- Email: [email protected]
The phase structure of the SU(2) × SU(2) spin system in two dimensions
Volume
109B, number
4
PHYSICS
THE PHASE STRUCTURE
25 February
LETTERS
1982
OF THE SU(2) X SU(2) SPIN SYSTEM IN TWO DIMENSIONS
M. CASELLE, F. GLIOZZI and R. MEGNA Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Turin, Italy Received
Strong rectangular transitions
15 December
1981
coupling expansions up tog -I6 of the correlation length for a mixed SU(2) X SU(2), SO(3) X SO(3) system in a lattice are considered. In addition to the asymptotic freedom fixed point, an unexpected line of second-order is discovered. The common critical exponent of these transitions is v = 1, like in the king model.
There is a close similarity between SU(N) lattice gauge theories in four dimensions and chiral SU(N) X SU(N) spin systems in two dimensions [ 1,2] : both systems are asymptotically free and have the same approximate renormalization group. Then one should expect similar phase diagrams in both theories. Recent Monte Carlo simulations have revealed that the phase structure of gauge systems is much richer than originally expected. In SU(N) gauge theories with the Wilson action, first-order phase transitions forN= 4 [3] andN= 5 [3,4] h ave been reported as well as lines of first-order transitions in the SU(2) model when the action is written as a sum of the fundamental and the selfadjoint representations [5]. Similar numerical simulations on the corresponding two-dimensional spin systems have revealed some difference between these two kinds of systems [6,7]. In particular, no sign of a first-order transition was found in the SU(4) X SU(4) model with the Wilson action nor in the SU(2) X SU(2) model with the variant form of the action. These results have led us to study in detail the strong-coupling expansion for the spin-spin correlation length in the SU(2) X SU(2) model. Indeed Monte Carlo experiments and perturbative expansions are somewhat complementary to each other: In the numerical experiments, first-order transitions give imponent hysteresis effects in the thermal cycles, while higher-order transitions, with their longrange fluctuations, yield a weak signal in the microscopic crystals stored in the computer. On the contrary 0 031-9163/82/0000-0000/$02.75
0 1982 North-Holland
the coefficients of a perturbative expansion are in general blind to the first-order transitions, while they are rather efficient in locating critical points. Actually we found in the SU(2) X SU(2) spin model a critical line in regions where Monte Carlo simulations seem to give no transition 16,7]. We dealt mainly with two different actions
(1)
(2) where i labels the sites of a two-dimensional square (1) or rectangular (2) lattice, e is a unit vector parallel to one of the axes x and y of the lattice, U is a SU(2) group element and trF (trA) denotes the trace in the fundamental (selfadjoint) representation. Our starting point was the spin pair correlation function r(Le)
= i (tr,~Z~l
Ui+Le) ,
where L denotes the relative separation between the two sites i and i + Le, and the angular brackets indicate the thermodynamic expectation value in an infinite 303
Volume 109B, number 4
u = Ql>
system described by eqs. (1) or (2). The correlation function is dominated, as L -+ 00, by an exponential decay law: r(Z,e) - exp(-l/t;
L -+ 00,
e )P(L1’2),
(4)
- 2u - u3 t 6u4 - 6u2v - $u5
+ 12u3v - 12uv2 t 26u6 - 36u4v + 18u2v2 - 6v3 t $?u7 + 20u5v - 26u3v2 + 16u8 t 10u6u + 16u4v2 - 22u2v3 t 12v4 - 8u5w - 12dwju - 12v5 - 24v2w2,‘u + 12v3w] ,
+ [-24uv4
a/gD = -log(2u) _ &4, + fgu4,2 -
(5)
- u2 t 4u4 - 3u2v - fu2 +Tu6
_ +2,2 _ Fu2u3
_ $3
+ ‘su8
+ qu4
t Fu6”
_ (ju3,,&
3”2W2/U2 - 2u5 w - 4u3vw - 2uv2w - [ 12”5] ) (6)
