Order and Disorder in Gauge Systems and Magnets

PHYSICAL REVIE% D VO LUME 17, NUMBER 10 15 MA Y 1978

Order and disorder in gauge systems and magnets

Eduardo Fradkin Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305

Leonard Susskind Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305 and Tel-Aviv University, Tel-Aviv, Israel (Received 13 January 1978)

We show how phase transitions in Abelian two-dimensional spin and four-dimensional gauge systems can be understood in terms of condensation of topological objects. In the spin systems these objects are kinks. and in the gauge systems either magnetic monopoles or fluxoids (quantized lines of magnetic flux). Four models are studied: two-dimensional Ising and XY models and four-dimensional Z2 and U(1) gauge systems.

I. INTRODUCTION We shall restrict our discussion in this paper to the first two analogies. These analogies were first In the present paper we discuss several underly- pointed out by Migdal, who discussed them in the ing analogies between four-dimensional gauge sys- framework of his recursion relations. ' tems and two-dimensional magnets. Our main The Z, gauge theory, which is a guage theory in purpose is to clarify the meaning of concepts such which the degrees of freedom are elements of the as confinement and its relationship with order and permutations group of two elements, was first dis- disorder in the system. Most generally we shall cussed and solved by Wegner. ' be interested in studying the long-distance behav- All our discussions will be done by putting the ior, i.e., long-distance correlations, of these sys- fields on a lattice with the time direction contin- tems in. order to understand their phase diagrams. uous and the space directions discrete. ' In Sec. It is well known that most four-dimensional gauge II we discuss the transfer matrix, ' a formulation theories are similar to their two-dimensional. spin which we shall use as a tool to build up the Hamil- system counterparts in the sense that the renor- tonian form of all the models. Later on in See. II, malization-group equations have the same struc- we discuss the one-dimensional quantum Ising ture in both systems', they exhibit the same kind model in a transverse field' and we introduce dual- of instantons, ' etc. Explicitly we show a remark- ity and dual order parameters. ' Dual order param- able analogy between the 3+1 Abelian gauge theory eters will be related to the existence of conden- and the 1+1XF ferromagnet. Roughly speaking, sates of kinks (in magnetic systems) and magnetic the long-distance behavior of both systems is sim- monopoles (in gauge thearies) which randomize ilar when proper analogous quantities are dis- the system. In Sec. III we discuss the Z, gauge cussed. For instance, the behavior of Wilson's theory, in Sec. IV the XP model and, finally, Sec. loop integral' (for the gauge theory) is similar to V is devoted to Abelian gauge theory.

the behavior of the two-point correlation function in / the XF model' once we recognize that a decay of II. HAMILTONIAN THEORY OF THE ISING MODEL the loop integral as the area of the loop means dis- A. One4imensional case order in a gauge theory. Ori the other hand, the XY model has a phase in which the correlation We shall begin our discussion with the Ising function falls off at large distances with a power- model (IM). Identify one of the lattice directions law behavior, 4 which is also true for the Abelian as the (Euclidean) time axis. We will look for a gauge theory. limit in which this direction can be considered In general the systems shown in the following continuous. The Ising model in this limit becomes table exhibit analogous behaviors: formally equivalent to a quantum-mechanical system with a well-defined Hamiltonian describing a Two-dimensional Four-dimensional continuous development in time. The method is magnets gauge theories most easily illustrated using the transfer matrix Ising model Z3 formalism. ' Let us construct the transfer matrix formalism XY model Abelian gauge for the one-dimansional" Ising model. The action Heisenberg Non-Abelian (Yang-Mills) is

17 2637 EDUARDO FRADKIN AND LEONARD SUSSKIND 17

Now we shall imagine that the axis of the lattice 8=-P Q [v, (i)o,(i+1)+ho,(i)], (2 1) sites is the time axis of quantum mechanics. Thus T carries information from one time to a neighboring where o, =+1 andi runs over all the sites. The time. We will in fact identify it with the time evol- parameter h represents an external magnetic ution operator for a quantum system of a single field. It is convenient to add a constant to the ac- spin. tion to-normalize the ground-state energy to zero We want to take a limit in which neighboring lat- when h =0. We also rewrite the term proportional tice sites are treated as infinitesimal transform- to k in a form which will prove more convenient. ations of the form Thus (2.8) 8= —g I[a,(i) — o(i +1)]' sites where 7 is infinitesimal and B is the Hamiltonian. ' Of course this is not true in general [see Eg. (2.6)], —h[v, (i)+ o,(i+1)]]. (2.2) but there exists a limit in which Eq. (2.6) has the Define form (2.8). The limit is — Z(z, 2+ 1) = p([o' (i) —0(i + P - ~ (low temperature), 1)]' (2.9) (2.3) Ph-Ae~~ (A, is any constant), so that aIld 8 = Q &(i,i+1). (2.4) (2.10) Then The partition function is (2.11) exp(-8). configurations — We can write T in terms of Pauli matrices along II exp[-z(i, i -1)]. (2.5) on the Hilbert space of the quantum spin configurations, i It is easy to. see that this is the trace" of the Nth T = 1+ (&v, + o, )7'' power of the transfer matrix, T, where the rows or (2.12) (columns) of T are labeled by the possible configurations of the initial (final) member of a neighbor- B=-01 —~03'. ing pair of spins: In taking the continuum limit we must imagine T = e x[p- Z(i, i + 1)] that the number of sites separating any two times e'~. con- 8gh increases as This dependence of coupling (2.6) stant (P) on lattice spacing is a simple example of 28 .e Sh) the renormalization group. The correspondences between classical statistical Z=TrT (2.7) mechanics and the equivalent Euclidean quantum where Ã is the total number of sites of the lattice. system are summarized in the following scheme:

Quantum system Statistical system

(1) Ground state Equilibrium state (2) Ground- state expectation Averages on the ensemble values of time-ordered operators (3) Ground-state energy Free energy

= — B. The two-dimensional case 8 Q (gP, [o,(r) v, (r n+, )]' The two-dimensional Ising model will be more — + (2.13) interesting than the one-dimensional case. The p,[o,(r)o,(r n, )]], action. of the anisotropic IM is where r runs over all the sites of a two-dimension- 17 ORDER AND DISORDER IN GAUGE SYSTEMS AND MAGNETS 2639

FIG. 3. The equal correlation contours for the symmetrical lattice,

the contours of the curves C(r) = const as circles in the case p, = p, (see Fig. 3). However, in the anisotropic case (P, &P,) the contours at large r Z are deformed into ellipses with major axis along FIG. 1. The space-time lattice. t (see Fig. 4). Now imagine rescaling the t axis in such a way alp'ectangular lattice and n, (n, ) is the unit vector that the ellipses are transformed back into circles.. in the f (z) direction (see Fig. 1). The coupling This is shown in Fig. 5. In this way we can ap- constants in the directions t, z are P„P, which are proximately compensate the effects of the aniso- not necessarily equal. We will not bother with an tropy by a rescaling of t relative to z. The form external field in this case. of. the correlation function in the new model is . Before dealing with the technical details of the ' similar to the symmetric case. transfer matrix we will qualitatively describe a We can repeat this process until we reach a lim- limit in which the t direction becomes continuous ~ it in which the lattice in the time direction becomes leaving discrete the z axis. In this limit the IM dense, This is the time-continuum limit. becomes equivalent to a Hamiltonian quantum system 'consisting of a one-dimensional (z) discrete C. Transfer matrix for the two-dimensional case system of interacting spins. We will now construct the time-continuum limit The tao-dimensional IM has a phase transition. in a precise way using the transfer matrix method. In the space of the parameters there is a P„P, Consider two neighboring rows of spins as in Fig. critical curve which separates the ordered (fer- 6. The spine on the "earlier" ("later" ) row are romagnetic) and disordered (paramagnetic) phases. denoted by s(n) [o(n)] where n labels discrete loca- This is shown in Fig. 2. The critical curve is giv- tion along the space z axis. The Lagrangian for en by' this pair of rows is = (sinh2P, ) (sinh2P, ) l. (2.14) ' 2= —Q [s,(n)- o,(n)]' We will illustrate the main ideas j.n terms of the n two-point correlation function (o,(0)o,(r)) = C(r) For has cubic symmetry (symmetry ——' (n)s, (n+I)+o', (n)cr, (n+1)]. (2.15) P, =P„C(x) 2 P[s, under rotations by v/2). To illustrate this we draw The rows and columns of the transfer matrix are labeled by the spin configurations of both layers. Since for N spins on a layer there are 2" config- ' urations, the 7. matrix is 2 x 2~. The diagonal elements of T are given by setting s,(n) =o,(n) for all n. Thus

