i P {' " 2NcDT(D/2-mil-D/2)[Y(D/2)YXri ( £ ) + ****} ( 3 J ) Once again the momentum integral was cutoff off at a scale p2 = A 2 . Exponentiat ing the R . H . S . of eq.(3.7) immediately yields the scaling dimension of the fermion propagator. Since both the scalar and fermion propagators are scale independent once the coupling Chapter 3. Four Fermi Models 34 is tuned to it's cri t ical , the theory must undergo a second order phase transition i f the field remains massless. The scaling dimensions 1 at this conformal point are given by, S(p) oc ~ \ P \ 2 A f + 1 - D —ip A 0(p) cx \ P r ° - D (3.8) where we have found, A B = 1+0(1/NC) D - l T(D - 1) nn/M*\ A f ~ 2NcDT(D/2-l)T(l-D/2)[T(D/2)]2+0[l/Nc) [ 3 ' 9 ) Notice that the fermionic exponent is Nc dependent, thus each choice of Nc corresponds to a theory in a different universality class. 3.2 A Four Fermi Theory with Gauged Flavor In this section a flavor index is introduced in the Gross-Neveu model and the simple fermion coupling is changed to an SU(NF) isovector coupling. The action is, S = j dDx JC7A< - ^«T>J#T>5) (3.10) where tp is a Nc x NF matr ix wi th spinor entries, and TA are the generators of SU(NF) with normalization trTATB = 8AB/2. This theory has a l l of the same symmetries of the basic one: C , P, T , and discrete chiral symmetry but has a larger internal symmetry: SU(NF) x U(Nc). As before one can introduce an auxiliary field 0, this time a NF x NF matrix field, which decouples the four fermi interaction. The action can then be written as, s = jdDx {<(<WyA + + f ^ 2 } ( 3 - n ) JThe scaling dimension, A , is related to the anomalous dimension, 7, via A = V + 7 where V is the canonical dimension of the field. The canonical dimensions are — 1 — e/2 and = (3 — e)/2. Chapter 3. Four Fermi Models 35 Notice that the (j> field must also be traceless and Hermitean to reproduce the SU(NF) flavor interaction. Once again the fermions can be integrated out leaving the effective action, S = -NCTR In {j^ + <£) + / dDx ^ t r 0 2 (3.12) where this time T R means a trace in matr ix indices as well as in function space. Since we art Figure 3.3: New corrections to the Fermion propagator. The lower graph is suppressed by a factor of —1/NF due to the SU(NF) group structure. are now dealing with matrix valued fields the double line representation of propagators and vertices w i l l be useful. The diagrams that correct the the scalar propagator are then exactly as in the Gross-Neveu model. However, the fermion propagator picks up an extra factor due to the occurrence of the two diagrams in fig.(3.3). This introduces a factor of NF — 1/NF into it's correction (see section 2.1 for details). Thus we immediately obtain the scaling dimensions of the fields, AB = \+0{\/Nc) » _ D-l NJ.-1 r ( D - l ) 2NCNF DT(D/2 - 1)T(1 - D/2)[T(D/2)}2 + 0(1/NC) (3.13) Chapter 3. Four Fermi Models 36 If we expand this expression around four dimensions, by setting D = 4 — e, AF becomes, We wi l l show in a later chapter that this four fermi theory lies in the universality class of one of the matrix valued field theories studied in chapter 2. Chapter 4 Spin Systems and Lattice Gauge Theories 4.1 Spin systems We wi l l consider a generalization of the quantum Heisenberg antiferromagnet in D d i -mensions on a square lattice. The Hamil tonian is ff = i E J(x)J{y) (4.1) L9 <x,y> where J(x) are quantum spin operators operating on a irreducible representation of SU(NF) at sites x, y,... on the lattice and with < x,y > the link between nearest neigh-bors. They have the algebra [jA(x),JB(y)]=ifABCJc(x)8xy (4.2) The Hamil tonian (4.1) does not have a coupling constant. The constant g2 simply sets the units in which one measures the energies of the quantum states (or one could consider it as a unit of time). For SU(Np) spin systems large Np corresponds to the quantum l imi t , as opposed to the l imit of large representations which is the classical l imit . In the former l imit , the ground state of the antiferromagnet has an SU(NF) version of Neel order. To get a handle on this theory we wi l l construct the spin operators explicitly. It is convenient to construct the spin operators for a given algebra in an irreducible representation using oscillators. These could be either fermionic or bosonic. Here, we shall use fermionic oscillators, wi th destruction and creation operators ip^{x) a n d i>a{x)i 37 Chapter 4. Spin Systems and Lattice Gauge