O (n ) tricriticality in two dimensions

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1 Front. Phys. Chna (007) 1: DOI /s y RESEARCH ARTICLE GUO Wen-an, Henk W.J. Blöte, Bernard Nenhus O (n ) trcrtcalty n two dmensons Hgher Educaton Press and Sprnger-Verlag 007 Abstract We present exact results for several unversal parameters of the trcrtcal O (n) model n two dmensons. The results apply to the range n 3/, and nclude the central charge and three scalng dmensons, assocated wth temperature, magnetc feld and the ntroducton of an nterface. Snce these results are based on an extrapolaton of known relatons between the O (n) and the Potts model, they cannot be consdered as rgorous. For ths reason, we perform an accurate numercal analyss of the central charge and the crtcal exponents. Ths analyss, whch s based on transfer-matrx calculatons on the honeycomb lattce, s n a full and precse agreement wth the theoretcal predctons. Keywords O(n) model, trcrtcalty, crtcal exponents PACS numbers 1 Introducton q, Cn, Fr, Hk The rapd development of the theory of crtcal phenomena and phase transtons has led to a consderable amount of exact results characterzng unversalty classes n two dmensons, ncludng results for models wth a contnuously GUO Wen-an ( ) Physcs Department, Beng Normal Unversty, Beng , Chna E-mal: waguo@bnu.edu.cn Henk W. J. Blöte Faculty of Appled Scences, Delft Unversty of Technology, P. O. Box 5046, 600 GA Delft, The Netherlands Insttuut Lorentz, Leden Unversty, P.O. Box 9506, 300 RA Leden, The Netherlands Bernard Nenhus Insttuut voor Theoretsche Fysca, Unverstet van Amsterdam, Valckenerstraat 65, The Netherlands Receved December, 006 varable symmetry parameter, such as the random-cluster representaton of the q-state Potts model and the loop representaton of the O (n) model. For a revew of these results, see the revew artcle by Nenhus [1]. Conspcuously mssng n ths collecton of exact results are, however, the unversal parameters of the trcrtcal O (n) loop model. Although numercal results are avalable [], thus far, only the magnetc exponent [3] has been conectured as a functon of the central charge c, whle the exact relaton between c and the symmetry parameter n remans unknown. In the present artcle we brdge ths gap by provdng exact formulas for the central charge and the man scalng dmensons of the trcrtcal O (n) model as a functon of n. These results are, n part, based on assumptons but t s stll reasonable to assume that they are exactly true, as we shall substantate below. The O (n) model s orgnally defned as a system of n-component spns on a lattce. The O (n) symmetry mposes full sotropy on the nteractons actng between the spns. Thus, par couplngs must have the form E = ε ( S S ) where and represent two neghborng lattce stes, and ε s some arbtrary functon that remans to be chosen. Graph expanson [4] of ths model transforms the partton sum nto a weghted sum of Euleran graphs, n whch n s no longer a dscrete parameter, but nstead assumes the role of a contnuous varable parameter. In ths context, a remarkable possblty appears f one chooses the model on the honeycomb lattce, and the par potental n the form ε ( p) log (1 + xp), where x s a measure of the nverse temperature. Then the graph expanson reduces to a gas of non-ntersectng and non- overlappng loops on the honeycomb lattce [5]. The partton sum of ths loop model can be subected to further mappngs on the Kagomé 6-vertex model and the Coulomb gas, and thus opens the possblty to derve exact results for the honeycomb O(n) model [6 9]. It s known that, n analogy wth the Potts model, trcrtcalty can be nduced n the O(n) model when a suffcent number of vacances s ntroduced. Ths was already confrmed for the case n = 0, whch descrbes the collapse of a polymer at the so-called theta pont, nduced by