T~,.„,„„=exp P, Q o,(n)o, (n+ 1) (2.16)

FIG. 2. Phase boundary for the anisotropic two- FIG. 4. The equal correlation contours become ellip- diinensional Ising morsel. ses for the anisotropical lattice. 2640 EDUARDO FRADKIN AND LEONARD SUSSKIND

of 0,. However, since w is infinitesimal we may ignore all terms of order -7', v', ... by comparison with the order-T term. The result is that the Ham- iltonian H contains only no-flip and single-flip FIG. 5. The ellipses are deformed back into circles terms: by squeezing the lattice in the time direction. H=-g a', (n)- & go, (n)o, (n+1). (2.22) The off-diagonal elements can be classified by The connection between the spacing T and e '~t the number of spin number of sites for flips [the provides a quantitative estimate of the amount of which The single-flip elements are a, (n) =-s~(n)]. rescaling of t which is required to compensate the T, ~, = exp(-2P, ) exp[-E (a, s )P,], (2.17) anisotropy when P, becomes large. The correlation functions will approach the lim- where s) is the sum of the energy of the two E(a, iting forms of the equivalent quantum system as independent rows. P, -~, P, -% ~~. Suppose for example Similarly the n-flip elements are Ta, (0, 0)a,(n t) I0) =r(n t) (2.23) T„=exp(-2nP, ) exp[-E(o, s)P,]. (2.18) for the quantum system. Then in original discrete the Now consider limit integer-valued coordinates of the lattice the cor- op relatiori function behaves like p (2.19) 2~t. (2.24) Pg This limit is similar to that used in the one-dim- where I ensional case. In fact the replacement P, —Ph is natural since the interaction with neighboring col- e'&t. umns exerts a field on each spin. X=Pg The limiting form of the matrix elements of T In particular as -~ the spatial correlation length are Pt [decay length of the function C(n, 0)] tends to a limiting function of A. . -1+8 '~~ T... pa, (n)a, (n+1), In the space P„Pt there exists a set of curves along which the spatial correlation length is con- -e~~~ +O(e 4'&), T, (2.20) stant. In particular, the critical curve is the curve T, -e '~~+0(e "~), where the correlation length is infinite. The above discussion shows us that for P, -~ these curves have the form (see Fig. 7) It is now possible to put T into the infinitesimal ~-28) J3 (2.25) form 8 T=1 —TH. The parameter ~ can be used to label the curves. We can relate any point on the symmetric line the e ~t v—the infini- We again identify quantity as P, = P, with a limiting theory by extrapolating along tesimal spacing along the t axis:

T= 1+ 7 'A. P a, (n)a, (n+ 1)+pa, (n) (2.21) +7 ~, na, m + ~ ~ ~

The Pauli matrices o, have been used to flip the spin so that the n-flip terms of T contain n factors

a. (n) ~ ~ ~ ~ ~ ~ 0 ~ s (n) FIG. 7. Asymptotic behavior of the equal correlation length curves in the Pt P, parameter space. Each curve FIG. 6. Two neighboring rows of spins. is labeled by a single value of A. . 17 ORDER AND DISORDER IN GAUGE S YSTEMS AND MAGNETS 2641 these curves. The qualitative long-range behavior in terms of the p, 's as is unchanged along a line of fixed ~. The -poirits H = — —& on the symmetrical line correspond to different g p, (n) p, (n+ 1) g p, (n) temperatures of the classical square Ising model. Thus we can relate each temperature of the iso- (2.29) tropic model to a unique quantum model with a corresponding value of ~. Generally large ~ The remarkable thing about the Hamiltonian is that (small X) corresponds to large P (small P). Thus it has the same form in terms of the p, 's and the we will refer to the large (small) X as low (high) 0's. The on).y differences are the overall factor temperature. of A, in (2.29) and the replacement A. —I/A. inside the brackets. We may summarize this property D. Lattice duality by the formula = The two-dimension3l IM and the equivalent one-di- H(o", X) AH(lJ. , V') . (2.30) mensional quantum problem' have the remarkable The self-duality of H is a very powerful result. property of self duality-. The dual of a cubic lat- It shows that the high-temperature behavior (X &1) tice is a new lattice whose sites are located at the and low-temperature behavior (X &1) are in a sense centers of the old cubes. In particular, for a one- equivalent. For example, we can map any eigen- dimensional lattice the sites of the dual lattice state of H(A) to a unique eigenstate of H(l/X). The correspond to the links of the original lattice (see energy spectrum has the property that if E(&) is the Fig. 8). energy of some state then E(X)/X is the energy of The original system can be redescribed by a a related state of H(l/X). new system with degrees of freedom attached to For example, the energy gap between the ground the dual lattice. For the one one-dimensional quan. — and first excited states satisfies tum Ising model with transverse field [Eq. (2.22)] G(X) = ZG(I/X). the dual lattice operators are called p. . They can (2.31) be written in terms of the original o's: The one-dimensional Ising model with transverse field is exactly soluble' for the spectrum. ' The gap p, (n) = (n + 1)o, , o, (n), G(A.) is given by (2.26) p, (n) = g o,(m). (2.32) which is easily seen to satisfy (2.31). The operators p, (n) describe the mutual state of The symmetry point of the duality transforma- two neighboring 0's. p,, flips all the spins to the tion is X=1. From (2.32) we see that the gap van- left of the site n. ishes at this point signaling the presence of mass- The 0 operators satisfy the following relations less excitations and a divergent correlation length. which s'pecify their algebra completely:. In other words, the point & =1 separates both the ordered and disordered phases. [o',.(m), o&(n)] = 0 for n em, (2.27) o,'(n) =1, E. Order and disorder parameters The duality transformation o,'(n) =1, (2.28) relates the high- and low-temperature behaviors of the system. We will o,(n)o, (n)o, (n) = -o,(n). discuss the properties of these phases now. (1) I ange X (small temPexature). For X»1 the From Eqs. (2.26), (2.27), and (2.28) it is easily term P[-Xo,(n)o, (n+ 1)] dominates the Hamiltonian. seen that the p, 's satisfy the same relations. Thus The ground state for ~= is doubly degenerate the variables on the dual sites are isomorphic to with all spins parallel either up or down (see Fig. the original variables. The Hamiltonian in Eq. (2.22) can be expressed filK&& ji Ground State (a)

n-i n n+I w n ~ e& w r& w r& w rw w vw w gw w Ground State ( b) n-I n ~ = Sites of Dual Lattice FIG. 9. The ground state of the quantum-mechanical x = Sites of Original Lattice Ising model in a transverse field at large values of A, is FIG. 8. Dual lattices in one dimension. doubly degenerate. 2642 ED UARDO FRADKIN AND LEONARD SUSSKIND 17

9). By picking boundary conditions at ~ we may Ising model. There the N dependence does not choose the ground state (a), Then the expectation disappear and t!he magnetization is not a smooth value of v, is 1. Defining IO&~ to be the ground function of T for N -~; state for given ~ we have It will prove to be interesting to define a dual order parameter or, "disorder parameter" which Io&„=1. (2.33) „&0Io, actually measures the degree of disorder of the More generally o, variables. To do this we perform the duality transformation on the order parameter, (v,&. &=,(OIo, IO) x0 (for X&l). (2.34) (o, From (2.26) we define the disorder parameter The quantity (o,) is known a,s the order parameter to be or magnetization. That the magnetization persists for noninfinite (6,(n)) = rr, (ml). (2.40) ~ is not completely trivial. For example, the or- (P dinary one-dimensional Ising model is ordered This object generally vanishes in the ordered phase for zero temperature but not for finite temperature. and has a nonvariishing expectation value in the dis- 6-'0 A. To see that (o,& for. large but finite we may ordered phase. To see intuitively why this is so, apply perturbation theory to see how (o,) changes we consider the action of the operator p,,(n) when with 1/X. For large X we write applied to a basis state in the 0, representation. The result is to flip all the spina up to the site n —H = ——1 -g o', (n)cr, (n+ 1) g o, (n) (Fig. 10). Therefore when applied to a magnetized state p, (n) reverses the sign of the magnetization 1 at an infinite number of sites. The resulting state =Ho+ —H~. (2.35) is obviously orthogonal to the original. Accordingly for any magntized state Applying standard perturbation theory we get (2.41) = ~ ~ (2.3 6) Io&, Io&„+ H, I 0&„+ 0 0 On the other hand, if the state is sufficiently disordered it may be possible for the state resulting H, flips one spin at a time. The state with the nth from an infinite number of spin flips to have a pro- spin flipped is called In): jection onto the original state. (2) Small & (high temPeratuxe) The grou.nd state (2.37) is that of H, =p„o',(n) for & «1. The o,'s are all aligned with positive value 1. Then we define the To order X ' we find ground state for &=0 as a state IO)o such that = a', (n) 0&o 0&o (all n). (2.42) ~(o,( ) o)~ I I (0I ( )I0) Evidently the average of v, (magnetization) satis- 2 A. fies + — n 0'3 tB (2.38) ,&0Iv, (n) Io&, =0. (2.43) The factor (1-Nj16A.') is the normalization factor ' (0 IO) to order X~ and N is the total number of sites. For m en, (n (m) =+1, while for I o, I n) Thus m=n, (nIo', (m) In) =-1. 0 ]q ]i]i]s]g]q]g]i N 1 1 ( 161' 161~ 16lP 1 ' (2.39) 8X2 (b) n The important feature of this result is that the ]s ](]c]c N dependence of the order ~ 2 correction cancels leaving a finite coefficient. This is true to all orders and therefore we expect a finite region of ~ to have a nonvanishing magnetization. This re- (n)10 ) sult may be contrasted with a calculation of the derivative of the magnetization with respect to FIG. 10. A kink applied at site n Qips all the spins temperature for the ordinary one-dimensional &rom-~ up to sitee, ORDER AND DISORDER IN GAUGE SYSTEMS AND MAGNETS 2643

