Phase diagrams of site-frustrated Heisenberg models on simple cubic, bcc, and fcc lattices

Transcription

1 PHYSICAL REVIEW B 73, Phase dagrams of ste-frustrated Hesenberg models on smple cubc, bcc, and fcc lattces A. D. Beath and D. H. Ryan Physcs Department and Centre for the Physcs of Materals, McGll Unversty, 3600 Unversty Street, Montreal, Quebec, Canada, H3A 2T8 Receved 15 March 2006; revsed manuscrpt receved 17 May 2006; publshed 29 June 2006 We have determned the phase dagrams of the ste-frustrated Hesenberg model n three dmensons over the entre concentraton regme of competng ferromagnetc F and antferromagnetc A stes for the three basc lattce types: smple cubc sc, body-centered cubc bcc and face-centered cubc fcc usng Monte Carlo methods. Long-range ferromagnetc FM or antferromagnetc AF order s establshed at a fnte temperature, second-order phase transton whenever F or A stes percolate. Snce ste percolaton thresholds n three dmensons are less than 1 2, FM and AF order coexst over a wde composton regme. The only apparent effect of frustraton s to cause the AF and FM order to occur perpendcular to one another. Strong evdence s found to suggest that all of the transtons reman n the Hesenberg unversalty class and that there exsts a tetracrtcal pont at x= 1 2 where T C=T N. DOI: /PhysRevB PACS number s : Lk, Uu, Hk, Mg I. INTRODUCTION Some of the most studed spn-glass materals are made by mxng ferromagnetc F and antferromagnetc A stes together n an amorphous sold, a typcal example beng 1 3 amorphous a- Fe 1 x Mn x G, where G s a glass former, such as S. The concentraton-temperature x-t phase dagram of ths materal s shown n Fg. 1. The man features are on ncreasng x from zero, the Cure temperature T C steadly decreases; above a crtcal concentraton x c1 0.17, a second transton develops at T xy below T C where spn-glass order coexsts transverse to the establshed FM order; above x c FM order dsappears and the materal exsts as a spn glass. A mnmal model, whch s thought to capture the essental physcs of these types of materals, s the stefrustrated Hesenberg model, where F and A stes are dstrbuted at random on a lattce. The frst to study ths model, to the best of our knowledge, was Aharony 4 who used renormalzaton group technques. Hs phase dagram, for bpartte lattces, s smlar to ours see Fgs. 2 a and 2 b, n that a tetracrtcal pont bounds a mxed phase consstng of coexstng FM and AF order. Further work, 5 however, cast doubt on hs orgnal result, although t s mportant to note that a true stefrustrated model was not beng consdered n ths later work. Instead, a ste-frustrated model was consdered to be equvalent to a bond-frustrated model 6 where, rather than a random dstrbuton of F and A stes, one consders a random mxture of F and A bonds; the well-known Edwards-Anderson spnglass model. 7 Wthn mean feld theory, the phase dagram of the bond-frustrated model s well understood 8 and s thought to capture the physcs of a dfferent class of materals, such as Refs. 1 and 9 a-fe 1 x Zr x. However, as we shall prove, the ste-frustrated model s not equvalent to a bond-frustrated model and the phase dagrams are fundamentally dfferent. The best results for the ste-frustrated Hesenberg model have been obtaned from Monte Carlo smulatons, yet the authors dsagree n ther nterpretatons of the phase dagram. Two studes 10,12 clam that a mxed noncollnear phase, smlar to spn-glass order, exsts n the vcnty of x= 1 2.On the other hand, we 13 and others 11 have clamed that the mxed phase conssts of coexstng FM and AF order, and that the AF- and FM-orderng vectors are mutually perpendcular. In addton, t has been reported 11,12 that the ferromagnetc transton belongs to the Hesenberg unversalty class for small x, and that wth ncreasng x the unversalty class of the transton changes, n contradcton to the Harrs crteron. 14 Bekhech and Southern 12 have also clamed that the tetracrtcal pont, where T C x =T N x, does not exst and nstead there exsts a regme 0.52 x 0.48 where T C x =T N x. Resolvng the dfferent nterpretatons s mportant snce one would lke to dentfy the mnmal physcs necessary to understand the phase dagrams of real materals. Here we present the results of extensve Monte Carlo smulatons on three dfferent lattce types: sc, bcc, and fcc. FIG. 1. Phase dagram for a- Fe 1 x Mn x 78 S 8 B 14 obtaned from SR, Mössbauer spectroscopy, and magnetometry. T C marks the lne separatng paramagnetc and ferromagnetc phases. T XY marks the onset of transverse spn freezng, occurrng for 0.17 x T SG marks the onset of spn-glass order, occurrng for x For detals see Ref /2006/73 21 / The Amercan Physcal Socety

2 A. D. BEATH AND D. H. RYAN PHYSICAL REVIEW B 73, FIG. 2. Phase dagrams of the ste-frustrated Hesenberg model for a smple cubc b bond-centered-cubc and c face-centeredcubc lattces. Transtons marked by sold symbols n a are obtaned wth 200+ confguratons of dsorder, and system szes L =4,6,8,10,12,16,20, and 24. Transton temperatures and errors are gven n Table II. Transtons marked by open symbols are obtaned from 16+ confguratons only, and smaller system szes. Lnes markng phase boundares are gudes to the eye. Our phase dagrams, shown n Fg. 2 for smple-cubc sc, body-centered-cubc bcc and face-centered-cubc fcc lattces, are somewhat dfferent from the phase dagrams gven n all prevous Monte Carlo studes If F and A stes do not nteract, the model can be readly understood from the theory of dlute magnetsm: 15 When a percolatng cluster of F A stes forms, a fnte temperature transton s found at T C T N and the transtons reman n the Hesenberg unversalty class. For bpartte lattces wth symmetrc nteractons, a decoupled tetracrtcal pont s found at x=0.5, snce by symmetry T C x =T N x. The tetracrtcal pont s referred to as decoupled snce the smultaneous FM- and AF-orderng events have no mpact on each other f the exchange nteracton between F and A stes s zero. The phase dagrams for ths model, wth no nteracton