Theories 38 respectively, and the algebra {tf{x),l{\v)}=6abSat,8(x-y) (4.3) We shall call the indices a, 6 , . . . = 1, . . . Nc "color" indices and a, j3,... = 1 , . . . , NF the "flavor" indices. The reason for this nomenclature wi l l become clear shortly when we discuss lattice gauge theory. The spin operators in the Lie algebra of SU(NF) are JA(x) = tf*{x)TAtf(x) (4.4) where the fundamental representation Hermitean matr ix generators TA of SU(NF) obey the algebra [TA,TB]=ifABCTc (4.5) and have the normalization condition Tr (TATB) = ^5AB (4.6) The spin operators, JA(x), then obey the commutation relations given in (4.2), and the space on which these generators operate is the Fock space which is created by operating creation operators il>^(x) tfWtf^i)" • • 1°) ( 4 - 7 ) on the empty vacuum which obeys C ( s ) |0> = 0 , Vx,a,a (4.8) For any Nc, this Fock space carries a reducible representation of the algebra. A n irre-ducible representation is obtained by projecting onto a subspace of the Fock space. This is accomplished by imposing a constraint. The representation wi th a rectangular Young tableau wi th k rows of Nc boxes is gotten by imposing the condition of gauge invariance Chapter 4. Spin Systems and Lattice Gauge Theories 39 under the U(Nc) transformation which is generated by gab(x) |phys.) =• ^E ltf{x)W(x)J bhys.) = 5ab k Iphys.) (4.9) The "physical states", |phys.) span the irreducible representation of the spin algebra given by the Young Tableau with Nc columns and k rows of boxes. The constraint operators obey the local algebra of U(Nc), [Gab(x), Qcdiv)] = {8ae8bc5df - 8ad5ce8bf) Qef(x) Sxy (4-10) and commute wi th the Hamil tonian, [gab(x),H] = 0 (4.11) and the "observables", [Gab(x),JA(x)} = 0 (4.12) They generate the gauge transformation, <(*) -+ Uab(x)^(x) (4.13) and the Heisenberg antiferromagnet in this formalism is gauge invariant. Antiferromagnets of this type were studied by Read and Sachdev [18] using semiclassi-cal methods. Their analysis was in two dimensions (D = 2) and the only free parameters are the integers Nc and NF. Nc » NF is the classical l imit of large representations, where the classical Neel ground state is stable wi th the staggered spin order parameter NC pab(x) = ( -1 )5>« < g tfLOOlM*) > (4-1 4) a=l On the other hand, the l imit NF » Nc is the quantum l imi t where fluctuations are important and the system is in a spin disordered state. For both Nc and NF large, they find a line of second order phase transitions in the (Nc, NF) plane at NF = const. • Nc Chapter 4. Spin Systems and Lattice Gauge Theories 40 where the constant is a number of order one. These results are somewhat insensitive to the value of fc.1 4.2 Hamiltonian lattice gauge theory In this Section, we shall review the Hamil tonian lattice formulation of Q C D in D + 1 dimensions. We shall see that the relationship between the antiferromagnet and Q C D is a very close one. This is a summary of the work reported in ref. [9] which maps the strong coupling l imi t of lattice Q C D onto the antiferromagnet wi th Hamil tonian (4.1) and in particular irreducible representations of the flavor algebra. The degrees of freedom of lattice Q C D are the gauge fields which are unitary operators U^ixy) and color electric fields which are Hermitean operators Eab(xy), both which live on links < xy > of the lattice and transform under the adjoint action of the gauge group as U(xy)^VxU(xy)Vj , E{xy) -> VxE(xy)V} (4.15) and obey the reflection conditions, U{yx) = rf(xy) , E(yx) = -rf(xy)E{xy)U(xy) (4.16) We shall assume that the gauge group is U(NC). There are also quark fields ip%(x) which live on sites, x and transform under the fundamental representation of the gauge group 1>a(x) -» V{x)il)a{x) (4.17) and which also transform under the fundamental representation of a global SU(NF) flavor group WW-> gafrfSix) , geSU(NF) (4.18) 1 In fact, they considered a slightly more general case than we have described here where there are different representations on even and odd sublattices. Even in that case, their results are insensitive to the representations on each sublattice. Chapter 4. Spin Systems and Lattice Gauge Theories 41 The Q C D Hamil tonian contains three terms, the quark kinetic energy and the electric and magnetic energies: H= E hl(x)Ua\xy)^y) + h.c. + -j:(EA(xy)) ) + <x,y> \ A A=\ ) +^Etr(n^ + n^ f) (4-19) where the first sum over links < xy > is the quark