Dual (o) (b)

Transformation ]C]L ]L]q]S i( ]L ]LL] ~ = . (0~) 0 ((T~) & 0

0 0 1 Kink at Dual Site n Antikink at Dual Site n

Fig. 11. The effect of the dual transformation. (c) ]L]L]s]S]L, n X X In fact Eq. (2.33) is true for all &&1. This is true n+1 because the transformation Kink- Antikink Pair at Dual Site 0'3 ~-0'3, (2.44) n and n+1 0'~. 0'~ FIG. 12. Kinks in action. is a symmetry of H. Unless this symmetry is spontaneously broken ~(0 le, l0)~ must vanish. Now = consider the disorder parameter for & O. Since menon (Fig. 13). Moreover, the kink-antikink l0), is an eigenvector of v, [see Eq, (2.42)] it fol- pairs become "ionized" at ~=1 without any cost lows that of energy. Kioks have very important features. Unpaired ,(Ol p, (n) l0)~=1. (2.45) kinks cannot be present in any ordered phase of the system since they violate the imposed boundary the same arguments as in 36)-(2.39) By Eqs. (2. conditions. They only can exist in the system we can prove that the disorder parameter is not ' paired with antikinks (in the ordered phase) or as vanishing for finite of ~: a range a condensate in the disordered phase where the system ignores boundary conditions. Moreover, LLL, o (0 l (n) l 0), 0; (2.46) they are topological objects because they are large we summarize these results in Fig. 11. perturbations of the system which change the boundary conditions. Finally they disorder the system, and above the critical temperature their F. Kink condensates and disorder presence as a condensate is responsible for the short range of the two-point correlation function. In the preceding section we showed, that (p. (n)) , Thus, if we are considering (0)o', an indef- measures the amount of disorder in.the system. (o, (n)), inite or random number of kinks occurring between However, we can reinterpret all these results in the two points will destroy the correlation between an interesting way. the two spins. The operator p, (n) acting on an ordered state creates a, spin configuration which we shall call a kink. This object has finite energy and the num-

'L LL L iLL ]L ig]L, ](]L ber of such an object is a conserved quantity. Thus ] i 4] C ]i] i ]& i( we can regard these configurations as ma, ssive par- (a) (b) XXX X ticles. Since a kink configuration is orthogonal to the ) =& No Kink X»I The paired Kink- ground state in an ordered phase there will be no Antikink Pairs Antikink Gas kinks present in this phase. However, we know (c) ~ but wiLL that if is finite there be LL IE very large a ] ] ]L.]i]a ]L] LiL ]L finite (and small) number of spine flipped. As we . XX X X X see from Fig. 12 a single spin flip is equivalent as a pair kink and antikink at two neighboring duaL ) = I+ ~ Star ts C ondensation sites. At lower vaLu~s of ~ there will be blocks of spins flipped, which are clearly equivalent to i LL] IL i i i c ]S ]LL]L ] 'L the kink-antikink pairs with some size. If ~ is large the distance between pairs will be much X X X X X X X X X larger than the size of the pair. However, as ~ 'LL I LL i LL I approaches its critical value 1 the interpair dis- The Pairs are Condensed tance becomes comparable to the pair size. Thus FIG. 13. The sequence shows how kinks disorder the the phase transition is a kink condensation pheno- system. The disordered state is the kink condensate. 2644 EDUARDO FRADKIN AND I EONARD SUSSKIND 17

III. GAUGE SYSTEMS IN 3+1 DIMENSIONS

A. Gauge invariance The Ising model studied in the previous sections has a global symmetry consisting of flipping all the cr, (n) simultaneously. We call this a global symmetry because the symmetry operation involves all the spins. Gauge symmetries"' are local symmetries in which the operation only involves degrees of freedom localized near some point. In this section we will consider the simplest example of a guage system in (3+ 1)-dimensional space- time. We will call it the Z, gauge system. Let us imagine a simple cubic lattice in (cf=4)- FIG. 15. In 3+ 1 dimensions, each lattice site is dimensional space. The elements of the lattice. are connected vrith eight links. sites labeled by four integers X= (x„x„x„x,) and links labeled by a site X and a unit vector n,. point- spins are flipped and 8b,„is again unchanged. ing in one of eight lattice directions. Alternatively Therefore Z„,„and 2 are invariant under local the links can be labeled by a pair of nearest-neigh- gauge transformations. A more general class of bor sites (X„X,). The spin degrees of freedom gauge-invariant objects can be formed by consid- for the gauge system are defined on the links (see ering arbitrary closed paths of links as in Fig. Fig. 14). Each site of the lattice is connected to 17. The products of o, 's on the links forming such eight links (Fig. 15) and therefore to eight spins. paths are gauge invariant. A local gauge transformation at the site X flip all Consider the expectation value of any gauge-in- eight spins leaving the remaining spins unchanged. variant object 1 I et us now build an action which is invariant under such gauge transformations. The terms of (I') = g I'(o') exp(-2) p exp(-2). (3.2) the action identified with the elemen- {fy) (ty) are faces or I tary boxes of the lattice (see Fig. 16). The sum is over all configurations of the o's. P», » For each box, define an action This means that we will add contributions corresponding to configurations which are identical 2, „=-Po, (1)o,(2)o.(3)cr, (4), . modulo a gauge transformation. Since both 2 and (3.1) I" are gauge invariant we are counting the same 2=-P a', cr, o, g o„ contributions boxes many times. , One way to avoid. that is to introduce a gauge-fixing condition or con- where a,o,a,o, represents the product of spins on straint which selects out from each gauge equival- the edges of the box. ence class a single configuration a. The sum Now consider the behavior of Zb, „under a local can be replaced by guage transformation at X. If X is not a corner of P», » "box" then none of the spins in Zb, „are flipped and Q N(cT) Zb, „is unchanged. If X is a corner of box then two where Q»;» means a sum over the unique repre- sentative of each class and N(cT) is the number of equivalent configurations to 0. For an infinite lattice N(o) is infinite but for any finite lattice N(o) is in fact independent of cr so that restricting the sum to 0 merely introduces an irrelevant mul-

= Spin.

