a spin model with non-abelian Π1

Volume 154B, number 1 PHYSICS LETTERS 18 April 1985

A SPIN MODEL WITH NON-ABELIAN I I t

Dan LEVY

Department of Nuclear Physics, Weizmann Institute of Science, Rehovot 76100, Israel

Received 28 January 1985

A spin model in two space dimensions with non-abelian fundamental topological group (H 1 = Q) is studied. The model has non-abelian vortices which change their relative sign when crossing each other. The phase structure of the model related to the presence of non-abelian vortices and their global symmetries is studied.

Since the pioneer work of Kosterlitz and Thouless on the X - Y model [ 1 ] which revealed the role of topological singularities (vortices) in deriving phase transitions much interest was drawn to various two-dimensional spin models having non-trivial II 1 (see refs. [ 2 - 4] and others cited therein). However, the models treated in these works have an abelian I11 . Non-abelian fundamental groups are known [5] to have special mathematical properties which lead to a qualitatively different behavior of the vortices. Poenaru and Toulouse [6] have demonstrated the physical implica- tions of these properties for the combination of line defects in appropriate media (biaxial nematics for instance). Bais [7] applied their ideas to flux tubes solu- tions in broken gauge field theories.

The smallest non-abelian H 1 one can have is the quaternion group Q (see ref. [1]), which is a subgroup of SU(2) and whose elements are:

O = {1 , -1 , io x, - iox , iOy, -iOy, iOz, - iOz} , (1)

where the o's are the Paull matrices. The vortices are identified with the conjugacy classes of the fundamental group. Q has five conjugacy classes:

C O = (1), CO = { -1 ) , C x = {+iox},

Cy = {+iOy), C z = {+iOz) , (2)

and therefore a topological space having H 1 = Q will have four non-trivial vortices, one of them abelian corresponding to C0, and three-non-abelian corresponding to C x , Cy, C z . The rule of combination of two vortices

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is given by multiplying the corresponding conjugacy classes. For an abelian I] I it is unambiguous since the multiplication table of the conjugacy classes is given by the group multiplication table. When the II 1 is non- abelian the case is different. For Q we get for the non- abelian vortices:

Ci .Ci=2C 0 +2C0, i = x , y , z , (3)

which means that the combination of two non-abelian vortices of the same type can result either in an annihi- lation of the two or in an abelian - 1 vortex. The final answer to what will be the result of the combination depends on the path along which the vortices will combine.

In order to formulate a two-dimensional spin mod-- el with a desired H 1 we need to form a nearest-neighbor interaction whose basic variables reside in a coset space G/H that has this H 1 . These variables should be spanned by G transformations and the nearest-neighbor interaction should be locally symmetric under H. The choice of G and therefore H is not unique. For our case we have:

III(SU(2)/Q ) = n1(so(3) /02) = Q, (4)

where D 2 is a four element group containing three 180 ° rotations around the three spatial axis, and the unit element. Using the SO(3)/D 2 form we can write down an interaction which resembles an ordinary spin -spin interaction. We choose as our basic variable a pair of three-dimensional mutually orthogonal unit vectors r l , r2, at each point in the plane. Altogether

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we have three degrees of freedom at each point which can be spanned by SO(3) transformations acting on a reference pair (r l = x, r2 =Y for example). A nearest- neighbor interaction which is locally symmetric under D 2, i.e., symmetric under each of the reflections ~1 -+ - r l , r2 ~ - r2 separately is:

Hi] = -2J[(~Xi'~lj) 2 + (r2i'J;2j) 2

-- (t~li'r2/)2 -- (r2i "rlj )2]" (5)

If the pair ~1, r2 is reorganized into a complex vector V = r l + if2 then (5) becomes

+ (V/*- V/) 2 + (V/" V/*)2]. (6)

An abelian -1 vortex is identified by the 21r rotation V makes ~iong a closed contour which surrounds it. The three non-abelian vortices will induce the changes V

- V o r V--~ V* or V-~ -V* along such a contour (graphical illustrations of all the vortices appear in ref. [51).