where a is the lattice spacing, u, v and w are the first few renormalized coefficients of the character expansion of the Boltzmann factor associated to a link: 304
w = t4it1 ,
v = t3it1,
(7)
where
where P(x) denotes a polynomial in x and &, is the correlation length, measured in units of lattice spacing a. Near a critical point, where the full rotational invariance is recovered, & becomes independent of e. A direct strong-coupling expansion of eq. (3) is not very effective in calculating .$, because it is difficult to single out the exponential factor from the other terms. However, Fisher and Burford [8], many years ago, have given a diagrammatic rule for calculating directly the character expansion of the decay factor. One has simply to compute the contribution of all the diagrams which join the site i with a straight line orthogonal to e located at a distance L from the site i. It is possible to show that the resulting expansion decreases exponentially with L for large L, and such a property can be used as a consistency check of the diagrammatic expansion. Another useful check is that the set of diagrams built with the character of the fundamental representation must coincide with the corresponding expansion of the Ising model, which is exactly known to all orders [8,9]. We calculated [ for two different choices of the unit vector e namely in a direction parallel to one coordinate axis e = x and along the principal diagonal D = (x + y)/2. We found a/& = -log(u)
25 February 1982
PHYSICS LETTERS
t,@F >P*) =+ Jd(Jtr,(u)
exp[&tr#J)
+ PAtrA(U)]
(8)
.
The square brackets terms in eqs. (5) and (6), which are of order greater than /3!, are added in order to have the correct expansion up to @A/&)5 and 8;. In the rectangular lattice described by the action (2) the expansion in the two variables X = u(&) and Y = u(flV) is
-~y5+2x4Y2t~x2Y4t~x4Y3t~x2y5
-~yl+~x6y2+~x4y4+~x2y6.
(9)
The analogous expression for the correlation length along they direction is simply obtained by exchanging X * Yin eq. (9): aY/&Y
=~&&,?@+
y).
(10)
We calculated also the correlation length { associated to trA(UiUl
t 20~3,
-
t 3ov2z2
75v4z
_ 2ouz2 -
1026
t yu6
)
(11)
where z = t5 /tl According to the theory of the continuous chiral two-dimensional models (see e.g. ref. [lo]), the SU(2) X SU(2) symmetry is realized algebraically through a particle multiplet with a common, dynamically generated mass. All the series we have constructed could be used to calculate such a mass, provided that there is a region where their behaviour is dominated by the asymptotic freedom fixed point at g = 0. In such a region the mass measured in units of lattice spacing should scale as ma = (mlA)f(g2),
(12)
wheref(g2) is the universal function derived by the Callan-Symanzik equation for the SU(2) X SU(2) model 1lOJ
Volume
a(d/da)g
109B, number
= (8n)-1g3
4
PHYSICS
+ (64i$-lg5
f’(g2) = (4?r/g2)1/2 exp(-4n/g2)[1
+ . .. , + O(g2)1 .
25 February
LETTERS
1982
(13) (14)
A is the mass scale set up by the asymptotic freedom and it depends on the details of the model. Chosing as reference the mass scale A, of the Wilson model (PA = 0), the ratio A,/A can be calculated from the oneloop weak-coupling expansion [ II]. For the model (1) we found AO/A = exp]10nflA/(3flF
+
SPA)1 .
0.5
(15)
0.4 I
For the rectangular lattice, the ratio A,/A is a nontrivial function [ 1l] of C = ax/a,. For C - 1 it is easy to find &,/A=-\Tc+
O[(l - C)2] .
by
(17) where the factors fi and C-l are required by rotational invariance, while the factor of l/2 in front of { occurs [7] because in such a model there are no spinorial multiplets (& = 0) and the lowest mass state consists of a pair of particles of mass m belonging to the fundamental representation of SU(2). The last equality in eq. (17), equipped with the relation [ll]: ]1 + 0(g2)]
(
0.2.
“;,\‘.