FIG. 14. The degrees of freedom of the gauge theory FIG. 16. The terms of the action are identified vvith are defined on the links of the lattice. the faces of the four-dimensional cubic lattice. ORDER AND DISORDER IN GAUGE SYSTEMS AND MAGNETS 2645

B. Hamiltonian form

. As in the Ising case, we will introduce two coupling constants, one for space-time boxes and one for.space-space boxes. The action for a given space-time box (see Fig. 19}is Z =-P,[cr,(1)(r,(2)o,(3)o,(4)] FIG. 17. A closed path of links. = -P,[o.(1)o.(3)] =+ —,'p, [o,(1)—cr, (3)]'—const. (3.4) tiplicative factor. Thus for each spatial link the sum over x4 is an In what follows we will impose such a restriction Ising-type action. We denote the space-time term on the configuration space. It can be shown that of the action by any corifiguration is gauge equivalent to a configuration in which the spins on timelike links are fixed 2P, [(r,(l,x,) - a, (l,x, + 1)]', (3.5) to be equal to 1. However, this condition does not (t, x4) determine a unique configuration. Consider an ar- where l labels spatial links. bitrary configuration of a', 's on spacelike links and - The space-space boxes contribute with a term a, =1 on time links. Now consider -a transformation which is composed of an infinite product of asasa'sa's (3.6) , ~ local gauge transformation-which is composed of a SS an infini. te product of local gauge transformations. where the sum is. over all spatially oriented. boxes. The product is over all the lattice sites which have Thus given spatial location (x„x„x,) and all values of Euclidean time x4. The relevant sites are shown in 8= ) —o,(l, + 1)]' The effect to reverse only those spins g 2P,[o,(l, x, x, Fig. 18. is [t,x4) on the six spatial links connected to (x„x„x,). In particular, no time link is flipped. Thus the gauge- ,~sas~s~s (3.7} fixing condition SS The passage to a Hamiltonian formulation is per- o, =1 (time links) (3.3) formed by the same limiting procedure as for the Ising case, namely does not uniquely define a configuration within each P ao gauge equivalence class. However, it can be seen (3.8) that the number. of configurations satisfying (3.3) P -Xexp(-2P, }. the same for each equivalence class. Thus im- is Thus, we find posing (3.3) on the configuration sum introduces a mere numerical factor. Henceforth Eq. (3.3) will H= — g cr, (l) —X g o,o,o,o„ (3 9) be assumed. links boxes where the sums are over spatial positions only. As in the previous case of the Ising model the a' s are Pauli spin operators acting in a Hilbert space. The Hamiltonian (3.9) has a local gauge invariance as a consequence of the original gauge in- variarice of the Lagrangian. Consider the spatial

X( Xp Xp Space FIG. 18. A time-independent gauge transformation at spatial site g&, x2 x3). FIG. 19. A space-time box. 2646 EDUARDO FRADKIN HAND LEONARD SUSSKIND

I site r= and define the operator X+A. (x„x„x,) I

G - = o', . 10) Q (f,), (3. FIG. 20. How to label a link. where l,. are the six links attached to x. G„ is a unitary operator which has the following action on so &g g& =0. Therefore there can be no mag- the 0's: fo, I netization in any state satisfying gauge invariance. G„'o,(l)G„=o, (l) all l, In particular, no phase transition can lead to a magnetized phase. Nevertheless we shall see that G„'o', )G„=—o, .) . attached to (3.11) (l, (l, (l, r), a phase transition exists. G„'o,(l)G„=o',(f) (l not attached to x). C. Lattice duality Thus, the action of G„ is to flip the 0,'s linked to site ~ and leave unchanged all 0,'s. Evidently In the first part of this paper we demonstrated the Hamiltonian (3.9) is invariant under G . the self-duality of the Ising model in the Hamilton- It may also be proved that the ground state of ian version. We shall now prove that the Z, gauge H is invariant under gauge transformations. " system is also self-dual in the Hamiltonian form. ' Calling the ground state IO), We shall explicitly construct the variables on the dual lattice. We show that the Hamiltonian takes 0&, (all r). (3.12) the same form in terms of the original and dual This is very different from the'global invariance variables. To carry out this discussion we will of the Ising model. In that case the ground state rice a compact notation to label spatial links. A for»&1 is doubly degenerate and the symmetry link may be labeled by a site and a unit vector. transformation takes one vacuum to the other. The link (x,n,. ) originates at x and ends at x+n, The stability of the spontaneously broken symmetry where i may be any of six unit vectors. The link lies in the fact that it takes an infinite number of (x, n,.) is evidently equivalent to ( +xn, , -n,.) (see steps in perturbation theory (powers of H, ~-~,) Fig. 20). The link variables will be denoted by to mix the degenerate states. This is not the case o(x, n). here. For example, sugpose the vacuum for The duality transformation turns out to be simp- X»1 was all o', =1. The perturbation -Q„.„„,o', lest in a different gauge the gauge we have used can act six times to flip the o spine linked to (x„ up to now. We define the "axial" gauge by thus mixing the ground state with another in x„x,) (3.16) the same class. Thus even for ~»1 the spontaneously broken ground state is unstable. on those links oriented along the spatial x, axis. Since we are interested only in gauge-invariant The independent variables in the axial gauge are operators acting on IO) the only states of interest the o, 's and o, 's on the x„x, (transverse) links. will also be gauge. invariant. Accordingly we con- The 0, 's on the x, links are defined in terms of the sider as physically interesting only those states independent variables by requiring Eq. (3.13) to be satisfying C

G(~) f(&= ft& (»1~) (3.13) Xp

. = . v, (l, ) I g& I g) .

Note that since H is gauge invariant, condition (3.13) is consistent with the. dynamics. Equation (3.13) has as a consequence the vanishing of expectation values of all o,(l). Thus con-

sider I 21

From (3.13) we write 5 ~ (X( X~ X~) ' = '(~)o. (3.14) l &elo.(f) le& &VIG (f)G(~) I e&, where r is one of the end points of the link /. But FIG. 21. The operator 0&(x&, x2, x3, n3) is defined in 's n, terms of the 0', on the & and n2 directions (broken G ', (x)a, (l)G(x) =-o,(l) , (3.15) lines) Og DER AND DISORDER IN GAUGE SYSTEMS AND MAGNETS 2647 true. This can be done by defining X2 4 = o', (x„x„x„n,) j[l gl o,(x'„x,x,', n, ),. (3.17) i n2 x'3 ~x3 g=l where o,(x„x„x„n,) is the (dependent) variable for the link (x, , x„x„n,) shown as solid in Fig. 21. The product is over all transverse links shown as Xi broken lines in Fig. 21. v, (x,n, ) satisfies the identity

o,(x„x„x„n,) = g o,(x„x„x„n,.) &«o', (x„x„x,—l, n, ). (3.18) The reader. can now easily prove that A=(Xj, Xp} 6 FIG. 23. on a dual link is defined as a product Q v, (x, n,.) =1 (3.18) g& j=1 of 0 f s on the solid links . is an identity. The Hamiltonian in the axial gauge takes exactly the same form as Eq. (3.9). The defined by infinite products of 0', 's from z =-~ only modification is that the o,(x,n, ) are set equal to the preceding site. The p3 on transverse links to 1 and v, (x,n, ) is defined by (3.17). are again infinite products of 0', 's. To define this Now we define the dual lattice. The sites of the product we note that each transverse link in dual lattice are placed at the body centers of the (say direction) may be identified with a box of the original lattice. (centers of cubes). The dual links thex, original lattice in the plane pierce the original boxes at their centers. The lying x~, (see Fig. 23). dual boxes correspond to the original links .(see Fig. 22). Now consider the product Next we must define dual lattice variables p., (n, ) = o, (x„x„n,); (3.21) and p, on the dual links. Each dual link corre- p, II 3 X3 ~x'3 sponds uniquely to an original box. The variables The links included in the product are indic3ted in p. are defined by , Fig. 23 by heavy lines. An identical procedure is used for p, (n, ) . ~so'Sonics ~ (3.20) &) =lI The following points can be proved very easily: (i) On transverse dual links the p, and p, satisfy where the four 0. 's belong to the edges of the box. , , , a Pauli algebra. For the p,3 variables we distinguish the x3.and (ii) Consider the six dua, l links originating at a. transverse links of the dual lattice. For the x3 dual site. Then links the defiriition of p, 3 is 1 since we are working in the axial gauge. For the transverse link the p, =l all dual sites. (3.22) definition of p3 is analogous to the Ising case. II , In the Ising model the dual variables p3 were (iii) For each dual box the product p, , p,, p, , p, , on the edges of the box is equal to the 0, on the corresponding original link. This, however, is only true if we impose Eq. (3.13). (iv) By definition p, , on a dual link equals 63030303 for 'the corre sponding box . Thus, it follows that the original Hamiltonian may be reexpressed in terms of the dual variables Dual Site Dual Link (a) as

P, 3 P,3 P.3 P.3 —~ P.l dual boxes dual link

Dual Face ——1 =~( P.l ~ P,3P,3 P3 P,3 (b) dual links dual boxes FIG. 22. The dual lattice. (3.23) 2648 KDVARDO FRADXrN AZD I. EOWARD SVSSKI~D

Thus H is self-dual. As for the Ising case the self-duality relates the physics of & &1 with ~(1.

