More on Algebraic Structure of the Complete Partition Function for the $ Z_n $ - Potts Model, Part 1

More on Algebraic Str ucture of the Complete P ar tition Function for the Z n - Pott s Model, P ar t 1 A. K. Kw a ´ sniewski ∗ , W. Ba jguz ∗∗ (*)the mem ber of the Institute of Co m binatorics and its Applications F a cult y of P hysics, Bia lystok Universit y ul. Lip owa 4 1, 15 424 Bia ly stok, Poland e-mail: kwandr@gma il.com ∗∗ Bia lystok Universit y , Institute of Computer Science, ul. Sosnow a 64 PL – 15- 887 Bia lystokk, Poland Abstract In this firs t pa rt of a larger rev iew undertaking the results of the first author and a part of the second author do ctor disser tation are pre sented. Next w e plan to give a survey of a now adays situation in the are a of inv estigation. Here we rep ort on what follows. Calculation of the partition function for any vector Potts mo del is at fir st re- duced to the calculatio n of traces of pro ducts of the gener alized Cliffor d algebra generator s. The formula for such traces is derived. The latter enables o ne , in principle, to use an explicit ca lculation algorithm for partition functions also in other models for which the transfer ma trix is an e lement from gener alized Clifford algebra . The metho d - simple for Z 2 case - b ecomes complicated for Z n , n > 2, howev er everything is controlled due to kno wledge of the cor resp onding alg ebra pro p erties and those of gener alized cosh function. Hence the work to g ain the thermo dynamics of the system, though po ssibly tanta- lous, lo oks now a reasonable, tangible task with help of co mputer symbolic calcu- lations. The discussion of the con ten t of the in statu nascendi second par t is to be found a t the end of this Part 1 prese nt ation. This constitutes the section V. P A CS n um b ers: 05.50. +q, Keyw ords: P otts mo dels, generalized Clifford Alge bras I. In tr o duction The main idea of all calculations to follo w (see [8]) is to consider the task of determining th e complete partition function for noncritical P otts m o dels - as a problem from the theory of generalized Clifford algebras C ( n ) 2 p classified in [15]. 1 As a matter of fact, the transfer matrix approac h h as led the authors of [1,14 except for 20,21] to th ese v ery algebras, although this observ ation d o es not seem to b e realize d by the men tioned authors. In general - the transfer m atrix tec h nique for a statistical system (or - a lattice field theory) with the most general translational in v arian t and glob- ally symmetric Hamiltonian (Action) on a tw o-dimensional lattice - do es generate ap propriate alge bras of op erators whic h are of th e typ e of algebra extensions of s ome group s [8,18,24]. If one considers Z n cyclic groups as symmetry group s of Hamiltonian (c hi- ral Action) then th e algebra generated by transfer matrix approac h is the corresp ondin g C ( n ) k algebra, and it is due to its p rop erties, that the giv en mo del h as the dualit y p rop erty [18]. F or dualit y prop erty w e refer the reader to the review [19], and for P otts mo dels, in general - to [2,23]. Calculations to b e carried out here for th e Z n - v ector Potts mo del sim plify tremendously in the case of n = 2 i.e. for the Ising mod el and there lead to the known complete partition function [5] (see also [22] for