between F and A stes, s shown n Fg. 3 for sc, bcc, and fcc lattces. Introducng an nteracton between F and A stes yelds the ste-frustrated Hesenberg model. We fnd that the orderng s essentally as FIG. 3. Phase dagrams of the nonfrustrated model wth J FA =0 for a smple cubc, b bond-centered-cubc, and c facecentered-cubc lattces. The phase dagrams are, asde from a rescalng of the transton temperatures, dentcal to those of the stefrustrated Hesenberg model shown n Fg. 2. Lnes markng phase boundares are gudes to the eye. one would expect f F and A stes dd not nteract, and as such there exsts a tetracrtcal pont at x=0.5, and the transtons reman n the Hesenberg unversalty class. The only notable effects ntroduced by couplng the AF and FM order together are a rescalng of the transton temperatures and perpendcular FM- and AF-orderng vectors. As the loss of order n the nonfrustrated model s assocated wth ste percolaton, and percolaton thresholds are determned by the lattce coordnaton, we have extended our study of the ste-frustrated model to nclude bcc and fcc lattces. Here we show that the same ste percolaton mechansm lkely controls the loss of order n ste-frustrated models as well. Wthn our pcture, the ste-frustrated Hesenberg model on bpartte lattces cannot account for the phase dagrams of materals lke a- Fe 1 x Mn x G snce a spn-glass phase s not realzed at any composton. The lack of a spn-glass phase s due to nsuffcent frustraton n the bpartte lattces. Increasng the concentraton of frustraton can be accomplshed by consderng a nonbpartte lattce, such as fcc. The ste

3 PHASE DIAGRAMS OF SITE-FRUSTRATED HEISENBERG¼ frustrated Hesenberg model on an fcc lattce, whch s geometrcally frustrated wth respect to AF order as are amorphous solds, s therefore more lkely to capture the orderng of glassy materals. II. MODELS, SYMMETRIES, AND METHODS A. Ste-frustrated Hesenberg model The ste-frustrated model we consder here s a classcal Hesenberg model wth the Hamltonan H = J j S S j, 1 j where the sum runs over all nearest-neghbor bonds J j. The dstrbuton of bonds n ste-frustrated models can be expressed as J j = J FF x x j + J AA 1 x 1 x j + J FA x 1 x j + x j 1 x, 2 where x =1 f ste s occuped by a F ferromagnetc ste and x =0 f ste s occuped by an A antferromagnetc ste. For random F/A occupancy, the probablty of ste beng F type P x =1 = 1 x and the probablty of ste beng A type P x =0 =x. Our ste-frustrated model corresponds to the choce J FF = J AA = J FA =1. 3 We have chosen J FA = 1 snce n our expermental work 2,3 on a- Fe 1 x Mn x G, bulk magnetzaton, Mössbauer spectroscopy, and SR measurements all show that for small x, the Mn moments orent opposte to the Fe-rch FM bulk. However, the phase dagram s nvarant wth respect to the sgn of J FA for bpartte lattces, and thus, ths choce of sgn s unmportant. We have consdered three dfferent lattce types, sc, bcc, and fcc wth lnear dmenson L contanng N=L 3, N=2L 3, and N=4L 3 stes, respectvely, wth perodc boundary condtons and L even. The phase dagrams for sc, bcc, and fcc lattces are shown n Fg. 2. We have also consdered a related and trval nonfrustrated model wth J FF = J FA = 1 and J FA =0. 4 The phase dagrams for ths model are shown n Fg. 3 for sc, bcc, and fcc lattces. As mentoned n the Introducton, when F and A stes do not nteract J FA =0, we expect fnte T C T N provded only that F A stes percolate. Snce for sc lattces the percolaton threshold s x p =0.312, 16 one expects ths model to have a mxed phase of coexstng FM and AF order for x 0.688, as s observed. For bcc lattces, 16 x p =0.245 and a mxed phase s expected for x 0.755, also observed. For the fcc lattce, 16 where x p =0.198 we observe a FM phase for 0 x For the fcc lattce, we have not measured the AF order, whch s more complex than n the bpartte lattces, and thus, we have not observed a mxed phase. Note that, asde from a mnor rescalng of the transton temperatures, the phase dagrams of ste-frustrated models Fg. 2 are dentcal to those of nonfrustrated models Fg. 3. PHYSICAL REVIEW B 73, It s mportant to note that we do not consder the choce J FF =J AA = J FA =1, a Matts model, 17 whch has no frustraton. The ste-frustrated model we consder here s, however, equvalent to Luttnger s ste-frustrated model 18 wth J 1 = J 2 =J 3 = 1 2 whch, for Isng spns, exhbts spn-glass orderng n the mean feld lmt. B. Gauge nvarance and symmetres In the presence of a random dstrbuton of bmodal nteractons, J j =±1, a Hamltonan of the type shown n Eq. 1 wll possess a number of useful symmetres. Frustraton s the property that, gven a set of spns S lnked wth a closed loop of nonzero J j,noset S exsts that can satsfy all of the nteractons J j. Frustraton s measured by consderng the value of the frustraton functon 19,20 = J j evaluated around a plaquette the smallest closed loop of S lnked by nonzero J j.if 0, then a plaquette s frustrated, whereas f 0, the plaquette s not frustrated. Snce Eq. 5 always contans even powers of J FA for a ste-frustrated model, frustraton s ndependent of the sgn of J FA. It follows that frustraton s only present f J FF = J AA. Model symmetres can be found by consderng a local transformaton of the spn and bond varables accordng to 20 5 S S, J j J j, 6 where a spn at ste s nverted as well as all of the bonds J j emanatng from ste. Any set of such local transformatons G S ; J j = S ; J j j 7 wth = ±1, called a gauge transformaton, preserves both the Hamltonan and the dstrbuton of frustraton. 20 Snce the Hamltonan s nvarant under any G, the partton functon and the free energy are also nvarant. It follows that any par of models wth dfferent dstrbutons of the varables J j, yet havng the same dstrbuton of frustraton, have the same phase dagram the order parameters are, n general, dfferent. A frst symmetry s revealed usng the gauge transformaton G A =±1; F = 1, 8 whch nverts the spns of ether all A or all F stes n the lattce. The bonds are transformed such that J FA J FA whle both J FF and J AA reman nvarant. Therefore, phase dagrams of ste-frustrated models provded J FF = J AA are nvarant wth respect to the sgn of J FA. A second symmetry occurs for bpartte lattces that can be decomposed nto two nterpenetratng sublattces and. The gauge transformaton G =±1; = 1 9 has the effect of flppng the sgn of all of the bonds so that J j J j. Ths s equvalent to takng a model wth a concentraton x of A stes 1 x of F stes and creatng a model