D. Small g phase FIG. 25. A disconnected graph. For ~ = 0 the Hamiltonian is Now suppose ~ is small but finite. The term (3.24) links H1 = —~ 0'30'30'30'3 (3.27) H has a well-defined nondegenerate gauge-invari- boxes ant ground state IO), , such that will cause modifications of the ground state and ' excitations. Evidently the action of H1 on the = &i(«) I0)g=o Io),=o (3.25) ground state is to create closed boxes of flipped spine (elementary gauge-invariant excitations). for all (r, n). The density of elementary gauge-invariant exci- The spectrum of excitations includes both gauge- tations in the perturbed ground state is -A'. Fur- invariant and gauge-noninvariant states. These thermore, the ground-state energy density is low- states are created by flipping the value of o; on ered. The ground-state energy per site is any combination of links. However, the gauge- — g =-3 — X ——X~+ ~ ~ ~ (3.28) invariant subspace sa,tisfying (3.13) corresponds to special configurations. To construct these More interesting is the-effect of the perturbation states we begin with an arbitrary closed path of on the strings. The perturbation in this case can links. The path may intersect itself and may con- act in two different ways. First it can excite an sist of several disconnected parts but it should elementary gauge-invariant excitation on a box have no ends. Now consider the state obta, ined by which is disconnected from the string (see Fig. slipping the o s on these links. The result is a 25). closed path of links with o; =-1. It is evident that These contributions are just renormalizing the as long as no end points occur then (3.13) is satis- vacuum. The other action is to deform the string fied at every vertex. These then are a complete by putting in a kink. This happens when the per- set of gauge-invariant excitations. The energy of turbatlon acts on a box containing a side on the an excitation for ~ =0 is simply string (see Fig. 26). Higher orders in X cause the string to fluctuate out of the straight line (see Fig. F =2n, (3.26) 27). Thus as we let the perturbation act on the where n is the total number of flipped 01 s. string a large number of times, the string w,ill In addition to the finite-energy excitations there start to percolate. - This effect will be more im- are a class of interesting excitations whose energy portant closer to the critical point. diverges linearly with the radius of the lattice. The fact that strings at ~ finite are not straight These consist of infinite lines of inverted 0, 's lines makes ambiguous our definition of the string called strings (see Fig. 24). The simplest such tension. We can define the string tension, for fin- object is a straight line of flipped spins along one ite values of &, as the energy of the string divided of the lattice axes. by N, the linear dimension of the lattice, i.e., the The energy of such a configuration is proportion- original string length. al to 2n where n is the linear dimension of the lat- Now we can see that the whole effect of the elemen- tice. The energy per unit length of such a line is tary gauge-invariant excitations acting on the string called-the string tension. For ~=0 the tension is is just to deform the string as well as to lower its ten- 2 sion. We find

= —'~2 '-2' &4+ - ~ ~ T 2 — —1536 (3.29) From (3.29) it appears that T might vanish for some finite ~. Suppose this occurs. Re argue that this signals a phase transition. The reason is that the tension cannot become negative. If it did the

FIG. 24. A stringlike excitation. FIG. 26. The string is deformed by the perturbation. ORDER AND DISORDER IN GAUGE SYSTEMS AND MAGNETS 2649

of (3.34) unchanged. Thus, there is a degeneracy due to the non-gauge-invariance of the state o, =1 (all r). In the Ising model an analogous condition occurs LX LL for ~= ~. Here the ground state is twofoM degen- FIG. 27. The string starts to percolate, erate. -This is connected with the global Z, invariance. There is, however, an important difference between the models In the Ising case, the two string would lower its energy by growing longer. vacuums cannot mix in any infinite order in per- The ground state would be unstable with respect turbation theory in & '. In other words, it requires to the creation af infinitely long strings which fill an infinite number of spin flips to go from one to space. Thus at the point where T vanishes global a the other. change in the behavior of the ground state must In the gauge case, a gauge transformation flips occur. 8, only six spins. Therefore, to go from one degen- If we ignore higher orders in ~ then T vanishes erate vacuum to another requires only six orders at about X'=2.1. However, for ~'=2.1 the serieS in ~ '. Accordingly sixth-order perturbation theory is not yet converging. We can improve the situa- lifts the degeneracy. tion by using Pade approximants to extrapolate The correct vacuum for ~=~ is a gauge singlet. (3.29). This gives It is formed by superposing symmetrically the 1 —0.67& state o,(r) = 1 with all its gauge-related counter- T 2 x (3.30) 1 0 42y2 parts. Of course for all gauge-invariant quantities we can ignore this subtlety and use the state v, (r) which vanishes at ~-1.22. A more refined method =1. is to compute thb logarithmic derivative of T and The lightest excitations of the ground state for use Pade approximants" to determine the pole of A = ~ are given by applying o,(link) on some link. 1 dT This flips the corresponding . — ~ ~ ~ o, = —'(1+1.1A.'+ ) T cjA, To give a gauge-invariant description of these 1 1 excitations we must the values of some (3.31) specify pave 4 1 —1.1A.2 complete set of gauge-invariant functions of the 's. Most we can give the value of every The pole (zero of T) occurs at o, simply box variable o,(1)a,(2)o,(3)o,(4) or equivalently p, . A, = 0.912. (3.32) For example, suppose we apply o,(link) on.the link shown as the heavy line in Fig. 28. This operation The exact position of the phase transition (assum- evidently inverts the four box variables on the four ing one occurs) must be at A. =1. This is because boxes containing the link. These are shown in the of the self-duality relating ~& 1 and ~&1. However, figure by unbroken light lines. The four boxes can the self-duality does not tell us whether the trans- be identified with four links of the dual lattice ition is first order or second order. The apparent shown as dashed lines. These four dual links form vanishing of T for ~=1 strongly suggests a second- a closed loop. Thus the resulting excitation is a order transition. closed ring of flipped p, ,'s dual to the excitations of the ~=~ ground state. E. Large& pham Now consider the limit »&l. We write H 1 0'3 1 0'3 2 0'~ 3 0'3 4 —— 0'~. 3.33 box links For X=~ the second term of (3.33) may be ignored. rt-~ the I In this case the ground state is determined by I I term

0'30'30'30'3. (3.34) box The lowest eigenvalue of (3.34) occurs when all 03 1 However, the ground state is inf initely degenerate. To see this consider a gauge transformation on the state with v, (r) =1. Such a transform- FIG. 28. An excited link (in dark) is dual to a box& ation flips various 0,'s but always leaving the value with inverted flux (dotted line). EDUARDO FRADKIN AND LEONARD SUSSKIND

FIG. 29. The string is dual to a tube of boxes with inverted Qux.

Next consider the dual of the infinite line of inverted o, 's. These excitations should be lines of FIG. 31. A closed loop on the lattice. boxes with the box product 0,0,o,o, =-l, namely lines of inverted p, 's (see Fig. 29). The small ~ excitations are lines of inverted 0, 's. A line of ever, will depend in general on the details 's Cr(R) inverted 0, is dual to a half plane of inverted of the loop F. But we shall be interested only in p3, as is shown in Fig. 30. However, this state its asymptotic behavior as 8 -. In this limit is not gauge invariant. We can get a gauge-invari- we shall want to know if there is a phase in which 0. 's ant state dual to the line of inverted , if we not- Cr(R) is asymptotically constant (this should be ice that, 0, is a gauge-invariant operator dual to the an ordered phase) or what kind of decay it exhibits box product p,, p, ,p., p., which is a gauge-invariant as R —~ otherwise. operator too. Thus the dual statement to o, =-1 We shall show that Cr(R) exhibits two different is p,, p,, p,, p, ,=-1 on the boxes which is dual to the behaviors for ~ large and small although there is link with an inverted 0,. Therefore a line of boxes nothing like an ordered phase behavior in both with the box product 0'30'30'30'3 = —.1 1s a gauge-in- cases. variant state which is dual to the line with p, , =-1. The asymptotic behavior is