kinetic and total electric energies, respectively and the second over plaquettes • is the magnetic energy. The lattice reg-ularization of the quark kinetic energy uses staggered fermions [19]. The electric fields E(xy) are L ie algebra valued operators and can be expanded in terms of the generators oiU(Nc), E(xy)= £ EA(xy)TA (4.20) A=0 with T ° the (unit matrix) generator of U ( l ) and TA are the generators of SU(NC). The electric energy in the Hamil tonian is the sum over gauge group laplacians which act on the color group degrees of freedom associated wi th each link. The magnetic term is the Wi l son energy function for a gauge field. The gauge fields and electric field operators satisfy the algebra [EA{xy), EB(zw)\ = ifABCEc(xy)6(xy, zw) (4.21) [EA{xy),U(wzj\ = U{xy)TA 5(xy,zw) . (4.22) The Hamil tonian is supplemented by the Gauss' law constraint, which we impose as a physical state condition, SaV)|phys.) E EA(xy)TA + ^^aa(x)^ba(x) |phys.) pj/6A/'(x) a=l = 5abNF/2 |phys.) (4.23) Chapter 4. Spin Systems and Lattice Gauge Theories 42 and which enforces gauge invariance of the physical states. Here the first summation is over al l links one of whose endpoints is the site x. This term is a lat t icization of the covariant divergence of the electric field. Staggered fermions have a relativistic continuum l imi t when their density is 1/2 of the maximum that is allowed by Fermi statistics, in this case NcNF/2 per site. Furthermore, the quark kinetic energy term in the Hamil tonian must have phases which produce an effective U(l) magnetic flux ir per plaquette [19, 9]. That the latter fact is necessary in order to obtain a relativistic spectrum in the continuum l imi t is easy to see i f one considers the naive latt icization of the Dirac Hamil tonian which has the correct lattice spectrum, but also has fermion doubling, h = ^2 (i^(x)aji/}(x + j) — iip(x)ajij)(x - j ))- (4.24) x,i • . where a* are the Dirac matrices. This operator describes a spinor wi th energy spectrum ,E{k) = J2smki (4-25) • i This dispersion relation has small energies where ki ~ 0 and where ki ~ ix. There are 2D such combinations, resulting in a fermion mult ipl ici ty of 2D for each component of the spinor in (4.24). Th i s mult ipl ici ty can be reduced by the spin diagonalization method. This method begins wi th the observation that the naively latticized Dirac Hamil tonian (4.24) resembles an ordinary lattice hopping problem where the Dirac matrices can be thought of as a background gauge field. This background gauge field can be diagonalized by a "gauge" transformation. This gives a number of independent copies of the staggered fermion Hamiltonians, one for each of the original components of the spinor. Choosing one component gives staggered fermions. These s t i l l have a mult ipl ici ty 2D in the continuum l imi t . That diagonalization is possible is a result of the fact that the curvature of the Dirac Chapter 4.' Spin Systems and Lattice Gauge Theories 43 matr ix background gauge field is a constant and is diagonal[10], OLiOt,jOL~^osj1 = — 1 (4.26) and there is a gauge in which the "gauge field" cVj is diagonal. In order to obtain the appropriate phases in the weak coupling continuum l imi t , we have chosen the sign of the th i rd , magnetic term in the Hamil tonian so that it is minimized by the configuration of U{Nc) gauge fields wi th the property < Yin U >= — 1- The constraint of half-filling is enforced by (4.23). If the lattice is 2 dimensional, the naive continuum l imi t yields 2+1-dimensional Q C D with gauge group U(Nc) and Np species of massless four component fermions. In 3 dimensions, there are 4 species of 4 component fermions. In both cases, the (chiral) flavor symmetry of the continuum l imi t is greater than that of the lattice theory. O n the lattice, for example, staggered fermions do not have any continuous chiral symmetry. On the other hand, they do have a discrete remnant of chiral symmetry (translation by one site) which forbids explicit fermion mass terms [9, 10]. A fermion mass term is a staggered density operator. j(x)TA^(x) ^'(-l)^y^x)T^^(x) = J ^ T ^ i x ) . (4.27) Thus, the antiferromagnetic order parameter and the order parameter for chiral symmetry breaking wi th a flavor-vector condensate are identical. In fact, in the strong coupling l imi t , the problem of finding the ground state of lattice . Q C D is identical to that of solving the