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how is the critical behaviour affected by the introduction of disorder ( usualy dilution or bond - randomness ) into a model ? the harris criterion @xcite states that @xmath5 , the scaling index of the operator corresponding to randomness at the pure system fixed point ( also called the crossover exponent ) is equal to @xmath6 ( @xmath7 and @xmath8 are the specific - heat and correlation length exponents of the pure model ) . thus the critical behaviour of the pure system is unaltered by disorder if @xmath9 , and altered when @xmath10 . if @xmath11 the situation is marginal . renormalization - group methods ( namely expansion in the parameter @xmath12 , where @xmath13 is the dimensionality ) , applied to n - component continuous spin models with weak quenched bond disorder@xcite , confirmed the harris criterion @xmath14 . moreover , it was found that for @xmath15 a new stable ` random ' fixed point with new exponents appears . for the case @xmath16 ( ising model ) the stability of an @xmath17 random fixed point was shown by jayaprakash and katz@xcite . for the @xmath18 ising model @xmath11 , and the operator corresponding to randomness is expected to be marginally relevant .    two partially conflicting theories by dotsenko and dotsenko@xcite ( dd ) and by shalaev@xcite , shankar@xcite , and ludwig@xcite ( ssl ) were suggested for the @xmath18 ising model with small randomness . while both theories agreed on the divergence of the specific heat , e.g. @xmath19 conflicting predictions were made for other quantities like the magnetization @xmath20 and susceptibility @xmath21 . extensive monte - carlo ( mc ) simulations of dilute and random - bond ising models ( see a review in ref . and references therein ) helped to confirm the prediction on the specific heat . the other quantities were seen to behave ( as predicted by ssl ) , in essence , as in the pure model . the disordered @xmath22 ising model has been studied by means of mc simulations as well@xcite . less attention has been paid in recent times to other disordered systems such as random q - state potts models . kinzel and domany @xcite used a real - space migdal - kadanoff renormalization transformation for the random - bond potts model on the square lattice in two dimensions and found ( when @xmath23 ) a stable random fixed point with a reduced ( with respect to the pure model ) but positive @xmath24 . using methods of the same spirit , andelman and berker @xcite also found a random fixed point . for @xmath25 their calculation suggests @xmath26 and @xmath27 . other workers@xcite too have suggested a negative @xmath28 for @xmath29 potts models in two dimensions . mc work on random q - state potts models has not been carried out , except for recent work on the bond - random two dimensional 8-state potts model@xcite , where randomness was shown to change the first order phase transition of the pure model into a second order one@xcite . novotny and landau ( nl)@xcite used mc simulations to study the two dimensional site - dilute baxter - wu@xcite model which is of the same universality class as the 4-state potts model . nl measured the exponents @xmath30 , @xmath31 , and @xmath32 which did not seem to depend on the amount of dilution . they could not determine whether the new stable fixed point is a @xmath18 random ising fixed point ( which is very similar to the pure ising fixed point ) or a new impure baxter - wu fixed point .    in this paper we address the effect of randomness on a critical line which connects the critical points of the @xmath18 ising and 4-state potts models . such is the critical line of the two dimensional ashkin - teller model@xcite on a square lattice . a convenient representation of the pure ashkin - teller ( at ) model is in terms of two ising spin variables , @xmath33 and @xmath34 , placed on every site of a lattice . denoting by @xmath35 a pair of nearest neighbor sites , the hamiltonian is given by @xmath36    \label{eq : at1}\ ] ] here @xmath37 is the strength of the interactions between neighboring @xmath38 and @xmath39 spins , and @xmath40 is a four - spin coupling ( throughout this article we absorb a factor of @xmath41 into the coupling constants ) . the phase diagram of the ferromagnetic at model is known in two dimensions from duality transformations and renormalization group studies@xcite . the three dimensional model has been studied as well@xcite . the phase diagram which we reviewed previously@xcite includes a line of critical points which connects the ising critical point at one end to the 4-state potts critical point at the other end . along this line the exponents vary continuously , and have been determined analytically @xcite , interpolating between their ising and four - state potts values . for instance , the crossover exponent connected with randomness @xmath14 changes smoothly from 0 at the decoupled ising point , to 1 at the 4-state potts point . therefore , according to the harris criterion , we expect randomness to be a relevant operator of varying strength in this regime of the at model , and expect the critical behaviour of the disordered model to differ from that of the pure model . the critical behaviour of the disordered at model has not been studied before , apart from a conjecture by alcaraz and tsallis @xcite as to the location of the critical manifold of the bond dilute at model . none the less the effect of disorder on the critical line of the baxter ( or symmetric 8-vertex ) model@xcite , which is isomorphic to the critical line of the at model@xcite , was considered . dd@xcite have extended their study of the disordered ising model to the baxter model , and found , for small disorder , the specific heat divergence to change from its pure form to the form of eq . ( [ eq : c(t)ising ran ] ) . matthews - morgan , landau and swendsen ( mls)@xcite have studied the site - dilute baxter model by means of a mcrg ( mc renormalization - group ) method . mls found @xmath42 and @xmath43 slightly below the ising value . both studies indicate that continuous variation of critical exponents of the pure model is substituted by flow to a random ising fixed point .    