I
I
(16)
which will also be derived shortly by a strong-coupling argument. The mass m and the various correlation lengths defined in eqs. (S), (6) (9) (10) and (11) are related
C= (/3/p,)1’2
0.3!
(18)
when combined with the 13, * ,r$ symmetry of eqs. (9) and (10) yields at once the mass scale ratio (16). In the Wilson direction of the multidimensional parameter space (/I’~ = 0, oY = 0, = flF:) we have at our disposal three different strong-coupling expansions to test the asymptotic freedom of the model; indeed the functions a/&, da/ED and 2(a/iK)aX/~Rx lc= 1, according to eqs. (I 6) and (17), should approach to the common limit ma. The three series, drawn in fig. 1, fit nicely to the scaling law (12). In particular the series a/& and the weak-coupling universal function f(g*) differ less than 2% in the range 1.75 < ,$ S 2.00 once we choose
Fig. 1. Mass gap fits to asymptotic broken and continuous curves are 2(a/aC)ax/~~lc=l,&fQ) and must have the same weak-coupling straight line of eq. (14).
m/A,
freedom. The dotted, the functions a/Ex respectively, which limit represented by the
= 24.6 5 0.2 ,
(194
i.e. m{Apv
= 1.98 + 0.02 ,
where A,, is the mass scale in the continuum with a Pauli-Villars regulator [ 1 l] : Ap,/Ao
=m
exp(rr/4).
(19bJ theory
(20)
Our results agree with the analogous calculations in the hamiltonian formalism which give [l l] m/b = 1.83 _+0.20. This point is to be contrasted with the situation in gauge theories where the roughening transition [ 121 is a natural barrer to extrapolate the strong-coupling series to the weak region and moreover there are consistency problems [ 131 in comparing the string tension calculated in the hamiltonian formalism with the one measured in Monte Carlo experiments. As CIAincreases from zero, the agreement with the scaling law (12) deteriorates rapidly. There is, however, a clear sign that the system lies again in a critical region because tX x fi$D > a (see fig. 2). 30s
Volume
109B, number
4
PHYSICS
25 February
LETTERS
of the function 2.000,
Fig. 2. Analysis of the sequence aNIaN+ for the function l/[ 1 - exp(-a/&)]. The two continuous curves are the maximal and minimal values of aN/aN+l with 2
In order to locate possible critical points in such a region, we found it useful to study the function [ 1 - exp(-Q/&)]-l
=F
Q&$,
(21)
which must diverge at a critical point as (SC - fi,)-V. The ratios aNIaN+ 1 of this series, which are polynomials in flA/@, are found to behave quite regularly for PA > 0 and converge to an almost common line for flA/flF = 0.55, where we approximately locate the endpoint of the critical line (see fig. 2). Such a behaviour suggests a line of second-order transitions with a critical exponent v - 1, which is just the value associated to the divergent correlation length of the conventional Ising model. Because of this value of v it is plausible that the critical line we have fund joins the second-order transition at fi - m reduces to the Ising model. A - ” where the system ’ It is worthwhile to note that such a critical curve is near the line of first-order transitions found in the corresponding SU(2) gauge model [5]. A similar analysis for the pure selfadjoint model (OF = 0) suggests the presence of a critical point for a finite value of PA. Indeed the strong-coupling expansion of { given in eq. (11) strongly deviates from the scaling behaviour (12) predicted by the asymptotic freedom. Actually the eq. (11) gives c = 00 for PA - 2.55 and a critical behaviour around this point is also supported by the sequences of ratios QN/QN+ 1
306
2.531,
[ 1 - exp(-a/[)] 2.295,
2.193,
1982
-l: 2.351 ,
(22)