generalized antiferromagnet wi th Neel order playing the role of chiral symmetry breaking. The argument of [9] can be summarized as follows: The strong coupling l imi t , e 2 —> oo suppresses fermion propagation (since the fermion kinetic term in the Hamil tonian is subdominant). In the leading approximation, the Hamil tonian is minimized by the states which contain as l i t t le electric field as possible Chapter 4. Spin Systems and Lattice Gauge Theories 44 and which are compatible wi th the gauge constraint (4.23). When NF is even [20], it is possible to solve Gauss' law wi th EA = 0. The occupation number of each site is NcNFj2 and < (— l) x '0L' i /w > = 0 in this state. This is a highly degenerate state - any gauge invariant state with NFNc/2 fermions has the same energy. Because they are required to be color singlets, this is the same set of states as occurs in the antiferromagnet when k = NF/2, i.e. in the representation of SU(NF) whose Young tableau has Nc columns and NF/2 rows. Furthermore, to resolve the degeneracy, one must diagonalize the matr ix of perturbations. These are non-zero only at second order and the diagonalization problem is equivalent [9, 10] to solving for the ground state of the antiferromagnet Hamil tonian (4.1) wi th — t2/e2. F inal ly , since the order parameters are identical, the Neel ordered states of the antiferromagnet correspond to chiral symmetry breaking states of Q C D . Thus, the infinite coupling l imit of Q C D is identical to the antiferromagnet. A main difference between Q C D with finite coupling add the antiferromagnet is that Q C D con-tains electric and gauge fields which allow a fermion kinetic energy and st i l l retain gauge invariance, whereas in the antiferromagnet, the fermions are not allowed to move. One could regard the corrections to the strong coupling l imit of Q C D as the addition of de-grees of freedom and gauge invariant perturbations in the antiferromagnet which allow fermion propagation. In fact, ref. [12] suggests even a stronger correspondence, that the additional degrees of freedom are generated dynamically. Chapter 5 Nature of the Phase Transitions 5.1 The Yukawa Matrix Model and Four Fermion Theories In this section we wi l l show that when a Yukawa interaction to fermions is introduced in the Hermitean matr ix theory of section 2.1, the existence of a non-trivial fixed point tends to make the phase transitions second order. The Yukawa matr ix model action is given by (2.17) and is re-displayed here, + l T r W . V 0 + 8 7 R V J | ( T r ^ + f W (5.1) The beta functions for the theory were derived in section (2.2) and are given in (2.21). A study of the second order behavior requires knowledge of the fixed points of the beta functions. The Yukawa coupling fixed points are easily found, y° = 0 V) 2 = , 8 N 2 p N c e (5.2) y Np + 2NFNC - 3 { { Second order behavior can be observed i f the beta functions support an I R stable fixed point. This is turn requires that the stability matr ix Wij = d/3i/dgj (g^ = y) has a l l positive eigenvalues at the relevant point. A fixed point of the coupled system (2.21) using y° could be U V stable but never I R stable, since By has negative slope there. The non-trivial fixed point y* must then be used when looking for I R stable fixed points. The beta functions for the matr ix self couplings are then reduced to, f(y*)2 \ ,' NF + 7 2 2N2F-3 NF + 3 2 45 Chapter 5. Nature of the Phase Transitions 46 * = {U-^+n***^^*^^  (5- 3) at this point. The existence of a non-trivial Yukawa fixed point has introduced a constant term in 62 (with respect to the Hermitean matr ix model, see (2.15) for the beta functions.) This serves to push the gaussian fixed point (g^g^) = (0,0), of the basic matr ix model, to some non-trivial value. The constant term steamed from the introduction of a box diagram (fig.