20June1996 a . __ f!ld ;B EISWIER PHYSICS LElTERS B Physics Letters B 378 (1996) 207-212 New universality class in three dimensions?: the antiferromagnetic RP2 model H.G. Ballesteros ‘, L.A. Ferntidez 2, V. Martin-Mayor 3, A. Mufioz Sudupe4 De,nariamento de F&en Tedrica I, Facultad de CC, Fisicas, Universidad Complutense de Madrid, 28040 Madrid, Spain Received 2 February 1996; revised manuscript received 4 March 1996 Editor: L. Alvarez-Gaume Abstract We present the results of a Monte Carlo simulation of the RP* model in three dimensions with negative coupling. We observe a second order phase transition between the disordered phase and an antiferromagnetic, unfrustrated, ordered one. We measure, with a finite size scaling analysis, the thermal exponent, obtaining v = 0.784(g). We have found two magnetic-type relevant operators whose related YJ exponents are 0.038( 2) and 1.338( 8) respectively. Keywords: RP2; Non-linear sigma model; Antiferromagnetism; Critical exponents; Phase transitions; Universality; Finite size scaling The theory of critical phenomena offers a common framework to study problems in Condensed Matter Physics (CMP) and in High Energy Physics (HEP) . In both areas, the concepts of Spontaneous Symmetry Breaking (SSB) and of Universality allow to relate problems in principle very different. The usual Heisenberg model, associated with the standard ferromagnetic Non-Linear CT Model (NLrM) , has a SSB pattern of type SO( 3) /SO( 2). With the introduction of nontrivial Antiferromagnetic (AF) interactions the SSB pattern normally changes completely and, usually, frustration is generated. In particular, a SSB pattern shared by several AF models is SO( 3) x SO( 2) /SO( 2). For instance, some frus- trated quantum AF Heisenberg models [ 11, or the helimagnets and canted spin systems [ 2,3] are exam- ’ E-mail: hector@lattice.fis.ucm.es. 2 E-mail: Iaf@Iattice.fis.ucm.es. s E-mail: victor@lattice.fis.ucm.es. 4 E-mail: sudupe@lattice.fis.ucm.es. ples of this behavior. Frustrated quantum spin models are specially interesting because of their possible re- lation with High Temperature Superconductivity [ 41. As a general consequence of the Weinberg the- orem the low energy physics of a system is com- pletely determined by its SSB pattern, the effective Lagrangian for the system being the corresponding NLaM. In this framework a study has been carried out for SO(3) xSO(2)/SO(2) in perturbation the- ory [ 31, where the main conclusion reached is that the only possible nontrivial critical point in three dimen- sions is that of 0( 4). In spite of that, we have found in a nonperturbative lattice formulation of a model with similar symmetry properties, a critical point with ex- ponents that seem to be different from those of O(4) computed in Ref. [ 51. From the HEP point of view, it is of great inter- est to understand whether AF interactions can gener- ate new universality classes. One could even hope that nontrivial antiferromagnetism would be the ingredient 0370-2693/96/$12.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved PII SO370-2693(96>00358-9 208 H.G. Ballesteros et al. /Physics Letters B 378 (1996) 207-212 needed in order to nonperturbatively formulate inter- and its Fourier transform T in a L x L x L lattice with acting theories in four dimensions [ 6,7]. periodic boundary conditions. In a previous work [ 81 we found that, on a three di- mensional AF 0( 3) model, the only new phase tran- sitions generated were first order. We will consider in this letter the RI” = g/22 (real projective space) spin model in three dimensions. We place the spins on a cubic lattice with a nearest neighbors interaction: The intensive staggered (nonstaggered) magnetiza- tion can be defined in terms of the tensorial field as the sum of the spins on even sites minus (plus) those on odd sites, or equivalently M, = $,,,,,,, l- (M = +o,o.o,) 3 (3) where V is the lattice volume. We have observed a phase transition at p N -2.41 for which M, is an order parameter (zero value in the disordered phase and a clear nonzero value in the L -+ co limit in the ordered one). The magnetization M is also an order parameter. As these operators correspond to different irreducible representations of the translations group, we will study the scaling properties of each observable independently. (1) (ii) where {vi} are normalized real three-components vec- tors. The local Z2 symmetry vi + -vi is preserved even after the SSB (Elitzur’s theorem), and so, the sense of a spin is irrelevant, it is only its direction that matters. In the AF case, ( 1) is related [9] with a lat- tice discretization of the action s tr[ P (R-‘a,Ru)*], where R E SO( 3), and P is the diagonal matrix {g, g, -g}, with g being the