in a previous article@xcite we described an efficient mc cluster algorithm@xcite for the at model . in this work we have used this algorithm to perform extensive mc simulations@xcite of the random - bond at model . the importance of using a cluster algorithm for the acquirement of reasonably accurate data can not be overlooked . this is especially true in the highly random regime where local algorithms are extremely slow@xcite while cluster algorithms are most efficient@xcite.the partial elimination of critical slowing down allows us to go to large lattice sizes ( up to l=256 ) and obtain accurate results on the finite size scaling of the thermodynamic quantities at criticality . our results seem to favor an ising like critical behaviour . this work is organized as follows . in sec . [ sec : rbat ] we define the random - bond at model and a related anisotropic at model . as will be explained , the anisotropic at model is used as a non - disordered reference model for the random - bond model . we use the duality transformation of the at model to locate exactly critical surfaces of the two models , which are related to the line of critical points of the pure model . in sec . [ sec : details ] we describe the mc cluster algorithm , give definitions of the measured quantities and describe our choice of simulation points.we have performed simulations of the anisotropic and random - bond at models at several points on their critical surface . in sec . [ sec : results ] we give our results which consist of the finite size dependence of thermodynamic quantities of the two models at criticality . analysis of our results shows that the anisotropic model exhibits varying critical exponents , i.e. the specific heat diverges as @xmath44 where at different points of the critical surface of the anisotropic at model we measured different values of @xmath28 . on the other hand , the random - bond model seems to exhibit a single type of behaviour , i.e. the specific heat diverges as @xmath45 this is identical with a pure ising behaviour and not with a random ising one . in sec . [ sum ] we summarize our results . the model we wish to study is the _ random - bond _ at model ( rbat ) on a square lattice , which is defined by the hamiltonian @xmath46 \;. \label{eq : rbat}\ ] ] the coupling constants @xmath47 and @xmath48 are chosen according to @xmath49    it seems quite natural ( the _ need _ for this comparison will be made clear at the end of this section ) to compare the rbat with the _ anisotropic _ at model ( aat ) with the same hamiltonian ( [ eq : rbat ] ) but with the couplings distributed as follows : @xmath50    spatial anisotropy is usualy known to be an irrelevant operator , that is , the critical behaviour of a model is unaltered by introduction of anisotropy . thus one expects that continuous variation of critical exponents will continue to exist on some critical manifold of the aat , as is the case for the isotropic ( or pure ) model . a duality transformation can be used to locate critical manifolds of the aat and rbat in a way we now present . consider an at system on a square lattice with spatially varying coupling constants ( the spatial variation can be due to anisotropy , or it can be due to randomness as we consider later ) . the dual lattice @xmath51 of the original square lattice @xmath52 is the square lattice whose sites are at the centers of the plaquettes of the original lattice . in figure [ fig : duality exp ] sites of @xmath52 are denoted by dots , while sites of @xmath51 are denoted by crosses . consider a bond of strength @xmath53 between two sites in @xmath52 . under duality it is mapped onto a bond of strength @xmath54 $ ] , where @xmath55 is the duality transformation of the at model@xcite which we list in the appendix . the dual bond lies in the dual lattice @xmath51 , intersecting the original bond . this situation is depicted in figure [ fig : duality exp ] . the original bond of strength @xmath53 is depicted by the solid line while its dual is depicted by the dashed line . suppose the only spatial variation of the couplings is one of anisotropy . then an at system with couplings of strength @xmath56 in the horizontal direction and strength @xmath57 in the vertical direction , transforms under duality into an at system on the dual lattice @xmath51 with couplings of strength @xmath58 in the vertical direction and strength @xmath59 in the horizontal direction . the free energies of the original system and its dual have the same singular part , and therefore we can write @xmath60= f_{s}[(\widetilde{k^{2}},\widetilde{\lambda^{2}}),(\widetilde{k^{1 } } , \widetilde{\lambda^{1 } } ) ] \ ; , \label{eq : fs anidual}\ ] ] where the first pair of couplings corresponds to the horizontal bonds , and the second pair to the vertical ones . the duality transformation @xmath55 is self - inverse . thus , an anisotropic at system with couplings @xmath61 $ ] such that @xmath62 , transforms under duality onto itself . so , there exists a two dimensional self - dual manifold ( it would help the reader to view the duality transformation as a mathematical mapping in the four dimensional space of couplings @xmath61 $ ] ) . the line of critical points of the isotropic at model lies within the subspace @xmath63 , defined by @xmath64 . @xmath63 is located between a decoupled ising model where @xmath65 and a four - state potts model with @xmath66 . two sub - areas of @xmath63 which map onto each other under duality are separated by a self dual line ( fulfilling @xmath67 ) . the assumption that @xmath63 contains only two phases , with a single phase transition between them(where @xmath68 is non - analytic ) leads to the conclusion that this phase transition must occur on the self - dual line . in this way the line of critical points of the isotropic at was located exactly@xcite . let us now consider the corresponding subspace , @xmath69 , of the aat for which @xmath70 and @xmath71 . @xmath69 lies between an anisotropic decoupled ising subspace where @xmath72 and an anisotropic four - state potts subspace with @xmath73 . @xmath69 is invariant under duality since for @xmath74 we get ( see appendix ) @xmath75 . it follows also that the self dual plane @xmath70 , @xmath62 is included in the subspace we are considering . thus , again assuming that @xmath69 contains only two phases , with a single phase transition between them , we conjecture that the self dual plane is a plane of phase transitions ( this kind of argument was first used by fisch@xcite for the random - bond ising model , and later for potts models by kinzel and domany@xcite ) . since the space of couplings is four dimensional , while the self dual plane is only two dimensional , the self dual plane is only _ part _ of the three dimensional critical manifold in @xmath69 . notice that this is unlike the case of the potts model or any model with a single coupling constant where the self dual line is _ identical _ with the critical line . the argument that leads to