RG treatment of an Ising chain with long range interactions

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1 RG treatment of an Isng chan wth long range nteractons Davd Ramrez Department of Physcs Undergraduate Massachusetts Insttute of Technology, Cambrdge, MA 039, USA An analyss of a one-dmensonal Isng model wth /r nteractons s conducted usng renormalzaton group technques. An explct example for a rescalng factor b 3 s presented, ncludng values for T c and the crtcal exponent ν. Results for general b, n partcular the lmtng case of b, are used to demonstrate the exstence of phase transtons at fnte temperature T c() 0 for <. I. INTRODUCTION The Isng model s one of the fundamental systems n statstcal mechancs. The model s useful for multple reasons; t s smplcty makes t an mportant pedagogcal example and allows for effcent numercal smulaton, yet t s also able to model varous types of physcal phenomena, such as ferromagnetsm, lattce gases, and even neuron actvty n the bran. The most famlar Isng models are those wth ferromagnetc, nearestneghbor nteractons n one and two dmensons. Both cases can be solved exactly, usng ether renormalzaton group or transfer matrx methods (see, for example, []). The three-dmensonal, nearest-neghbor Isng model remans unsolved, as do most one- and two-dmensonal models wth more complcated nteractons. An mportant category of Isng models are ferromagnetc models wth long range nteractons, by whch we mean a d-dmensonal lattce of N spn-/ Isng spns σ ± subject to the Hamltonan H J (,j) σ σ j. () Here and J are postve constants, wth (, j) denotng dstnct pars of lattce stes, and r j s the dstance between stes and j. We take r j to be measured n unts of the lattce spacng a. The lmt yelds the nearest-neghbor Isng model, whle for 0 d, t can be shown that the system exhbts nonextensve behavor and accordngly the thermodynamc lmt s not well-defned. We shall restrct to the case where > d; nonextensve systems are dscussed n the references [] and [3]. In ths letter, we consder the one-dmensonal model of Eq.. In ths case, we have r j j. We study the system usng a real-space renormalzaton group (RG) scheme usng the Nemejer-van Leeuwen (NvL) cumulant expanson (see []) to obtan the crtcal temperature T c as well as the crtcal exponent ν governng the decay of the correlaton length. The paper s outlned as follows; r j Electronc address: d ram@mt.edu frst, we explctly perform RG for a rescalng factor of b 3 and obtan T c and ν as functons of. We then generalze to arbtrary b, and n partcular, we look at the lmt b to obtan exact results. II. NVL CUMULANT EXPANSION FOR b 3 The partton functon for the system n consderaton s gven by Z e βh[{σ}] exp σ σ j, r {σ }± {σ }± (,j) j () where we have ntroduced the dmensonless couplng βj J/k B T. To perform RG, we group the spns nto cells α of sze b 3, as depcted n Fg.. The spns σ α are defned usng the majorty rule σ α sgn[σ α + σ α + σ 3 α]. (3) The new spns {σ α} are well-defned and two-valued, snce b s odd. We construct the renormalzed nteractons between the σ α by summng over the nternal spns {σ α},.e. e βh [{σ α }] {σ α }± δ σ α,σ α e βh[{σ α }], (4) where δ σ α,σ α s f sgn[σ α + σα + σα] 3 σ α and s otherwse. The bass for the NvL expanson s to separate the Hamltonan nto two parts H H 0 + V, where H 0 contans only ntracell nteractons whle V contans the ntercell nteractons. We then treat V perturbatvely. We frst defne the expectaton value O 0 Z 0 {σ α }± δ σ α,σ α e βh0[{σ α }] O, (5) where the unperturbed partton functon Z 0 s defned as Z 0 [{σ α}] α }]. (6) {σ α }± δ σ α,σ α e βh0[{σ