2 104 polymer at the so-called theta pont, nduced by attractve nteractons between the polymer segments [10, 11], and for the Isng case n = 1 [1, 13] where the exstng results for the trcrtcal q = Potts model are applcable [1]. For the general O (n) loop model, the exstence of trcrtcal ponts was revealed by transfer-matrx analyses [, 15] for several values of n n the range n of n [, 15]. Whereas ths work yelded reasonably accurate values for some unversal parameters, no exact formulas were found for these parameters as a functon of n. However, as noted recently by Janke and Schakel [3], the conformal classfcaton of the magnetc exponent n terms of the Kac formula [16, 17] as X h = X m/, m/ (as explaned below n Eq. (10) and the accompanyng text), whch s known for the two cases n = 0 and 1, s also applcable to other values of n. Ths classfcaton s consstent wth numercal data obtaned for the trcrtcal O (n) model. But the relaton between n and the central charge c remans thus far unknown. It s our purpose to provde the mssng nformaton, namely the relaton between n and the central charge c. Ths wll enable a proposal for the exact expressons for the central charge and three crtcal exponents of the trcrtcal O (n) loop models as a functon of n, whch apply n the range n 3/. A less extensve report on ths research has already appeared n Ref. [18]. The outlne of the present paper s as follows. In the next secton, Secton, we ntroduce the model and ts underlyng theory. In Secton 3, we perform the numercal analyss of the unversal quanttes predcted by the theory, and we conclude the paper wth a short dscusson n Secton 4. represents the occuped faces of the honeycomb lattce. The product over L ncludes all spns except those on the vertces of the vacant hexagons. Nv s the number of vacances, N s the total number of faces (occuped or not), w s a parameter descrbng the spn-spn couplng, and represents all pars of nearest-neghbor stes on the honeycomb lattce. Snce the reduced spn-spn nteracton energy (1+ w S S ) as mpled by Eq. (1) retans the O (n) symmetry, we expect that the unversal propertes are applcable not ust to ths model, but to a whole class of models wth par nteractons of a smlar nature. As before, we apply an expanson [5] of the partton sum n powers of the couplng constant w, and thus obtan the loop representaton of the model, but ths tme vacances are ncluded. The confguratons of ths loop gas are the graphs G consstng of any non-negatve number of non- ntersectng closed loops coverng an arbtrary number of edges of the honeycomb lattce, whle avodng the edges of the vacant hexagons. The partton sum follows, completely analogous as n Ref. [5], as v v w l Z = v (1 v ) w n () L GL N N N N where N w s the number of vertces vsted by a loop, and N l the number of closed loops. The frst summaton s over all possble confguratons L of occuped faces. The second sum s over all graphs G allowed by the vacancy confguraton L. An example of a possble confguraton s shown n Fg. 1. N Model and theoretcal background As a representatve of the supposed trcrtcal O (n) unversalty class, we choose a generalzed verson of the O (n) spn model on the honeycomb lattce studed by Domany et al. [5]. The generalzaton concerns the ntroducton of vacances. Ths dluton s ntroduced by means of face varables that st n the center of the elementary hexagons of the honeycomb lattce. These face varables have two possble states: vacant wth weght v, or occuped wth weght 1 v. Furthermore, there s, as before, an n-component vector spn S on each vertex, provded that t s surrounded by three occuped hexagons. The edges of a vacant face,.e., a vacancy, cannot be vsted by a loop. The one-spn weght dstrbuton satsfes the O (n) symmetry. For reasons of smplcty, t s normalzed accordng to ds = 1 and dss S =n. Therefore, the partton functon gven by [6, 7] generalzes to N N N L L v v Z = v (1 v ) d S (1 + ws S ) (1) where the sum s on all allowed confguratons of ste and face varables, and L s a subset of the dual lattce and Fg. 1 A typcal confguraton of honeycomb O(n) model wth vacances. A possble soluton for the problem concernng the relaton between n and the central charge c, may be suggested by the hypothess that the generc crtcal O (n) model corresponds wth a trcrtcal q = n -state Potts model [6, 7]. Perhaps the soluton can be found by brngng the trcrtcal Potts model nto an even hgher multcrtcal state. It s known that ths can be realzed [19, 0] by the smultaneous ntroducton nto the Potts model of vacances and ther dual counterparts. The latter nteractons appear as four-spn couplngs that freeze the four Potts varables nto

