On the phenomenon of symmetry enhancement in abelian gauge theories

Nuclear Physics B (Proc. Suppl.) 17 (1990) 585-589 North-Holland

585

ON THE PHENOMENON OF SYMMETRY ENHANCEMENT IN ABELIAN GAUGE THEORIES Mihail Marcu II . Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, D-2000 Hamburg 50, FRG and Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7 ZN spin sysytems in 2 dimensions and gauge theories in 4 dimensions (clock models) have an intermediatecoupling massless phase, which very much resembles the massless phase of the U1) theories obtained in the N -+ oo limit. We investigate the question of whether this phenomenon, sometimes referred to as symmetry enhancement, really means that the superseleetion sectors of the ZN model are labelled by the representations of U(1) rather than those of ZN C U(1) . For the gauge theory case, we discuss possible constructions of charge N finite energy states that are not in the vacuum sector . An actual proof for the existence of charged states labelled by the representations of a larger group than the defining gauge symmetry can be given in a technically simpler situation: we consider the massless phase of a Z-gauge theory (discrete Gaussian model) ; there a charged state of charge 27r exists, which has the vacuum quantum numbers with respect to the group Z of integers, but not with respect to the "enhanced" group R of real numbers. ZN clock models have been proven to have a massless intermediate phase for spin systems in 2 dimensions and for gauge theories in 4 dimensionsl . 2 (for N > 5). Let us denote by a. = 0, 1, . . . , N - 1 the ZNspin at the site x, and by ax,, = 0,1, . . . , N -1 the gauge variable at the link starting from x in direction Et . The clock model action is

S = -/3 E cos

21r

N(a., - a--+,ù)

for the spin system (# is a lattice-unit vector in the it direction), and

S = -,Q

cos

27r

N(a=,#& +

ay,W) (2)

for the gauge theory. Both at small and at large values of 8, standard convergent expansion techniques can be used to show that these models are in a massive phase. At intermediate values of ,Q, the 2-d ZN spin system and the 4-d ZN gauge theory have massless phases, which are very similar to the massless phases of the U(1) models obtained as N-+oo. For large N, the variables N a are a good approximation 0920-5632/90/$3.50 © Elsevier Science Publishers B.V . North-Holland

of the U(1) fields ; moreover, the vortices (2-d spin case) and the monopoles (4-d gauge case), which are known to trigger the transition in the limiting U(1) models,3, 4 can be well approximated using these discrete variables. Let us further notice that the intermediate massless phase exists precisely for the case of self-dual models' , 2 (for self-duality we also need the minor modification of the Villain action instead of (1-2) ). As N --+ oo, the large-# massive phase disappears, and the duality relationships are different: the U(1) models are dual to Z-models (integer valued fields). Technically, the massless phase is found by showing that for N > 5 the correlation functions of the discrete model closely resemble those of the U(1) case, both for original and dual variables. This similarity between ZN and U(1) models was named symmetry enhancement 2: a model with symmetry group ZN behaves, in the massless phase, like a model with a larger symmetry group U(1) D ZN . The conventional meaning of a symmetry is however somewhat different. The quantum mechanical Hilbert space is a direct sum of different superselec tion sectors that are labelled by the representations of the model's symmetry group.5- 6 In this spirit, we

586

M. Marcu /Symmetr__y enhancement in abelian gauge theories

ask the following question : can the concept of symmetry enhancement be taken literally, i.e. are the superselection sectors in the massless phase of the Zlv models labelled by the characters of U(1) ? This question is of special interest in view of recent progress in the theory of superselection sectors :6 roughly speaking, for massive theories there is now a method of reconstructing the symmetry from the knowledge of the superselection sectors. Massless theories are however much more complicated, and it is useful to have examples where the situation is possibly different. In order to understand the physics of the massless phases in ZN models, it is helpful to employ the block-variable picture. After a few blocking steps, the block spins or block link variables are a much better approximation of the U(1) fields than the original ZN fields were (linear combinations of ZN variables have more than N possible values). For the effective block-variable action, the U(1)-breaking terms vanish in the continuum limit. The question we raised about the meaning of symmetry enhancement cannot however be answeresd within such a framework. We are interested here in finding U(1) superselection sectors, in particular charged states, in the lattice ZN models as defined by (1-2), not in some continuum limit. A possible strategy to follow is to embedd the ZN model into a larger model that posesses the new sectors as a consequence of an explicit U(1) symmetry. To achieve this, additional fields have to be introduced, and the extended model has to be such that in some limit the original ZN model is recovered. A particularly pointed question to ask in the limit of the ZN model is that of the existence of a charge N state that is not in the vacuum sector. If the sectors were labelled by ZN alone, this would be impossible, since N=0 modulo N. If the symmetry is in reality U(1), then the charged states are labelled by unrestricted integers (characters of U(1) ), and a charge N state is not in the vacuum sector . Unfortunately, it is not known how to embedd the 2-d ZN spin system; into a U(1) model that fits our strategy. For the 4-d gauge theory case however, we can couple a U(1) gauge field to a charge N, scalar, fixed length, complex matter ffield l