An alternative model which is studied in this work uses a "Villain type" form of interaction with G = SU(2) and H = Q. The action of this model is given by [3]

S[g,h] =/3(1 sit~es i Tr(gihi,~g~+;,+ Tr(gihi,fig~+5) )

+ ~ ~ Tr(hi,ychi+£,fihi+~+~,_~hi+fi,_fi) plaq

--~ ~ Si[g,h ] + ~ ~ -~Tr(hp), (7) sites i plaq

where g E SU(2) and h E Q are in the fundamental representation of SU(2) and hp is a shorthand nota- tion for the multiplication of four h variables along the links surrounding an elementary plaquette. (7) is invariant under the local gauge transformation

gi-~gih, hi,~-~h?hi,~, h=+~,+_fi, (8)

where h i _~ =- h~h.,, The vorticity of a given region is deFmed t'o be the multiplication of the link variables along its border. Therefore the second term in (7) is a sort of chemical potential that controls the vorticity of elementary plaquettes of the lattice. Actually, the "real" vorticity should be found from the orbits the g

matrices describe in the parameter space of G, but it seems that "link" vortices reflect quite well the dy- namics of the "true" vortices through all the range of the coupling. It is easy to see that also with this defini- tion the vorticity is determined by the conjugacy classes of Q. Assume we are given a closed path of N links, then

V = h l~2h2~ 3 ... hN~ 1 . (9)

Gauge transformations (8) at sites that are not on the path do not affect the h's participating in (9) at all, and transformations at sites on the path always change two of them simultaneously. These changes cancel un- less the transformation is at site 1. In this case

V ~ h t Vh, (10)

which leaves V in the same conjugacy class. As a result we cannot change the type of the vortex by any gauge transformation which is of course a desired property analogous to the stability of continuum vortices against continuous deformations.

A single vortex in the plane is generated by a single plaquette which has a non-trivial hp. We can always use gauge transformations to pass to a reference frame in which the vortex will appear as one singular plaquette carrying an infinite string of trivial plaquettes that have non-trivial hp links, stretching from it to some direction. All other links (if we deal with only one vortex) can be set to the identity. This picture is convenient for illustrating the special effect of non- abelian II1 resulting from (3) when two x-type vortices are to be combined in the presence of a thirdy-vortex [5]. Suppose that the singular plaquettes of the x-vortices are located at x 1 and x 2. The signs of these plaquettes are not gauge invariant quantities, however, the quantity hp(Xl)hAhp(X2)hA 1 , where A is some path connecting x 1 with x 2, is gauge invariant since it is equal to +1 or -1 which belong to different conjugacy classes. Once we choose a path A and move x 1 to x 2 along it the result of their combination is clear. It is easy to see that paths which pass through the y- string will give the opposite result to paths which avoid the y-string since

iOyiOxiOy 1 = - io x. (11)

An action of the form (7) is symmetric under two kinds of global transformations:

(i) Continuous transformations:

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g i ~ U g i , U G G . (12)

(ii) The combined similarity transformations:

g i ->UgiU-1 , hi ,h--rUhi,hU -1 , U E G ' . (13)

(12) cannot break in two dimensions because of the Mermin-Wagner theorem, but (13) can be spontaneously broken if it contains some discrete part. The set G' is defined by the following demands:

G'C__G, g ' h g ' - l E H ~thEH, (14)

which are sufficient and necessary to prove that G' is a subgroup of G. It is also clear that H itself is the maximal normal subgroup of G'. G' is called in group theory the normalizer of H in G. Under the transformation U E G' the vorticity V of a closed region transforms like

V ~ V '= UVU -1 . (15)

Under (15) a conjugacy class of H transforms to a conjugacy class of H which need not be the same (though it must be of the same dimension and must have the same character). Therefore G' can act non-trivially on the vortices. For G = SU(2) and H some discrete abelian subgroup of it generating a non-trivial II 1 we can immediately find out G'. Taking the element of H to be V = exp(OdOz), we get by taking the trace on both sides of(15);

c o s ( ~ ) = c o s ( ~ ' ) ~ ~ ' = _+,~.