(2.3)) to the correction of the g2 vertex. This diagram is not a planar graph and is therefore suppressed by factors of NF. Consequently in the l imi t NF » 1 1 the Yukawa matr ix model reduces to the basic model since, (V*)2 n , ,. : 3 l im -^ - f - = 0 and, l im n : " r (y*) 4 = 0 (5.4) NF^OO 2NF NF^OO 8NCNF K ' K ' In the basic model second order behavior was found to occur only i f NF < y/Ef as a result the large NF l imit produces a theory that undergoes a first order phase transition. However, in the opposite l imi t 2 , Nc ~> 1, the Yukawa interaction adds new structure. Under such circumstances the non-gaussian fixed points of the simple matrix theory are stabilized by the Yukawa term. It is t r iv ia l to show that in the large Nc l imi t , an I R stable fixed point of the coupled system, (5.3), occurs at, 9 2 = + ^ + ° ( 1 / ^ c ) 2 ) (y*)2 = WFe + 0(l/Nc) (5.5) The matr ix self couplings are suppressed by factors oiNc and are consequently consistent wi th a one-loop calculation. In this l imit the scaling dimensions 3 of the bosonic and 1This is known as the quantum limit in the case of the generalized Heisenberg antiferromagnet 2This is the limit of large representations in the antiferromagnet, and corresponds to the classical limit 3The scaling dimension, A , is related to the anomalous dimension, 7, via A = V + 7 where V is the canonical dimension of the field. The functions 7 were calculated previously and are given by (2.24). Chapter 5. Nature of the Phase Transitions 47 fermionic fields are, A 0 = 1 + 0(1/NC) = ^ + rae + ° ( 1 / ^ ( 5 - 6 ) We now understand the quantum and classical l imits of this theory, it is possible to obtain results for intermediate number of colors and flavors only through computer calculations. In [21] the analysis was carried out, however the yamagishi requirements for first order behavior were neglected, as such the region of second order behavior as claimed by [21] is slightly larger than the actual regime. As the arguments in section 1.4 show, a theory can show second order behavior only if the coupled system of equations (2.21) admit a I R stable fixed in the stability wedge given by (1.42), 0, NF £ even 0, NF £ odd 0, VJV/r > 1 Once this is kept in mind the numerical results of [21] are modified slightly, but the remaining arguments are valid. The crit ical relation between-Nc and NF for second order behavior is shown in fig.(5.1). In addition we have found that varying epsilon does not alter the character of the renormalization flow. The flow diagrams scale wi th epsilon (as dictated by a one-loop calculation), however the stability surface does riot. Figure 5.2 show some flows of al l three couplings from the U V l imi t near the origin to either the IR stable point shown, or off to infinity. Only the projection down to the (<?i, g2) coupling space is shown so that lines that appear to cross in the diagram do not do so in the full space. The stars, circles and crosses show the points where that particular flow crossed the stability surface in the region that indicates first order behavior (see (1.41) and related ones) for 0i + 9i + 92 > 0 , NF 92>0 , N F - 1 NF 9I + ^~92 > 0 , 92 > 92 > 92 < Chapter 5. Nature of the Phase Transitions 4 8 Figure 5.1: Each line corresponds to a fixed choice of Nc, as one moves from the lower region to the top NF is decreased from infinity down to it's lowest allowable value, NFrit(Nc) for a second order phase transition to occur. c — 1, 0.5 and 0.25 respectively. Notice that all flows that lead to first order behavior do so regardless of the value of epsilon, the points at which the flows hit the stability surface simply move in closer to the stability wedge as e —> 0, but no curve that leads to second order behavior suddenly hits the stability surface as epsilon is varied. This suggests that the critical behavior is insensitive to the choice of e. From this point on choose the pair (Nc, NF) such that the corresponding theory allows second order phase transitions. With this aside we will to discuss the universality class of this matrix valued field theory. This model was built as a generalization of <fi4 theory, where the symmetry <f> —> U^lP, with U G C/(l) was generalized to U G SU(NF) and a Yukawa interaction to fermions was introduced, in the present model there are alot more symmetries: C, P, T , discrete chiral symmetry and global SU(NF) x U(Nc) symmetry. These symmetries are shared by another theory, the four fermi theory described by the Chapter 5. Nature of the Phase Transitions 49 Yukawa Matrix Model N_f = 80 N_c = 50 g _ l / e Figure 5.2: These are the renormalization trajectories projected down to the coupling constant space (^I , ^ ) - The dotted line represents the boundary of the stability wedge. The stars, circles, and crosses indicates where that particular flow hit the stability surface wi th e = 1, 0.5, and 0.25 respectively. Chapter 5. Nature of the Phase Transitions 50 action, S = J dDx j < V W - ^M^XA) ' .