9'x

= exp iv-_ :

S = -lQ

cos (A.,, + A.+,û .., - Ax+t,,» - A.,,,)

m,p
lis cos (NA.,, + V.+,

ù

.T,là

- v=)

(3)

with A,,, and Vz E As the "hopping parameter" rc -+ oo, the "unitary gauge" variable Az ,~ = A.,, + i(SPz+~ - V.) is frozen to multiples of and we recover the Z,v gauge theory. The model (3) has the following phase diagram:

iv,

Higgs

(0,0)

Confine ment

Coulomb

The confinement and Higgs mechanism phases are easily seen to be massive, by convergent expansion techniques . Furthermore, there is good evidence, from inequalities combined with simulations, that there is no extra phase transition line inside the massless Coulomb phase.8 As usual,9 we can construct energy-minimized sources (states with external charges) V¢_") as the limit of source-antisource "dipole" states : gpz" 1 = llm llm

y-oo t- c»

II

1 I exp{ -tH) exp

in

x

At

I0) (4)

where H is the Hamiltonian (logarithm of the transfer matrix), t is Euclidean time (an integer), Q are spatial (i .e. time-zero) links on a straight line connecting the spatial points x and _y, A are the quantized time-zero gauge fields, and II - II is the norm of the vector following this symbol in the formula . Let us first discuss the case n 36 0 modulo N. Since the matter field of the model (3) has charge N, a source of such strength cannot be completely screened ; the state T(n) will have a nontrivial total electric flux at infinity . By acting on it with local operators - gauge invariant or not - and taking the completion of the resulting vector space, we

M. Marcu /Symmetry enhancement in abelian gauge theories will obtain superselection sèctors which are different from the vacuum sector . These superselection sectors are somewhat special: they do not contain any gauge invariant states . Nevertheless, they are superselection sectors in the sense that .they carry a representation of the field algebra (and, of course, of the observable algebra) that is inequivalent to the vacuum representation . 5

In the confining phase, the strength-n Wilson loop obeys an area law for n 0 0 modulo N . This means that the states C~n) have infinite energy and are therefore unphysical . In the Coulomb and Higgs phases, the perimeter law obeyed by Wilson loops means that the energy regularization by Euclidean time translations has produced a finite energy state. In the Higgs phase, the U(1) gauge symmetry is "broken", in the sense of the Higgs mechanism, up to the ZN that becomes the gauge group in the ic-a oo limit. Thus the sectors are labelled by an integer n modulo N (we can use the convergent expansion to proove this). In the Coulomb phase however, we expect the U(1) not to be "broken", so for finite rc the sectors should be labelled by an unrestricted integer n (at least for small re, this can be seen using the hopping parameter expansion) .

Now let us turn to the discussion of the charge

N sector in the Coulomb phase. One possible con-

struction for a gauge invariant charged state is : 9 4t (N) = exp q,(N) a . ji0a)

(5)

(SP are the time-zero matter field operators) . By Griffith inequalities its scalar product with the vacuum, i.e . the vacuum overlap order parameter (VOOP), is zero for large ,3 and small ic . This indicates that 4,(N) is not in the vacuum sector . Since there seems to be no additional phase transition inside what we called Coulomb phase,$ we would expect the VOOP to remain zero for large rc too. This is however not proven, and in the large K region proofs are not easy to do . 2 To see that we actually do not know what happens, let us note that the VOOP becomes identically 1 if we interchange the limits rc -i oo and y -i oo, t -+ oo .