So in this case there can be only one non-trivial transformation which takes a vortex into its anti-vortex. The U's for this transformation are the two other gen- erators of SU(2), which together with the U(1) generated by io z form G'. For the SU(2)/Q model G' is found to be:

G' = (exp(+ ¼rtioi) exp(+ ~rrlcrj),l • • exp(+ ¼rtioi);-1;1),

i ,] = 1,2, 3. (16)

G' has a simple geometrical interpretation: it contains all possible rotations which take a three-dimensional cube into itself, the cube being actually represented by Q. These transformations include rotations by 90 degrees around the x , y , z axis and around the princi- pal diagonals. The action of G' on the vortices is to permute between the three non-abelian ones. For example a permutation between x and z vortices is given

by:

exP(¼~" iO y) io x exp ( - ¼rr iOy) = io z ,

i i exP(¼,riOy) io z exp ( - ~rr Oy) = - i o x, (17)

while the rest of the elements of Q are left invariant under this transformation. This is to be compared with the case of an abelian 1I 1 discussed previously in which the only permutation possible is between a vortex and its anti-vortex.

Two more things should be noted about G'. The first is that it is separable from the local symmetry in the sense that we can define a model with the same 111 which does not have the same global symmetry. For example, a term of the form )`x~plaq Tr2(hhhhiox) will suppress or promote x-type vortices, according to the sign of )`x ,and therefore will explicitly break G' if it is added to (7). The second thing is that this symmetry is not unique to the Villain form of the model. We can introduce a third vector r3 = rl X r2 at each point in the plane which does not carry any new degree of freedom, and then add terms to (5) to make it totally symmetric with respect to ~1, ~2, r3" Now global vortices permutations will be expressed by the global transformations ri -+ i/'.

An interesting phenomenon which we want to point out is the spontaneous breaking of the permutation symmetry between the vortices. It should be stressed that the breaking of G' which generates this symmetry does not necessarily imply the breaking of the actual symmetry between the vortices. We will dis- cuss this subject in the combined limit )` ~ _~o,/3

oo keeping/a = ),//3 finite, which can be dealt with analytically following the technique and ideas of ref. [3], used for an SU(2)/Z 2 model. In this limit entropy does not play any role and only ground state energies should be considered. There are three competing ground states: For large enough/a the trivial ground state with no vortices wins, for small enough/a the - 1 abelian vortices condense and between these two phases there will be a phase in which one type of the non-abelian vortices will condense. The ground state of this condensation is calculable in the same way as that of the -1 vortices in the sense that the minimizing configuration for the whole lattice is the replica of the configuration that minimizes a single plaquette. The energy per link that we get from the/3 term of the action in this case is cos0r/8). The phase in which the

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non-abelian vortices condense breaks G' and the symmetry between the vortices. The transition between the trivial ground state and the non-abelian one occurs at

- 2 cos(rr/8) = - 2 - / a 1 =~/.t 1 = -0 .1522 , (18)

and the transition between the non-abelian and the - 1 phases occurs at

- 2 cos0r/8) = - 2 cos(Tr/4)+/~2 ='/a2 = -0 .4335 . (19)

These two values of/a are supposed to be the start of first-order phase transition lines. For a model with G = SU(2) and an abelian discrete H we will get in this limit a breaking of G' without breaking of vo r t ex - anti-vortex symmetry. In this case all the elements of H besides the unit element represent non-trivial vortices and their anti-vortices, and taking the above limit we get (n/2) + 1 different phases, where n is the number o f elements in H, each phase corresponds to a condensation of one type o f vortex together with its anti- vortex, including the trivial vortex. All o f these phases except for the - 1 and the trivial phases break the discrete part o f G' which is generated by io x and iOy though the vortex and the anti-vortex continue to appear in equal numbers. This comes from the fact that when H 1 is abelian the vortex-anti-vortex condensation breaks translational symmetry, and the lattice is symmetric only to translations by even number of lattice spacings a. The non-abelian ground state, on the other hand, does not break translational symmetry because a non-abelian hp is not gauge invariant and we can always cancel the result of a global transformation acting on it by a local gauge transformation. To sum- marize: In the non-abelian case the symmetry between the vortices breaks and the translation symmetry of the lattice is preserved in the/3 ~ oo, ~, ~ ___oo limit while in the abelian case the opposite is true.