F. Correlation functions exp(-A), X &1 C ,(R) (3.36) In the preceding sections we stated that only exp(-P), ~ & 1 gauge-invariant operators will have a nonvanish- where A and I' are the area and the perimeter of ing expectation value. Thus it is clear thai the " two-point correlation function vanishes identically the loop. Clearly e is analogous to the exponen- in and therefore we since o,(r, n)o, (r', n') is not a gauge: tial decay Ising-type systems regard the ~&1 phase as a disordered phase. How- (a,(P, n)o', (r', n')) =—0 (all A). (3.35) ever, the phase»l is not ordered since C~(R) but slow- What we need is a suitable definition of the correla- vanishes as R -~, it decays much more tion functions. ly than in other phase. It is clear that the phase but The correct object to study is the ground-state transition is not order-disorder, it is a change in the of the correlation function. expectation value of the product of 0,'s along a behavior closed loop" on the lattice (see Fig. 31). Let It is easy to understand the behavior of the ~&1 phase. Since the unperturbed ground state us call Cr(R) such a magnitude for a, loop 1" with ~0) typical size R. Since Cr(R) is a gauge-invariant is orthogonal to the state II„,',o, ~0), then at zeroth quantity it may have a nonvanishing value. How- order in perturbation theory =0, ~=0. (3.37)

I ~ ~ ~ ~ 0 ~ ~01$ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ IO ~ ~ ~ OO ~ ~ I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 However, we may get a nonvanishing result if we go to some higher order in perturbation theory. The lowest order needed to get a nonvanishing result is equal to the least number of elementary loops enclosed by F, which is exactly the area of I I I I F. Thus = — o. I yn 8-nIinXI l r (R) (3.38) for the lowest order in ~. Here n is the area of ~& FIG. 30. A half plane of inverted p, 3 is dual to the F and 1. string, although is not a gauge-invariant state. Let us consider the other behavior, ~&1. Now ORDER AND DISORDER IN GAUGE SYSTENIS AND MAGNETS . 2651 the operator IIro; just counts the number of inverted spins along the loop I". More precisely, it coun-ts if there is an odd or even number of inverted o,'s. The ground state at first order in ~ ' is 1 )x (1 &Ng-2)1/2

x, 0 -g~+~& n (3 39) boxes where in) is the state with all the boxes with flux (v,v, o,o,) equal to 1 but the nth box with flux -1. Then ' Boundary of inverted links g(r, ~0)~-g= i0)~-i + p X Q ~n) r boxes FIG. 32. The kiriks of the theory in 3 + di- unlinked Z2 gauge 1 to I' mensions are lines of inverted links which are closed at infinity. —2~' In) (3 4o) boxesZ 1 + 4N~-2)&g. linked to I' the system. These kinks change the behavior of A box is considered to be linked to the loop I' if the correlation function when they condense. They its inverted spin lies on I'. There are four linked play here exactly the same role as in the Ising boxes per spin flipped lying on 1". model. in the strongly coupled phase (A, & 1) they Thus if 4n is the number of boxes linked to 1", are massive. However, at the critical point they we can write become massless and therefore they condensate randomizing the system. It is clea,rly seen from their definition that they are large topological objects which change the boundary conditions and cannot be removed by any finite number of spin flippings. 1 (1+-,'NX ') ' (3.41) IV. THEXYMODEL Cr(R) =—1 —2n& ' =—e '"~ A. Construction of the Hamjltonian But if P is the perimeter of F, we have n =P. In the next systems we shall study, the symmetry Then operations, both global and gauge, are continuous. The group Z, is replaced by the continuous rotation 2P The Ce pe ) —= exp (- (3.42) group 0,. first example with 0, symmetry is the two-dimensional XY model of a, magnet. 4 At which is the "perimeter decay" behavior. There- each site of a two-dimensional square lattice there fore we can regard the phase transition as a change is a unit two-vector, a, described by an angle in the behavior of the correlation function since it Q(r). All physical quantities are pe'riodic in Q. decays exponentially as A for ~&1 while as R' for The interaction between sites is of the form o(x) ~ o(x+1) or cos[Q(r) —Q(r')]. Thus for an an- We can understand the perimeter behavior as a isotropic lattice the action is defined as boundary effect. However, as ~ is decreased more cos r — r+nt and more box fluxes may be inverted, and when ~ t is close enough to the critical value 1 the regions with inverted flux become of the same size of the — P, cos[Q(r) —Q(r+n, )]. (4.1) area of I". Thus a boundary effect is turned into gr an area effect. These large regions of inverted Evidently the action is minimized by configura- flux have boundaries which are lines of flipped tions in which all the spins are parallel as for the spins. Near the critica, l point long lines of in- Ising model, although this state is not unique. verted spins go through the loop (see Fig. 32). However, unlike the Ising case the degeneracy is These lines are closed at infinity and they are infinite corresponding to the continuously. variable topological objects which cannot be removed from direction of magnetization. The fact that the de- 2652 EDUARDO FRADKIN AND LEONARD SUSSKIND 17

generate states are infinitely close to one another makes the two systems essentially different. This (4.9) will be evident when we discuss the long-range correlations in the system. - where L is the canonical momentum to To construct a time-continuum limit we can fol- conjugate Quantum mechanically it satisfies low the method used for the Ising case. The same, result can be obtained from a simpler and more [I e44)] g (s4441 (4.10)

, familiar argument. Let us first allow -. We P, Since the variables are periodic it follows also introduce a time lattice spacing a which in the P that L has a discrete spectrum which consists present case behaves like 1/P, . The reason for re- of all the integer numbers. scaling the time direction is again to make the theory finite as P, -~. B. Large- &behavior We write the time-like term as For X large the term -AZ cos[(t)(z) —Q(z+1)] . 4(r) —4(r+s, cosa (4.2) forces the field to b'e very smooth. For the -p, a )) p low-energy configurations the field differences As p, -~ the important values of (t(r) and (j)(r+n, ) at neighboring sites will be so small that we may will be those for which Q(r) —p(r+n, ) -0 as 1/p„ expand the cosine. Thus which is equal to a. Thus we replace (4.2) by gL(z)' (4.3) 2 &[y(z) —y(z+1)]' &[y(z) —y(z+1)]' We may of course ignore the constant term. The 2 4f sum over space-time location may be approximated + ~ ~ ~ (4.11) by I For the lowest-energy states it is a good approx- imation to truncate the series after the second term. To see this explicitly a change of variables Thus the timelike terms in a are replaced by will be useful. Define U= )('~4y (p, a) (4 4) (4.12) P=~ "4L The space terms can, be written Evidently U and P are conjugate variables. In terms of the new variables H becomes a —cos r — r +Pc, (4.5) r X" iH= — — If P,P, is allowed to remain finite as P, -~ then Pz + Uz Us+1 P,/a is a constant & and (4.5) becomes 1 — (II(r) —U(s+1)]'+ . . (4.14) dt g cos[(t'(r) (j)(r+n, )]. ( (4.6) ) Thus the full action is In this form it is plausible that the quartic and higher terms can be neglected for the lowest-en- 4'(z)' ergy states. It is also evidently useful to rescale rr =+ dr —1 - Q cos(4'(r) 4(s+ ())). the energy so that ~ ' 'H becomes the new energy. Let us consider the ground-state correlations (4. 7) (vacuum expectation values) defined by In this case the trajectories in P„P, space cor- (0 ~e4(4(0)& i(I(s) ~0}- (4.14) responding to time-continuum limits are hyper- which can be written as bolas

(4.8) II — s*p S.r. (&(II) &(r)i 4) (4.15) Finding the Hamiltonian from (4.7) is just the usual problem of passing from the Lagrangian We will approximate this quantity by the corre- to the Hamiltonian in mechanics except that the sponding free field value defined by truncating time variable is Euclidean. Thus we obtain (4.13). For a free field we can write