mo dern pre- sen tation) whic h after carrying out the therm o dynamic limit, goes in to the Onsager formula [16,17]. The m etho d w e choose to pro ceed with, is an appropriate generaliz ation of the one used in [22 ] whic h c onsists there in reducing the problem o f find ing of the partition function for the Ising mo del to calculation of T r ( P 1 ...P s ), where T r is the n ormalized trace wh ile P ’ s are linear com bin ations of γ matrices - generators of usual Clifford algebra n aturally assigned to the lattice. Then the observ ation that T r ( P 1 ...P 2 s ) is just a P faffian [3] of an an tisym- metric matrix formed with s calar p ro du cts of P ’s leads one to calculation of the determinant from this very matrix. The metho d prop osed in [5,22] is purely algebraic, and though pr obably not the shortest one, it lac ks ambiguities of other metho ds and u ses well established, simple language of Clifford algebras. F or other asp ects and connotations of s uc h an approac h see [9,10,1 1,12]. Our p ap er is organized as follo w s: In the second section w e write do wn the Z n -v ecto r P otts mo del in a form resem bling (and generalizing!) the Is ing m o del without external field an d then w e represent the transfer matrix (an elemen t of C ( n ) 2 p !) as a sum of expressions p rop ortional to T r ( γ i 1 ...γ i s ), where this time γ ’s are generalized γ matrices. Equiv alen tly , the Hamiltonian for th is P otts mo del can b e lo ok ed up on as an Action for th e Z n c hiral mo del on the squ are lattice. In th e third section we derive the formula for T r ( γ i 1 ...γ i s ). The last section is to supply an inevitable information on C ( n ) k algebras for the Reader’s conv enience as w ell as some calculations av oided in the 2 current text in order to make th e pr esen tatio n more tr ansparent . W e w ould like to end this introdu ction by quotation from R. J . Baxter’s b o ok (see [2] p. 454): ” Th e only hop e that o c curs to me is just as Onsager (1944) and Kaufman (1949) original ly solve d the zer o-field Ising mo del by using the algebr a of spinor op er ators, so ther e may b e similar algebr aic meth- o ds for solving the eight-vertex and Potts mo dels ”. Our su ggestion then is that these v ery algebras are just generalized Clifford algebras, and the presen ted pap er is aimed to deliver arguments in f a v or of that p oint of view. I I. The transfer matrix as a p olynomial in γ ’s In the f ollo win g the tran sfer matrix M f or the Z n -v ecto r P otts mo del is repre- sen ted in a form of ”multi-sum” of expressions prop ortional to T r ( γ i 1 ...γ i s ). Let us assign to the s et of states for this Z n v ector P otts mo del on a p × q torus lattice ( p rows, q columns), a set Def: S = { ( s i,k ) = ( p × q ) ; s i,k ∈ Z n } , ♦ where we hav e chosen a multiplica tiv e r ealizatio n for the cycli c group Z n =  ω l  n − 1 l =0 and s ik denotes a matrix elemen t of the ( p × q ) matrix. Here, naturally , ω denotes the p rimitiv e r o ot of unit y . The total energy E is then giv en b y: − E [( s i,k )] k T = a p,q X i,k =1  s − 1 i,k s i,k +1 + s − 1 i,k +1 s i,k  + b p,q X i,k =1  s − 1 i,k s i +1 ,k + s − 1 i +1 ,k s i,k  (2 . 1) while the partition function is defined to b e: Z = X ( s i,k ) ∈ S exp  − E [( s i,k )] k T  (2 . 