The existence of phase transitions for zero or positive ), is a long withstanding controversial problem. The question is general and concerns all two-dimensional G/H models which are locally isomorphic to globally symmetric O(n) models. There are claims [2] that the non-abelian nature of G induces in these models a renormalization o f the contribution of vortices to the free energy, which modifies the famous Kosterlitz -Thouless estimate [1 ] for pair dissociation point and leads to the existence of free vortices at any non-zero

coupling. Several works have been done on models with abelian topology using either Monte Carlo meth- ods or strong coupling expansions. A summary of the present situation and references for these works can be found in a recent paper o f Caselle and Gliozzi [4]. This paper also presents an argument which shows that if a vortex pair dissociation process occurs at some finite non-zero coupling then certain correlations will undergo a drastic change in their path dependence and therefore it is expected that such a process will induce a real thermodynamic phase transition. This argument does not basically depend on the nature o f the II 1 of the model but only on the existence of non-trivial vortices. However, for non-abelian Il I an extra source for sharp change in thermodynamic behavior can be thought of provided that vortex pair dissociation does occur. Using the picture presented earlier of vortices as non-trivial plaquettes carrying infinite strings, it can be seen that a full dissociation of pairs means that some of the strings are forced to cross each other. Configura- tions of non-abelian vortices with crossed strings are not gauge equivalent to the same configurations where the vortices stay in their places but the strings do not cross each other. This effect may cause non-analytic behavior at the point of dissociation. If such a mecha- nism exists we can expect the following:

(i) A first order transition is not ruled out because this crossing can cause a discontinuity in energy.

(ii) The properties of the transition should depend heavily on the concentration of non-abelian vortices at the transition point. For example, increasing ~ towards positive values suppresses the vortices so we should expect the signs of the transition to become weaker and also/3 c to decrease since the mean separa- tion between vortices grows when they are diluted.

Another indication to a possible difference between models with abelian and non-abelian topology can be seen by considering the following plaquette-plaquette correlation operator:

--1 Pcorr = ~ [Tr(hp(n)hnmhp(m)hmn)

- Tr(hp(n)) Tr(hp(m))], (20)

where hnm denotes the multiplication of link variables going along a particular path from plaquette n to plaquette m and hmn = hnlm . This operator has the full gauge and global invariances. It has a non-zero value only if

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hp(n) = +hp(m) = ioi, i = x , y , z . (21)

In the high-coupling limit the thermodynamic average of Pcorr has a strong dependence on the path nm cho- sen. This can be easily seen from the fact that the first diagram contributiong to (Pcorr) must contain all the links on the path. Now, for a model with H = (1, - 1 , +iOz) which has an abelian HI, we can define the same operator which has all the properties mentioned besides the obvious fact that it is path independent from the start *~ . This difference between the abelian and non-abelian models will certainly exist in the strong- coupling regime. It can pursue through all couplings if vortices stay free for all couplings and it may indicate a difference in the nature of the phase transition if a phase transition does occur. (In this case the path dependence of the non-abehan Pcorr should change dras- tically at the dissociation point for similar reasons to those given in ref. [4] .)

Several Monte Carlo (MC) simulations were performed on the SU(2)/Q model defined by (7), using the "heat bath" algorithm [8]. The aim was to verify the existence of global symmetry breaking discussed above and to get a qualitative picture of the phase diagram. The simulations were run partially on an IBM 3081 and partially on a special processor [9].

The points in the/3 - k plane where possible signs of phase transitions were detected are plotted in fig. 1. The points that are marked by stars and crosses indicate clear hysteresis loops in all quantities measured and clear signs of changes in the symmetry of the vortices. The question marks indicate a much weaker hysteresis which weakens towards positive values o f k and no symmetry breaking. The crosses for k = ( -9 .0 ) , ( -6 .0 ) , ( -5 .0 ) , ( - 4 .0 ) are constructed from the theoretical//1 values since the hysteresis loops for these transitions were huge. However it was verified that the loops lie on a more or less constant / / l ine and that the theoretical values fall inside them. Interesting features of the MC results in the large and negative k region are as follows:

(i) The second transition line which is marked with stars appears with a much smaller hysteresis loop compared to the other line. It can be argued that this is

,1 This abelian model seems to provide a "natural control" for the SU(2)/Q model. For instance, it has in the/3 ~ *% k --, -** limit the same phase structure.