which, however, is not sufficiently regular to give any estimate of V. It should be noted that, even in such a case, the location of the transition point is very near the first-order transition at PA = 2.50 f 0.03 found in the SO(3) gauge model (14,1]. In conclusion, we found two different critical behaviours for the SU(2) X SU(2) spin model. One is described by the asymptotically free continuum theory and it allows an accurate estimate of the dynamically generated mass; the other one is dominated by the center of the group and looks like the Curie transition of the Ising model. It must however differ from it in the low-temperature region, because the MerminWagner theorem 1151 forbids a spontaneous breaking of SU(2) x SU(2). A renormalization group analysis should reveal nontrivial, interesting properties of such a critical behaviour. References [l] A.A. Migdal, Zh. Eksp. Teor. Fiz. 69 (1975) 810. [2] A.M. Polyakov, Phys. Lett. 59B (1975) 79. [3] M. Creutz, Phys. Rev. Lett. 46 (1981) 1441. [4] H. Bohr and K.J.M. Moriarty, Phys. Lett. 104B (1981) 217. [5] G. Bhanot and M. Creutz, BNL preprint (May 1981). (61 S. Duane and M. Green, Phys. Lett. 103B (1981) 359. [7] J. Kogut, M. Snow and M. Stone, University of Illinois preprint (1981). [8] M.E. Fisher and R.J. Burford, Phys. Rev. 156 (1967) 583. 191 L. Onsager, Phys. Rev. 65 (1944) 117. [lo] A. McKane and M. Stone, Nucl. Phys. B163 (1980) 169. [11] J. Shigemitsu and J. Kogut, Nucl. Phys. B190 FS3 (1981) 365. [ 121 C. Itzykson, M.E. Peskin and J.B. Zuber, Phys. Lett. 95B (1980) 259; M. Liischer, G. Mtinster and P. Weisz, Nucl. Phys. B180, FS2 (1981) 1; A. Hasenfratz, E. Hasenfratz and P. Hasenfratz, Nucl. Phys. B180, FS2 (1981) 353. [ 13) P. Hasenfratz, Talk EPS Intern. Conf. on High-energy physics (Lisbon, 1981), CERN preprint TH.3157. [ 141 I.G. Halliday and A. Schwimmer, Phys. Lett. 1OlB (1981) 327; J. Greensite and B. Lautrup, Niels Bohr Institute preprint NBI-HE-814; Phys. Lett. 104B (1981) 41. 1151 N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133.
109B, number
4
PHYSICS
THE PHASE STRUCTURE
25 February
LETTERS
1982
OF THE SU(2) X SU(2) SPIN SYSTEM IN TWO DIMENSIONS
M. CASELLE, F. GLIOZZI and R. MEGNA Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Turin, Italy Received
Strong rectangular transitions
15 December
1981
coupling expansions up tog -I6 of the correlation length for a mixed SU(2) X SU(2), SO(3) X SO(3) system in a lattice are considered. In addition to the asymptotic freedom fixed point, an unexpected line of second-order is discovered. The common critical exponent of these transitions is v = 1, like in the king model.
There is a close similarity between SU(N) lattice gauge theories in four dimensions and chiral SU(N) X SU(N) spin systems in two dimensions [ 1,2] : both systems are asymptotically free and have the same approximate renormalization group. Then one should expect similar phase diagrams in both theories. Recent Monte Carlo simulations have revealed that the phase structure of gauge systems is much richer than originally expected. In SU(N) gauge theories with the Wilson action, first-order phase transitions forN= 4 [3] andN= 5 [3,4] h ave been reported as well as lines of first-order transitions in the SU(2) model when the action is written as a sum of the fundamental and the selfadjoint representations [5]. Similar numerical simulations on the corresponding two-dimensional spin systems have revealed some difference between these two kinds of systems [6,7]. In particular, no sign of a first-order transition was found in the SU(4) X SU(4) model with the Wilson action nor in the SU(2) X SU(2) model with the variant form of the action. These results have led us to study in detail the strong-coupling expansion for the spin-spin correlation length in the SU(2) X SU(2) model. Indeed Monte Carlo experiments and perturbative expansions are somewhat complementary to each other: In the numerical experiments, first-order transitions give imponent hysteresis effects in the thermal cycles, while higher-order transitions, with their longrange fluctuations, yield a weak signal in the microscopic crystals stored in the computer. On the contrary 0 031-9163/82/0000-0000/$02.75