2) One sees th at the total energy of the s ystem as repr esen ted by (2.1 ) is at the same time - a generalizati on of its own Z 2 Ising case and - the Action of the corresp onding c hir al mo d el (for connection b etw een lattice gauge theory and s pin systems see [6] and references therein). The p artition fu nction could b e written in terms of transfer matrix and for that p urp ose w e introdu ce the follo wing notation: 3 Notation: ~ s · ~ s ′ = p P i =1 s i s ′ i , ~ s k =      s 1 ,k s 2 ,k . . . s p,k      , ~ s ∗ k =      s ∗ 1 ,k s ∗ 2 ,k . . . s ∗ p,k      ( s i,k ) = ( ~ s 1 , ~ s 2 , ..., ~ s q ) (2 . 3) With the notation (2.3) adopted, the partition function Z ma y b e n o w rewritten in a form Z = X ~ s 1 ,...,~ s q exp ( a q X k =1  ~ s ∗ k · ~ s k +1 + ~ s ∗ k +1 · ~ s k  + b q X k =1  ~ s ∗ k · X 1 ~ s k + ~ s k · X 1 ~ s ∗ k  ) , (2 . 4) after th e natural p erio d icit y conditions h a v e b een imp osed. P e rio dicit y conditions: ~ s q +1 = ~ s 1 , ( ~ s k ) 1 = ( ~ s k ) p +1 ; k = 1 , ..., q , (2 . 5) where ( ~ x ) i denotes the i -th comp onent of ~ x . The matrix P 1 is a p × p generalized Pauli matrix w ith matrix elemen ts δ i +1 ,j , where i, j ∈ Z ′ p = { 0 , 1 , ..., p − 1 } and ”+” is understo o d as the Z ′ p group action. W e introdu ce also the σ 1 generalized P auli matrix, which is one of the three σ 1 , σ 2 , σ 3 - p la yin g the same role in repr esen ting C ( n ) 2 p generalized Clif- ford algebras as the usu al ones in r epresent ing the ordinary C (2) 2 p Clifford algebras via w ell kno wn tensor p ro du cts of σ matrices [4]. It is no w obvio us that Z ma y b e represen ted as Z = T r M q , (2 . 6) as we ha v e Z = X ~ s 1 ,...,~ s q M ( ~ s 1 , ~ s 2 ) M ( ~ s 2 , ~ s 3 ) ...M ( ~ s q , ~ s 1 ) , where m atrix elemen ts of th e transfer matrix M are giv en by: M  ~ s , ~ s ′  = exp n 2 b Re  ~ s ∗ · X 1 ~ s o exp  2 a Re  ~ s ∗ · ~ s ′  . (2 . 7) It is con v enien t to consider the matrix M as a p ro du ct M = B A , where the corresp ondin g matrix eleme n ts are identified as A ( ~ s ′′ , ~ s ′ ) = exp { 2 a Re ( ~ s ′′∗ · ~ s ′ ) } and B ( ~ s, ~ s ′′ ) = exp { 2 b Re ( ~ s ∗ · P 1 ~ s ) } δ ( ~ s, ~ s ′′ ) . (2 . 8) 4 As all these A, B , M m atrices are multiindexed it is ob v ious that they might b e repr esen ted either as tens or pr o ducts of ( n × n ) matrices ( p times) or as ( n p × n p ) m atrices. It is not difficult then to see that A = ⊗ p b a i . e . (2 . 9) A is the p -th tensor p o w er of the ( n × n ) matrix b a , wh ic h has the form of a circulan t matrix W [ σ 1 ]: b a = ( b a I ,J ) =  exp  2 a Re  ω J − I  = n − 1 X l =0 λ l σ l 1 ≡ W [ σ 1 ] , (2 . 10) where I , J ∈ Z ′ n = { 0 , 1 , 2 , ..., n − 1 } and λ l = exp n 2 a Re  ω l o . (2 . 11) In ord er to see that A and B matrices are ju st some elemen ts of C ( n ) 2 p w e shall express them in terms of op er ators X k and Z k ; k = 1 , 2 , ..., p i.e. matrices t yp ical for tensor pro duct r epresent ation of generalized Clifford algebras via generalized Pauli matrices (see (A.3)). Def: X k = I ⊗ ... ⊗ I ⊗ σ 1 ⊗ I ⊗ ... ⊗ I ( p − terms) , Z k = I ⊗ ... ⊗ I ⊗ σ 3 ⊗ I ⊗ ... ⊗ I ( p − terms) , ♦ where σ 1 and σ 3 are situated on the k -th site, coun ting from the left hand side. The matrix A may b e therefore no w rewritten as a p ro duct of ( n p × n p ) matrices A = p Y k =1 W [ X k ] , where W [ X k ] = n − 1 X l =0 λ l X l k . (2 . 