3 105 the same state. It s possble to map the random-cluster representaton of ths model on a loop model on the the surroundng lattce. The latter model appears to allow, for specal choces of the nteracton parameters, solutons of the Yang-Baxter equatons [19]. There appears to be four branches of exact solutons parametrzed by q. One of these branches could be nterpreted n terms of tr-trcrtcal Potts transtons [19, 0]. The exact central charge and exponents of ths model follow as a functon of q from an alternatve representaton n terms of a Temperley-Leb model [19]. These results are confrmed by subsequent numercal analyses [0]. It s now very temptng to dentfy the loop weght q of the equvalent loop model wth the loop weght n of the model (), and to assume that the unversal propertes of the q-state tr-trcrtcal Potts model are equvalent wth those of the trcrtcal O( n = q) loop model. The central charge derved n Ref. [19] s expressed n n = q, determned by the followng equatons: 6 π 1 c = 1, cos =, = n (3) mm ( + 1) m+ 1 Furthermore, Ref. [19] yelded scalng dmensons of whch we quote three as: X k 1 = mm ( + 1) where we ntroduced an ndex = 1, or 3, and k s gven by wth cos[ kπ /( m+ 1)] = (5) = 1/ 1 = 1/ (6) 3 = 3 Numercal verfcaton In ths secton we provde a test of the valdty of the relaton proposed above between the trcrtcal O (n) model and the tr-trcrtcal Potts model. Ths test s numercal n nature and employs transfer-matrx calculatons for the honeycomb loop model wth vacances. The geometry of our loop model s chosen as a model wrapped on a cylnder, orented such that one of the lattce edge drectons corresponds wth that of the axs of the cylnder. We fx the unt of length such that the smaller dameter of the elementary hexagons equals 1. The weght factors appearng n the transfer matrx account for the change of the numbers of loops, vacances, and vertces covered by a (4) loop segment when a new layer of stes s appended to the cylnder. The largest egenvalue of the transfer matrx determnes the free energy densty. The fnte-sze dependence of the latter quantty allows a numercal determnaton of the central charge [1] of the correspondng conformal feld theory. The numercal analyss yelded three more egenvalues λ. These correspond wth the correlaton lengths assocated wth three dfferent three correlaton functons. These results allow fnte-sze estmates X (v, w, L) of the correspondng scalng dmensons X []. In ths way, the temperature dmenson X t s assocated wth the second egenvalue of the transfer matrx. The magnetc dmenson X h s assocated wth the largest egenvalue of a modfed transfer matrx descrbng a system wth a sngle loop segment runnng n the length drecton of the cylnder. The nterface exponent X m follows from the transfer matrx descrbng a system wthout such a sngle loop segment, but wth a modfed column of edges wth bond weghts of the opposte sgn. For further detals about the transfer-matrx technque, we refer the reader to Refs. [, 15, 3]. Let the vcnty of the trcrtcal fxed pont be parametrzed by a leadng relevant temperature-lke feld t 1, a subleadng relevant temperature feld t, and an rrelevant scalng feld u. The correspondng renormalzaton exponents are yt 1, yt and y u respectvely, wth yt 1 > yt. The trcrtcal pont s numercally approxmated by solvng the unknowns v and w n the two equatons: X (, v w, L) = X (, v w, L 1) = X (, v w, L ) (7) where the functons X ( = h, t, m) are provded by the