4 A. D. BEATH AND D. H. RYAN PHYSICAL REVIEW B 73, wth a concentraton 1 x of A stes x of F stes, whle also flppng J FA, whch, as we have already shown, s rrelevant. Therefore, for bpartte lattces, the phase dagrams must be symmetrcal about x= 1 2. The symmetry between ferromagnetc and antferromagnetc models on bpartte lattces found by usng the gauge transformaton gven by Eq. 9 only apples to lattces wth L even. If L s odd, as n the smulatons of Matsubara et al., 11 then two equvalent sublattces do not exst, and for ths reason we beleve these results should be consdered wth cauton. One effect of Eq. 9 s to transform the FM-order parameter m f nto the AF-order parameter m st, and vce versa. Thus, f the model orders as a collnear FM, a collnear AF, or a mxture of the two whose order need not be parallel, then gauge symmetry mples the followng symmetry between the order parameters m f and m st : m f x,t;j FA = m st 1 x,t; J FA m st x,t;j FA = m f 1 x,t; J FA for all x and T. Ths symmetry also apples to all hgherorder powers of m f and m st, and therefore the susceptbltes are also symmetrc under the exchange x 1 x, J FA J FA. Ths symmetry perssts even when the orderng does not consst of a mxture of collnear FM and AF order, as, for example, spn-glass order. However, n the case of spn-glass order, m f and m st would scale to zero n the lmt of large L. The fact that we observe a nonzero m f and/or m st n the lmt of large L for bpartte lattces excludes any possblty that the mxed phase s a spn-glass phase. C. Monte Carlo methods Our study of the ste-frustrated Hesenberg model has proceeded n two parts. In the frst, we have employed a smple Metropols Monte Carlo algorthm wth smulated annealng to obtan the phase dagrams shown n Fgs. 2 and 3. The system szes studed range from 4 L 16 sc, from 4 L 14 bcc, and from 4 L 12 fcc for the concentratons shown n the phase dagrams by open symbols. The range of system szes for sc lattces are the same as those used n Refs. 11 and 12. For the sc and bcc lattces wth J FA =0 Fgs. 3 a and 3 b, only x 0.5 were studed and T C,N x 0.5 are obtaned by symmetry. The number of Monte Carlo updates per lattce ste MCS ranged from to after dscardng the frst to MCS at each T to ensure equlbraton. The number of MCS s chosen such that we exceed the measured sample ndependence tme, gven by twce the ntegrated autocorrelaton tme 21,22, by orders of magntude. The large value of, whch occurs near and below T C,N, lmts the lattce szes and the number of dsorder confguratons studed. In ths ntal survey, a mnmum of 16 realzatons of dsorder were used to obtan confguratonal averages. In order to ncrease L, and the number of confguratons C, we must reduce. To do so, we have employed an overrelaxaton scheme, 23 where the spns evolve accordng to FIG. 4. Autocorrelaton functon for the magnetsaton Eq. 13 usng Metropols MCS and Metropols plus over-relaxaton OR-MCS dynamcs for T T C L=10,x=0.45. The nset shows the ntegrated ntegrated autocorrelaton tme at T=0.95 usng overrelaxaton plus Metroplols dynamcs. S 2 S B B B B S, 12 where B = j J j S j s the nternal feld experenced at ste due to the couplng wth nearest-neghbor spns S j. Followng each Metropols MCS we use fve overrelaxaton MCSs, whch then comprses a sngle OR-MCS. The OR-MCS update has reduced by about a factor of 100. However, there remans a dvergence of accordng to L z wth z unaltered from the value found usng only Metropols dynamcs z 0, z 2, and z 3 for T T C, T T C and T T C, respectvely. Overrelaxaton has been found useful n studes of the ste-frustrated Hesenberg model, 11 the bond-frustrated Hesenberg model, 24 and fcc Hesenberg antferromagnets. 25 The dramatc reducton n that occurs when employng the overrelaxaton update s llustrated n Fg. 4, where we have plotted the normalzed autocorrelaton functon of the magnetzaton near T C for MCS and OR-MCS updates. The autocorrelaton functon of the magnetzaton or staggered magnetzaton A f t and A st t, respectvely at tme t s gven by A f,st t = m f,st 0 m f,st t m f,st 2, 13 where the angular brackets represent a tme average and the square brackets represent a confguratonal average m f and m st are defned below. The decays A f,st t are comprsed of a dscrete sum of exponental decays, 22 and the derved autocorrelaton tme s shown n the nset of Fg. 4 for T T C usng the OR-MCS update. The OR-MCS update has enabled us to equlbrate larger sc lattces wth 4 L 24. The number of confguratons used at each x and L s lsted n Table I. The number of OR-MCS used here ranges from at hgh temperatures

5 PHASE DIAGRAMS OF SITE-FRUSTRATED HEISENBERG¼ TABLE I. Number of dsorder confguratons C used for each concentraton x and system sze L for sc lattces usng the OR- MCS Monte Carlo update as explaned n the text. L x= TABLE II. Crtcal temperatures for the ste-frustrated Hesenberg model for the sc lattce type. Note that by symmetry T C x =T N 1 x. x T C x T N x to at low temperatures, after dscardng between to OR-MCS to ensure equlbraton. Agan, the number of OR-MCS s chosen to be much larger than any measurable sample ndependence tme. The resultng phase dagram s shown n Fg. 2 a by sold symbols, and summarzed n Table II. These results are more accurate than our orgnal survey, although the form of the phase dagram remans unaltered. Two characterstcs of the A f,st t decays have been extensvely checked to ensure that our results are both equlbrated and representatve of the lmt t. Equlbrum requres that the A f,st t decays