A different construction for a gauge invariant

587

charge N state is : 10 e.N) =exp {i~p=} exp i -

t

E(N) ~4r_

10)

(6)

where E(N) is the value at t of the discretized spatial (3-d) classical Coulombic electric field of a source of strength N located at x. The basic idea here is to act on the vacuum with the Coulomb-gauge matter field operator . Unfortunately, unsolved problems connected to the Gribov ambiguity do not allow us to use this construction for the model (3). 11 Let us describe a possible alternative to the constructions (5-6). We will try to give a discrete ap-

proximation to the creation of a Coulombic electric field. Consider a set L_= of N oriented spatial

lines starting at the common point x and ending at yi, i = 1, . . . , N, and assume that the directions of these lines are distributed as isotropically as possible in the 3-d space. Create from the vacuum a flux of strength 1 on each of these lines. The resulting state has a total electric flux of strength N through any surface containing x - . As in the construction (4-5).9 the flux is concentrated on thin lines, so this state will have an infinite energy as the yi--+oo; Euclidean time translations can however be used to regularize the energy. To sum up, we have constructed the state 1p(N) = lim £,yi t--oo

II 1 II exp{

-tH} exp i F, Âl 9 -L

1 10)

that has a source of strength N at x and sources of strength -1 at yi . To obtain the new candidate for =N), we a gauge invariant charge N state, denoted C let the _yi go to infinity along the N lines starting at a, and we act with the appropriate matter field at x: (N)

- exp (io.l ~lim

=v)

(

=N) We can of course take the limit is -+ oo of C to be a candidate for a charge N state in the ZN model. It is however interesting to also know whether we can define such a charged state without enlarging the ZN model. For rc = oo there is no matter field left, but the state with the N flux lines is already

M. Marcu /Symmetry enhancement in abelian gauge theories

588

gauge invariant at x. In the notation of (2), we can define the following candidate for a charge N state: ,&(N) = lim lim Fi- t-oo

i=1,. .,N

1 II

II

exp{-tH) exp

r

l_E~

Consider the noncompact scalar QED (fixed length matter field), with the local gauge symmtery group R (real numbers), defined by : S

ât

10)

4=N) Actually, 4(N) differs from the re --~ oo limit of by the order in which we take the limits r;, --+ oo and yi --+ oo, t --+ oo . This is not a mere technicality. In the case of 4(N), the point x is singled out as the origin of the electric flux because in the construction we used the state T(v) containing a source at x (the location and strength of a source is obviously not affected by Euclidean time translations) . x plays a similarly special role in (5-6). For 4( N) on the other hand, we started with a state that is gauge invariant at x too; after the energy regularization we can no longer expect that it is precisely the point x that plays a special role . In the end it might still turn out that the two states have similar properties for the electric flux at infinity, but they certainly are not equal. The problem of whether CN) is a charge N state is related to the stability of the configurations of electric flux centered in the region around x and directed towards the N sources at _yi. We believe that, in the presence of a sufficient number of isotropically distributed sources at a distance r from x, it is not possible to lower the energy by removing the flux from a sphere of, say, radius r/2 around x. Therefore we conjecture that 4(N) is indeed charged. Let us note that a state of charge N would be dual to a monopole of charge N.1 If the one exists, then the other exists too, and the massless phase of the ZN gauge theory would have photons, charges and monopoles as its particles. Interesting questions, like the relationship between the electric and magnetic fine structure constants, could be investigated . Unfortunately we were not yet able to proove the existence of the charge N state. It is not even clear whether the (rather involved) expansion of 2 is the only tool we need . In order to make the whole discussion of symmetry enhancement more plausible, we end by describing a situation where everything can be rigorosly checked.

= 2 E (A=, + x,,a
- i£ E s,,à

A=+A,,, - A.+i,,,, - A.,,)'

cos (A=,, + Vx+j - p.)