12) Similarly , for the matrix B we deriv e: B = exp ( b p X k =1  Z − 1 k Z k +1 + Z − 1 k +1 Z k  ) , (2 . 13) where Z p +1 = Z 1 . The form ula (2.13) follo ws fr om the simple observ ation that matrix elemen ts of 5 Z − 1 k Z k +1 + Z − 1 k +1 Z k (m ultiindexed b y − → s and − → s ′′ ) giv e exactly ln of the cor- resp ond ing term of (2.8) expression for B . The δ function arises d ue to th e fact that σ 3 =  δ I ,J ω I  (see Ap p end ix) and the exp onen tiation of matrix elemen ts is p ossible b ecause B is simply prop ortional to unit matrix. Once A and B ha v e b een represen ted as in (2.12) and (2.13) it is easy to express them in terms of generalize d γ matrices. In tr o ducing then the tensor pr o duct represen tation (A.3) w e get: X k = ω n − 1 γ n − 1 k γ k Z − 1 k Z k +1 = γ n − 1 k γ k +1 for o d d n , (2 . 14) and X k = ξ ω n − 1 γ n − 1 k γ k Z − 1 k Z k +1 = ξ γ n − 1 k γ k +1 for eve n n , (2 . 15) where k = 1 , 2 , ..., p − 1 and ξ 2 = ω . The corresp onding exp ression on the b ound aries - read: Z − 1 p Z 1 = U γ n − 1 p γ 1 for o d d n , (2 . 16) and Z − 1 p Z 1 = ξ − 1 U γ n − 1 p γ 1 for ev en n , (2 . 17) where ω · U = ⊗ p σ 1 . (2 . 18) F or the p ro of of (2.14 )-(2.17) use (A.6 ) and (A.7). F rom no w on we shall proceed with formulas for n b eing o d d, loosing nothing from generalit y of considerations while corresp onding form u las for the case of n ev en are easily deriv able from th ose for the o d d case. This in mind w e get A = p Y k =1 W  ω − 1 γ n − 1 k γ k  , (2 . 19) B = exp ( b p − 1 X k =1  γ n − 1 k γ k +1 + γ n − 1 k +1 γ k  ) × exp  bU γ n − 1 p γ 1 + bU − 1 γ n − 1 1 γ p  . (2 . 20) Our fi rst goal is then ac hieved if one n otes that U = p Y k =1 γ n − 1 k γ k (2 . 21) i.e. the trans fer matrix M is no w expressed in terms of generalized γ matrices. 6 Before pro ceed, it is rather trivial and imp ortant to note that U n = 1 , Z n k = 1 , X n k = 1 , with obvious implication of the same pr op erty for the n -th order p olynomials in (2.19) and (2.20). Our second and the main goal of this section is to rep resen t the transf er matrix in a form - reducing the T r M q problem to calculation of T r ( γ i 1 γ i 2 ...γ i s ) for some collect ions of γ ’s. (Note that for n = 2 the wa y to get the complete partition function is shorter as there, it is enough to r educe the T r M q problem to calculation of T r ( P 1 P 2 ...P s ) w here P ’s are linear com binations of γ ’s. Hence the n um b er of necessary sum mations is m u c h, muc h sm aller than in th e case n > 2, where it is r ather useless to try to repr esen t A and B matrices in th at con venien t form). F or that to do w e sh all deal n o w with the matrix B , to rev eal its ad equate for the purp ose - structure. Let u s start with observ ations follo win g from U n = 1 prop ert y of U . Define V ± k matrices (see (A.11)) to b e V + k = 1 n n − 1 P i =0 ω − k i U i V − k = 1 n n − 1 P i =0 ω − k i U − i (2 . 22) and also e B + k = exp  b ω k γ n − 1 p γ 1  e B − k = exp  b ω k γ n − 1 1 γ p  , where k = 0 , 1 , ..., n − 1 . (2 . 23) Then we ha v e exp  b U γ n − 1 p γ 1  = n − 1 X k =0 e B + k V + k (2 . 24) and exp  b U − 1 γ n − 1 1 γ p  = n − 1 X k =0 e B − k V − k . (2 . 