locating the corresponding critical plane of the rbat is very similar to the one presented above , up to a subtle difference . consider a specific realization of a rbat on a square lattice with couplings @xmath76 distributed at random with probabilities @xmath77 and @xmath78 , respectively . replacing each bond by its dual we get ( on the dual lattice ) a system with couplings @xmath58 and @xmath59 , distributed at random with probabilities @xmath77 and @xmath78 . this defines a one to one mapping under duality from the ensemble of realizations of @xmath79 $ ] rbat systems , onto the ensemble of the dual systems @xmath80 $ ] . under duality each bond is mapped onto the bond on the dual lattice which intersects the original bond ( see fig . [ fig : duality exp ] ) . thus , the spatial distribution of the two types of bonds on the dual lattice is different from that of the original lattice . but since all spatial distributions have the same weight in their ensemble , there is a one to one correspondence of bond configurations in the two ensembles , and we can write : @xmath81= f_{s}[(\widetilde{k^{1}},\widetilde{\lambda^{1}}),(\widetilde{k^{2 } } , \widetilde{\lambda^{2}}),p ] \ ; ,   \label{eq : fs rbdual}\ ] ] where it is understood that in calculating @xmath68 we average over all spatial bond distributions . choosing @xmath82 we find that there is a self dual manifold at @xmath62 . our ability to locate a self dual manifold for the case @xmath82 makes this choice of @xmath77 a convenient case for studying the rbat . viewing the duality transformation as a mathematical mapping in the space of couplings @xmath83 $ ] , it is identical to the duality transformation in the space of couplings @xmath84 $ ] of the anisotropic at model . thus all conclusions which were based on duality in sec . [ sec : duality ani ] apply here equally . following the logic of the previous subsection we conjecture that the subspace of the rbat for which @xmath70 , @xmath85 and @xmath82 includes in it a critical plane defined by @xmath70 , @xmath62 .    in order to study the effect of randomness on the at model we need a reference non - disordered system for comparison . when studying a specific rbat model with couplings @xmath83 $ ] it is not clear with which specific non - random at model with couplings @xmath53 should one compare . none the less a comparison with the corresponding aat model with couplings @xmath84 $ ] seems to be quite natural . consider the anisotropic model with bonds @xmath56 in the horizontal direction and bonds @xmath57 in the vertical direction . next , consider as a perturbation , changing a small fraction @xmath77 of the bonds in the horizontal direction into bonds of strength @xmath57 and the same fraction of bonds in the vertical direction into bonds of strength @xmath56 . when @xmath77 increases all the way to @xmath82 the anisotropy disappears and the model is completely random . so , in a sense , the perturbation of randomness can take us continuously from the critical plane of the anisotropic model to the critical plane of the random model . in view of this simple picture we have performed simulations of the anisotropic and random - bond at models at several points on their self - dual planes described above . we indeed find the two planes to be critical . in the next section we give some details of the simulations . in ref . we described a cluster algorithm for the at model and tested its performance ( on the isotropic model ) . as explained there , the algorithm can be regarded from two different points of view . the conceptually simpler and faster ( though less general ) one is that of the wolff embedding idea@xcite . it is faster in the sense that its implementation on the computer runs faster since the freeze - delete procedure is less complicated . the main idea is to embed into the at model an ising model of space dependent couplings @xmath86 and simulate it using the sw@xcite or wolff@xcite procedure for the ising model . to be explicit , consider the hamiltonian ( [ eq : rbat ] ) , and take the @xmath39 variables as fixed , so we can write @xmath87 @xmath88 represented by the second sum is a constant , and remembering that we are considering @xmath89 for all ( i , j ) ,  @xmath90 is a ferromagnetic ising model in the @xmath38 variables with couplings @xmath91 . simulating this ising model with any procedure that will maintain detailed balance with respect to @xmath90 and will not change the value of @xmath88 , will also maintain detailed balance with respect to @xmath92 . so we can use for example the sw@xcite or wolff@xcite procedure for the ising model . this by itself will maintain detailed balance but will not be ergodic since the @xmath39 variables will not be updated . obviously to update the @xmath39 variables the same process should be repeated , holding the @xmath38 variables fixed and simulating an ising hamiltonian with the @xmath39 variables . to summarize , our procedure goes as follows : choose at random whether to embed into the at hamiltonian an ising hamiltonian in the @xmath38 ( or @xmath39 ) spins . pick a random site in the lattice , grow a cluster of @xmath38 ( or @xmath39 ) spins , using the wolff single cluster procedure with the ising hamiltonian ( ( [ eq : at - is ] ) or the opposite one in case of a @xmath39 embedding ) , and flip it . all our simulations were performed at the critical plane defined by @xmath93 : \ ; ( k^{2},\lambda^{2})=(\widetilde{k^{1}},\widetilde{\lambda^{1 } } ) \;\;,\;\ ; 0 \leq \lambda^{1 } \leq k^{1 } \;\ }   \ ; ,   \label{eq : def rbs}\ ] ] as explained in sec . [ sec : rbat ] ( for the anisotropic model just omit the @xmath82 ) . for convenience we also use the following notation for the coupling constants : @xmath94    the first series of measurements were performed at five points @xmath95 , which we label as @xmath96 . these were chosen so as to interpolate between @xmath97 , which is an anisotropic ( or random - bond ) decoupled ising critical point , and @xmath98 , which is an anisotropic ( or random - bond ) four - state potts critical point . the points @xmath99 interpolate in a similar manner to the way in which the critical line of the pure at connects the pure decoupled ising critical point with the pure four - state potts critical point . thus we chose the five points @xmath100 to lie equidistantly on the line @xmath101 the slope @xmath102 was chosen so that the line ( [ eq : line ] ) lies parallel ( in the @xmath103 plane ) to the critical line of the pure at model , @xmath104 . note that once @xmath105 are defined , @xmath106 are defined through ( [ eq : def rbs ] , [ eq : def xz ] ) . we clearly have some freedom in our choice of @xmath107 , the decoupled ising couplings ; setting @xmath108 in ( [ eq : def rbs ] ) one gets @xmath109 as well . the remaining parameter , @xmath110 , is chosen so that ( see ( [ eq : def rbs ] ) )    @xmath111    since the extent of deviation from pure behavior is obviously determined not only by @xmath77 but also by the difference between the two sets of couplings , the ratio of @xmath112 was chosen so that randomness will be pronounced@xcite .    