are ndependent of the orgn of tme. For selected concentratons and temperatures, we have verfed that the decays are repeatable, and thus equlbrated, by measurng A f,st t whle varyng the number of dscarded MCSs or OR-MCSs pror to the measurement. Indeed, accurate measurement of A f,st t requres that the number of MCSs exceeds by orders of magntude, whch essentally guarantees ths result. To ensure our results are equlbrated at each temperature, we dscard several s worth of MCSs or OR-MCSs pror to our measurements. To check that our results are representatve of the long tme lmt, we have compared our results when varyng the numbers of MCS or OR-MCS updates. The A f,st t decays provde a measure of for the scalar quantty m f,st, whle our demonstraton, presented later, that the FM and AF order are mutually perpendcular, requred us to exceed for the vector quanttes m f,st. Snce for vector order parameters s much larger than for scalar order parameters, the long tmes necessary to demonstrate that the FM order and AF order are perpendcular also allows us to compare m f,st measured n typcal smulatons wth MCSs to that measured n a very long and atypcal smulaton wth MCSs. The comparson confrms that exceedng by orders of magntude yelds results that are characterstc of the lmt t. The most mportant quanttes we measure are powers of the fnte lattce magnetzaton and staggered magnetzaton. At tme t, the nstantaneous values of the nth power of the magnetzaton or staggered magnetzaton, m f and m st, are n m f,st = N 1 + N 1 PHYSICAL REVIEW B 73, L f,st S x 2 + N 1 L f,st S z 2 n/2, L f,st S y 2 14 where x,y,z denote Cartesan components and L f,st s an operator wth the symmetry of the ferromagnetc or antferromagnetc state. We have not made an attempt to measure the AF order for the fcc lattce, whch s more complex than the AF order n bpartte lattces. 25 Average values are obtaned by averagng over tme angular brackets and dsorder square brackets to yeld m n f and m n st. For brevty, we wll henceforth refer to m f,st as m f,st wth the understandng that averages over tme and dsorder have been taken. Other quanttes of nterest are the connected ferromagnetc and antferromagnetc susceptbltes, c f and c st, gven by c f,st = N m 2 f,st m f,st 2 15 and the dsconnected ferromagnetc and antferromagnetc susceptbltes, d f and d st, gven by d f,st = N m 2 f,st 16 Lastly, we have measured the Bnder cumulant for the ferromagnetc and antferromagnetc state, B f,st, gven by B f,st = m 4 f,st m 2 f,st 2, 17 whch s helpful n locatng T C and T N. The normalzaton s chosen such that B f,st =0 at T= and, n the case of the pure x=0,1 models, B f,st =1 at T=0. III. RESULTS The phase dagrams we have found for J FA = 1 Fg. 2 and J FA =0 Fg. 3 bear a strkng smlarty: FM order occurs provded F stes percolate and AF order occurs for sc and bcc lattces provded A stes percolate. At low temperatures wthn the mxed phase, m f and m st account for the majorty of the total spn at x=0.5 m f +m st 0.8. Furthermore, bpartte lattces always show ether ferromagnetc order, antferromagnetc order, or a mxture of the two, and thus, the possblty of a spn-glass phase whch lacks any perodc order can be conclusvely ruled out. The mportant dfference between our phase dagram on the sc lattce and the phase dagrams found by others

6 A. D. BEATH AND D. H. RYAN PHYSICAL REVIEW B 73, are: Both Matsubara et al. 11 and Bekhech et al. 12 have clamed that for large enough x, T C x, and by symmetry T N 1 x, do not reman n the Hesenberg unversalty class and the exstence of a tetracrtcal pont has been challenged; 12 nstead they suggest that there s a range of concentratons for whch T C x =T N x. Our results are at odds wth both snce we fnd that all transtons reman n the Hesenberg unversalty class and there s no evdence to support the exstence of a range of concentratons for whch T C x =T N x. Instead, we conclude that the model, for bpartte lattces, possess a tetracrtcal pont. The dfferences are related, we beleve, to a msdentfcaton of the transton temperatures. In all cases where Bekhech and Southern 12 clam that the model remans n the Hesenberg unversalty class, our estmates of T C,N are n agreement wth thers. By constrast, n all cases where they clam the model does not reman n the Hesenberg unversalty class, our estmates of T C,N are not n agreement. To determne the form of the phase dagram, we have extracted the system-sze dependent pseudotranston temperatures T C,N L from the peak n f,st and the maxmum c slope n B f,st. Accordng to fnte sze scalng theory, we expect that T C,N L should, for large enough L, scale accordng to T C,N L = T C,N + al 1/, 18 where s the exponent of the correlaton length. Equaton 18 allows us to locate T C and T N. We begn by assumng that the model remans n the Hesenberg unversalty class, for whch the exponent takes the value = ,27 Scalng plots for several concentratons are shown for the sc lattce n Fg. 5 for both T C and T N. Wth ncreasng L, the data begn to fall onto straght lnes see also Fg. 9 b, ndcatng that correctons to scalng are mportant for the smaller lattces. For L 8 f,st and L 10 B f,st, the T C,N L are ln- c ear n L 1/ and yeld the same estmates for T C and T N wthn error. In the case of the pure model x=0, the asymptotc scalng regme beyond whch correctons to scalng become neglgble s also 27 L 10, and so the devatons at small L are to be expected. Straght-lne fts for L 10 yeld two ndependent estmates for both T C and T N, and the weghted average of the two estmates are summarzed n Table II for x 0.5. Although the estmates for T C and T N so obtaned are our most precse results, they do depend on the assumpton of unversalty. However, estmates found assumng unversalty are consstent wth other estmates, n partcular, those found from crossngs of the Bnder cumulant, whch make no reference to a partcular unversalty class. The analyss that led to the phase dagram shown n Fg. 2 a assumed that the