The Information Geometry of the One-Dimensional Potts Model

a Xi :cond-ma /0207180 1 6 Jul 2002 The In o ma ion Geome y o he One-Dimensional Po s Model B.P. Dolan Depa men o Ma hema ical Physics Na ional Uni e si y o I eland Maynoo h, I eland D.A. Johns on Dep . o Ma hema ics He io -Wa Uni e si y Ricca on Edinbu gh, EH14 4AS, Sco land and R. Kenna School o Ma hema ical and In o ma ion Sciences Co en y Uni e si y Co en y, CV1 5FB, England Janua y 20, 2004 Abs ac In a ious s a is ical-mechanical models he in oduc ion o a me ic on o he space o pa ame e s (e.g. he empe a u e a iable, β, and he ex e nal ield a iable, h, in he case o spin models) gi es an al e na i e pe spec i e on he phase s uc u e. Fo he one-dimensional Ising model he scala cu a u e, R, o his me ic can be calcula ed explici ly in he he modynamic limi and is ound o be R= 1 + cosh(h)/qsinh2(h) + exp(−4β). This is posi i e de ini e and, o physical ields and empe a u es, di e ges only a he ze o- empe a u e, ze o- ield “c i ical poin ” o he model. In his no e we calcula e R o he one-dimensional q-s a e Po s model inding an exp ession o he o m R=A(q, β, h) + B(q, β, h)/pη(q, β, h), whe e η(q, β, h) is he Po s analogue o sinh2(h) + exp(−4β). This is no longe posi i e de ini e, bu once again i di e ges only a he c i ical poin in he space o eal pa ame e s. We ema k, howe e , ha a nai e analy ic con inua ion o complex ield e eals a u he di e gence in he Ising and Po s cu a u es a he Lee-Yang edge. 1 In oduc ion The no ion o a dis ance be ween configu a ions in s a is ical-mechanical and la ice-field models has been discussed by se e al au ho s [1, 2, 3, 4, 5, 6], and he geome y ha his endows on he mani old, M, o he pa ame e s cha ac e ising he models explo ed. This dis ance is measu ed using he Fishe -Rao me ic [7], which is calcula ed om he Fishe in o ma ion ma ix o he sys em o in e es . When we conside a spin model in field, Mis a wo-dimensional mani old pa ame ised by (θ1, θ2) = (β, h). In his case, he componen s o he Fishe -Rao me ic ake he pa icula ly simple o m Gij =−∂i∂j , (1) whe e is he educed ee ene gy pe si e and ∂i=∂/∂θi. I has been sugges ed ha in such a geome iza ion o s a is ical mechanics he scala cu a u e, R, o Mplays a cen al ole [1, 2, 3]. Fo a spin model in field, wi h he me ic gi en in equ.(1), Rmay be calcula ed succinc ly as R=1 2G2 ∂2 β ∂β∂h ∂2 h ∂3 β ∂2 β∂h ∂β∂2 h ∂2 β∂h ∂β∂2 h ∂3 h  ,(2) whe e G= de (Gij). Since he only scale p esen nea c i icali y o a model displaying a highe -o de ansi ion is he co ela ion leng h, ξ, i has been hypo hesised on dimensional g ounds ha R ∼ ξd, whe e dis he dimensionali y o he sys em [1, 2, 3]. I we assume ha hype scaling holds, νd = 2 −α, his leads o R ∼ |ξ|(2−α)/ν.(3) To es he beha iou o R, one equi es models which a e sol able in field in o de o ob ain analy ic exp essions, and hese a e a he hin on he g ound. Indeed, Rhas been calcula ed o he mean-field and Be he-la ice Ising models [4] and he abo e scaling beha iou e ified. I has also been calcula ed o he one-dimensional Ising model [2] whe e i akes he ema kably simple o m, RIsing = 1 + cosh h qsinh2h+e−4β .(4) In his case Ris posi i e defini e and di e ges only a he ze o- empe a u e, ze o-field “c i ical poin ” o he model. The co ela ion leng h is gi en by ξ−1=−ln( anh(β)),(5) so ha ξ∼exp(2β) nea c i icali y, and (3) holds he e wi h α= 1, ν = 1 as expec ed1. A second no ewo hy ea u e o he Be he-la ice and 1D Ising models conce ns he me ic associa ed wi h (1). This me ic is diagonalized using he co esponding eno maliza ion g oup in a ian . 1The Be he la ice model also sa is ies he pos ula ed scaling, al hough he e a e some sub le ies coming om he exponen αbeing ze o [4]. Gi en he sho age o explici ly calculable examples, any u he addi ions o he lis would be e y wo hwhile in o de o see which ea u es in he models a e gene ic and which a e pa icula o he models conce ned. In his pape we discuss ano he example, he 1D q-s a e Po s model, whe e an exp ession o Rmay be ob ained in a e y simila manne o he Ising model (which is he q= 2 case o he Po s model). We compa e he p ope ies o he cu a u e, R, as well as he me ic in he Po s model and he Ising model, highligh ing bo h hei s uc u al simila i ies and diffe ences in de ail. 2 The 1D Po s Model The pa i ion unc ion o he 1D q-s a e Po s model is gi en by ZN(y, z) = X {σ} exp  β N X j=1 δ(σj, σj+1)−1 q!+h N X j=1 δ(σj,1) −1 q! ,(6) whe e he spins, σj∈ {1, . . . , q}, a e defined on he si es, j∈ {1,...N}, o he la ice and whe e we ha e defined y= exp(β) and z= exp(h) o la e calcula ional con enience. The model may be sol ed by ans e ma ix me hods [8], jus as he 1D Ising model. Fo gene al q he ull ans e ma ix T(y, z) may be w i en as q−2 diagonal elemen s, (y−1)(yz)−1/q, and a 2 ×2 ac o (y, z): (y, z) = 1 (yz)1/q yz z1/2√q−1 z1/2√q−1y+q−2.(7) The pa i ion unc ion is ZN(y, z) = T T(y, z)Nand he eigen alues o T(y, z) a e λ0, λ1,...,λq−1, whe e λ0 λ1)=1 2y(1 + z) + q−2±q(y(1 −z) + q−2)2+ (q−1)4z(yz)−1 q(8) and λ2,...λq−1= (y−1)(yz)−1/q. The educed ee ene gy pe si e in he he modynamic limi , N→ ∞, is hus gi en by =−lnλ0. I is s aigh o wa d o use his exp ession o he ee ene gy in equ.(2) o ob ain he cu a u e, R. In he cu en no a ion we e-de i e he exp ession o he Ising model 2as RIsing = 1 + y(1 + z) √y2−2y2z+y2z2+ 4z.(9) The exp ession o gene al qis simila in o m o his, and is RPo s =A(q, y, z) + B(q, y, z) qη(q, y, z),(10) whe e he coefficien s may be u he b oken down as A(q, y, z) = α(q, y, z)/γ(q, y, z)2and B(q, y, z) = β(q, y, z)/γ(q, y, z)2and a e smoo h unc ions o yand zand do no di e ge o fini e (physical) empe a u e o field. Fu he mo e η(q, y, z) = [y(1 −z) + q−2]2+ (q−1)4z. (11) 2The e is a ac o o wo di e ence in he de ini ions o β, h be ween he Ising and Po s no a ions coming om he di e en spin de ini ions. The exp essions o α(q, y, z), β(q, y, z) and γ(q, y, z) a e e y leng hy o gene al q(al- hough s ill easily ob ained) so, o he sake o compac ness, we w i e down only hose o q= 3, which al eady display he gene ic ea u es. We ha e α(3, y, z) = −1 4h−6−12 z+ 116 z2+ 80 z3−16 z4 +y(−44 −68 z+ 400 z2+ 552 z3−192 z4) +y2(−113 −64 z+ 138 z2+ 476 z3−1004 z4) +y3(−178 + 30 z+ 384 z2−1884 z3−2896 z4+ 8 z5) +y4(−188 + 458 z−222 z2−276 z3−3400 z4+ 64 z5) +y5(−106 + 592 z−474 