while ( [ eq : ratio1 ] ) defines the first simulation point @xmath113 , the fifth simulation point @xmath114 is chosen to be a four - state potts point lying at the intersection of the line ( [ eq : line ] ) with the four - state potts line @xmath115 ( or @xmath116 ) . the other three simulation points @xmath117 lie equidistantly on the line ( [ eq : line ] ) between @xmath118 and @xmath119 : @xmath120 thus the first series of points @xmath99 at which we made measurements @xmath121 is defined through equations ( [ eq : def rbs],[eq : def xz],[eq : ratio1],[eq : x1place ] ) and the definition of @xmath119 . two additional measurement points were intended to monitor the effect of varying the amount of randomness in the model on the critical behaviour . in order to define these two points we define two series of five points each , @xmath122 and @xmath123 . similarly to the series @xmath121 , these two series are defined through equations ( [ eq : def rbs],[eq : def xz],[eq : x1place ] ) . but in contrast with the series @xmath121 , eq . ( [ eq : ratio1 ] ) is substituted by @xmath124 for the series @xmath122 , and by @xmath125 for the series @xmath123 . though we defined here ten additional points we actually performed measurements only at the points @xmath126 and @xmath127 . the points @xmath126 , @xmath127 , @xmath128 represent three rbat ( or aat ) models with coupling ratios @xmath129 and @xmath130 respectively . we calculated the energy@xcite per site : @xmath131=-\frac{1}{l^{2}}[{\langle   \sum_{<i , j>}\{k_{i , j}(\sigma_{i}\sigma_{j}+\tau_{i}\tau_{j})+ \lambda_{i , j}\sigma_{i}\tau_{i}\sigma_{j}\tau_{j}\ }   \rangle } ]      \label{eq : e}\ ] ] where @xmath132 is the linear lattice size ; the angular brackets denote the usual thermal mc average whereas the square brackets denote the quenched average over random - bond configurations . the specific heat per site follows from the energy fluctuations @xmath133 . \label{eq : cv}\ ] ] we define @xmath134 to compute the magnetization per site , @xmath135 , we used @xmath136 \ , ,      \label{eq : m}\ ] ] taking advantage of the symmetry between the @xmath38 and @xmath39 spins to increase accuracy . another order parameter of the transition plane we are considering is the polarization@xcite , @xmath137 , defined as @xmath138 \ , ,      \label{eq : p}\ ] ] which measures the magnetization of the @xmath139 spins . we calculated the magnetic susceptibility of the @xmath38 spins using@xcite @xmath140    \label{eq : xi}\ ] ] and also measured @xmath141 , the size of the cluster flipped at each step , and calculated @xmath142 . there is a connection between the size of the clusters and the susceptibility @xcite which is common for all algorithms that generate non - interacting clusters of spins ( if all spins in a cluster have the same value ) . for the wolff single cluster procedure it has the simple form @xcite @xmath143 lastly we measured the susceptibility of the @xmath139 spins , @xmath144 : @xmath145    \label{eq : xip}\ ] ]    for each sample we estimated the effective autocorrelation times@xcite of the energy , @xmath146 and the magnetization , @xmath147 using the method outlined in ref . . the length of an individual run was often too short for a precise estimate , but averaging over all samples gave a rough estimate of @xmath148 $ ] . these estimates of @xmath149 allowed us to verify that all samples were equilibrated well enough before we started collecting data . typically we discarded at least @xmath150 spin configurations . the fluctuations in the random bond configurations introduce , in addition to the usual mc error , another source of statistical error . thus the total error for the specific heat , @xmath151 , for example , is given by @xmath152 where @xmath153 is the number of random bond samples , @xmath154 is the length of the mc runs and @xmath39 is the autocorrelation time of the algorithm ( which we can estimate by @xmath149 ) . @xmath155 is the variance of @xmath1 within a given sample and is due to thermal fluctuations , while @xmath156 is the variance of the exact @xmath157 within the ensemble of random bond samples . thus both the thermal fluctuations and the sample to sample fluctuations contribute to the error . we found that both terms in ( [ eq : p toterr ] ) imply that simulations of the at models and the potts model are much more expensive in computer time than simulations of the ising model . first there is an increase in @xmath39 in consistence with our findings in the pure models@xcite . e.g. at @xmath158 we find for the random ising model @xmath159 compared with @xmath160 for the random four - state potts model . second we found that for all thermodynamic quantities @xmath137 the relative variance @xmath161 was larger for the random four - state potts model than for the random ising model . this effect was strongest for the specific heat , e.g. for @xmath158 @xmath162 of the random four - state potts model was approximately an order of magnitude larger than that of the random ising model . in accordance with these findings , the number of samples simulated for the ising model varied between @xmath163 for @xmath164 and @xmath165 for @xmath158 , while for the four - state potts model it varied between @xmath166 for @xmath164 and @xmath167 for @xmath158 . we expect the critical specific heat of the aat to follow the finite size scaling@xcite form of ( [ eq : c(t)ani ] ) , @xmath168 the exponent ratio @xmath6 is predicted analytically@xcite to vary continuously along the critical line of the pure at model between @xmath169 at the decoupled ising critical point and @xmath170 at the four - state potts critical point . one expects the same variation to hold for the corresponding critical manifold of the aat at which the simulations were performed . in figure [ fig : cvacft1 ] we plot the critical specific heat , @xmath1 as a function of @xmath171 for the five critical aat models @xmath172 . fits are made to the form ( [ eq : c(l)ani ] ) , and the estimated values of @xmath173 are listed in table [ tab : aat1 ] . since the model @xmath174 is an anisotropic ising model , its specific heat does not follow the form ( [ eq : c(l)ani ] ) , but rather @xmath175 . the anisotropic four - state potts model @xmath176 should have @xmath177 with a logarithmic correction@xcite to scaling @xmath178 . in the range of @xmath132 considered , the logarithmic correction results in an estimate for @xmath173 which is too low . the results for the specific heat of the aat models clearly demonstrate the variation of @xmath173 from its ising value , @xmath179 to its four state potts value . as will be seen shortly this is quite different from the results for the rbat models .    in order to analyze the specific heat results of the rbat we first tried to fit the results to ( [ eq : c(l)ani ] ) , but unlike the specific heat of the aat , this was not possible for the full lattice size range @xmath180 . thus in fig . [ fig : cvrcfc1 ] we show the specific heat as a function of @xmath171 for the five critical rbat models @xmath172 along with fitting curves to ( [ eq : c(l)ani ] ) , where only data for @xmath181 were used to fit . the estimated exponents @xmath173 obtained from these fits are 0 within errors indicating a possible logarithmic behaviour . in addition , fig . [ fig : cvrcfc1 ] indicates a crossover of the effective exponents ( the slope between every two successive data points ) from higher to lower values . physically we expect that this is due to a crossover from the pure models exponents to some new random critical behaviour . we are thus led to the following finite size scaling ansatz for the specific heat @xmath182\ ; , \label{eq : c fss cross}\ ] ] where @xmath183 is the critical exponent ratio of the corresponding pure model and @xmath184 . an important point to note is that for each random bond at model , following the discussion at the end of sec . [ sec : duality ran ] , @xmath183 is taken from our results ( listed in table [ tab : aat1 ] ) for the corresponding anisotropic at model . @xmath185 is a measure of the randomness , and is related through the relation @xmath186 to a crossover length @xmath187 , at which crossover from the pure model s power law behaviour to the random logarithmic behaviour occurs . thus for @xmath188 our eq . ( [ eq : c fss cross ] ) reduces to ( [ eq : c(l)ani ] ) while for @xmath189 and @xmath190 a logarithmic behaviour is attained , @xmath191    apart from crossing over to the correct pure result ( [ eq : c(l)ani ] ) when @xmath192 , eq . ( [ eq : c fss cross ] ) was constructed so that in the ising model limit , @xmath193 , it becomes @xmath194 this is the finite size scaling form of ( [ eq : c(t)ising ran ] ) which was shown by wang et al.@xcite to fit the random bond ising model well . this form will be examined below as another candidate for describing the rbat models .    in fig . [ fig : cvrcaat ] the specific heat of the five critical rbat models @xmath172 are plotted again but with fits to ( [ eq : c fss cross ] ) using the full lattice size range @xmath180 , on a semilogarithmic scale . the fitting parameters are listed in table [ tab : cv fit ] . for these points ( with large randomness , @xmath195 ) , the crossover lengths @xmath187 are found to be 1 .    in fig . [ fig : cvrdaat ] the specific heats of three critical rbat models @xmath196 are plotted with the fitting functions of ( [ eq : c fss cross ] ) . @xmath183 is taken from the corresponding anisotropic models and the fitting parameters are listed in table [ tab : cv fit ] . @xmath183 of the three models @xmath196 is very similar ( .40 , .37 , .37 respectively ) but they differ in their amount of randomness , @xmath130 respectively . the form ( [ eq : c fss cross ] ) seems to describe all three models adequately with the main change being in the crossover length @xmath187 which decreases as randomness increases . the estimates for @xmath187 can be compared with results from random bond ising models@xcite with the same coupling ratios . for @xmath197 we obtain @xmath198 compared with @xmath199 and @xmath200 for the respective ising models . according to the harris criterion one expects the crossover length to scale roughly as @xmath201 where @xmath202 is the crossover exponent and @xmath203 ( @xmath204 , @xmath205)is a measure of the randomness . for the rbat models considered here @xmath206 , while for the random bond ising models randomness is marginal ( @xmath207 ) , so that the much smaller crossover lengths that we find are consistent with the harris criterion . since there is no theoretical prediction for the critical behaviour of the rbat model , we examined possible scaling forms for the specific heat , other than ( [ eq : c fss cross ] ) . a natural candidate is a double logarithmic behaviour as in eq . ( [ eq : c(t)ising ran ] ) . this form was predicted by dd for the disordered baxter@xcite and ising@xcite models and confirmed for the latter model . in its asymptotic limit , the finite size scaling form of ( [ eq : c(t)ising ran ] ) is @xmath208 we compare this form with ( [ eq : c fss log ] ) as both these forms are asymptotic and do not include crossover from the pure behaviour . in figure [ fig : cvrclg1 ] we plot again our specific heat results for all the models tested except for @xmath126 which is not considered due to its large crossover length . the results are fitted by ( [ eq : c fss log ] ) in fig . [ fig : cvrclg1 ] and by ( [ eq : c fss log log ] ) in fig . [ fig : cvrcll1 ] , for @xmath209 .    as found by wang et al . , for the random bond ising model @xmath174 , agreement with ( [ eq : c fss log log ] ) is much better than with ( [ eq : c fss log ] ) , according to our error analysis . for the model @xmath210 too agreement with ( [ eq : c fss log log ] ) is better but this can be attributed to slow crossover behaviour from the small power law @xmath211 ( the condition @xmath212 is not fulfilled for such small @xmath183 and for the range of @xmath132 considered ) . for the models @xmath213 the quality of both fits is comparable while for the random bond four state potts @xmath176 the fit ( [ eq : c fss log ] ) is better . for the model @xmath127 we have obtained larger statistics ( more than 900 samples for @xmath214 and 300 samples for @xmath158 ) and the result is conclusively in favor of ( [ eq : c fss log ] ) . last we considered the possibility that the crossover from the power law behaviour to a logarithmic behaviour is yet followed at even larger @xmath132 by a crossover to a double logarithmic behaviour . this was done by fitting the specific heat data for @xmath215 according to ( [ eq : c fss dotz ] ) . though the fits were of good quality , the estimated crossover lengths @xmath216 did not make sense physically . for the @xmath127 model we found @xmath217 which actually implies a logarithmic behaviour for all accessible values of @xmath132 . this should be compared against our estimate @xmath218 and the value @xmath219 for the corresponding ising model ( see above ) . for the models @xmath121 we find that @xmath220 increases monotonically from the ising model @xmath174 where @xmath221 to the four state potts model @xmath176 where @xmath222 . this is contrary to the expectation that the crossover length should decrease due to the increase in the crossover exponent @xmath202 , as implied from ( [ eq : lc harris ] ) . we conclude that our results are in favor of a logarithmic divergence of the specific heat and not a double logarithmic one for the rbat model , excluding the random bond ising model . the finite size scaling form ( [ eq : c fss cross ] ) gives the best agreement with our results for all the models , including those with weak randomness , and for the full lattice size range @xmath180 . thus we predict for @xmath223 the scaling form : @xmath224\ ; , \label{eq : c scaling cross}\ ] ] a possibility which we tend to rule out is that both types of critical behaviors ( that is ( [ eq : c scaling cross ] ) and ( [ eq : c(t)ising ran ] ) ) occur and that there is some bifurcation plane which separates the two types of behaviors . according to finite size scaling theory one expects that the critical susceptibility diverges as @xmath225 for large enough @xmath132 . the exponents ratio @xmath226 is predicted analytically@xcite to be constant , @xmath227 , all along the critical line of the pure at model . this is expected also to hold for the corresponding critical manifold of the aat . figure [ fig : xtacall ] shows results for the critical susceptibility as a function of @xmath171 for the five critical models , @xmath172 , of the aat . so that the points wo nt fall on top of each other , @xmath21 for the model @xmath99 has been multiplied by @xmath228 . the solid lines are linear fits to the form ( [ eq : xfss ] ) for @xmath229 , yielding estimates for critical exponents @xmath43 which are listed in table [ tab : aat1 ] . all results seem to be consistent with the analytic prediction , giving a combined best estimate of @xmath230 . an effective exponent analysis ( or a trend analysis ) shows that as @xmath171 increases , @xmath231 approaches the value @xmath232 from above . the finite size scaling form ( [ eq : xfss ] ) and the value of @xmath233 have been predicted@xcite for the disordered decoupled ising model and confirmed in mc simulations@xcite . figure [ fig : xtrcall ] shows results for the critical susceptibility as a function of @xmath171 for the five random critical models @xmath172 , of the rbat . the same analysis has been carried out as for the aat and the estimated critical exponents @xmath43 are listed in table [ tab : rand1 ] . the estimates for the decoupled ising critical point @xmath174 and for the models @xmath234 are consistent with @xmath233 . on the other hand for the model @xmath235 and the random - bond four - state potts model @xmath176 we obtain @xmath236 and @xmath237 respectively . we also tried fitting the results with the form @xmath238 for @xmath229 . the obtained estimate for @xmath239 is listed in table [ tab : rand1 ] . for the random - bond four state potts we find @xmath240 while for the models @xmath241 the error in @xmath239 is at least of the same order as the estimate for @xmath239 itself .      the critical magnetization of the five anisotropic models is expected@xcite to follow the form @xmath242 with @xmath243 . applying straightforward finite size scaling analysis , as described above for the susceptibility , to the magnetization results , we obtained estimates for @xmath244 which are listed in table [ tab : aat1 ] . the results are slightly below the expected value , but an effective exponent analysis ( or a trend analysis ) shows that as @xmath171 increases @xmath245 approaches the value @xmath246 from below . the finite size scaling form ( [ eq : mfss ] ) and the value of @xmath247 have been predicted@xcite for the disordered decoupled ising model and confirmed in mc simulations@xcite . figure [ fig : m1rcall ] shows results for the critical magnetization as a function of @xmath171 for the five critical models , @xmath172 , of the rbat . a fit to the form ( [ eq : mfss ] ) yields the estimated critical exponents @xmath244 listed in table [ tab : rand1 ] . the estimates for the decoupled ising critical point @xmath174 and for the models @xmath234 are consistent with @xmath247 . on the other hand for the model @xmath235 and the random - bond four - state potts model @xmath176 we obtain @xmath248 and @xmath249 respectively . we also tried fitting the results with the form @xmath250 for @xmath229 . the obtained estimate for @xmath251 is listed in table [ tab : rand1 ] . for the random - bond four state potts model we find @xmath252 while for the models @xmath241 the error in @xmath253 is at least of the same order as the estimate for @xmath251 itself . since the polarization is an order parameter of the transition , the critical polarization , @xmath137 , and @xmath144 should follow the forms    [ eq : pxp1 ] @xmath254 @xmath255    the critical exponents @xmath256 and @xmath257 are known analytically@xcite all along the critical line of the pure at model . for example the ratio @xmath258 varies continuously between its value at the decoupled ising critical point , @xmath259 , and its value at the four state potts critical point , @xmath260 . these two extreme values can be obtained , through their relations with @xmath244 , for any ( pure , anisotropic or disordered ) decoupled ising or four - state potts model . for the decoupled ising models @xmath72 ( or @xmath65 for the pure model ) , the @xmath38 spins system and the @xmath39 spins system are decoupled , hence @xmath261 this implies @xmath262 for all the decoupled ising models . for the four state potts models @xmath263 and @xmath264 ( or @xmath265 for the pure four state potts ) so that @xmath38 spins , @xmath39 spins and @xmath139 spins are equivalent , and @xmath266 it follows that @xmath267 . thus interpreting our results as @xmath247 for the critical plane of the rbat implies @xmath268 for the random - bond decoupled ising model , and @xmath269 for the random - bond four - state potts model . the question still remains whether the variation between these two values along the critical plane of the rbat will be continuous , as is the case for the pure model , or whether it will become discontinuous under the influence of disorder . the results presented below are in favor of a continuous variation . the values of @xmath270 can be obtained through the scaling relation , @xmath271 which our results fully confirm . this leads to @xmath272 for the ising models and @xmath273 for the four - state potts models . figures [ fig : xpacall ] and [ fig : p3acall ] show the critical @xmath144 and polarization , @xmath137 , respectively , as a function of @xmath171 for the five critical models , @xmath172 , of the aat . the solid lines are linear fits to the forms ( [ eq : pxp1 ] ) for @xmath229 , yielding estimates for critical exponents @xmath270 and @xmath258 which are listed in table [ tab : aat1 ] . as expected we find that @xmath270 varies between @xmath274 for the decoupled ising model and @xmath275 for the four state potts model . @xmath258 varies accordingly fulfilling the scaling relation ( [ eq : scaling1 ] ) . figures [ fig : xprcall ] and [ fig : p3rcall ] show the critical @xmath144 and polarization , @xmath137 , respectively , as a function of @xmath171 for the five critical models , @xmath172 , of the rbat . the same analysis has been carried out as for the aat and the estimated critical exponents @xmath270 and @xmath258 are listed in table [ tab : rand1 ] . the results for the random - bond four - state potts , @xmath176 , agree with @xmath276 and @xmath277 to a high precision and thus need no further explanation . there is a clear mismatch between the estimated values of @xmath270 and @xmath258 and their expected values for the random - bond ising model @xmath174 . even the left side of the relation ( [ eq : pising ] ) , @xmath278 , which implied @xmath279 , is far from being met , but rather we find @xmath280 both for the anisotropic and for the random - bond ising models . this is probably connected to the fact that we are measuring , as is usual in a mc calculation , @xmath281 and @xmath282 which are larger than @xmath283 and @xmath284 respectively . both in the random - bond and in the anisotropic ising model the relative variance of @xmath137 is larger than the relative variance of @xmath135 . this makes @xmath282 a worse estimator of @xmath285 than @xmath281 is of @xmath283 . the discrepancy is a finite size effect@xcite and should decay as @xmath132 increases . indeed if we estimate @xmath258 from lattices of size @xmath286 we obtain @xmath287 which is consistent with the exact value @xmath259 . an estimate of @xmath270 from lattices of size @xmath288 yields @xmath289 which is consistent with the exact value @xmath272 . the results for @xmath144 and @xmath137 seem to favor a continuous variation of @xmath258 and @xmath270 for the rbat models as is the case for the aat models . as explained above , for symmetry reasons one expects that if @xmath290 then @xmath291 . this is consistent with the variation of exponents we find . none the less since we found the specific heat of all models to diverge logarithmically one could expect the rbat models to be more universal . a possible scenario to expect could be @xmath292 for all rbat models but the ising model . then assuming @xmath293 with ( [ eq : scaling1 ] ) would imply @xmath294 for the ising model and @xmath295 for all the other rbat models . such a behaviour would of course be masqueraded by crossover , but none the less we do not find evidence for such a behavior in our results . further investigation of this problem is clearly needed . we ve examined the effect of bond - disorder on the line of critical points of the at model . a duality transformation was used in order to locate a critical manifold of the random bond at model which corresponds to the line of critical points of the pure model . an anisotropic at model was used as a convenient reference pure model . our results consist of the finite size dependence of @xmath296 at criticality of several random bond and anisotropic at models . the models @xmath297 interpolate between ising and four - state potts models with large disorder , thus also monitoring the effect of changing the crossover exponent @xmath183 . the models @xmath196 serve to monitor the effect of changing the amount of disorder while keeping the crossover exponent approximately the same ( @xmath298 ) . the results of the critical specific heat of all the rbat models agree very well with the crossover behaviour ( [ eq : c fss cross ] ) , according to which for @xmath299 , @xmath1 diverges with the same exponent @xmath183 as the corresponding aat model , while for @xmath300 , @xmath1 grows as @xmath171 . our estimates for the crossover lengths of the models @xmath301 are much smaller than estimates of wang et al . for random bond ising models with the same amount of disorder . this is consistent with the harris criterion and the crossover exponent @xmath302 of the former models and the crossover exponent @xmath207 for the latter models . for the random bond ising model we find a double logarithmic divergence ( [ eq : c fss dotz ] ) , or ( [ eq : c fss log log ] ) asymptotically , as was found by wang et al . we view the adequacy of ( [ eq : c fss cross ] ) for all the rbat models ( with different @xmath183 and different amounts of randomness ) for the full lattice range @xmath303 , together with the consistency of the crossover lengths with physical understanding ( harris criterion ) , as a strong indication that the rbat models ( including the four state potts model ) exhibit a a logarithmic divergence of @xmath1 . we tend to rule out the possibility of a double logarithmic divergence of @xmath1 as predicted by dd@xcite for the baxter model , for reasons explained in sec . [ sec : results ] . although @xmath304 and @xmath305 do not vary for the pure model , our results indicate a possibility that @xmath304 and @xmath305 do vary for the random models . none the less this very small variation might be an artifact of a correction to scaling , e.g. of a logarithmic type . the exponents connected with the polarization @xmath137 and the susceptibility @xmath144 , @xmath306 and @xmath307 which vary continuously for the pure at model seem to do so also for the rbat model . but on the basis of universality the possibility that these exponents are actually the same ( with some non - universal corrections to scaling and a crossover effect ) for all rbat models excluding the ising model , seems to be a plausible scenario in need of further investigation . very recently an experimental study of a two dimensional four - state potts system with quenched impurities was done@xcite and the values of @xmath308 , @xmath309 , and @xmath310 were obtained . these are consistent with our findings for the four - state potts system @xmath176 but with a value of @xmath311 which is smaller than ours . sw would like to thank h.j . heermann and the many particle group of the hlrz jlich for their warm hospitality and the generous allocation of computer time on the intel ipsc/860 and the paragon . this research has been supported by the us - israel bi - national science foundation ( bsf ) , and the germany - israel science foundation ( gif ) . we list here the duality transformation @xmath312 $ ] of the at model and a few of its properties . for convenience we use instead of @xmath53 different variables for the coupling constants , @xmath313 ; @xmath314 under a duality transformation @xmath55 , a bond of strength @xmath315 transforms into a bond on the dual lattice of strength @xmath316 given by @xcite @xmath317 we list several properties of the transformation . it is self - inverse , i.e. , @xmath318=\vec{x}$ ] . it maps the zero temperature point ( @xmath319 ) onto @xmath320 ( @xmath321 ) and vice versa . the ising subspace @xmath322 and the 4-state potts subspace @xmath323 are invariant under @xmath55 . the subspace @xmath63 , defined by @xmath324 ( or @xmath325 ) , is also invariant under @xmath55 . the line @xmath104 is a self dual line and its intersection with @xmath63 is the line of critical points of the at model . it is also true that if @xmath326 then @xmath327 and vice versa . 99 email address : shai@elect1.weizmann.ac.il email address : fedomany@weizmann.weizmann.ac.il a.b . harris _ j.phys . c_,*7*:1671,(1974 ) . r.b . stinchcombe , in _ phase transitions and criticalphenomena_,edited by c. domb and j.l . lebowitz(academic , new york , 1983 ) , vol w.kinzel and e. domany _ phys . b_,*23*:3421 , ( 19801 ) . t.c . lubensky and a.b . _ , * 24*:311,(1974 ) . c. jayaprakash and h.j . rev . b_,*16 * : 3987,(1977 ) . vik s. dotsenko and vl s. dotsenko , adv . * 32 * , 129(1983 ) . shalaev , sov . phys . solid state * 26 * , 1811 ( 1984 ) . r. shankar , phys rev . lett . * 58 * , 2466 ( 1987 ) ; phys rev . lett . * 61 * , 2390 ( 1988 ) . ludwig , phys rev . lett . * 61 * , 2388 ( 1988 ) . w. selke , in computer simulations in condensed matter physics iii , edited by d.p . landau , k.k . mon , and h.b . schttler ( springer , heidelberg , 1991 ) . d. andelman and a. nihat berker _ phys . b_,*29*:2630,(1984 ) . b. derrida and e. gardner , _ j. phys a_,*17 * , 3223 ( 1984 ) . ludwig , _ nuc . b_*285 * , 97 ( 1986 ) . s. chen , a.m. ferrenberg and d.p . lett._,*69*:1213,(1992 ) . a.n . berker _ physica a _ * 194 * , 72 ( 1993 ) . novotny and d.p . rev . b_,*24 * : 1468,(1981 ) . baxter and f.y . lett._,*31*:1294,(1973 ) . j. ashkin and e. teller , _ phys . rev._*64 , * 178 ( 1943 ) . f.y . wu and k.y . lin , _ j.phys . c_*7 , * l181 ( 1974 ) . e. domany and e.k . riedel , _ phys . b_*19 , * 5817 ( 1979 ) . ditzian , j.r . banavar , g.s . grest and l.p . kadanoff , _ phys . b_*22 , * 2542 , ( 1980 ) . s. wiseman and e. domany , _ phys . rev e_*48 , * 4080 ( 1993 ) . . baxter _ exactly solved models in statistical mechanics _ , academic press ( 1982 ) . b. nienhuis , in _ phase transitions and critical phenomena _ , edited by c. domb and j.l . lebowitz(academic , new york , 1987 ) , vol . f.c . alcaraz and c. tsallis , j.phys . a * 15 * , 587 ( 1982 ) . baxter , ann . * 70 * , 193 ( 1972 ) . vik s. dotsenko and vl s. dotsenko _ j.phys . a_*17*:l301,(1984 ) . d. matthews - morgan , d.p . landau and r.h . lett._,*53*:679,(1984 ) . for a review see : a.d . sokal , _ nuc . suppl.)_,*20,*55 , ( 1991 ) k. binder , d.w . heermann , _ monte carlo simulations in statistical physics . _ springer - verlag , berlin , ( 1988 ) . heuer , phys . b * 42 * , 6476 ( 1990 ) . wang , w. selke , vi.s . dotsenko and v.b . physica a_,*164*:221,(1990 ) . m. hennecke and u. heyken , _ j. stat . phys._*72 * , 829 ( 1993 ) . f.y . wu and y.k . wang , j. math . * 17 * , 439 ( 1976 ) . r. fisch _ j. stat . phys._,*18*:111,(1978 ) . u. wolff , _ phys . lett._*62 , * 361 ( 1989 ) . s. caracciolo , r.g . edwards , a. pelissetto , and a.d . sokal , _ nuc . suppl.)_*20 , * 72 ( 1991 ) . swendsen and j.s . wang , phys . lett . * 58 * , 86 ( 1987 ) . andreichenko , vl.s . dotsenko , w.selke , and j .- s . wang , nucl . phys . b * 344 * , 531 ( 1990 ) . a factor of @xmath328 has been omitted , but is irrelevant since all measurements are at a single temperature . b_,*300*:501,(1988 ) . sokal , in _ computer simulation studies in condensed matter pysics : recent developments , _ ed . landau , k.k . mon and h barber , in _ phase transitions and critical phenomena _ , edited by c. domb and j.l . lebowitz , vol . li and a.d . sokal , _ phys . lett._*63 , * 827 ( 1989 ) . l. schwenger , k. budde , c. voges , and h. pfnr , _ phys . rev . lett._*73 * , 296 ( 1994 ) .    . critical exponents ratios and fitting parameters from the first series of simulations , @xmath172 , and the second series @xmath329 of the anisotropic at model . the exponent ratios and parameters are defined in the text . [ cols="<,<,<,<,<,<",options="header " , ]
the critical behaviour of a bond - disordered ashkin - teller model on a square lattice is investigated by intensive monte - carlo simulations . 
 a duality transformation is used to locate a critical plane of the disordered model . 
 this critical plane corresponds to the line of critical points of the pure model , along which critical exponents vary continuously . along this line 
 the scaling exponent corresponding to randomness @xmath0 varies continuously and is positive so that randomness is relevant and different critical behaviour is expected for the disordered model . 
 we use a cluster algorithm for the monte carlo simulations based on the wolff embedding idea , and perform a finite size scaling study of several critical models , extrapolating between the critical bond - disordered ising and bond - disordered four state potts models . 
 the critical behaviour of the disordered model is compared with the critical behaviour of an anisotropic ashkin - teller model which is used as a refference pure model . 
 we find no essential change in the order parameters critical exponents with respect to those of the pure model . 
 the divergence of the specific heat @xmath1 is changed dramatically . 
 our results favor a logarithmic type divergence at @xmath2 , @xmath3 for the random bond ashkin - teller and four state potts models and @xmath4 for the random bond ising model .