0 1982 North-Holland
the coefficients of a perturbative expansion are in general blind to the first-order transitions, while they are rather efficient in locating critical points. Actually we found in the SU(2) X SU(2) spin model a critical line in regions where Monte Carlo simulations seem to give no transition 16,7]. We dealt mainly with two different actions
(1)
(2) where i labels the sites of a two-dimensional square (1) or rectangular (2) lattice, e is a unit vector parallel to one of the axes x and y of the lattice, U is a SU(2) group element and trF (trA) denotes the trace in the fundamental (selfadjoint) representation. Our starting point was the spin pair correlation function r(Le)
= i (tr,~Z~l
Ui+Le) ,
where L denotes the relative separation between the two sites i and i + Le, and the angular brackets indicate the thermodynamic expectation value in an infinite 303
Volume 109B, number 4
u = Ql>
system described by eqs. (1) or (2). The correlation function is dominated, as L -+ 00, by an exponential decay law: r(Z,e) - exp(-l/t;
L -+ 00,
e )P(L1’2),
(4)
- 2u - u3 t 6u4 - 6u2v - $u5
+ 12u3v - 12uv2 t 26u6 - 36u4v + 18u2v2 - 6v3 t $?u7 + 20u5v - 26u3v2 + 16u8 t 10u6u + 16u4v2 - 22u2v3 t 12v4 - 8u5w - 12dwju - 12v5 - 24v2w2,‘u + 12v3w] ,
+ [-24uv4
a/gD = -log(2u) _ &4, + fgu4,2 -
(5)
- u2 t 4u4 - 3u2v - fu2 +Tu6
_ +2,2 _ Fu2u3
_ $3
+ ‘su8
+ qu4
t Fu6”
_ (ju3,,&
3”2W2/U2 - 2u5 w - 4u3vw - 2uv2w - [ 12”5] ) (6)
where a is the lattice spacing, u, v and w are the first few renormalized coefficients of the character expansion of the Boltzmann factor associated to a link: 304
w = t4it1 ,
v = t3it1,
(7)
where
where P(x) denotes a polynomial in x and &, is the correlation length, measured in units of lattice spacing a. Near a critical point, where the full rotational invariance is recovered, & becomes independent of e. A direct strong-coupling expansion of eq. (3) is not very effective in calculating .$, because it is difficult to single out the exponential factor from the other terms. However, Fisher and Burford [8], many years ago, have given a diagrammatic rule for calculating directly the character expansion of the decay factor. One has simply to compute the contribution of all the diagrams which join the site i with a straight line orthogonal to e located at a distance L from the site i. It is possible to show that the resulting expansion decreases exponentially with L for large L, and such a property can be used as a consistency check of the diagrammatic expansion. Another useful check is that the set of diagrams built with the character of the fundamental representation must coincide with the corresponding expansion of the Ising model, which is exactly known to all orders [8,9]. We calculated [ for two different choices of the unit vector e namely in a direction parallel to one coordinate axis e = x and along the principal diagonal D = (x + y)/2. We found a/& = -log(u)
25 February 1982
PHYSICS LETTERS
t,@F >P*) =+ Jd(Jtr,(u)
exp[&tr#J)
+ PAtrA(U)]
(8)
.
The square brackets terms in eqs. (5) and (6), which are of order greater than /3!, are added in order to have the correct expansion up to @A/&)5 and 8;. In the rectangular lattice described by the action (2) the expansion in the two variables X = u(&) and Y = u(flV) is
-~y5+2x4Y2t~x2Y4t~x4Y3t~x2y5
-~yl+~x6y2+~x4y4+~x2y6.
(9)
The analogous expression for the correlation length along they direction is simply obtained by exchanging X * Yin eq. (9): aY/&Y
=~&&,?@+
y).
(10)
We calculated also the correlation length { associated to trA(UiUl
t 20~3,
-
t 3ov2z2
75v4z
_ 2ouz2 -
1026
t yu6
)
(11)
where z = t5 /tl According to the theory of the continuous chiral two-dimensional models (see e.g. ref. [lo]), the SU(2) X SU(2) symmetry is realized algebraically through a particle multiplet with a common, dynamically generated mass. All the series we have constructed could be used to calculate such a mass, provided that there is a region where their behaviour is dominated by the asymptotic freedom fixed point at g = 0. In such a region the mass measured in units of lattice spacing should scale as ma = (mlA)f(g2),
(12)
wheref(g2) is the universal function derived by the Callan-Symanzik equation for the SU(2) X SU(2) model 1lOJ
Volume
a(d/da)g
109B, number
= (8n)-1g3
4
PHYSICS
+ (64i$-lg5
f’(g2) = (4?r/g2)1/2 exp(-4n/g2)[1
+ . .. , + O(g2)1 .
25 February
LETTERS
1982
(13) (14)
A is the mass scale set up by the asymptotic freedom and it depends on the details of the model. Chosing as reference the mass scale A, of the Wilson model (PA = 0), the ratio A,/A can be calculated from the oneloop weak-coupling expansion [ II]. For the model (1) we found AO/A = exp]10nflA/(3flF
+
SPA)1 .
0.5
(15)
0.4 I
For the rectangular lattice, the ratio A,/A is a nontrivial function [ 1l] of C = ax/a,. For C - 1 it is easy to find &,/A=-\Tc+
O[(l - C)2] .
by
(17) where the factors fi and C-l are required by rotational invariance, while the factor of l/2 in front of { occurs [7] because in such a model there are no spinorial multiplets (& = 0) and the lowest mass state consists of a pair of particles of mass m belonging to the fundamental representation of SU(2). The last equality in eq. (17), equipped with the relation [ll]: ]1 + 0(g2)]
(
0.2.
“;,\‘.
I
I
(16)
which will also be derived shortly by a strong-coupling argument. The mass m and the various correlation lengths defined in eqs. (S), (6) (9) (10) and (11) are related
C= (/3/p,)1’2
0.3!
(18)
when combined with the 13, * ,r$ symmetry of eqs. (9) and (10) yields at once the mass scale ratio (16). In the Wilson direction of the multidimensional parameter space (/I’~ = 0, oY = 0, = flF:) we have at our disposal three different strong-coupling expansions to test the asymptotic freedom of the model; indeed the functions a/&, da/ED and 2(a/iK)aX/~Rx lc= 1, according to eqs. (I 6) and (17), should approach to the common limit ma. The three series, drawn in fig. 1, fit nicely to the scaling law (12). In particular the series a/& and the weak-coupling universal function f(g*) differ less than 2% in the range 1.75 < ,$ S 2.00 once we choose
Fig. 1. Mass gap fits to asymptotic broken and continuous curves are 2(a/aC)ax/~~lc=l,&fQ) and must have the same weak-coupling straight line of eq. (14).
m/A,
freedom. The dotted, the functions a/Ex respectively, which limit represented by the
= 24.6 5 0.2 ,
(194
i.e. m{Apv
= 1.98 + 0.02 ,
where A,, is the mass scale in the continuum with a Pauli-Villars regulator [ 1 l] : Ap,/Ao
=m
exp(rr/4).