(5-7) where a, B = 1 , i V F , a, b == 1 , N c , ip is a Nc x i V F matr ix of four component Dirac spinors and TA are the generators of SU(NF) wi th the standard normalization. The fact that this theory obeys the same symmetries as the Yukawa matr ix model suggests that they might lie in the same universality class. A s the study in section 3.2 showed, the crit ical exponents coincided exactly with those of the present theory, to this order in Nc (compare (5.6) wi th (3.13 and (3.14).) Thus the connection between the matr ix model and the four-fermi theory has been established. Numerical calculations have shown that for Np large, the cri t ical number of colors is given by Nc « 0.27NF. For intermediate values the results are consistent wi th those obtained from equivalent, four-fermi theories [22]. One can speculate whether the upper crit ical value of NF « 3.7 Nc for the existence of chiral symmetry breaking in the four-fermi theory is related to the upper cri t ical Np ~ conts. x Ncoior for the existence of chiral symmetry breaking in (2 + 1) dimensional Q C D [23]. A correspondence of this type was the intention of [24]. However, one cannot naively identify U(NQ) wi th the color group, since i f Q C D is confining U(Nc) symmetry is absent and i f it is not confining then massless glouns should contribute to the crit ical behavior. In the next section a theory in which the identification can be made is discussed. 5.2 The Chiral Phase Transition and The Gauged Yukawa Matrix Model It has been observed that, in both 2+1-dimensional Q E D [25] and Q C D [23], there exists a cri t ical number of flavors such that i f Np < Npnt- the model breaks chiral symmetry spontaneously and i f NF > Npnt- the theory is in a chirally symmetric, deconfined phase. For large Np and for large number of colors Nc the equation of the cri t ical line is Chapter 5. Nature of the Phase Transitions 51 approximately 1?8 N F - — N c = 0 (5.8) A heuristic argument for this behavior is that when NF » Nc internal gluon exchanges and the gluon self-coupling are suppressed by factors of Nc/NF. Resummation of leading order diagrams, which are chains of bubbles, produces an effective interaction which falls off like 1/r, rather than the tree level In |r | . The weak coupling of order Nc/NF and mi ld infrared behavior of this resummed theory result in a chirally symmetric, de-confined phase. When NF is small , the effective coupling is large and can generate a condensate, which is already seen in Q E D [25]. In fact, in Q C D , when NF « Nc a l l planar diagrams contribute to processes, making the effective interaction string-like [26] and the theory is in a confining and chiral symmetry breaking phase [1]. Numerical simulations [27] of 3-dimensional Q E D support this scenario wi th 7 Y p n t - ~ 4. Mass operators for basic 2-component fermions in 2+1 dimensions are pseudoscalars and break parity explicitly [28]. Massless 2+1 dimensional Q C D wi th an odd number of flavors of 2-component fermions is afflicted wi th the parity anomaly [29] which generates a parity violat ing Chern-Simons term and also fermion mass term by radiative corrections. W i t h an even number of flavors, there exists a parity and gauge invariant regularization and Q C D is the 2+1-dimensional analog of a vector-like gauge theory in 3+1 dimensions. In particular a kind of chiral symmetry can be defined. It is known that, in this case, parity cannot be broken spontaneously [30] and therefore to study chiral symmetry breaking it is necessary to seek parity conserving mass operators. Following [25, 23, 5] we shall use NF species of 4-component fermions. The flavor symmetry of massless Q C D in this case is actually SU(2NF). We wi l l add operators to the action which reduce the symmetry to SU(NF), for example, the gauged Nambu-Jona Lasinio ( N J L ) model wi th Chapter 5. Nature of the Phase Transitions 52 four-fermion interaction, (5.9) where TA is a generator of SU(NF) in the fundamental representation. Notice this is just the gauged form of the four-fermi theory (5.7). The 4-fermi operator, which is renormalizable in the 1/Nc expansion [6], can drive the chiral phase transition wi th condensate <$>A = < ij;TAif; >• The results of [25, 23] indicate that i f NF < NFrit the order persists even when A = 0. Gauged N J L models, when analyzed by solving the gap equation [31], exhibit second order behavior at a surface in the space (NF, Nc, X). Our analysis w i l l indicate that for a large range of parameters fluctuations make this transition first order. Our results do not apply to the hypothetical case where NF or Nc are varied to drive the transition [32]. In chapter 4 we have reviewed the arguments that show that the strong coupling l imi t of (2+1) dimensional Q C D is equivalent to the generalized Heisenberg antiferromagnet on a 2 dimensional lattice. One common feature of these theories is that besides the external parameters, NF and Nc, they have no free parameters. In these theories once NF and Nc are given, either the vacuum is symmetric or the symmetry is spontaneously broken. One could imagine adding operators such that i f their coupling constants are varied, it can induce the chiral transition. We conjecture that these transitions fall into a universality class which can take into account a l l such modifications, as long as they respect the symmetries of the theory. We argue that the universality class is described by the 4 — e dimensional Euclidean field theory given by 2.25 and redisplayed here, Chapter 5. Nature of the Phase Transitions 53 (5.10) The evidence that (5.10) describes the universality class comes from the work of the last section, where the gauge fields are absent. Recall that the anomalous dimensions of the fermion and matr ix fields were identical to leading order in 1/NC and e to those of a 2 < D < 4 dimensional four-fermi theory. However, that model suffered from a lack of U(Nc) gauge invariance, and as such could not represent the universality class of 2 + 1 dimensional Q C D . The model (5.10) does however have a l l the required symmetries, and we conjecture that it represents the universality class of lower-dimensional four-fermi theories wi th U(Nc) gauge invariance. We wi l l show that, as a consequence, the chiral phase transition is a fluctuation induced first order transition for a large range of values of (NC,NF). When it is second order, the crit ical exponents are computable and are presented here. In the last section'we showed that the introduction of a Yukawa coupling to the Hermitean matrix model opened a window in (Nc, NF) space for second order phase transitions. When two more couplings, to a gauge field, is introduced we wi l l show that the window closes (but not completely). This is to be expected since at each fixed point there is now an entire plane of directions in which instabilities can develop. This result can be shown explicit ly by repeating the analysis of the last section. The fixed points of the beta functions for this model have a similari ty to the Yukawa model. In particular, al l but the matrix self couplings can be t r ivia l ly removed from the coupled system (2.29). The zeros of 3 e i and 8e2 can be solved independently and allows the potentially I R stable non-zero solutions, Chapter 5. Nature of the Phase Transitions 54 Using these zeroes in 3 y gives the potentially I R stable Yukawa coupling fixed point, [ V } ~ [\ 9UNc-2NF)•NF + 2NFNc-3t ( 5 > 1 2 ) Notice that in the large NF l imi t the result obtained without the gauge couplings is reproduced, however large Nc introduces an extra numerical factor of | . This difference is a result of the gauge fields coupling to the color indices of the matr ix field and not the flavor. Since the beta functions for the matr ix self couplings are identical in the gauged model and non-gauged model, the I R fixed points are obtained through the solution of the coupled equations (5.3) where the y* is now the one appearing in (5.12). In 4 dimensional Q C D i f Np < b.hNc the theory is asymptotically free, however in this regime e* corresponds to an U V stable fixed point and not an I R one. The entire system is then IR unstable in the asymptotically free region. To obtain IR.stable solutions, one must move into the non-asymptotically free region. This sets a lower bound for our conformal window. Recall that the arguments in section 1.4 forced the fixed point to remain in the stability wedge, gi+cx(NF,g2)g2>0 (5.13) • for second order transitions to occur (where cx(NF,g2) is given by (1.35).) Let us consider the case a(NF G even, g2 > 0) = 1 so that the symmetry breaking pattern is the one where half the eigenvalues of the < (j) > groundstate a l l have equal magnitude but half are positive and the other half negative. W i t h this symmetry breaking pattern in mind we must impose