ste-frustrated Hesenberg model remans n the three-dmensonal Hesenberg unversalty class. That the transton temperatures found assumng ths unversalty are correct s demonstrated n Fg. 6, where we have plotted the crossng of the Bnder cumulant B f for x=0.45. Accordng to fnte sze scalng theory, B f,st should scale as FIG. 5. Fnte sze scalng of the pseudotranston temperatures T C L and T N L for the ste-frustrated Hesenberg model on sc lattces. Lattce szes L=4,6,8,10,12,16,20, and 24. The exponent =0.704 s taken for the Hesenberg unversalty class. Sold lnes are fts, for L 8, to T C,N L Eq. 18, obtaned from the peak n c. Dashed lnes are fts, for L 10, to T C,N L obtaned from the maxmum slope n B f,st. Where error bars are not apparent, they are smaller than the symbol sze. B f,st = B f,st tl 1/, 19 where t= T T C,N /T C,N s the reduced temperature. Thus, a plot of B f,st for dfferent L should exhbt a crossng at T C,N. Note that the locaton of ths crossng s ndependent of the unversalty class of the transton. The crossng near T C = s clear and agrees well wth our estmate T C = found from Fg. 5, where we assumed that the model was n the Hesenberg unversalty class n order to extract T C. Our value s both far from the earler estmate 12 T C = , and more precse. At x=0.45, they estmated that However, as shown n the nset of Fg. 6, the Bnder parameter scales very well usng Eq. 19 wth =0.704 provded L 8. More evdence to support our conjecture that the model remans wthn the Hesenberg unversalty class s found n the scalng of the order parameters and ther fluctuatons

7 PHASE DIAGRAMS OF SITE-FRUSTRATED HEISENBERG¼ PHYSICAL REVIEW B 73, FIG. 6. Bnder cumulant B f of the ferromagnetc order near T C for frustrated sc lattce at x=0.45. Lnes are gudes to the eye. Inset shows collapse of the data accordng to Eq. 19 usng the exponent = Where error bars are not apparent, they are smaller than the symbol sze. Fnte sze scalng theory predcts that n the vcnty of T C,N, m f,st and f,st scale accordng to m f,st = L / M tl 1/ f,st = L / X tl 1/ for large enough L. In Fg. 7, we show scalng plots for m f,st d and f,st at x=0.45, where t has been estmated 12 that / 0.35 and, from hyperscalng, / =2.3. In the four plots we have used the exponent ratos / =0.514 and / =1.973, wth =0.704, as found for the pure Hesenberg model 26,27 x=0 n three dmensons. The values of T C,N are taken from our Table II. The collapse s excellent for 8 L 24, whch strongly suggests that the transtons reman wthn the Hesenberg unversalty class. Undong the collapse of the x axs shown n Fg. 7 provdes further support that the ste-frustrated model remans n the Hesenberg unversalty class. If we plot m f,st L / or d f,st L / vs T for dfferent L, then Eqs. 20 and 21 predct that we should observe a crossng of the data at T C,N smlar to that observed for B f,st, as shown n Fg. 6. In Fg. 8, we show such plots for the ferromagnetc transton at x=0.45 the same data used n the left panels of Fg. 7. A clear crossng s observed just below T=0.95. We now turn to the form of the phase dagram near x =0.5. Aharony 4 frst postulated the exstence of a tetracrtcal pont where the lnes of second-order phase transtons, T C x and T N x cross. The phase dagram of Nelsen et al. 10 also exhbts a tetracrtcal pont, as does the phase dagram of Matsubara et al., 11 although n the latter case t was assumed. In contrast, t has been suggested 12 that the tetracrtcal pont FIG. 7. Fnte sze scalng collapse of m f and d f wth T C = and m st and d st wth T N = for L=8,10,12,16,20, and 24. We have used the ratos / =0.514 and / =1.973 of the three-dmensonal Hesenberg unversalty class. does not exst and nstead there exsts a regme from 0.48 x 0.52 where T C x =T N x. We have recently reported 13 that the scalng of T C,N L accordng to Eq. 18 at x =0.49,0.495, and 0.5 for L 20 and C=100 dsagrees wth ths conjecture. Our updated results at x=0.49 and x=0.5 wth L 24 and C 200, as lsted n Table II, confrms that T C x T N x wth the excepton of x=0.5. FIG. 8. Scalng plot of m f L / and f d L / vs temperature at x =0.45. Exponents are the same as n Fg. 7. A clear crossng s observed near T C = A further scalng the x axs accordng to tl 1/ results n the collapse shown n Fg. 7. Where error bars are not apparent, they are smaller than the symbol sze

8 A. D. BEATH AND D. H. RYAN PHYSICAL REVIEW B 73, FIG. 10. Examples of frustrated and satsfed plaquettes for the ste-frustrated model for a four-spn plaquette geometry approprate for sc and bcc lattces and for a three-spn plaquette geometry approprate for fcc lattces. FIG. 9. T C and T N vs. J FA wth J FF = J AA = +1 for the sc lattce at x=0.49. Inset a shows the dfference, T C T N. Inset b shows the scalng of the pseudotranston temperatures for J FA = 1. Where error bars are not apparent, they are smaller than the symbol sze. Further evdence n favor of a tetracrtcal pont s provded by examnng the behavor of the transton temperatures whle varyng the magntude of the nterste couplng J FA. Settng J FA =0 decouples F and A stes, gvng the nonfrustrated model and, hence, T C =T N only at x=0.5. Elsewhere, T C T N and the phase dagram possess a decoupled tetracrtcal pont at x=0.5. If the orderng scenero n Ref. 12 were correct and T C =T N at x=0.49 when J FA =1, then T C and T N would evolve wth ncreasng J FA such that the two transton temperatures merged. Conversely, f the transton temperatures reman dstnct, as one expects f the stefrustrated model possess a tetracrtcal pont, then T C and T N must reman dstnct as J FA ncreases. In Fg. 9, we show the evoluton of both T C and T N at x =0.49 for the sc lattce wth J FA =0, 1, 2, and 3. It s clear from the plot that both T C and T N ncrease wth ncreasng J FA n ths regme, and that T C s never equal to T N.In Fg. 9 a, we show the dfference, T C T N, whch clearly ndcates that the transton temperatures do not merge. Instead, the dfference steadly ncreases the dfference between T C and T N