I ORDER AND DISORDER IN GAUGE SYSTEMS AND MAGNETS 2653

'! /'/'/'/'/'/'/'/'/'/' f t t t t t t t I f 0 exp — „,[P(0) P(e)]} 0): Z FIG. 33. The kink creation operator rotates all the spins from - up to site & by the same . =exp —,z, 0 U0 —U g 0, 4.16 angle@

and for large z one easily finds Thus define — = x &0 I [U(0) U(z)]'10 const 1 =exp L(z) (4.21) Thus the Kz(z,) if P correlation function behaves like SgÃp (4.17) I.et us consider the action of this operator on Thus for large X the correlation decays as a the classical ground state for which {t}(z)=0. Since ~ power of the distance. This type of behavior is L(z) and ({()(z) are conjugate, K& rotates all the somewhat unusual. Qn the one hand, the fact that spins for z& z, by angle f (see Fig. 33). the correlation goes to zero implies a lack of in- We shall now show that finite range order or what is equivalent, no spon- &0 IK,(z) io& (4.22) taneous magnetization. On the other hand, the order is not of the usual short-range type which is a suitable parameter to describe the phase decays as an exponential. Note that the power transition and since it is related with the dual behavior depends on X. of the XF model, we call it the dual order parameter. Let us consider first the larg(e-X phase. C. The small-&behavior We calculate the correlation function For the limit X-0 the dominant term in H is &0 = — [K~(s)K*(z) [ 0) C (z s), (4.23) L(z)' (4.18) where z) s, and we want to compute this function g 2 in the limit —s » 1. In order to out this ~z ~ carry calculation it is useful to make a spin wave ex- and the ground state is the product state annihilat- pansion of the field p(z) . . I.et a'(k), a (k) be the ed by each I.(z): creation and destruction operators of a spin wave ~0)=0 (all of momentum k and let ~~- ~k be the dispersion L(z) z) . (4.19) ~ relation for small k. Since For this ground state the correlation function is identically zero for any nonvanishing separation. Z(z) = y(z) = ~'"L(z) This is because exp'~' ' increases the value of we get the following expansions for p(z) and p(z): I.(0) by one unit which is not compensated by exp[-i&t(z)] if z wO. dk {I)(z)= [a'(k)e'"+ a (k)e '"], For small X we may use perturbation theory )t(d, to compute the correlation function. To get a non- (4.24) vanishing contribution to the correlation the per- (t}(z)= -i dkv &@~[a'(k)e'~' —a (k)e '~']. turbation must act at least z times. Accordingly the correlation will behave like The magnitude we want to compute is

8 -I lnXI g (4.20) if 04{x-e}=(0 exp „, Q 0(x) . 0), (4.25) S cXC For this phase the correlation decays in the conventional way of a disordered system. The energy which is the same as spectrum for small X consists of a ground state and massive excitations created by applying the z operators exp[i&(z)]. C&(z —s) = exp —,~, 0 (t)(x) 0 ( g ) free f ield D. Kinks (4.26) We have seen that the X» 1 phase is riot charac- for the same arguments given above. terized by a nonvanishing order parameter. We After some algebra one finds that the leading will now show that a dual order parameter exists contribution to which also vanishes for X»1 but which is nonzero for X» The existence of this dual 1. order (4.27) can be used to characterize the phase transition. 2654 EDUARDO FRADKIN AND LEONARD SUSSKIND as L = ~z —s to infinity is V. ABELIAN GAUGE THEORY ~

&L model 0 Q(x) 0 =81n (4.28) A, The g 2 ( p&g& I free As in Sec. III a simple cubic three-dimensional then the correlation function Cz(L) falls off as lattice replaces space. Time is continuous. The degrees of freedom are attached to the links 4 and c~(c)=exp(- ln— consist of planar rotators or phase angles p(r, n). „, The conjugate variables L(r, n) [= -L(r+ n, -n)] have integer spectrum. Z/2 = const & L (4.29) The Hamiltonian for this system is given by the sum of hvo terms which we call electric and in leading terms in L. This means that the order magnetic, parameter (0 ~K&(z) ~0) is exactly zero in this phase since 2 Ifelectrge o L(ri n) (5.1) links~ ~~ ~0)'=lim =0. (4.30) (0 ~Kz Cz(L) where g is a dimensionless coupling constant and a is the lattice spacing in some arbitrary units. However, the behavior of the correlation function The magnetic term is a sum of interactions, is not of the type for a disordered phase, i.e., each associated with an elementary square box of exponential decay. the lattice. Let us consider a given box as shown Let us finally consider the small-X phase where in Fig. 34. The sides of the box are labeled 1, 2, 3, (0~K&~0) is finite. Since the ground state for A. =O 4 and are thought of as oriented. The magnetic verifies that L 0)=0 for all lattice sites, it is interaction for this box is given by clear that ag',, cos[P(l)+ p(2)+ Q(3)+ Q(4)]. (5.2) (4.31) Thus the Hamiltonian is at A. =O. Once again we may ask the question: Is it nonzero only at X=0 or is there a finite neighborhood of &=0 where has a nonvanishing (0 ~K& ~0) L( -)2 value? This question can be answered by com- in perturbation puting (0 jK& ~0) theory. 1 The calculation gives the result cos[p(1)+ ~ ~ ~ + p(4)] . (5.3) bpxes g

= ——,'X2(1 — O(X') (0 iK& i 0) 1 cosf)+ (4.32) B. Gauge invariance for the lowest nontrivial order in perturbation For each site of the lattice we can define a gauge theory. Since the coefficient of X' is finite we transformation which rotates the phase angles of argue again, the same as in the Ising model, that all six links radiating from that site. Thus (0 ~Kz ~0) is not only nonzero at &=0 but it has a n& +iX nonvanishing value at a finite neighborhood of X=O. g(fo~ eig(rO'n (5.4) This result makes it impossible to have a first- These transformations are expressed as unitary order phase transition in X at X= 0 as is the case of the one-dimensional Ising model as a function of temperature. As a conclusion we summarize the results just obtained drawing a qualitative picture of both phases. For large A. the system has massless spin waves and exhibits an almost-ordered phase 4]N 5/2 in terms of the original system. In this phase the system has heavy kinks. Qn the other hand; for small ~ the spin waves become massive and 'the kinks become massless and, what is much morA important, they condense giving rise to a nonvanishing dual order parameter (0 ~Kz ~0) which FIG. 34. The magnetic terms are associated with characterizes the phase. the boxes. ORDER AND DISORDER IN GAUGE S Y STEMS AND MAGN ETS 2655

operators Later we will see that this is the lattice form of Gauss's t.aw V' E =0. It is easy to show that for G(4o)=exp i&Qf (r„n,) (5.5) any gauge-invariant state the expectation value of e'~'""' vanishes. The Hamiltonian is invariant under these trans- Moreover, the two-point correlation function formations. Furthermore, as in the Z, gauge theory we will consider the of eiLe(oin) 4(R, n)1 physical space (P lP& states to consist of the gauge-invariant states. Froxn (5.5) we see that a physical state is defined by vanishes. The vacuum is defined to be a gauge- invariant state lp) where we have lp)= G lp) for - I. = 0. (r„n,) l g& (5.6) Qi=i all G. Thus

(o le"&'"'e-*" "'lo&=(o lG '(o)e"""'G,(o)G -'(o)e-"&R ")G,(o) lo& lee&4(O, n)eX')e-ie(R, n) (0 Io) (5.V)

since I and (t) commute at different lines. Then 4(R'I) n) (P le&i)&o'n)e- lP) =e'x(0 le'e)(o, n)e-is(R, lo& ) A (r, n) = —(r, n) (5.12) ag y (5.8) and the conjugate electric field E(y, n), for all values of X. Equation (5.8) implies that" E(r, n) = —,f, (r, n), (5.13) (P leis(o'")e &4&R n) lP)=0 . (5.9) so that This means that the two-point correlatio'n function is not the right object to look at. LA(r, n), E(r', n')]= —,lS„- -„, 6;;, (5.14) Again, as in the Z, gauge theory, the correct magnitude is the loop integral, namely The Hamiltonian now takes the form i p2 exp (5.10) H=a'. (link)+a' (Q A) Q ' Q,2a links boxes