the condition g\ > —g2. B y setting g* = —pg exactly, (5.3) can be used to obtain a crit ical relation between NF and Nc which separates those theories that allow second order transitions from those that do not. The two equations for g{ are, 0 = -(U-^+^^2-mN-F^ ^ Chapter 5. -Nature of the Phase Transitions 55 Wri t ing 7 = NF/NC and taking the l imi t Nc —> 0 0 , one obtains the constraint, 7<8 .3 (5.15) for a solution to exist. Figure 5.3 shows how the fixed points in the (51,^ 2) space vary wi th Nc and Np. Notice that even though I R fixed points exists for NF > 8.3iV c, they correspond to fixed points that have no flows which miss the stability surface, and al l lead to first order phase transitions. Also, the flows in this model, like the Yukawa model, are insensitive to e in the domain 0 < t < 1. Near e = 1 the coupling constants are large and one would not expect the one-loop approximation that we have used to be accurate there. One can also find the cri t ical exponents for large NF and Nc, 2 T 2 - n 7 - 18 € A c = 1 -( 2 7 - l l ) ( 7 + 2 ) 2 _ 3 2 7 2 - 15 7 - 50 e A F - 2 " ( 2 7 - I I X T , ) "I ( } where the bound 5.5 < 7 < 8.3 is imposed by asymptotic freedom and the above ar-guments. A n important note can be made here, in the asymptotic free regime of the theory, where Np < HNc/2, there should be nonperturbative behavior associated with confinement which is inaccessible to our computation. So that second order behavior in the region 7 < 5.5 is not entirely ruled out. If is possible to obtain some physical results from this analysis. In particular, the physical quantum spin j antiferromagnet corresponds to Np = 2 and Nc = 2j. W i t h Nc = 1 ( j — 1/2) the non-abelian field must be removed, and the resulting theory has no I R fixed point, indicating a first order transition. For NF = 2, Nc > 2 (i.e. j > 1) the theory is in the asymptotically free regime and hence these antiferromagriets cannot be analyzed by these techniques. We speculate that confinement is associated wi th a nonperturbative I R fixed point of the gauge coupling. In that case, since we have noticed Chapter 5. Nature of the Phase Transitions 56 12.0 Running Coupling Constants ( N_C f i xed , N_F v a r i e d ) 10.0 Figure 5.3: Each line corresponds to a fixed choice of 7V C , as one moves from the top left region to the bottom, NF is increased from 11/2JV C up to it 's lowest allowable value. The dotted line indicates.where gY + g2 = 0, and also specifies the upper crit ical N^Nc) for a second order phase transition to occur. Chapter 5. Nature of the Phase Transitions 57 that non-trivial fixed points tends to stabilize the fixed points, it is likely that these antiferromagnets would have a second order transition. •• J Bibliography [I] S. Coleman and E . Wi t t en , Phys. Rev. Lett . 45, 100 (1980). [2] see D . A m i t , Field Theory, the Renormalization Group method, and Critical Phenom-ena London 1978. [3] H . Yamagishi , Phys. Rev. D23, 1880 (1981). [4] R . Pisarski and F . Wilczek, Phys. Rev. D 2 9 , 1222 (1984); F . Wilczek, Nuc l . Phys. A 5 6 6 , 123c (1994); Int. J . M o d . Phys. D 6 3 , 80 (1994); Int. J . M o d . Phys. A 7, 3911 (1992). [5] R . Pisarski , Phys. Rev. D29, 2423 (1984). [6] H . Hamidian, G . Semenoff, P. Suranyi and L . C . R . Wijewardhana, Phys. Rev. Lett . • 74, 4976 (1995). . [7] J . SmhV, Nuc l . Phys. B 175, 307 (1980); T . Banks et. a l . , Phys. Rev. D 15, 1111 (1977). [8] Salmhofer and E . Seiler; C o m m . M a t h . Phys. 139,395 (1991). [9] E .Langmann and G.Semenoff, Phys. Lett . B297 (1992), 175. [10] M.C .Diaman t in i , P.Sodano, E .Langmann and G.SemenofF, Nuc l . Phys. B406 (1993), 595. [II] I. Affleck and B . Marston, Phys. Rev. B 3 7 , 3774 (1988); L . B . Ioffe and A . I. La rk in , 1 Phys. Rev. B 3 9 8988 (1989) [12] P. Beran, D . Poilblanc and R . B . Laughl in , preprint cond-mat/9505085; Lectures at this conference. [13] L . Onsager, Phys. Rev. 65, 117 (1944). [14] C . G . Cal lan , Jr., Phys. Rev. D 2 1541 (1970), and independently K . Symanzik, Comm. Ma th . Phys. 1 8 227 (1970). [15] R . Pisarski , Phys. Rev. D 4 4 , 1866 (1991). 58 Bibliography 59 [16] T . M u t a , Foundations of Quantum Chromodynamics: A n Introduction to Pertur-bative Methods in Gauge Theories, Wor ld Scientific Lectures Notes in Physics - V o l . 5, 1987. [17] D . J . Gross and A . Neveu, Phys. Rev. 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