transfer-matrx algorthm. Expanson of the fnte-sze-scalng functon at the trcrtcal pont ndcates that the soluton v(l) of Eq. (7) converges to the trcrtcal value v (tr) of v accordng to (tr) yu yt v( L) = v + al + (8) where a s an, n prncple, unknown ampltude. Smlarly w(v, L) converges to ts trcrtcal value w (tr). Furthermore, the values X (L) taken at the solutons of Eq. (7) are found to converge to the trcrtcal scalng dmenson X as : y u X ( L) = X + bul + (9) where b s another unknown ampltude. Ths procedure to locate the trcrtcal ponts and to estmate the trcrtcal exponents from Eq. (9) was performed both for X = X h and for X = X m. These calculatons were performed along the same lnes as n Ref. [], but here we use larger fnte szes up to L = 14, and moreover we nclude several values for n < 0. The results for the trcrtcal ponts are lsted n Table 1, together wth the estmated error margns and are shown n Fg.. In general they agree well wth those reported n Ref. [], but n some cases, we found that the apparent fnte-sze convergence was less rapd than suggested by the smaller range of system szes used n Ref. [], so that the estmated

4 106 error margns had to be enlarged. The new analyses usng X h and X m generated consstent results and thus provdes a consstency check for the numercal uncertantes. Table 1 Trcrtcal ponts determned from the scalng equatons for the magnetc and the nterface correlaton lengths. The estmated numercal uncertanty n the last decmal place s gven between parentheses. n v w (1) (1) (1) (1) (1) (1) (1) (1) (3) (1) (1) () () () () () 0 1/ (1) (1) (1) (1) () () (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) Fg. Trcrtcal lne of the O(n) model as a functon of n. The data ponts show the numercal data. The curve and ts proecton on the w-n plane are added to gude the eye. Furthermore, we used the second egenvalue of the transfer matrx at the trcrtcal ponts thus calculated, n order to obtan fnte-sze estmates of X t. These data were extrapolated; the results are, together wth the central charge and the two other exponents, lsted n Table. The estmated error margns are added between parentheses. A comparson of the numercal results for the central charge wth Eq. (3), as gven n Table, appears to be n good agreement wth the exact classfcaton of the trcrtcal O (n) model proposed above. Our numercal results for X t match X n Eq. (4), those for X m agree wth X 1. Usng the value of the central charge and m as a functon n, we confrm that the numercal results for the magnetc scalng dmenson agree wth the entry ( = m /, = m / ) n the Kac formula Table Transfer-matrx results for the central charge and three trcrtcal exponents. Estmated error margns n the last decmal place are added between parentheses. The numercal results are ndcated by (num). For comparson, we nclude theoretcal values obtaned from Eqs. (3), (4), and (10). For n < 3/, the temperature exponent X t becomes complex. n c (num) c (exact) X m (num) X m (exact) () (1) () () () (3) (1) (3) (1) 6/ () 1/ (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) 7/ (1) 3/ (1) () () (5) (4) (5) () 0.10 (1) n c (num) c (exact) X m (num) X m (exact) (1) (1) (1) (1) () (1) () / (1) 7/ () (3) () (1) (1) (1) /4 1/ (1) (1) (1) (1) () (1) (1) 1/ () 3/ (1) () (5) (1) 1/ (10) 0.13 (1) (10) 0.15 () X, [( m+ 1) m] 1 = mm ( + 1) (10) The O(n) model wth n > 0 appears to correspond wth branch 1 as defned n Ref. [19], and those for n < 0 wth branch. The numercal results and theoretcal values of the