s oa' ( where Z„(r) means the sum of all the angles &t) over rA), 4) , g 4' O( (5.15) all the links lying along the closed loop 1. boXes ' Following Qlilson's criterion. we calculate where I' is an elementary box of the lattice. For the long wavelengths we can approximate 0 expi. 0, (5.11) \ a' P- d'x which is known as the loop integral. %'e shall (5.16) show that this function is able to distinguish —gA-VxA between two phases. First of all as the opera- 0 r tor (5.10) is gauge invariant (5.11) may not be The long-wavelength physics is approximated zero. Furthermore, we shall show that in the by the continuum field theory small coupling phase it decays as e '"' ""and ' in the large-coupling phase it goes as e "in a = —, a'x[E'(x)+ (V )& A)']+ O(g'a') . (5.17) agreement with the results already discussed in the Z, gauge theory. In addition we shall see a Thus the first two terms define conventional free remarkable parallel between the behavior of the field electrodynamics and the higher-order non- 3+ 1 Abelian gauge theory and the 1+ 1 XE' ferro- .renormalizable term's are believed to be unim- magnet. portant for distances very much larger than the 4 lattice spacing. C. Small-g phase We now would like to calculate the loop integral As in the XF model we have to rescale the variables in order to describe the small-g phase. Thus C„= 0 T expig A dl 0 (5.18) we define the components of the, vector potential 2656 EDUARDO FRADKIN AND LEONARD SUSSKIXD I' for a circular path of radius B. In the limit@- ~ the vacuum state ~0) is de- It can be easily proved that termined by the first term in the Hamiltonian and therefore satisfies g 2 C~=exp — 0 [A(l) dl][A(l') ~ dt'] =0 all 2 0)I, L(r, n) ~0) links. (5.26) The second term will be treated as a perturbation (5.19) and it wBl excite boxes with circulating electric where A(I) indicates the vector potential at the flux. This phase is very similar to the small-X point l along the closed curve. phase of the Z, gauge theory. Here we have stable Since C~ is gauge invariant we choose to evalu- lines of electric flux whose energy is proportional ate (5.19) in the Feynman gauge. The free field to their length. propagator is e shall now calculate the loop integral in this phase. As in the Z, case the lowest contribution (5.20) in powers of I/g~ is proportional to (I/g )", where N is the number of boxes of the minimal surface The integrals can be most easily evaluated de- bounded by the loop. fining angular coordinates on the circle. The Thus final result is A C~ = const& —4 Cr=exp — I (5.21) 4 2 = const x e ~"'~ (5.2 7) where A is the area of the loop. where I is the integral We have seen that for weak'"' coupling"" the correla- 21' tion function behaves as e while here it j. cos(e, —62) I=- is e ". These two behaviors characterize two sin'[(8 —9 )/2] different phases of the theory which must be sep- arated a phase transition. However, I is singular since the integrand has by singularities at 8, —6I2=0, 2m. Thus the integral E. Monopoles has to be cut off by constraining both angles and angular differences to be larger than a/R. With The loop integral'characterizing the two phases this cutoff procedure the integral I can be ex- can be rewrittin, for small g, as pressed as exp ig 8 do (5.28) R 2m+ 0, I=+ 2 ln —+ + const (5.23) 0 where the integral indicates the total magnetic for R»a. flux passing through the loop. The phase trans- Thus the loop integral has the asymptotic form ition is caused by the increasing large scale fluc- I tuations of the magnetic flux which randomize the const g'R (5.24) integral for arbitrary large loops. The source 2/2 ll'2] 2](a (ft/ )(d of these fluctutations can be traced to the condensation of magnetic monopoles. " This is essentially a perimeter law modified by a We shall first discuss what is a magnetic mono- falling power at large distances. This is very pole on a lattice. Following Dirac'7 we define a similar to tPe behavior of the XY model. In both magnetic monopole of the field configuration cre- cases the asymptotic behavior at large distances ated by an infinitely thin solenoid with one end of the correlation function is a power-law modifi- placed at infinity. To define a monopole on a lat- cation of an ordered-phase behavior. tice" we embed the solenoid along the Z axis with the finite end at the center of a cube. D. Large-g phase The monopole is described by the classical vec- To study the g» 1 phase we write the Hamiltoni- tor potential A„(r —r,), where r, is the position an, of the monopole. The classical lattice monopole, a Fig. 35, is defined by assigning a phase to each H' =—2II link according to the rule L r+n (r, d) —— cos(P„A). (5.2l) A„~ dl (5.29) I inks 'g boxesI box ORDER AND DISORDER IN GAUGE SYSTEMS AND MAGNETS 2657

/ vacuum. For this purpose we write

M'= exp, i A,'", ' E(r)d'r

The'electric field can Qe expanded in creation and annihilation operators for free photons and the I l~ ~ expectation value can be computed to be

/ I ' ~M'(r) = exp ——, A'„( r)A~„(r ') (0 ~ 0)

) ' && (E'(r)Z'(r '))d'r d'r FIG. 35. The classical lattice monopole. (5.34) Explicit calculations with free fields show -the Let us consider the magnetic energy of the clas- exponent to be logarithmically divergent of the sical lattice monopole. Since the field is static volume. This result is analogous to the behavior there is no electric contribution to the energy of the kink creation operator in the XF model. (Z =A). The magnetic energy is Next let us consider the behavior of this quantity i,n the g»1 phase. For g=~ the vacuum state z .,=, P 1 —cos(&PI (5.30) satisfies = 0 all links I. ~ 0) where we have subtracted the of the clas- energy and therefore sical vacuum. The (1/g)Eg is the flux E of the classical field through the box. I.et us divide the (5.35) into two a term for those boxes energy parts, For weak coupling the order parameter through which the solenoid passes and all others. (0~M'(0) ~0) vanishes, but for g-~ we have seen the class of boxes the flux box is For first per that it is equal to one. To show that it is con- The just the total monopole charge p. . energy. nected to a phase transition we have to show that stored in those boxes is the order parameter is nonzero for some range L of the coupling constant. To test this statement [1—cos(p, (5.31) ag2, g) ], we have to compute the derivatives of (0 ~M' ~0) with respect to 1/g' by strong-coupling perturba- where I. is the length of the solenoid in lattice tion theory. If these derivatives axte finite we can units and it is infinite. say that in the neighborhood of g= ~ the ground- Accordingly the magnetic monopole energy can state expectation value of M' is nonzero. Indeed be finite if

p,g = 2'&. (5.32) This is the famous Dirac quantization relation" which in the lattice formulation expresses the condition of the energy of the monopole ta be fin- I ite. In order to define the monopole condensate phase (g» 1) it is convenient to introduce a monopole / creation operator M'(r, ), / z IVI onopo I e M'(r )=exp i rf&„(link)1(link) (5.33) rQ linksg r ] ~Loop This operator has the effect of translating the / phase of every link by an amount p„. Thus it also / shifts the magnetic Qux on each box by E„. Acting /. on the vacuum of the samll g phase it creates a / state with a monopole with position V.o. Let us consider (0 ~M'(r) ~0) in the weak-coupling FIG. 36. A loop and the Qux of a nearby monopole. 2658 EDUARDO FRADKIÃ AND LEONARD SUSSKIND strong-coupling perturbation expansion gives the due to the monopole condensate. In fact monopoles result near the loop will change the phase of the loop integral in about m per monopole (Fig. 36). In the 1 (0~M'~0)=l [I —cosE(box)] phase where the monopoles form a condensate, —,64g g they will change the loop's phase wildly or, what is the same, they will randomize the loop integral. (5.36) The total effect will be to make the loop integral fall off very fast. Thus, the phase in which mono- This result shows that for g» I the system looks poles form a condensate is the disordered phase like a monopole condensate with a finite monopole of the system. density. In the large-g phase the loop integral C~ has the asymptotic behavior ACKNOWLEDGMENT C„-exp(—area, ) . This work was supported by the Department of We can now understand this result as an effect Energy and the National Science Foundation.

~A. A. Migdal, Zh. Eksp. Teor. Fiz. 69, 810 (1975) ~H. A. Kramers and G. H. Wannier, Phys. Bev. .60, 252 [Sov. Phys. -JETP 42, 413 (1976);'69, 1457 (1975)] (1941); see also Ref. 5. 142, 743 (1976)]; L. P. Kadanoff, Ann. Phys. (N, Y.) ' The object we are calling the action is actually the 1OO, 359 (1976). energy function of classical statistical mechanics. A. M. Polyakov, . Nucl. Phys. B120, 429 (1977); R. ~'This is exactly correct for periodic boundary condi- Savit, Phys. Rev, Lett. 39, 55 (1977). tions. ID the thermodynamic limit the precise bound- 3K. Q. Wilson, Phys. Bev. D 10, 2455 (1974); T. Banks ary conditions are unimportant. ' et al. , Nucl. Phys. B129, 493 (1977). S. Elitzur, . Phys. Rev. D 12, 3978 (1975). 4J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 G. Baker, Jr., Essentials of Pade Appxoximants (Aca- (1973); J. Jose et al. , Phys. Bev. B 16, 1217 {1977). demic, New York, 1975). 5I'. J. Wegner, J. Math. Phys. 12, 2259 (1971);B. Balian, '4Each lattice link can be denoted in two ways, (r, n) J. M. Drouffe, and C. Itzykson, Phys. Rev. D 11, 2098 and (r+n, -n). It is convenient to define Q{r,n) (1975). = -Q(r + n;-n) . 6J. Kogut and L. Susskind, Phys. Rev. D 11, 395 (1975). ~5Equation (5.9) holds for all gauge-invariant states. ~T. Schultz, D. Mattis, and E. Lieb, Bev. Mod. Phys. ~6S. Mandelstam, LBL report (unpublished). 36, 856 (1964). ~VP. A. M. Dirac, Proc. B. Soc. London A133, 60 (1931). P. Pfeuty, Ann. Phys. {¹Y.) 57, 79 {1970). A. M. Polya, kov, Phys. Lett. 59B, 82 (1975).