2 the /r nteractons; a smple computaton yelds: σ α 0 σ 3 α 0 σ α Z 0,α ( e +/ + e +/) σ α 0 σ α Z 0,α ( e +/ + e / e +/). () FIG. : Groupng of spns nto blocks of sze b 3. We note that Z 0 depends on the confguraton of the σ α. Wth these defntons, we can wrte Eq. 4 as e βh [{σ α }] Z 0 [ βv 0 + ] (βv) 0 Z 0 e βv 0. (7) From ths we obtan the renormalzed Hamltonan: βh [{σ α}] ln Z 0 [{σ α}] + ln e βv 0. (8) For the Hamltonan used n Eq., H 0 s gven by βh 0 α [ σασ α + σασ α 3 + ] σ ασα 3. (9) The correspondng partton functon s then [e +/ + e / + e +/] Z 0 [{σ α}] α [e +/ + e / + e +/] N/3 Z N/3 0,α, (0) where we have defned the partton functon for a partcular block Z 0,α for smplcty. Note that Z 0 does not depend explctly on σ α, and thus the ln Z 0 n Eq. 8 s smply an addtve constant. The frst term n the cumulant expanson Eq. 8 s smply β V 0 σασ j β 0, () r (α,β) α j j β where the sum over (α, β) ndcates summng over dstnct pars of blocks and, j label the spns n blocks α, β respectvely. Snce the weght of the expectaton values n Eq. 5 s a block-ndependent probablty dstrbuton, we have σ ασ j β 0 σ α 0 σ j β 0. The averages σ α 0 wll clearly be ndependent of α, but wll depend on, due to We can smplfy notaton by wrtng σα 0 a ()σ α/z 0,α, wth a () gven by the coeffcents above. We now let r αβ be the dstance between the center stes,.e. the dstance between spns σα and σβ, n unts of the rescalng length b 3. The dstances r j, between the spns σα and σ j β, then can take one of fve values: 3r αβ, 3r αβ ±, or 3r αβ ±. Thus, for r αβ, we can approxmate r j 3r αβ. (3) Ths approxmaton has the effect of weakenng the local nteractons between blocks; for example, the worst case s when r j, so accordng to Eq. 3, we are approxmatng r j by 3, and therefore the nearest-neghbor nteractons between the spns at the edges of the block are weghted by /3 rather than. Snce the approxmaton only dffers drastcally for the local nteractons between blocks, there should not be a sgnfcant effect on the propertes of the crtcal phenomena, whch are exctatons of long-wavelength modes of the system. Wth ths, we can now evaluate the sums n Eq., obtanng β V 0 3 where 3 s defned as 3() Z 0,α b [ 3 a () r (α,β) αβ ] σ ασ β, (4) Z 0,α b [ 3e +/ + e / + e k+/]. (5) Therefore, to frst order n V, the renormalzed Hamltonan s gven by βh N 3 ln Z 0,α + r (α,β) αβ σ ασ β. (6) As desred, βh has the same form as βh. We now seek to determne the crtcal ponts of the system by analyzng the fxed ponts of the recurrence relaton Eq. 5. A fxed pont has 3( ). A plot of 3 as a functon of s shown n Fg., where the ntersectons of the curves wth the straght lne ndcate fxed ponts. The values of can be calculated numercally and are shown n Table I for a few select.