s 10. We have observed 28 the same behavor for the bcc lattce at x=0.4 up to J FA =30. As shown n Fg. 9 b, the scalng of the pseudotranston temperatures as n Fg. 5 at x=0.49 wth J FA = 1 s typcal of all our measurements and does not ndcate anythng that mght suggest that T C =T N. We conclude that the ste-frustrated model possess a tetracrtcal pont at x=0.5 and that T C T N for all other x. IV. DISCUSSION A. Frustraton densty and dstrbuton To understand the phase dagrams of the ste-frustrated Hesenberg model, t s necessary to consder both the densty and dstrbuton of frustraton wthn the model. The densty of frustraton can easly be calculated from the probablty that a plaquette exsts n a frustrated confguraton. In Fg. 10, we show examples of frustrated plaquettes for a four-spn plaquette, approprate for sc and bcc lattces, and for a threespn plaquette, approprate for fcc lattces. In the case of sc and bcc lattces, ste-frustrated models produce frustrated plaquettes only when the plaquette contans two neghborng F stes and two neghborng A stes. For fcc lattces, a plaquette s frustrated when t contans ether two or three A stes. The densty of frustrated plaquettes x f s x f =4x 2 1 x 2 22 for sc and bcc lattces and x f =3x 2 1 x + x 3 23 for fcc lattces. In the same way, the densty of frustrated plaquette n the bond-frustrated model s x f =4x 1 x 3 +4x 3 1 x 24 for sc and bcc lattces and x f =3x 1 x 2 + x 3 25 for fcc lattces, where for bond-frustrated models x s the concentraton of antferromagnetc bonds, whle for stefrustrated models x s the concentraton of antferromagnetc stes. The denstes of frustrated plaquettes gven by Eqs are plotted n Fg. 11. If ste and bond-frustrated models were gauge equvalent, as has been prevously assumed, 5 then x f would be the same for both the ste-frustrated model and ts gauge equvalent bond-frustrated counterpart, snce x f s gauge nvarant. The ste-frustrated model on sc and bcc lattces has a maxmum of x f =0.25, and thus, the assumed gauge equvalent bondfrustrated model must have x 0.08 or x 0.92, so that x f 0.25 see Fg. 11. At ths concentraton of antferromagnetc ferromagnetc bonds, 3d bond-frustrated models are

9 PHASE DIAGRAMS OF SITE-FRUSTRATED HEISENBERG¼ PHYSICAL REVIEW B 73, FIG. 11. The densty of frustraton for ste- and bond-frustrated models on sc, bcc, and fcc lattces. Note that the densty of frustraton s much less for the ste-frustrated model on sc and bcc lattces as compared to the bond-frustrated model. FIG. 12. Two-dmensonal square lattces showng varous possble nterfaces between F and A stes where frustraton occurs. a Flat 1 1 nterface, no frustraton. b Flat 1 0 nterface, nfnte number of frustrated plaquettes crosses on nterface. c Two frustrated plaquettes separated by a fnte dstance, energy s proportonal to the length of the dual strng broken lne lnkng them. d Rough 1 0 nterface, random lnear dstrbuton of frustraton. Only one of the two shortest lnkngs s shown. ordered ferromagnets 24,29,30 antferromagnets. Assumng gauge equvalence between ste- and bond-frustrated models n three dmensons shows that the ste-frustrated model conssts 31 of FM or AF order for all x, and rules out the possblty of spn-glass order. Ths result s n drect contradcton of the assumpton 5 that the equvalence of bond and ste frustraton leads to spn-glass order n ste-frustrated models. It s easy, however, to prove that ste and bond frustraton are not gauge equvalent on bpartte lattces. To do so, consder the probablty that, gven a frustrated plaquette, all of the neghborng plaquettes exst n a frustrated confguraton. For a bond-frustrated model, ths probablty s fnte for 0 x 1, whereas for ste-frustrated models ths probablty s zero for 0 x 1, whch s suffcent to prove that ste- and bond-frustrated models are not gauge equvalent. It s also nterestng to note that the ste-frustrated model on an fcc lattce exsts as an ordered ferromagnet untl x f 0.9 x 0.8 whle for the sc bond-frustrated model ferromagnetsm s lost by x f 0.44 x= for Hesenberg 24 spns. Ths serves to emphasze that the densty of frustraton alone s not responsble for the destructon of perodc order, but the dstrbuton, whch, as we have shown, s qute dfferent for ste- and bond-frustrated models. For sc and bcc lattces, frustrated plaquettes exst only when a plaquette contans two neghborng F stes and two neghborng A stes. If the lattce s consdered to be a collecton of F and A clusters, t follows that all of the frustrated plaquettes exst on the surfaces separatng the clusters. Snce all of the frustraton exsts at the nterface, whch separates clusters of F and A stes, unsatsfed bonds are lkely to also resde near that nterface. Gven ths stuaton, the phase dagrams of ste frustrated models Fg. 2 can be expected to mmc those of the nonfrustrated models Fg. 3, as we have observed. The effect of concentratng all of the frustrated plaquettes onto the surfaces separatng clusters of F/A stes wll be explored by consderng smple Isng spns snce algorthms exst for determnng exact ground states. In two dmensons, ground states for Isng spns wth ±J nteractons are determned by parng frustrated plaquettes wth a lne called a dual strng. Bonds traversed by the dual strng are broken wth the remanng bonds satsfed. It then follows that the ground state s determned by mnmzng the total length of all dual strngs. 20 We frst consder a dagonal, 1 1, surface separatng clusters of F and A stes. Ths confguraton of bonds, shown n Fg. 12 a, has no frustraton, and the orderng whch takes place belongs to the pure, ferromagnetc, two-dmensonal Isng unversalty class. A 1 0 nterface, shown n Fg. 12 b, whle superfcally smlar to 1 1 nterface, actually creates a confguraton wth an nfnte number of frustrated plaquettes. The lowest energy state then conssts of all F stes ordered ferromagnetcally and all A stes ordered antferromagnetcally. Satsfed and unsatsfed bonds alternate along the nterface leadng to a two fold degeneracy, decouplng the two regons. The orentaton of the FM order s free to pont up or down wth respect to a fxed AF order, ths effectvely mmcs the model where J FA =0. The thrd confguraton we consder s two frustrated plaquettes separated by a fnte dstance shown n Fg. 12 c. The lowest energy state conssts of F stes ordered FM and A stes ordered AF wth a lne of unsatsfed bonds along the shortest path connectng the two frustrated plaquettes. In ths case, a preferred relatve orentaton undrectonal ansotropy s mposed on, for nstance, the FM order f the AF orderng has already taken place. The last confguraton we consder conssts of an nterface wth randomly placed frustrated plaquettes, shown n Fg