coupling. This is just a particular case of the NLcrM considered in Ref. [ 31. For /? positive this model presents a weak first order phase transition which has been used to describe liquid crystals [ IO]. The ordered phase corresponds to states where all spins are aligned. For p negative there is also an AF ordered phase with a more complex structure [ 111. There is a second order phase transition between the disordered phase and an ordered AF one. Let us call a site even (odd) when the sum of its coordinates x + y + z. is even (odd). A state where, for instance, all spins on even sites are aligned in a given direction, and those on odd sites lie randomly in the orthogonal plane, has zero energy. So, at T = 0 the ground state is highly degenerate with a global O(2) symmetry. However, when fluctuations are taken into account, the scenario is much more complex. We will show, with Monte Carlo simulations, that the O(2) symmetry is broken in the critical region. In a forthcoming paper [ 91 we shall discuss the very low temperature regime as well as further details of the vacuum structure on the critical region. In order to discuss the observables measured, let us construct the (traceless) tensorial field T with com- ponents TyP = $$ - $jcyp) (2) To measure in a Monte Carlo simulation on a finite lattice we have constructed scalars under the O(3) group. For the magnetization and the susceptibility we compute respectively M=(m), X=V(trM*), (4) and analogously for the staggered observables. We have also measured the second momentum cor- relation length, which is expected to have the same scaling behavior at the critical point as the exponential (physical) one, but it is much easier to measure [ 121 (5) where F is the mean value of the trace of T squared at minimal momentum (2rr/ L in direction x, y or z ) . To define cs we use xs and compute F, from T at momentum (27r/ L + 7r, P, P) and permutations. The action ( 1) is suitable for cluster update meth- ods by using Wolff’s embedding algorithm [ 131. We have checked the performance of both the Swendsen- Wang [ 141 and the single cluster methods [ 131. LJn- fortunately, due to the AF character of the interaction, the critical slowing down is not reduced in any case, as there is always a large cluster that contains most of the lattice sites. In fact, we have measured a dynamic exponent z M 2 for both methods. We have also developed a Metropolis algorithm. Near the transition the spin fluctuations are large, and a H.G. Ballesteros et al/Physics Letters B 378 (1996) 207-212 209 Table 1 Number of Monte Carlo sweeps performed for different lattice sizes. Measures have been taken every 10 sweeps. The integrated autocorrelation times (in sweeps) for both susceptibilities are also displayed. They have been computed using a selfconsistent 67 window. The statistical errors am below the 5% level. We have discarded in each case about 2007,, iterations for thermalization. L MC sweeps ( x 106) TXS TX 6 6.71 7.4 5.8 8 17.07 11.4 7.4 12 6.51 24.8 12.8 16 22.14 44.1 21.4 24 8.77 107 48 32 10.13 179 87 48 3.93 430 205 spin proposal uniformly distributed over the sphere is accepted with nearly 30% probability. So we have used a 3 hit algorithm, reaching a mean 70% acceptance. Regarding the performance of the three methods mentioned for a given lattice size, the differences are very small in terms of the CPU time. We have selected the Metropolis method which is slightly faster. In Table 1 we display the number of Monte Carlo sweeps performed for the different lattice sizes as well as the integrated autocorrelation times for the observ- ables x, xs. The total CPU time has been the equiva- lent of 12 months of DEC Alpha AXP3000 distributed over several workstations. Every 10 Monte Carlo sweeps we store individual measures of the energy and of the Fourier transform of the tensorial magnetization at suitable momentum values. We have used the spectral density method [ 151 to extrapolate in a neighborhood of the critical point. The data presented here correspond to simulations at two p values (-2.41 and -2.4). We compute the quantities referred above as well as their P-derivatives through the connected correlations with the energy. We have firstly analyzed several quantities that present a peak near the transition point. As we have found that the specific heat does not diverge, we have to limit ourselves to study quantities related with magnetization operators. The advantage of measuring a peak height is that its position also defines an ap- parent critical point allowing for a very simple and accurate measure. Unfortunately, quantities like the P-derivatives of the magnetizations or the connected susceptibilities ( xcon = x - K!4*) present their peaks