3 3 III. RESULTS FOR ARBITRARY b FIG. : Plot of recurrence relaton Eq. 5 for the dmensonless couplng. The black lne corresponds to b. c c ν TABLE I: Approxmate values of c, c, and ν for varous, wth a rescalng length of b 3. From Eq. 5, t s clear that for every there exst the two trval fxed ponts at 0 (T ) and (T 0). As can be seen from Fg., there exsts some c such that for < c, d 3/d s greater than one at 0, mplyng that 0 s a repulsve fxed pont, and thus T c. However, f > c, then d 3/d s less than one at 0, mplyng that 0 s an attractve fxed pont. Furthermore, snce 3 scales as 3 3 for, f c < <, there exsts another fxed pont at fnte c () > 0. Snce the fxed pont at 0 s attractve and the monotoncty of 3 as a functon of requres that 3 can cross only once at fnte, c () must be a repulsve fxed pont, ndcatng a phase transton at a fnte temperature. For >, there s no fnte fxed pont due to the asymptotc scalng of 3, so must be a repulsve fxed pont and therefore T c 0. The exponent ν governng the dvergence of the correlaton length can be calculated from d 3 d b /ν ν() c ln 3 ln(d 3 /d c ). (7) The results from calculatng ν() numercally are shown n Table I. We now consder performng the same RG transformaton for b k +, where k s an arbtrary postve nteger. Snce b s odd, the majorty rule Eq. 3 can be generalzed to σ α sgn( b σ α). Generalzng to larger b s of nterest because the nth cumulant of e βv 0 wll be of order b n ; therefore, the cumulant expanson s becomes a seres expanson n powers of b, mplyng that H V 0 s a better approxmaton for b large. The general formalsm of the prevous secton generalzes n the obvous way; to frst order n the cumulant expanson we have the followng generalzaton of Eq. H V 0 σα 0 σ j β 0, (8) r (α,β) α j j β except here, j range from to b, labelng the b spns n the block α, β respectvely. Smlarly, the averages n Eq. 8 have the form σα 0 a ()σ α, where a () does not depend on the block α. Followng Eq. 3, we approxmate r j br αβ. Wth ths, the recurson relaton Eq. 5 generalzes to b() Z 0,α b [ b a ()]. (9) The dscusson of fxed ponts proceeds as before; t s easy to see that for, a ()/Z 0,α for all (see appendx), and therefore Eq. 9 mples that b ( ) b. There are stll two trval fxed ponts 0 and, and there exsts an c (b) such that 0 s repulsve f < c (b) and attractve f > c (b). If c (b) < <, then by monotoncty and the asymptotc scalng, there must exst a fxed pont at fnte. We now calculate c (b). c (b) s defned by the followng condton d b d [ b Z0,α ( 0)b c a ( 0)]. 0 (0) As shown n the appendx, a (0)/Z 0,α s gven, for all, by a ( 0) Z 0,α ( 0) Wth ths, Eq. 0 gves b c γ (b) c (b )! γ(b) b ( b [ +!). () ] ln γ(b). () ln b For b 3, ths gves c (3) 0.738, consstent wth the results n Table I. Usng Strlng s formula for n! as n, we see that γ(b) b / for b large and thus c n the lmt b. Therefore, we expect that the

4 4 exact soluton to the model exhbts a phase transton at fnte temperature for < <, a result proven rgorously n [4]. As from below, we expect that c. In ths lmt, expandng Eq. 9 yelds (see appendx) b() b [b 4e B(b)], (3) where B(b) b n /n. As dscussed n the appendx, B(b) s the energy dfference between the ground state and the frst excted state of a sngle block of b spns. Rearrangng ths expresson, we obtan α A(b)e B(b)c, (4) where A(b) 8/(b ln b). As b, we have A 0 and B π /3. Snce A(b) governs the regon where Eq. 4 holds and A 0 as b, Eq. 4 suggests that there s a phase transton at a fnte temperature for,.e. T c ( ) 0. IV. CONCLUSION In summary, we have seen that there exst three regmes for the one-dmensonal Isng model wth /r nteractons: () : The system s non-extensve and tradtonal statstcal mechancs cannot be