10 A. D. BEATH AND D. H. RYAN PHYSICAL REVIEW B 73, d. The two shortest ways of connectng these plaquettes wth dual strngs A B, C D, E F, etc., and B C, D E, F G, etc. need not have the same energy, leadng to undrectonal ansotropy. However, for a very long nterface, the dfference n energy assocated wth these two possble lnkngs becomes nfntesmally small, and the FM-ordered F stes and AF-ordered A stes are agan decoupled, mmckng the model wth J FA =0. The examples consdered here demonstrate that frustraton at the nterface of F/A clusters tends to ether decouple the FM and AF orderng, Fgs. 12 b and 12 d, or to ntroduce undrectonal ansotropes, Fgs. 12 c and 12 d. Ths behavor wll only occur f the dual strngs, whch lnk frustrated, plaquettes, resde near the nterface and do not pass through the volume of the cluster. However, f we constran our nterest to percolatng F/A clusters, the densty of frustraton at the nterface s large 25% so that the total length of the assocated dual strngs lnkng the frustrated plaquettes wll be mnmzed by usng short elements traversng the nterface. Equvalently, bonds that compose the volume of a percolatng cluster, L d, are satsfed at the expense of the bonds at the nterface of the percolatng cluster, L d 1 leadng to the concluson that percolatng clusters of F or A stes order FM or AF, dentcal to the respectve pure models. Decouplng of the FM- and AF-ordered clusters wll then occur when the number of satsfed and unsatsfed bonds at the nterface are equal so that the ordered clusters are free to take any possble orentaton. Gven the confguraton of bonds n Fg. 12 b, where for Isng spns there s a decouplng of the AF and FM order, t can be magned that for Hesenberg spns the energy of the system can be further reduced: If the AF and FM orderng drectons are chosen to be mutually perpendcular, there s a net reducton n the nterface energy f the AF-ordered spns near the nterface cant ether up J FA =+1 or down J FA = 1 wth respect to the FM order, mnmzng exchange energy from the J FA bonds at the expense of the J AA bonds. Ths net energy change would then gve rse to mutually perpendcular FM and AF order. 11 The perpendcular nature of the FM and AF order s shown n Fg. 13, where we depct the parallel and perpendcular components of m st wth respect to m f for both the frustrated and nonfrustrated models. 32 The stuaton s analogous to sotropc antferromagnets n a unform external feld 33 where the AF orderng s transverse to the appled feld, wth the moments cantng upward along the appled feld n order to mnmze the total energy. The dfference n phase dagrams between sc and bcc lattces compared to fcc lattces leaves open the possblty that spn-glass orderng s permssble for fcc lattces n the concentraton regme 0.8 x 1. However, we do not beleve that spn-glass orderng occurs for fcc lattces. Rather, based on the observed phase dagrams for sc and bcc lattces, the percolatng structure of A stes lkely decouples and orders dentcally to the approprate pure model: the geometrcally frustrated fcc antferromagnet. Orderng of the fcc Hesenberg antferromagnet x=1 occurs at T=0.223, 25 and thus, our phase dagram for fcc lattces lacks a lne of lowtemperature transtons T N x from 0.2 x 1, as we have not attempted to measure t. The FM orderng for fcc lattces, FIG. 13. Parallel m st and perpendcular m st components of the total staggered magnetzaton m st relatve to the magnetzaton m f for the ste-frustrated Hesenberg model wth x=0.45 for a sc lattce wth L=10. The total staggered magnetzaton s also shown. Inset shows the same data for the nonfrustrated model J FA =0. Its evdent that the effect of the J FA bonds s to orent the AF order transverse to the FM order n the mxed phase. whch survves for an extremely hgh level of frustraton x f 0.9 almost twce that of the sc bond-frustrated model, whch s a spn glass 34 demonstrates agan that t s not the amount of frustraton that determnes spn-glass order but ts dstrbuton. When frustraton s pushed out to the surface of a percolatng cluster, perodc orderng takes place. B. Expermental consequences The prmary motvaton for ths work has been our expermental study of the amorphous alloy a- Fe 1 x Mn x G. 2,3 In ths alloy, Fe moments couple to neghborng Fe moments ferromagnetcally and Mn moments couple to both Fe and Mn antferromagnetcally, hence, our choce J FA = 1. The magnetc response of ths alloy, therefore, may be consdered prototypcal of a ste-frustrated magnet. An mportant dfference between the model and the alloy s the dsorder nherent to the glass structure. Despte ths dfference, many mportant detals of the a- Fe 1 x Mn x G phase dagram can be understood based on the model dscussed here. For small x, the materal remans ferromagnetc and the small concentraton of Mn moments order antparallel to the Fe-domnated FM order. Beyond a concentraton of Mn stes x 0.2, spnglass-lke orderng transverse to the magnetzaton occurs, whch s domnated by Mn moments. Ths concentraton corresponds to ste percolaton of Mn moments for an amorphous alloy 35 wth 12 nearest neghbors. Because of structural dsorder, the percolatng cluster of Mn moments cannot order as a perodc AF and orders as a spn glass. The phase dagram of ths alloy s lkely to be captured by the fcc phase