z2−68 z3+ 1232 z4+ 120 z5) +y6(−17 + 294 z−368 z2+ 250 z3−127 z4−32 z5) +y7(4 + 66 z−136 z2+ 60 z3+ 2 z5+ 4 z4)i,(12) and β(3, y, z) = −1 4h−30 + 12 z+ 468 z2+ 768 z3+ 240 z4 +y(−82 −82 z+ 664 z2+ 3604 z3+ 1744 z4−16 z5) +y2(−171 + 12 z+ 1366 z2+ 2292 z3+ 1764 z4−160 z5) +y3(−261 + 213 z+ 1710 z2+ 846 z3−7584 z4−756 z5) +y4(−278 + 386 z+ 400 z2−3804 z3−8804 z4−2488 z5+ 8 z6) +y5(−254 + 570 z−1094 z2−5654 z3−1584 z4−3712 z5+ 64 z6) +y6(−165 + 504 z−1494 z2−1008 z3+ 891 z4+ 1152 z5+ 120 z6) +y7(−51 + 263 z−866 z2+ 986 z3−203 z4−97 z5−32 z6) +y8(−4 + 66 z−182 z2+ 188 z3−72 z4+ 2 z5+ 2 z6)i,(13) o he nume a o s, and γ(3, y, z) = −3−4z−2z2−4y−16 y z −16 y z2 −7y2+ 2 y2z−31 y2z2−4y3+ 4 y3z2,(14) and η(3, y, z) = 1 + 8 z+ 2 y−2y z +y2−2y2z+y2z2,(15) o he e ms in he denomina o s. In ze o-field (z= 1) he exp ession o Ris much mo e compac and is w i en o gene al qas R=(y+q−1)(4y2+ (q−2)y−(q−2)(q−1)) (q−1)(2y+q−2)2.(16) We see ha as y anges om 1 o ∞,R anges om (4−q)/(q−1) o ∞. In pa icula , he sign o he y= 1 (β= 0) limi o Rchanges a q= 4, al hough he gene al mo phology o Ras a unc ion o yand z emains he same o all q > 2 as we see below. The co ela ion leng h o he one-dimensional Po s model is defined in a simila manne o ha o he Ising model, ξ−1=−ln λ1 λ0!,(17) so ξ∼y o z= 1, y → ∞. We hus e ie e he expec ed scaling o R o he one- dimensional Po s model om equ.(3), namely R ∼ yas y→ ∞. The exponen s, as o he Ising model, a e α= 1, ν = 1. The gene al ea u es o Ra non-ze o empe a u e and field a e pe haps easies seen in a con ou plo as a unc ion o yand z. In Fig.1 we show he Ising (q= 2) case which has ce ain non-gene ic ea u es. The ±hsymme y o he Ising model is mani es as a z→1/z symme y in he plo o R. In addi ion, one can see ha Ris posi i e o all y and z. The maximum o R o a gi en y alue lies along he ze o field line a z= 1. In Fig.2, Ris plo ed o he 3-s a e Po s model, using he exp essions gi en abo e o A(3, y, z) and B(3, y, z). We see ha he e is no longe a z→1/z symme y and ha Ris no posi i e defini e. This beha iou is ypical o he q > 3 case, as can also be seen om he q= 10 esul s plo ed in Fig.3. I is also clea om Fig.3. ha he z= 1, y = 1 limi is nega i e unlike he 3-s a e model, as indica ed by equ.(16). When q > 2 he maximum o Rno longe lies on he ze o-field line hough i s locus is s ill simply de e mined, as we discuss in he nex sec ion. 3 Rema ks on Co-o dina es and Geodesics We ha e concen a ed so a on he cu a u e R, which is o cou se a geome ic in a ian and independen o he pa icula co-o dina e scheme we use in he calcula ion. The p ope ies o he pa icula me ics and line elemen s used in he calcula ion a e also o some in e es [5]. The me ic o he 1D Ising model calcula ed om equ.(1) gi es he line elemen ds2=e−4β (sinh2h+ e−4β)3/2×  4e−4βcosh h+ 8 sinh2h(cosh h+qsinh2h+ e−4β) (cosh h+qsinh2h+ e−4β)2dβ2 + 4 sinh h dβdh + cosh h dh2i.(18) Al hough his is clea ly non-diagonal, he choice o βand has co-o dina es has he ad an age o gi ing he simplified exp ession, equ.(2), o Rsince hey appea as linea mul iplie s in he Hamil onian. The me ic can be diagonalized by inspec ion in his case, and his p e e ed choice o co-o dina es u ns ou o be a he in e es ing om he physical poin o iew. I one conside s a decima ion ype eno maliza ion ans o ma ion on a 1D Ising chain (o ing) whe e e e y o he e ex is decima ed and he leng h scale sui ably adjus ed, hen one can exac ly de e mine he ela ions be ween he eno malized pa ame e s β′, h′and he o iginals β, h om ZN/2(β′, h′) = ANZN(β, h) (19) whe e ZN/2is he decima ed pa i ion unc ion and ZN(β, h) is he o iginal one. In addi ion o he pa ame e ans o ma ion he e is also an unimpo an o e all scaling ac o , AN. This ans o ma ion has an in a ian ρ= e2βsinh h, which is in effec encoding he in a iance o he magne iza ion unde he ans o ma ion since M=ρ √1 + ρ2.(20) I he co-o dina es a e ans o med om β, h o β, ρ hen, using dρ = 2e2βsinh hdβ + e2βcosh hdh, (21) one finds he diagonalized exp ession ds2=1 q(1 + ρ2)(e4β+ρ2)  4 e4βdβ2 (√1 + ρ2+qe4β+ρ2)2+dρ2 (1 + ρ2) .(22) The use o he eno maliza ion g oup in a ian , o al e na i ely he magne iza ion, also diagonalizes he me ic o he Ising model on a Be he la ice. I is he e o e na u al o ask whe he his ea u e pe sis s in he Po s models conside ed he e. Al hough he Po s me ic and line elemen s a e a he mo e complica ed han hose o he Ising model, he same gene al s uc u e is once again appa en ds2=(q−1) η(q, y, z)(3/2) ×  C(q, y, z) + D(q, y, z)qη(q, y, z) ˜ λ(q, y, z)2dβ2+ 4[yz(z−1)]dβdh +z[y(1 + z) + q−2]dh2i(23) whe e η(q, y, z) is defined in equ.(11) and ˜ λ(q, y, z) = y(1 + z) + q−2 + q(y(1 −z) + q−2)2+ (q−1)4z(24) is p opo ional o λ0, he dominan eigen alue. The unc ions C(q, y, z) and D(q, y, z) a e bo h easily calculable, bu a he long, and a e no ep oduced he e. I we ca y ou a decima ion on he 1D Po s model ZN/2(y′, z′) = ANZN(y, z),(25) hen he in a ian is gi en by ˜ρ=[y(1 −z) + q−2] √z,(26) which is again ela ed o he magne iza ion. Following he Ising p ocedu e, we can change a iables β, h o β, ˜ρusing d˜ρ=y(1 −z) √zdβ −1 2 [y(1 + z) + q−2] √zdh (27) and find ha he me ic is, indeed, diagonalized. In he Ising model i can be shown ha he line ρ= 0 (i.e. h= 0 o z= 1) is a geodesic o he me ic (1). In Fig.1 his is he line unning along he idge o R. Fo q6= 2 Po s models he local maximum in Rno longe lies a z= 1, as is clea om Fig.2 and Fig.3, howe e an analysis o he geodesic equa ions dV β ds + Γβ ββVβVβ+ 2Γβ βhVβVh+ Γβ hhVhVh=λ(s)Vβ dV h ds + Γh ββVβVβ+ 2Γh βhVβVh+ Γh hhVhVh=λ(s)Vh(28) whe e spa ame e izes he flow lines, Vβ=dβ/ds,Vh=dh/ds, he Γ a e he a ious Ch is offel symbols o he me ic o equ.(1), and λ(s) allows o non-affine pa ame e s, shows ha he line z= 1 is s ill a geodesic. To p o e his conside a ec o field wi h a flow line along z= 1 (h= 0). This flow line has Vh= 0 so equa ions (28) become dV β ds +Γβ ββh=0VβVβ=λ(s)Vβ(29) Γh ββh=0VβVβ= 0.(30) The fi s o hese equa ions is a second o de o dina y diffe en ial equa ion o β(s) which always has a solu ion. The second equi es Γh ββh=0 = 0 and i can easily be shown, using he me ic (23), ha his is indeed he case. Hence he line z= 1 is a geodesic o he me ic (23) o any alue o q. A diffe en change o a iable in he Po s models b ings one back o some hing e y simila o he Ising pic u e. I we ans o m z o w=yz y+q−2(31) and lea e yun ouched, we find ha he local maximum in Rlies on he line w= 1. We show Rplo ed agains win Fig.4 o he 3-s a e Po s model o compa ison wi h Fig.2. The pic u e is simila o highe q. This ans o ma ion does no g ea ly simpli y α(q, y, z), β(q, y, z) o γ(q, y, z) and we do no ep oduce he exp essions he e. In summa y, al hough he choice o β, h as pa ame e s is na u al in any calcula ion o R, he me ic is diagonalized o bo h he 1D Ising and Po s models when he eno - maliza ion g oup in a ian , o equi alen ly he magne iza ion, is used as a co-o dina e ins ead o h. The line z= 1 is a geodesic o he me ic (1) o any q. 4 A Lee-Yang Di e gence Some yea s ago Lee and Yang [9] add essed he ques ion o how he singula i ies associ- a ed wi h field-d i en phase ansi ions in Ising-like spin models on la ices a ose in he he modynamic limi . This was la e ex ended by a ious au ho s o o he models and o empe a u e-d i en ansi ions [10, 11]. Lee and Yang obse ed ha he ze oes o he pa i ion unc ion o a spin model in a complex ex e nal field on a fini e la ice would gi e ise o singula i ies in he ee ene gy. In he he modynamic limi hese complex ze oes mo e in o pinch he eal axis, signalling he he onse o a physical phase ansi ion. Typically, he loci o ze oes a e lines in he complex field o empe a u e plane and when he endpoin s o such lines occu a non-physical (i.e. complex) ex e nal field alues hey can be conside ed as o dina y c i ical poin s wi h an associa ed edge c i ical exponen , usually dubbed he Lee-Yang edge exponen [10]. The Lee-Yang ze oes o he one-dimensional Po s model on a pe iodic chain wi h N si es a e gi en by he solu ions [8, 12] o ZN= (λ1)N+ (λ0)N= 0 ⇔λ1= exp(inπ N)λ0(32) whe e λ0,1a e he eigen alues gi en in equ. (8) and nis odd. In he he modynamic limi he locus o ze oes is de e mined by |λ0|=|λ1|o η(q, y, z) = [y(1 −z) + q−2]2+ (q−1)4z= 0 (33) which can be sa isfied o complex (in he q= 2 Ising case, pu ely imagina y) alues o h. Unlike he Ising model case, he Lee-Yang ze oes o he 1D q6= 2 Po s models do no lie on he uni ci cle in he complex zplane. Howe e , p ecisely he change o co-o dina es we used in discussing he Po s me ics in he p e ious sec ion, w=yz/(y+q−2), places hem on he uni ci cle in he complex wplane. I we associa e a “field” wi h w ia w=exp(˜ h), hen pu ely imagina y alues o ˜ hgi e he ze oes, as in he Ising case3. F om hese conside a ions, i is clea ha Rwill also di e ge as he locus o ze oes is app oached o bo h Ising and Po s models i we allow ou sel es he libe y o an analy ic con inua ion o he field o complex h alues once Rhas been calcula ed, since R=A+B/√ηand A, B a e fini e as η→0. The p esence o he squa e oo means ha he di e gence is cha ac e ised by an exponen σ=−1/2 which is he Lee-Yang edge exponen o he one-dimensional Po s (and Ising) model [10]. The s a us o hese obse a ions a e a li le unclea o us, since he calcula ion o R has assumed a eal me ic geome y h oughou and such an a bi a y con inua ion in he final exp ession migh be a he dange ous. One is on sligh ly fi me g ound wi h he Ising model, since in ha case he equi ed con inua ion is o pu ely imagina y fields which co espond simply o a change in he signa u e o he me ic in β, ρ co-o dina es. The use o he wco-o dina e in he Po s case does, howe e , sugges a simila in e p e a ion since he e he ze oes occu o imagina y alues o ˜ h. Wi h hese ca ea s, i is none heless in e es ing ha he Lee-Yang edge ansi ion is s ill isible as a di e gence in R. 5 Conclusions We ha e seen ha he scala cu a u e, R, o he Fishe -Rao me ic may be ob ained o he 1D q-s a e Po s model in a e y simila ashion o he 1D Ising model since he ee ene gy can be calcula ed in field in bo h cases using ans e ma ix me hods. Al hough R o he q-s a e Po s model canno be exp essed as succinc ly as o he Ising model i s ill has he same gene al o m, R=A+B/√η, and di e ges only a he ze o empe a u e c i ical poin o he Po s model o physical empe a u e and field alues, i.e. y= 1 ...∞and z= 0 ...∞. We w o e down Rexplici ly he e o q= 3, bu no ed i was a simple ma e o calcula e i o a bi a y q. We obse ed ha he e we e some ea u es o he Ising model Rwhich did no pe sis o gene al q, as was clea om 3This is no , o cou se, he ield happea ing in he Hamil onian. con ou plo s. In pa icula , Rwas no longe posi i e defini e and he z→1/z symme y o he Ising model was no longe p esen . The choice o co-o dina e scheme used in he calcula ion o Rwas also discussed. We no ed ha al hough β, h we e a na u al choice om he poin o iew o simpli ying his calcula ion, he use o he eno maliza ion g oup in a ian in place o hdiagonalized he me ic o bo h he Ising and Po s models. A diffe en choice o coo dina es, in ol ing a escaling o zwas also employed o iden i y he local maximum in R. Fo any alue o q he line z= 1 (h= 0) is a geodesic o he Fishe -Rao me ic. Finally, we obse ed ha i complex field alues a e pe mi ed he e is a di e gence in Ra he Lee-Yang edge wi h an exponen σ=−1/2. We ha e confined ou discussion he e o physical alues o q, i.e. 2,3,..., in he Po s models. Since qappea s as a pa ame e in he ans e ma ix solu ion he e is in p inciple no ba ie o ex ending i o gene al non-in ege qand also o q < 2. The q→1 limi o he Po s model is ela ed o pe cola ion, o example, so he beha iou o Rmigh be o in e es in his limi oo. I would also be in e es ing o calcula e R o models wi h genuine ansi ions a fini e β, an exe cise ha has so a been confined o mean-field o mean-field-like spin models. Possibili ies which sugges hemsel es in his con ex a e he Ising model on plana andom g aphs [13] and he sphe ical model. 6 Acknowledgemen s D.J. was pa ially suppo ed by EC IHP ne wo k “Disc e e Random Geome ies: F om Solid S a e Physics o Quan um G a i y” HPRN-CT-1999-000161. D.J. and R.K. we e also pa ially suppo ed by an En e p ise I eland/B i ish Council Resea ch Visi s Scheme g an . Re e ences [1] G. Ruppeine , Re . Mod. Phys. 67 (1995) 605; Phys. Re . A 20 (1979) 1608; ibid A 24 (1980) 488. [2] H. Janyszek and R. M uga la, Phys. Re . A 39 (1989) 6515; H. Janyszek, Rep. Ma h. Phys. 24 (1986) 1; ibid 11. [3] D. B ody and N. Ri ie , Phys. Re . E 51, 1006 (1995); D. B ody and A. Ri z, Nucl. Phys. 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