0

- 2

- 4

- - 6 - -

- - 8 - -

- 1 0 '

0

9

_ Q

-~:" +

i , , , , i _ ~

I I I - ~

60 I I I I I I I

20 40

Fig. 1. Observed signs of phase transitions.

only an artifact of the MC simulation because the//1 transition occurs at higher/3 values and perhaps con- vergence is much slower there, but it seems that a more careful and detailed explanation is needed.

(ii) In the region k = ( - 3 . 5 ) - ( - 5 . 0 ) the second transition line occurs at an almost constant 13 rather than a constant// . The reason % that for this range of X values, the system moves from the phase of one non- abelian type condensation to a phase in which the three non-abelian types dominate with equal quantities, before//reaches/12 region. In the three-types phase as in the - 1 phase the global symmetry is of course returned. The/3 term in the action gets larger values in this phase compared to the one-type phase but since we are at finite 13 entropy is also important and in the three-type phase we gain a factor of three in the multiplicity o f a single vortex. Because the one- type and the three-types have equal contributions to the k term T c in this limited k region, should not have a significant k dependence. At k = - 9 . 0 the transition occurs from the non-abelian one-type phase directly to the - 1 phase, as in the ideal limit. Indeed, the value of //is very close to the theoretical value//2-

(iii) At lower absolute values of X the two phase transition lines seem to join somewhere between k = - 3 . 2 and k = -3 .5 . This can also be expected since the phase with the broken symmetry is well defined and therefore should be bordered by the phase transition lines. Indeed, the two lines have a common point at k = _oo and 13 = o¢. They should have a second common point of meeting at finite values of/3 and k since other possibilities are excluded.

Several runs were also performed for a set of k values between ( -3 .2 ) - (+1 .0 ) . The results of these runs

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can be summarized as follows: (i) No breaking of global symmetry. (ii) The density of - 1 plaquettes falls rapidly to

zero. At large values of/3 66 ~ 14 for X = 0) the surviv- ing non-abelian plaquettes appear as pairs where the two members of the pair are of the same non-abelian type but with negative relative sign (the vorticity around the pair is zero). This is found directly by look- ing at several equilibrium lattice configurations.

(iii) The behavior of energy and non-abelian vorticity is similar at large/3.

(iv) There is a weak hysteresis-like region in all quantities measured in all values of X.

(v) For )~ = 0 no apparent peak in C o and no significant change between N = 60 and N = 30 lattices are seen.

Since the hysteresis-like signs are quite weak more numerical work with better statistics is needed to es- tablish, from pure numerical point of view, their signi- ficance or insignificance. Also some sort of proof should be given, for any numerical work on G/H models, that the observed behavior of the vortices is not in- duced by artifacts like finite size effects. Nevertheless, we can say that the properties o f the question marks line are consistent with the expected properties of the non-abelian string crossing process discussed earlier.

Summary. The results of this work seem to point out that the nature of the 1-I 1 of the model (abelian or

non-abelian) does indeed influence its phase structure. In particular it was shown that the global permutation symmetry of non-abelian vortices is richer compared to an analogous abelian model and that this symmetry can break down spontaneously for a large enough chemical potential of the vortices. Another region of interest, that was discussed, for which non-abelian effects can cause qualitative differences is the region of X = 0. However, further investigation is needed to find out whether a pair dissociation process with non- abelian strings crossing does actually occur.

I am greatly indebted to Prof. Adam Schwimmer who introduced me to the subject of this work and fol- lowed all its stages with advice and discussion.

References

[1] J.M. Kosterlitz and D.J. Thouless, J. Phys. C6 (1973) 1181 [2] S. Solomon, Phys. Lett. 100B (1981) 492. [3] S. Solomon, Y. Stavans and E. Domany, Phys. Lett. lI2B

(1982) 373. [4] M. Caselle and F. Gliozzi, Phys. Lett. 147B (1984) 132. [5] N.D. Mermin, Rev. Mod. Phys. 51 (1979) 591. [6] V. Poenaru and G. Toulouse, J. Phys. (Paris) 8 (1960) 887. [7] F.A. Bals, Nucl. Phys. B170 (1982) 32. [8] M. Creutz, L. Jacobs and C. Rebbi, Phys. Rep. 95 (1983). [9] H. Brafman et al., A fast general purpose IBM hardware

emulator, Review of the impact of specialized processors in elementary particle physics (Padua, 1983).

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a spin model with non-abelian Π1

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