used, () < : The system exhbts a phase transton at fnte T c () 0, () > : The phase transton occurs at T c 0. These results were obtaned by studyng the RG recurrence relatons n the b lmt, and reproduce the exact results, provdng confdence n the methods used. Wth more sophstcated analyss, t s possble to extract an estmate of c ( ) π / 0.83; such a method s demonstrated n [5]. Ths approxmaton falls wthn the numercal estmate of c 0.79 ± 0.05, gven n [6]. The case s partcularly nterestng as t can be mapped nto the spn-/ ondo problem, a model of electrcal resstvty [6]. The methods used n ths letter, modeled after those used n [5], could be generalzed to hgher dmensons or more complex nteractons, such as the one-dmensonal Potts model wth long range nteractons [7]. Appendx In ths secton, we demonstrate how the values for γ(b) and the asymptotc scalng form of b (), Eq. and Eq. 3 respectvely, were obtaned. To smplfy some of the equatons, we recall our defnton of k, b k +. To obtan Eq., we frst fnd the form of Z 0,α at 0. For a gven σ α, there are k (b )/ possble arrangements of the spns σα: one arrangement where σα σ α for all n block α, ( b ) arrangements wth one spn opposte of σ α, ( b ) arrangements wth two spns opposte, etc. Snce 0, each confguraton gves the same weght e 0, so we have Z 0,α ( 0) k ( ) k + 4 k b. (5) 0 We now need the form of a ( 0), whch s ndependent of (snce 0) and gven by the dfference between the number of confguratons wth σα σ α and the number of confguratons wth σα σ α. Consder the arrangement wth spns opposte of σ α. There are ( b ) possble confguratons of ths form. ( b ) ( k ) of these arrangements have σ α opposte of σ α. Therefore, usng some standard factoral denttes (see, for example, [8]), we have a ( 0) Z 0,α ( 0) k ( ) k k + ( ) k 0 ( ) ( ) k πk! + k (k /)! 0 4 k + (k)! (b )! (k!) ( b. (6)!) Ths combnes wth Eq. 5 to yeld Eq.. We now obtan the asymptotc expresson Eq. 3 for b (). We consder only the two lowest energy terms n Z 0,α and a (). The ground state s the unque confguraton wth all spns algned wth σ α, whle the frst excted state has two possble confguratons: σα σ α or σα b σ α, whle the rest of the spns are algned wth σ α. The dfference n energy between these states s gven by β(e 0 E ) b /, where E 0 denotes the ground state energy and E denotes the energy of the th excted state. Therefore, the frst two terms of Z 0,α () are Z 0,α ( ) e βe0 ( + e P / + O(e β E )), (7) where E E 0 E. We look for a smlar expanson of a (). If, b, then both of the frst excted state confguratons have σα σ α, and therefore the coeffcent of e βe wll have a factor of. However, f, b, then one confguraton has σα σ α whle the other has σα σ α; therefore the contrbutons from these two cancel, yeldng { e βe0 ( + e P / + O(e β E )), b a () e βe0 ( + O(e β E )), b. (8)

5 5 We note that for large enough, a ()/Z 0,α, as clamed n the prevous secton. Therefore, summng over the a and dvdng by Eq. 7, we obtan In the lmt, we can replace wth two n the exponent, obtanng Eq. 3. a ( ) Z 0,α ( ) P b + (b / )e + O(e β E ) + e P / + O(e β E ) b 4e P / + O(e β E, e 4 ). (9) [] M. ardar, Statstcal Physcs of Felds (Cambrdge Unversty Press, New York, 007). [] S. Cannas and F. A. Tamart, Phys. Rev. B 54, R66 (996). [3] F. Nobre and C. Tsalls, Physca A 3, 337 (995). [4] R. S. Ells, Entropy, Large Devatons, and Statstcal Mechancs (Sprnger, New York, 985). [5] S. Cannas, Phys. Rev. B 5, 3034 (995). [6] P. W. Anderson and G. Yuval, J. Phys. C 4, 607 (97). [7] S. Cannas and A. de Magalhaes, J. Phys. A 30, 3345 (997). [8] M. Abramowtz and I. A. Stegun, Handbook of Mathematcal Functons wth Formulas, Graphs, and Mathematcal Tables (Dover, 970).