fig. I. Deviations from a power law fit of the P-derivative of the staggered (upper side) and nonstaggered (lower side) correlation lengths at the critical point, using data from lattice sizes with L 2 16. The dotted lines correspond to a fit using all data sizes. far away from the critical point suffering from large corrections to scaling. For example, xcon in the E = 16 lattice peaks at p = -2.29 where the correlation length 5 is one half of its value at the critical point. This is due to the weakness of the tensorial ordering: the staggered magnetization, for instance, does not reach one-half of its maximum until p < -3.5. Another possibility is to obtain the infinite vol- ume critical point by other means and, then, to measure the different quantities at this point. By studying the matching of the scaling function ((trMz)*)/((trM?)*), as well as that of ts ( L, /?) / L and the corresponding nonstaggered quantities, we conclude that ,& E [ -2.415, -2.4051 . (6) To improve the above determination it is necessary a careful consideration of the corrections to scaling. This subject will be discussed elsewhere [ 91. In the case of quantities that change rapidly at the critical point as the magnetizations do, the errors in the determination of the critical point affect very much the results, and this method is not accurate. Nevertheless, we have found important quantities, like the P-derivatives of the correlation lengths, which are very stable. Both d[,/dp and d(/dp should scale as L’+‘/“; fitting the data from all lattice sizes we obtain an acceptable fit: v = 0.793(2) with X2/dof = 2.0/5 and v = 0.787(2) with x*/dof = 5.5/5 respectively. However, if we discard the L = 6 data the fitted pa- rameters change significantly and change again after discarding the L = 8 ones. The fits for L > 12 and L 2 16 agree within errors. We, thus, choose the L 2 12 data for computing ZJ but take the statistical error from the fit with L > 16 (see Fig. 1) : 210 H.G. Balksterm et al. /Physics Letters B 378 (1996) 207-212 Table 2 Critical exponents obtained from a Finite Size Scaling analysis using data from lattices of sizes L aad 2L. In the second row we show the operator used for each column. L 6 8 12 16 24 v &I@ 0.78616) 0.785(4) 0.789(S) 0.787(9) 0.77(2) @I@ 0.790(6) 0.781(4) 0.782(9) 0.781(8) 0.77(2) r)s rl xs MS X M 0.0431(10) 0.0474(9) l&2(2) 1.44712) 0.0375(7) 0.0409(8) 1.413(2) 1.416(2) 0.0357( 17) 0.0382( 18) 1.391(3) 1.393(3) 0.0371(19) 0.0390( 19) 1.379(4) 1.381(4) 0.038(5) 0.038(5) 1.362(S) 1.365(9) 5s : v = 0.788(7) 5 : v = 0.779(6) . (7) To estimate the errors, not considered in (7), as- sociated with the uncertainty in the determination of the critical point we repeat the fits with p at the lim- its of the interval (6). We observe that v changes by an amount of a 1%. To avoid the problems reported above, we have also used a method directly based on the Finite Size Scaling ansatz, that allows to write the mean value of any operator 0 as (O(L,P)) = LXfo(S(L,P)/L) +. . . * (8) where t( L, j3) is the correlation length measured at coupling p in a size L lattice, fo is a smooth operator- dependent function and x depends also on 0. The dots stand for corrections to scaling. Measuring (0) at the same coupling in lattices 2L and L and using (8) we can write for their quotient (9) Considering the dependence of Qo on p = 5( 2L, p) / ,$( L, p), we just have to measure at the point where p = 2 to obtain Qo = 2X up to corrections to scaling. x can be written in terms of the critical exponents: x = Y/V for x, x = -P/V for M, etc.. To compute u we can use 0 = d[/dp for which x = 1 + I/Y. As the quantity p is an observable and not an external param- eter, like p, this procedure does not require a previ- ous determination of PC. Another advantage is that the result depends only on measures in just two lattices which allows a better error estimation. In columns 2 and 3 of Table 2 we report the results for the ther- mal exponent Y obtained from data in lattices L (first column) and 2L, using as operators the &derivatives of the correlation lengths cs and 5 respectively. Even in the smaller lattices we do not observe any correc- tions to scaling. The data from columns 2 and 3 are very correlated statistically and so, by taking the mean value, the errors are only slightly reduced. We select as our best estimation the mean of the results for the lattices 16-32: v = 0.784( 8). (10) To be compared with the values Y(O( 3)) = 0.704(6) [ 161 and v(O(4)) = 0.748(9) [5]. In the case of the magnetic exponents, due to the large slope of Q as a function of p (see Fig. 2), a correct determination of the errors requires to take into account the statistical correlations of the whole data. We obtain for ys and &, in the most favorable cases, errors as small as a 0.1%. From these we compute, using the scaling relations ys/v = 2 - vs and 2/3,/v = D - 2 + qs, the values of 77, with acceptable accuracy (see columns 4 and 5 of Table 2). In this case, the corrections to scaling are only significative for the L = 6 lattice. We quote as our preferred value rls = 0.038(2). (11) For the nonstaggered sector, we observe that the usual susceptibility diverges much more slowly than the staggered one (ys - y - 1). The results for v. using the corresponding scaling relations, are reported in the last two columns of Table 2. In this case the corrections to scaling are nonnegligible for all lattice sizes. As the data fit very well to a linear function of 1 /L we take as the L -+ co value H.G. Ballesteros et al./Physics Letters B 378 (1996) 207-212 211 1.96 1.98 2.00 2.02 2.04 sSw$Ys&P) . Fig. 2. Quotients of several observables as a function of the quotient of the staggered correlation lengths. The growing function of the lattice size. 6; is the P-derivative of the staggered correlation length. sizes of the symbols are a 7) = 1.338(8), (121 where the error is half statistical and half due to the possible deviations from linearity. Notice that the large value of 17 means that, at the critical point, the spatial correlation function (G(r) lp+ N Ir]-‘-~), in the nonstaggered sector, decreases much faster than in the staggered case. Our results show that the O(3) symmetry is fully broken in the ordered phase near the critical point. If we discard order tp terms (t being the reduced tem- perature), M is zero, so the magnetization tensors of the even and odd sublattices are opposite, and the eigenvectors corresponding to the maximum eigenval- ues are orthogonal. Studying the finite size scaling of the operator [M,, M] = $ [ mven, Modd] we observe a behavior of type L-(p+ps)I”. We thus conclude that considering the order tp terms the orthogonality will only hold approximately what implies that the O(2) symmetry of the ground state does not remain. We have studied a spin model in three dimensions with the symmetries of the 0( 3) group but with very interesting new properties. It presents an ordered vac- uum where the O(3) symmetry is fully broken. The model has two odd (magnetic type) relevant opera- tors with different associated q exponents. Finally, the values obtained for the exponents Y (Eq. ( 10) ) and 17 for the staggered channel (Eq. ( 11) ) are very differ- ent from those of the 0( 3) model, and they are 3 stan- dard deviations apart from those computed in Ref. [ 51 for the O(4) model. From the analysis using differ- ent lattice sizes we are confident that the systematic errors due to finite size effects are small and it seems difficult that they could make our results compatible with those obtained in Ref. [ 51. We think that in addition to the interest of the model by itself, the results suggest further studies of related models, like the addition of vector interactions, or four dimensional systems. We thank Alan Sokal for many enlightening discus- sions at the beginning of this work, specially regarding the finite size scaling techniques. We are also indebted to Jose Luis Alonso, Juan Jesus Ruiz-Lorenzo and Alfonso Tarancon. This work has been partially sup- ported by CICyT AEN93-0604, AEN95-1284-E and 212 H.G. Ballesteros et al./Physics Letters B 378 (1996) 207-212 AEN93-0776. H.G. Ballesteros and V. Martin-Mayor are MEC fellows. References [ 11 T. Dombre and N. Read, Phys. Rev. B 39 (1989) 6797. 12 1 H.T. Diep, Phys. Rev. B 39 (1989) 397. 13 1 P. Azaria, B. Delamotte and T. Jolicoeur, Phys. Rev. Lett. 64 (1990) 3175; P. Azaria, B. Delamotte, E Delduc and T. Jolicoeur, Nucl. Phys. B 408 ( 1993) 485 . 14 1 M. Inui, S. Doniach and M. Gabay, Phys. Rev. B 38 (1988) 6631. 15 1 K. Kanaya and S. Kaya, Phys. Rev. D 51 (1995) 2404. [ 61 G. Gallavotti and V. Rivasseau, Phys. Lett. B 122 (1983) 268. t71 [81 191 1101 1111 [I21 [I31 [I41 [I51 [I61 C. Holm and W. Janke, Phys. I_&. A 173 (1993) 8. J.L. Alonso et al., hep-lat/9507003 preprint, to appear in Phys. Lett. B. J.L. Alonso, A. Tarancdn, H.G. Ballesteros, L.A. Femdndez, V. Martin-Mayor and A. Mutioz Sudupe. Phys. Rev. B 53 (1996) 2537. H.G. Ballesteros, L.A. Femandez, V. Martin-Mayor and A. Mufioz &dupe, work in preparation. PA. Lcbwohl and Cl. Lasher, Phys. Rev. A 6 (1972) 426. G. Korhing and R.E. Shrock, Nucl. Phys. B 295 (1988) 36; S. Romano, Int. J. Mod. Phys. B 8 (1994) 3389. S. Caracciolo, R.G. Edwards, A. Pelissetto and A.D. Sokal, Nucl. Phys. B 403 (1993) 475. Il. Wolff, Phys. Rev. L&t. 62 (1989) 3834. R.H. Swendsen and J.S. Wang, Phys. Rev. Lett. 58 (1987) 86. A.M. Ferrenberg and R.H. Swendsen, Phys. Rev. L&t. 61 (1988) 2635.