(19bJ theory
(20)
Our results agree with the analogous calculations in the hamiltonian formalism which give [l l] m/b = 1.83 _+0.20. This point is to be contrasted with the situation in gauge theories where the roughening transition [ 121 is a natural barrer to extrapolate the strong-coupling series to the weak region and moreover there are consistency problems [ 131 in comparing the string tension calculated in the hamiltonian formalism with the one measured in Monte Carlo experiments. As CIAincreases from zero, the agreement with the scaling law (12) deteriorates rapidly. There is, however, a clear sign that the system lies again in a critical region because tX x fi$D > a (see fig. 2). 30s
Volume
109B, number
4
PHYSICS
25 February
LETTERS
of the function 2.000,
Fig. 2. Analysis of the sequence aNIaN+ for the function l/[ 1 - exp(-a/&)]. The two continuous curves are the maximal and minimal values of aN/aN+l with 2
In order to locate possible critical points in such a region, we found it useful to study the function [ 1 - exp(-Q/&)]-l
=F
Q&$,
(21)
which must diverge at a critical point as (SC - fi,)-V. The ratios aNIaN+ 1 of this series, which are polynomials in flA/@, are found to behave quite regularly for PA > 0 and converge to an almost common line for flA/flF = 0.55, where we approximately locate the endpoint of the critical line (see fig. 2). Such a behaviour suggests a line of second-order transitions with a critical exponent v - 1, which is just the value associated to the divergent correlation length of the conventional Ising model. Because of this value of v it is plausible that the critical line we have fund joins the second-order transition at fi - m reduces to the Ising model. A - ” where the system ’ It is worthwhile to note that such a critical curve is near the line of first-order transitions found in the corresponding SU(2) gauge model [5]. A similar analysis for the pure selfadjoint model (OF = 0) suggests the presence of a critical point for a finite value of PA. Indeed the strong-coupling expansion of { given in eq. (11) strongly deviates from the scaling behaviour (12) predicted by the asymptotic freedom. Actually the eq. (11) gives c = 00 for PA - 2.55 and a critical behaviour around this point is also supported by the sequences of ratios QN/QN+ 1
306
2.531,
[ 1 - exp(-a/[)] 2.295,
2.193,
1982
-l: 2.351 ,
(22)
which, however, is not sufficiently regular to give any estimate of V. It should be noted that, even in such a case, the location of the transition point is very near the first-order transition at PA = 2.50 f 0.03 found in the SO(3) gauge model (14,1]. In conclusion, we found two different critical behaviours for the SU(2) X SU(2) spin model. One is described by the asymptotically free continuum theory and it allows an accurate estimate of the dynamically generated mass; the other one is dominated by the center of the group and looks like the Curie transition of the Ising model. It must however differ from it in the low-temperature region, because the MerminWagner theorem 1151 forbids a spontaneous breaking of SU(2) x SU(2). A renormalization group analysis should reveal nontrivial, interesting properties of such a critical behaviour. References [l] A.A. Migdal, Zh. Eksp. Teor. Fiz. 69 (1975) 810. [2] A.M. Polyakov, Phys. Lett. 59B (1975) 79. [3] M. Creutz, Phys. Rev. Lett. 46 (1981) 1441. [4] H. Bohr and K.J.M. Moriarty, Phys. Lett. 104B (1981) 217. [5] G. Bhanot and M. Creutz, BNL preprint (May 1981). (61 S. Duane and M. Green, Phys. Lett. 103B (1981) 359. [7] J. Kogut, M. Snow and M. Stone, University of Illinois preprint (1981). [8] M.E. Fisher and R.J. Burford, Phys. Rev. 156 (1967) 583. 191 L. Onsager, Phys. Rev. 65 (1944) 117. [lo] A. McKane and M. Stone, Nucl. Phys. B163 (1980) 169. [11] J. Shigemitsu and J. Kogut, Nucl. Phys. B190 FS3 (1981) 365. [ 121 C. Itzykson, M.E. Peskin and J.B. Zuber, Phys. Lett. 95B (1980) 259; M. Liischer, G. Mtinster and P. Weisz, Nucl. Phys. B180, FS2 (1981) 1; A. Hasenfratz, E. Hasenfratz and P. Hasenfratz, Nucl. Phys. B180, FS2 (1981) 353. [ 13) P. Hasenfratz, Talk EPS Intern. Conf. on High-energy physics (Lisbon, 1981), CERN preprint TH.3157. [ 141 I.G. Halliday and A. Schwimmer, Phys. Lett. 1OlB (1981) 327; J. Greensite and B. Lautrup, Niels Bohr Institute preprint NBI-HE-814; Phys. Lett. 104B (1981) 41. 1151 N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133.