5 107 central charge and the three scalng dmensons are shown as a functon of n n Fgs. (3) and (4). Fg. 3 Central charge (+) and temperature dmenson ( ) of the trcrtcal O (n) model ( ) of the trcrtcal O (n) model vs n. The numercal results are ndcated by the data ponts, and the theoretcal predctons by the two curves. 3/ < n are not suggestve of a dscontnuous transton, and allow at most a weak dscontnuty. But at the same tme t s clear from Tables 1 and that the estmated errors are ncreasng wth n for n > 3/, as a result of deteroratng fnte-sze convergence. Ths s suggestve of the possblty that an operator becomes margnal at n = 3/, n lne wth c = 1 (see Table ). Ths s remnscent of the q > 4 Potts model, where the margnal operator leads to anomalously slow fnte-sze convergence, whch obscures the weak frst-order character n a range of q near 4. The scenaro sketched above ndcates that the crtcal and trcrtcal O (n) branches are not connected. Furthermore, t s not suggestve of a relaton between the trcrtcal O (n) model and the crtcal Potts model, such as was recently quoted [3]. The results presented above apply to the non-ntersectng loop model. Loop ntersectons are rrelevant n the crtcal O (n) model, but they are relevant n the low-temperature phase [5]. Whle the possble relevance of such ntersectons could modfy the unversal behavor, ths appears not to be the case for the n = 1 trcrtcal O (n) loop model, snce ts exponents are known to agree wth those of the correspondng spn model,.e., the trcrtcal Blume-Capel model. Acknowledgements Ths research was supported by the Natonal Natural Scence Foundaton of Chna (Grant No ), by a grant from Beng Normal Unversty, and, n part, by the Lorentz Fund. References Fg. 4 Scalng dmensons X h ( ) and X m (+) of the trcrtcal O(n) model vs n. The data ponts represent the numercal results, the curves the theoretcal predctons. 4 Dscusson It appears that the formulas for X 1 and X, as gven by Eq. (4), do not correspond wth entres n the Kac table, at least not wth pars of ndces that are lnear n m. Ths explans the apparent dffculty to conecture the exact values of X m and X t from numercal data alone, even f supplemented by data for c. Ths problem dd not apply to X h whose classfcaton n terms of the Kac table was already gven above. It s noteworthy that, unlke the crtcal branch, whch extends to q = 4, the Potts tr-trcrtcal branch ends at q = 9/4. For q > 9/4 the exact soluton s no longer crtcal, and the transton probably turns frst-order [19, 0]. The correspondence q = n thus yelds the result that the trcrtcal O(n) branch ends at n = 3/, possbly wth a dscontnuous transton for n > 3/. At frst sght, the numercal results for 1. Nenhus B., n Phase Transtons and Crtcal Phenomena, edted by C. Domb and J.L. Lebowtz, London: Academc Press, 1987, Vol. 11. Guo W. -A., Blöte H. W. J., and Lu Y. -Y., Commun. Theor. Phys., 004, 41: Janke W. and Schakel A. M. J., Phys. Rev. Lett., 005, 95: Stanley H. E., Phys. Rev. Lett., 1968, 0: Domany E., Mukamel D., Nenhus B., and Schwmmer A., Nucl. Phys. B, 1981, 190 [FS3]: Nenhus B., Phys. Rev. Lett., 198, 49: Nenhus B., J. Stat. Phys., 1984, 34: Baxter R. J., J. Phys. A, 1986, 19: Baxter R. J., J. Phys. A, 1987, 0: Duplanter B. and Saleur H., Phys. Rev. Lett., 1987, 59: Blöte H. W. J., Batchelor M. T., and Nenhus B., Physca A, 1988, 51 : Blume M., Phys. Rev., 1966, 141: Capel H. W., Physca A, 1966, 3: Capel H. W., Phys. Lett., 1966, 3: Guo W. -A., Blöte H. W. J., and Nenhus B., Int. Mod. J. Phys. C, 1999, 10: Fredan D., Qu Z., and Shenker S., Phys. Rev. Lett., 1984, 5: Kac V. G., n Group Theoretcal Methods n Physcs, edted by W. Beglbock en A. Bohm, Lecture Notes n Physcs, Sprnger, New York, 1979, 94: Guo W. -A., Blöte H. W. J., and Nenhus B., Phys. Rev. Lett., 006,

6 108 96: Nenhus B., Warnaar S. O., and Blöte H. W. J., J. Phys. A, 1993, 6: Knops Y. M. M., Blöte H. W. J., and Nenhus B., J. Phys. A, 1993, 6: Blöte H. W. J., Cardy J. L., and Nghtngale M. P., Phys. Rev. Lett., 1986, 56: 74. Cardy J. L., J. Phys. A, 1984, 17: L Blöte H. W. J. and Nenhus B., J. Phys. A, 1989, : For revews, see e.g. Nghtngale M. P., n Fnte-Sze Scalng and Numercal Smulaton of Statstcal Systems, edted by V. Prvman, Sngapore: World Scentfc, 1990, and Barber M. N., n Phase Transtons and Crtcal Phenomena, Vol. 8, edted by Domb C. and Lebowtz J.L., Academc, New York, Jacobsen J. L., Read N., and Saleur H., Phys. Rev. Lett., 003, 90: