One-Dimensional Ising Chain with Competing Interactions: Exact Results and Connection with Other Statistical Models

Transcription

1 Journal of Statstcal Physcs, Vol. 25, No. 1, 1981 One-Dmensonal Isng Chan wth Competng Interactons: Exact Results and Connecton wth Other Statstcal Models S. Redner ~ Receved March 20, 1980 We study the ground state propertes of a one-dmensonal Isng chan wth a nearest-neghbor ferromagnetc nteracton Jl, and a kth neghbor antferromagnetc nteracton J~. When Jk/J = -- 1/k, there exsts a hghly degenerate ground state wth a resdual entropy per spn. For the fnte chan wth free boundary condtons, we calculate the degeneracy of ths state exactly, and fnd that t s proportonal to the (N + k - l)th term n a generalzed Fbonacc sequence defned by, F~ ~)= F~ l + F}~. In addton, we show that ths one-dmensonal model s closely related to the followng problems: (a) a fully frustrated two-dmensonal Isng system wth a perodc arrangement of nearestneghbor ferro- and antferromagnetc bonds, (b) close-packng of dmers on a ladder, a 2 oo strp of the square lattce, and (c) "drected" self-avodng walks on fnte lattce strps. KEY WORDS: One-dmensonal Isng chan; competng nteractons; ground state degeneracy; Fbonacc sequence; close packng of dmers; drected self-avodng walks, 1. INTRODUCTION Consder a one-dmensonal Isng chan wth nearest- and kth-neghbor nteractons Jl and Jg, respectvely. The Hamltonan for ths system s ~C = --Jls Jk~aSS+ k (1) Work partally supported by grants from AFOSR and ARO. 1Center for Polymer Studes and Department of Physcs, Boston Unversty, Boston, Massachusetts /81/ / Plenum Publshng Corporaton

2 16 Redner For J1 > 0 and Jk < 0, the two nteractons compete n determnng the ground state of the system. Ths competton effect (1) s of current nterest because of ts possble relevance as the mechansm for both spn glass behavor 2 and helcal orderng n many magnetc systems. To descrbe helcal order n a d-dmensonal system, one can consder a chan of (d - l)-dmensonal ferromagnetcally coupled layers, wth nteractons along the chan gven by Eq. (1). Generally, the case of nearest- and second-neghbor nteractons (k = 2) has been studed most often. The resultng model, whch we call the RS model, (3) was frst consdered by Enz (4) and Ellott (5) to descrbe the magnetsm of the rare earths. Recent nvestgatons of ths model have focused on the Lfshtz pont, where there s a coexstence of dsordered, ferromagnetc, and helcal phases, (3'6'7) and on the strkng features at low temperatures. (8'9'1~ Here an nfnte sequence of commensurate phases occurs as J2/Jl vares, and as ToO these phases coalesce nto a sngle nfntely degenerate phase at J2/Jl = - 1/2. Ths degenerate phase exhbts some very strkng features; therefore we have studed the one-dmensonal system where we can obtan exact results. In the thermodynamc lmt, the exact soluton for the k = 2 case (RS model) has been found prevously, (11) but we are able to calculate exactly the ground state propertes for both the fnte and nfnte chan for all values of k. We fnd that when Jk/J 1 = --1/k, the ground state s hghly degenerate for any number of spns N >/2k. Moreover, for fnte N, we can express ths degeneracy n terms of a smple generalzaton of the Fbonacc sequence. From the asymptotc behavor of ths sequence, we derve very smply a closed-form expresson for the entropy n the thermodynamc lmt. In addton, we fnd that the Isng chan can be mapped onto several apparently dfferent statstcal models, leadng to further exact results. 2. THE GROUND STATE AND ITS DEGENERACY To begn, we consder the nature of ground state for J1 > 0 and varyng Jk (see Fg. 1). For Jk < 0 but suffcently small, ferromagnetsm occurs. For Jk < 0 but large, every kth spn orders antferromagnetcally, and to obtan the ground state, the energy n the remanng ferromagnetc bonds must be a mnmum. Ths happens when there are k spns 1", then k spns $, etc., and we denote ths phase as (k) J Now consder the case Jk/J1 ~- -- 1/k. In an nfnte chan, the energy per spn of the ferromagnetc and (k) phases, Erect o =-J1--Jk and E<k > = (2/k)J,, + [(k - 2)/k](-Jl + Jk), respectvely, are equal. Thus the 2We use the notaton of Ref. 8.

3 One-Dmensonal Isng Chan wth Competng Interactons 17 m ~,....,x ferro tttt... (k) t... t,+... +~,*... t,~... k 1 ~.kf... I.,..,..~... ~. t... t.~-~. ->k ->k ->k ->k D -Jk/J1 Fg. 1. The zero temperature phase dagram for the system descrbed by the model Hamltonan of Eq. (1). The three ordered phases that occur are ndcated schematcally. ground state conssts of a random mxture of ferromagnetc and (k) phases, leadng to a large degeneracy. We can descrbe ths degenerate phase by the followng smple pcture: Startng wth the ferromagnetc phase we may ntroduce a doman wall by flppng all the spns to one sde of a gven spn. Any number of such walls can be successvely created wth no energy cost as long as the walls are /> k lattce spacngs apart (and also /> k lattce spacngs from the end of the chan wth free boundares). Therefore ths phase has domans of >/ k spns I", followed by domans of /> k spns $, etc., as ndcated n Fg. 1. To fnd the degeneracy assocated wth ths phase, we frst descrbe our calculaton for the RS model (k = 2), and then outlne a smple generalzaton for arbtrary k. In what follows we employ free boundary condtons, and we also assume that the number of spns N n the chan s /> 2k. (For N < 2k, the ground state s only twofold degenerate.) Let a N be the total number of degenerate ground state spn confguratons of a chan of N spns. Ths number equals bs, the number of confguratons n whch the last two spns are parallel, plus CN, the number of confguratons n whch the last two spns are antparallel. Now b u can be obtaned by addng an addtonal spn parallel to the (N - 1)th spn for any confguraton n b u_ 1 and c N_. Hence bn = bn-1 + CN-1 = an-1 We can obtan c N by addng a spn antparallel to the (N- 1)th spn for any confguraton n b N_ 1" Therefore CN = bn- 1 (2a) (2b) (Notce that we cannot add an antparallel spn to a confguraton n c N because ths would lead to an energetcally unfavored doman of a sngle spn.) Combnng (2a) and (2b) yelds an = bn + C N = ajv_ 1 + an-2 (2c)

4 18 Redner Ths s just the recurson relaton for the Fbonacc sequence defned by FN (2)= FN(~I + FN(~2, and F(2) = F2 (2) = 1. We therefore obtan an= 2 FN(2+) 1. For arbtrary k, we may add the Nth spn parallel to the (N - 1)th spn for any confguraton n a N_ I. On the other hand f we add the Nth spn antparallel to the prevous spn, then at least spns N- 1 through N- k must all be antparauel to the Nth spn. Because of the determnancy n these k spns, any such state of N - 1 spns corresponds unquely to a state n a N_k. Hence we fnd an = an-1 + an-k (2d) Ths recurson relaton defnes a "generalzed" Fbonacc sequence FN (~) = F~(~I + F~(~, wth F1 (~) = F2 (k)... Fff ) = 1. From ths we obtan an = 2F(Nk)+ k- l " It s nterestng to examne the k dependence of the entropy per spn S (~) n the thermodynamc lmt. To obtan ths lmt, we frst note that the Fbonacc sequence becomes purely geometrc as N--> oc. Ths follows by wrtng FN(~I, FN(~2... FN(~k as the elements of a vector v N_ 1, and then wrtng the k k transfer matrx transformng v N_ 1 to v N. Ths matrx has only one egenvalue > 1, and ths gves the multple n the geometrc sequence. Equvalently we can fnd ths egenvalue more easly by usng Fff)~x N n the defnng relaton for the Fbonacc sequence. Ths yelds the characterstc equaton, x N = X N- 1 "1" X N- k, or x k = X ~- L + 1. From the largest root of ths equaton, we fnd S (2~ = log[(1 + V3-)/2]. Ths agrees wth the entropy found from takng the exact expresson for the free energy, (11~ and evaluatng t at J2/J~ /2. For k > 2, the largest root of the characterstc equaton yelds S (3) =1og( ), S (4) = log( ), S (5) = log( ), etc. As k---> o0, the spn domans become nfntely large, and ths corresponds to the entropy approachng 0. In ths lmt, the degenerate and (k) phases become dentcal to the ferromagnetc phase. 3. CONNECTION WITH OTHER STATISTICAL MODELS In addton to elucdatng the strkng features of the degenerate phase, we show that the Isng chan wth competng nteractons has close connectons wth several apparently dverse problems. (a) A fully frustrated two-dmensonal Isng model: The pled up domnoes (PUD) model ntroduced by Andr6 et a/. ~12) s a two-dmensonal Isng model wth a perodc arrangement of nearest-neghbor ferro- and antferromagnetc nteractons of dfferent strengths J and J', respectvely (see Fg. 2a). Every elementary plaquette conssts of an odd number of

5 One-Dmensonal Isng Chan wth Competng Interactons 19 J J J' (a) (b) (c) (d) 4- *- a-.-4 /' -H-.., 1 - [-; -. H., -! - (e), / 1, ---; /- Fg. 2. (a) The PUD model: The sngle and double bonds represent ferromagnetc nteractons J, and antferromagnetc nteractons J', respectvely. (b) A secton of the PUD model wth the dual lattce shown dashed. The + and - sgns ndcate the spns n a typcal ground state for [J'l > J. The antferromagnetc vertcal chans order "n phase," n one of two possble states. On the other hand, there s a large degeneracy assocated wth the possble states of the vertcal ferromagnetc chans. (c) Under the transformaton s ~ tt+ ], the vertcal ferromagnetc chan n (b) maps nto a chan wth alternatng nearest-neghbor nteractons (zg-zags), and ferromagnetc second-neghbor nteractons (vertcal). The + and - sgns now ndcate the sgns of the new spn varables t. These are defned only up to an overall sgn; consequently there s a two-to-one correspondence between the ground states of the RS model and a ferromagnetc chan n the PUD model. The open crcles ndcate the spns to be redefned n order to map the Isng chan to the RS model through a gauge transformaton. (d) The RS model, wth ferromagnetc nearest-neghbor nteractons, and antferromagnetc second-neghbor nteractons. The + and - sgns ndcate the spn states obtaned after the gauge transformaton from (c). Shown dashed s the dual lattce. (e) The dual ladder of the PUD model and the spns states of the orgnal lattce from (b). The close-packed dmer confguratons that are derved from ths spn state are ndcated by the full lnes. (f) The dual of the RS model wth the close-packed dmer confguraton correspondng to the spn state of (d). From ths dmer confguraton we can obtan the geometrcally equvalent confguraton (e) as follows: Frst we "straghten out" the dual lattce, and then slde alternate horzontal bonds to one sde of the vertcal chan. Fnally, we add a second vertcal dmer chan dentcal to the frst one to obtan the dual ladder n the dmer state (e). antferromagnetc bonds, hence each plaquette s frustrated n the ground state.(l) Ths model s nterestng because t s equvalent to a spn-glass-lke model n whch the J and J' bonds can be rearranged n an apparently random way by a gauge transformaton.( ]3) In the ground state, the vertcal chans of antferromagnetc bonds order "n phase" when IJ't > J (see Fg. 2b). Consequently the system s

6 20 Redner essentally one dmensonal because each vertcal ferromagnetc chan feels only a staggered magnetc feld of strength 2J n addton to the nearestneghbor nteracton. Thus we may wrte the followng effectve Hamltonan for the ferromagnetc chans: ~)~eff ~- - J2 ss+ l- 2J~ ( - 1)s (3a) We can show that ths system s equvalent to the RS model by frst usng the well-known transformaton s ---> tt+ 1 to rewrte (3a) as ~eff = - - J~ tt+ 2-2J~,, (- 1)tt~+ l (3b) Now we have a spn chan wth alternatng ferro- and antferromagnetc nearest-neghbor nteractons, and a second-neghbor ferromagnetc nteracton as shown n Fg. 2c. Next we perform the gauge transformaton ndcated n the fgure. We reverse the sgns of the spns at the crcled stes, and reverse the sgns of the bonds ncdent on these spns. Thus all nearest-neghbor bonds become ferromagnetc, and all second-neghbor bonds become antferromagnetc, and we have obtaned the RS model when J2/Jl = -I/2 (see Fg. 2d). (b) Dmer statstcs: Another way to understand the equvalence of the PUD and RS models, s to map the degenerate spn ground states nto close-packed dmer confguratons on the respectve dual lattces. That s, we place a dmer on a dual bond whch crosses a frustrated bond on the orgnal lattce (Fg. 2e). In the ground state of the PUD model, exactly one bond per plaquette s frustrated; ths mples that the dmers must be close packed on the dual. (t2) Moreover, because the antferromagnetc chans are all "n phase," frustrated horzontal bonds occur n pars. Consequently, the vertcal dmers between any two antferromagnetc chans must also occur n pars. Thus evaluatng the ground state entropy of the PUD model s equvalent to countng the number of correspondng close-packed dmer confguratons on a "ladder," a 2 ~ strp of the square lattce. Now consder the RS model: The mnmum energy spn states also map to close-packed dmer confguratons on the dual, a strp of the hexagonal lattce (Fg. 2f). By followng the smple geometrc manpulatons ndcated n the lower porton of Fg. 2, we deduce that countng the close-packed dmer confguratons for the two problems are dentcal. (c) "Drected" self-avodng walks: We defne "drected" self-avodng walks (SAW) as random walks whch may vst a partcular lattce ste only once, and whch have the addtonal constrant that steps n one drecton are prohbted (see Fg. 3a). To see the connecton between such walks on a ladder and the Isng chan, we magne placng a spn up at each vsted ste on the upper edge of the ladder, and vce versa on the lower edge. Upon

7 One-Dmensonal Isng Chan wth Competng Interactons 21 1` f 1` 1` 1` 1` ' 'spn up' lllllllj ~ ~ ~ ~ J, ~ ~ 'spn down' (a) (b) Fg. 3. (a) A drected SAW on the ladder of wdth two lattce spacngs. Steps to the left are prohbted. At each ste, we place a spn up f the walk passes through a ste on the upper edge of the ladder, and vce versa for the lower edge. As the walk proceeds we trace out one of the spn confguratons n the degenerate ground state of the RS model. (b) A typcal SAW on a 3 oo strp of the square lattce, wth perodc boundary condtons. The vertcal bonds drectly jonng the top and bottom rows are not drawn. The walk shown belongs to the class b N as defned n the text. followng the path of a typcal drected SAW, a sequence of spns s traced out. By constructon, domans of parallel spns must be at least two lattce spacngs n sze. Consequently we have obtaned one of the degenerate ground states of the RS model (k = 2). We can generalze ths constructon to arbtrary k n a straghtforward manner to SAWs n whch there must be at least k - 1 horzontal steps between vertcal steps. These drected SAW confguratons are somorphc to the degenerate ground states of the Isng chan when J~/J~ = - 1/k. These results lead us to consder the problem of drected SAWs on lattce strps of arbtrary wdth l (see Fg. 3b). In the nfnte twodmensonal lmt (l~oo), ths has been solved exactly by Fsher and Sykes (14) n connecton wth obtanng rgorous bounds on the connectve constant or effectve coordnaton number for random SAWs on the square lattce. Our soluton complements ther work by fndng the generatng functon for drected SAWs on fnte strps of arbtrary wdth. We consder here perodc boundary condtons n the transverse drecton by wrappng the strp onto a cylnder. The free boundary case can also be treated by the method descrbed here, but the detals are consderably more complcated. We derve the generatng functon for the l-- 3 case explctly, and generalzaton to arbtrary l follows drectly. Let a N be the total number of N-step-drected SAWs. Ths number conssts of all walks wth the last step horzontal (=--bn), walks wth the second-to-last step horzontal and the last step vertcal (-- Cu), and walks wth the thrd-to-last step horzontal and the last two steps vertcal (-- dn). Now b u can be obtaned by addng a horzontal bond to any walk n the

8 22 Redner three classes. That s, bn = bn-1 -t- C N_ 1 + dn-1 We obtan c N by addng a vertcal bond n two possble drectons to each walk n b N_ 1 (ths holds only for perodc boundares). Smlarly we obtan d u by addng a vertcal bond to each walk n e N_. Thus (4a) cu = 2b u_, and du = cu- 1 (4b) Combnng these yelds for the total number of N-step-drected SAWS, an = bn + CN + d N = an_ l + 2au_ z + 2aN_ 3 (4c) For the case of a strp of l lattce spacngs we fnd smlarly a N = an_ 1 + 2(aN_ 2 + an_ an_l) (48) In the lmt N~ 0% an~t~, where the connectve constant ~z s the largest root of the characterstc equaton derved from (4d), x t= x t-1 + 2xt-2+ 2xl , and ths gves /~2 = 2, /z 3 = and ]L 4 = etc. As l~ oo, /~z approaches a fnte lmt whch s the soluton of x t=x t-i (l+2/x+2/x 2+...), or x=(l+l/x)/(1-1/x). From ths we fnd that lmt~ ~/h = 1 + ~ = In ths lmt our result duplcates that of Fsher and Sykes. In summary, we have studed the propertes of a fnte onedmensonal Isng chan wth competng nteractons Jl and Jk. At zero temperature there exsts an nfntely degenerate ground state for a partcular value of Jk/J1. We have calculated ths degeneracy exactly for the nfnte chan and for the fnte chan wth free boundary condtons. In the latter case the degeneracy can be expressed as the terms n a generalzed Fbonacc sequence. We have also shown that the Isng chan s closely related to the followng problems: the PUD model, a two-dmensonal Isng system wth a perodc arrangement of ferro- and antferromagnetc bonds, close packng of dmers on a ladder, a 2 ~ strp of the square lattce, and drected self-avodng walks on the ladder. Fnally we have generalzed, and calculated exactly, the propertes of drected SAWs on lattce strps of arbtrary wdth. ACKNOWLEDGMENTS I am especally grateful to H. E. Stanley for proposng the approach for some of the dervatons presented n ths paper. I wsh to thank hm, and R. Bdaux, P. J. Reynolds, and T. Wtten for many llumnatng dscussons. I also thank A. Huber and S. Muto for a crtcal readng of the manuscrpt and many constructve suggestons.

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