( 10)

with AI,l, E R and JP, E (-7r, w) . The phase diagram differs from that of the model (3) by the absence of the confining phase (the Coulomb phase starts at 0 = 0) . As ic --+ oo, the "unitary gauge" variable A' ,,, = A.,,, + Sp.+,a - ,pd is frozen to multiples of 2r, and we get the Z gauge theory (discrete Gaussian model) : S = 27rZ,0

x,,p
(az,,,+aM+iA."-az+v,,,-a=,,)2 (11)

which as,,, unrestricted integers . The representations of Z are labelled by the real numbers modulo 27r. For the Z gauge theory, the question of the meaning of symmetry enhancement can be most pointedly posed as follows: is there, in the mass less small Q phase, a charged state of charge 21r ? If the answer is positive, then the superselection sectors are labelled by unrestricted real numbers rather than by the representations of Z C R. Using the techniques of 11, 1 for the noncompact scalar QED, we prooved that, for small ,0 and independently of rc, the state C.i) (defined as in (6) ), is orthogonal to all local excitations of the vacuum 12 (notice that no problems with the Gribov ambiguity arise here ll ). In particular, for rc=oo this state becomes exp 2ri

r t

Esl)âe

10)

(12)

which, in the language of the Z gauge theory, is a charged state of charge 27r . Thus for the Z gauge theory the superselection sectors are indeed labelled by the representations of R D Z. Let us remark that the charged state (12) corresponds to a Dirac magnetic monopole state of the U(1) pure gauge theory, which in 4 dimensions is dual to the Z gauge theory. It has been for a long

M. Marcu /Symmetry enhancement in abelian gauge theories

589

time accepted that the existence of the monopole as a charged state (on the lattice it of course has finite energy) is due to stability properties, this time for configurations with magnetic flux.

5. S. Doplicher, R. Haag and J. E. Roberts, Commun. Math . Phys. 23 (1971) 199, and 35 (1974) 49 .

ACKNOWLEDGEMENTS I am indebted to Klaus Fredenhagen, Florian Nill and Kornél Szlachânyi, who frequently and at great length discussed with me the concepts and ideas presented here.

7. See e.g. J. Jersâk, in Lattice Gauge Theory a Challenge to Large Scale Computing, Wuppertal 1985, edited by B. Bunk, K. H. Mutter and K. Schilling, Plenum (1986) p. 133.

REFERENCES

1. S. Elitzur, R. Pearson and J. Shigemitsu, Phys . Rev. D19 (1979) 3698; D. Horn, M. Weinstein and S. Yankielowitz, Phys. Rev. D19 (1979) 3714; A. Ukawa, P. Windey and A. H. Guth, Phys . Rev. D21 (1980) 1013 . 2. J . Fr6hlich and T. Spencer, Commun . Math . Phys . 81 (1981) 527, and 82 (1982) 411, and in Scaling and Selfsimilarity in Physics, J. Fr6hlich editor, Birkäuser (1984) p. 29 ; J . Fröhlich, in Unified Theories of Elementary Particles, Muenchen 1981, Lecture Notes in Physics 160, Springer (1982) p. 117. 3. J. M . Kosterlitz and D. J . Thouless, J. Phys. C6 (1973) 1181 ; J . M. Kosterlitz, J. Phys . C7 (1974) 1046 . 4. A. H . Guth, Phys . Rev. D21 (1980) 2291 .

6. S. Doplicher and J. E. Roberts, preprint University of Rome I. (1989).

8. J. M. F. Labastida, E. Sanchez-Velasco, R. E. Shrock and P. Wills, Stony Brook preprint ITPSB-85-42 (1985), Phys. Rev. D34 (1986) 3156, and Nucl . Phys. B264 (1986) 393. 9. K. Fredenhagen and M. Marcu, Phys. Rev. Lett . 56 (1986) 223; M . Marcu, in the Wuppertal proceedings (quoted above), p. 2673 ; K. Fredenhagen, T. Filk, M. Marcu and K. Szlachânyi, in Lattice 88, Fermilab 1988, A. S. Kronfeld and P. B. Mackenzie editors, Nucl . Phys . B (Proc. Suppl.) 9 (1989) p. 40. 10. K. Szlachânyi, in the proceedings of the Brighton High Energy Physics conference 1983 p. 13, and Commun . Math . Phys. 108 ~1987~ 319 . 11 . C. Borgs and F. Nill, Commun . Math . Phys. 104 (1986) 349, and Nucl . Phys . B270 (1986) 92; F. Nill, PhD Thesis, München preprint MPIPAE-PTH 87-31 (1987) . 12 . A similar proof was briefly outlined in J. Fr6hlich and P. A. Marchetti, Europhys . Lett . 2 (1986) 933, where the authors also point out some as pects of the particle properties of the state (12) .