11 PHASE DIAGRAMS OF SITE-FRUSTRATED HEISENBERG¼ dagram Fg. 2 c, where, n addton to FM orderng at T C, a lne of AF-type orderng characterstc of the fully frustrated Hesenberg antferromagnet may occur at low temperatures. Ths transton, observed for the pure model 25 at T N =0.223, s analogous to the spn-glass orderng occurrng at T xy for x 0.2 n a- Fe 1 x Mn x G. The transverse nature of the FM and AF order appears to be the only sgnfcant consequence of couplng the two types of stes together va J FA. The FM and AF orderng s n no way weakened or destroyed by the frustrated couplng between the two types of stes. Ths result stands n stark contrast to clams that the orderng behavor of ron-rch a-fe x Zr 100 x alloys s due to competton between antferromagnetc clusters embedded n a ferromagnetc matrx, 36 and that orderng of the AF clusters destroys the preexstng FM order. Although there s now strong expermental evdence aganst ths vew, 37 the model studed here actually has ths specfc structure. For approprate compostons, t could be descrbed as a FM matrx wth AF clusters, but the orderng of these AF clusters clearly does not destroy the preexstng FM order. V. CONCLUSIONS We have used Monte Carlo methods to nvestgate the orderng of Hesenberg spns on sc, bcc, and fcc lattces for a ste-frustrated model. At all concentratons, the system forms at least one percolatng network of F or A stes and, n the mxed phase, both F and A stes percolate. When both F and A stes percolate on bpartte lattces, they both order as one would expect f the frustraton were removed, a fact readly PHYSICAL REVIEW B 73, apparent from the smlarty between Fgs. 2 and 3. In the case of the fcc lattce, AF orderng of A stes lkely occurs for 0.2 x 1, and thus, we expect that a tetracrtcal pont should occur here as well. The resultng phase dagrams for frustrated Fg. 2 and nonfrustrated models Fg. 3 are closely related, sharng smlar, f not the same, crtcal concentratons. The unversalty class of the transtons reman n the three-dmensonal Hesenberg unversalty class, n agreement wth the Hars crtera. The J FA bonds couplng the A and F stes lead only to mnor changes n the orderng: The transton temperatures are slghtly ncreased see Fg. 9, and the F and A clusters have mutually perpendcular orderng drectons see Fg. 13. For bpartte lattces, there s a tetracrtcal pont at x= 1 2 where FM and AF order develop smultaneously n the nfnte percolatng F and A clusters. The presence of long-range perodc order at all concentratons allows us to rule out spnglass orderng n these models for bpartte lattces. In the case of fcc lattces, we have attempted to detect the ferromagnetc transton only. However, by analogy wth the sc and bcc lattces, we expect that the percolatng A cluster undergoes the orderng characterstc of the fully frustrated fcc model. ACKNOWLEDGMENTS Ths work was supported by grants from the Natural Scences and Engneerng Research Councl of Canada, and Fonds pour la formaton de chercheurs et l ade à la recherche, Québec. We would also lke to thank Juan Gallego for help wth the Shre beowulf cluster D. H. Ryan n Recent Progress n Random Magnets, edted by D. H. Ryan World Scentfc, Sngapore, 1992, p.1. 2 D. H. Ryan, A. D. Beath, E. McCalla, J. van Lerop, and J. M. Cadogan, Phys. Rev. B 67, A. Kuprn, D. Warda, and D. H. Ryan, Phys. Rev. B 61, A. Aharony, Phys. Rev. Lett. 34, S. Fshman and A. Aharony, Phys. Rev. B 19, The ste frustrated model of Ref. 5 was consdered to be dentcal to a bond frustrated model, apart from short-range correlatons whch were consdered unmportant n determnng the longrange order. 7 S. F. Edwards and P. W. Anderson, J. Phys. F: Met. Phys. 5, M. Gabay and G. Toulouse, Phys. Rev. Lett. 47, D. H. Ryan, J. van Lerop, M. E. Pumarol, M. Roseman, and J. M. Cadogan, J. Appl. Phys. 89, M. Nelsen, D. H. Ryan, H. Guo, and M. Zuckermann, Phys. Rev. B 53, F. Matsubara, T. Tamya, and T. Shrakura, Phys. Rev. Lett. 77, S. Bekhech and B. W. Southern, Phys. Rev. B 70, R A. D. Beath and D. H. Ryan, J. Appl. Phys. 97, 10A A. B. Harrs, J. Phys. C 7, R. B. Stnchcombe n Phase Transtons and Crtcal Phenomenon edted by C. Domb and M. S. Green Academc Press, New York, 1983, Vol D. Stauffer, Introducton to Percolaton Theory Taylor & Francs, London, D. C. Matts, Phys. Lett. 56A, J. M. Luttnger, Phys. Rev. Lett. 37, G. Toulouse, Commun. Phys. London 2, E. Fradkn, B. A. Huberman, and S. H. Shenker, Phys. Rev. B 18, H. Muller-Krumbharr and K. Bnder, J. Stat. Phys. 8, R. G. Brown and M. Cftan, Phys. Rev. B 54, M. Creutz, Phys. Rev. D 36, A. D. Beath and D. H. Ryan, J. Appl. Phys. 97, 10A J. L. Alonso, A. Tarancón, H. G. Ballesteros, L. A. Fernández, V. Martín-Mayor, and A. Muñoz Sudupe, Phys. Rev. B 53, C. Holm and W. Janke, Phys. Rev. B 48, K. Chen, A. M. Ferrenberg, and D. P. Landau, Phys. Rev. B 48, A. D. Beath and D. H. Ryan, J. Appl. Phys. 95, A. Ghazal, P. Lallemand, and H. T. Dep, Physca A 134,

12 A. D. BEATH AND D. H. RYAN PHYSICAL REVIEW B 73, J. R. Thomson, H. Guo, D. H. Ryan, M. J. Zuckermann, and M. Grant, Phys. Rev. B 45, The FM/AF order n the equvalent ste frustrated model may be hdden, n the sense that the FM order n Matts models s hdden. However, snce bond frustrated models reman n the Hesenberg unversalty class for x 0.2, the hdden order would also be wthn the Hesenberg unversalty class. 32 The data shown n Fg. 13 are obtaned usng Metroplols dynamcs. To observe the correct behavor, we must exceed for the vector quanttes such that m f and m st complete many orbts. In addton, these long runs usng MCSs at low temperatures provde a good check for convergence of the scalar quanttes where, typcally, we use OR-MCSs. 33 L. Neél, Ann. Phys. Pars 18, L. W. Lee and A. P. Young, Phys. Rev. Lett. 90, M. Ahmadzadeh and A. W. Smpson, Phys. Rev. B 25, S. N. Kaul, J. Phys.: Condens. Matter 3, D. H. Ryan, J. M. Cadogan, and J. van Lerop, Phys. Rev. B 61, Informaton on our computer cluster, named the shre, can be found at