25) Equations (2.2 4) and (2.25) follo w from examining exp { U x } via series ex- pansions mod ulo n , as in (A.8 )-(A.10) . The m atrices V ± k ha v e an imp ortant pr op erty (see (A.11))  V ± k  n = V ± k (2 . 26) hence from (2 .20), (2.24), (2.25) and the commutat ivit y of matrix arguments of B ( U , γ n − 1 k γ k +1 ... etc.) it follo w s that: B = n − 1 X l,k =0  B + k V + k B − l V − l  (2 . 27) 7 where B + k = exp ( b p − 1 X α =1 γ n − 1 α γ α +1 ) e B + k , (2 . 28) and B − k = exp ( b p − 1 X α ′ =1 γ n − 1 α ′ +1 γ α ′ ) e B − k . (2 . 29) Expression ( 2.27) for B b ecomes sti ll simpler d ue to the r emark able pr op ert y of V k ’s: V k V l = 0 ; k 6 = l (2 . 30) where, for the momen t, V k = V ± k , (see App en dix for pro of ). Hence B = n − 1 X k =0 B + k B − k V + k V − k , (2 . 31) - all terms of the k -th summand - comm uting. As f or the comm uting of v arious matrices inv olv ed in repr esen ting of the transfer matrix, note that [ A, B ] 6 = 0 , [ U, A ] = 0 and  V + k , A  =  V − k , A  = 0 . All this is sufficien t to write: M q = n − 1 X l =0  B + l B − l A  q  V + l V − l  r (2 . 32) where 1 ≤ r ≤ n − 1 and V q = V r as V n = V . It is not d ifficult to see how ” r ” arises. Namely ” r ” is th e residu al of th e quotien t ( q − n ) ( n − 1) . This n − 1 m ultiplicit y in formula (2.3 2) mak es the pr oblem of thermo- dynamic limit more in teresting and inv olve d. If one assumes how ev er that assumption: q = n + l ( n − 1) , (2 . 33) with l an arbitrary in teger, then M q = n − 1 X l =0  B + l B − l A  q V + l V − l . (2 . 34) Note that th ere is no multiplici t y in (2.32) for the Z 2 case (Ising mo del) and also for that case V + k = V − k ≡ V k , k = 0 , 1. The m ore: V k V k = V k and (2.34) reduces its form co nsiderably . 8 Comparison: In order to compare (2.34) w ith the known expression for M q in the case of Ising mo del [22] one s hould note that, for Z 2 , (2.15) and (2.17) differ only by i and − i from (2.14) and (2.16) corresp ondingly , hence we hav e M q = ( B − A ) q V + + ( B + A ) q V − , (2 . 35) where B − = B + 0 B − 0 = exp  2 bi  p − 1 P α =1 γ α γ α +1 − γ p γ 1  , B + = B + 1 B − 1 = exp  2 bi  p − 1 P α ′ =1 γ α ′ γ α ′ +1 − γ p γ 1  and V + = V 0 = 1 2 ( 1 + U ) , V − = V 1 = 1 2 ( 1 − U ) . (2 . 35 a ) The formula (2. 35) coincides then with the one kno wn for Ising mo del [22] apart fr om the ob vious (see (2.1 )) and in significan t scaling of constan ts a and b b y factor 2. Using the form u la (2.35), th e notion of Pfaffian and its relation to d etermi- nan t - the author of [22] r eobtained the complete partition function leading to the famous O nsager formula. ♦ Ha vin g the same goal in m ind w e are go ing at fir st to examine the expression (2.34) in order to see ho w (the p olynomial in γ ’s !) M q is represen ted as a m ultisum of summand s prop ortional to T r ( γ i 1 ...γ i s ). F or that purp ose we write: e B + k = exp  bρu + k  , (2 . 36) and e B − k = exp  bρ − 1 u − k  , (2 . 37) where !  u + k  n =  u − k  n = 1 i.e. u + k ≡ ω k ρ − 1 γ n − 1 p γ 1 , (2 . 38) u − k ≡ ω k ργ n − 1 1 γ p , (2 . 39) while ρ ≡ ρ ( n ) = ω n 2 − 1 2 ( k = 0 , 1 , ..., n − 1) . (2 . 40) Therefore b oth e B + k and e B − k b ecome n − 1 order p olynomials in u + k and u − k corresp ondin gly (see (A.10 )). One also sho ws easily that v + α and v − α defined b elo w 9 exp  b γ n − 1 α γ α +1  = exp  b ρ v + α  , (2 . 41) exp  b γ n − 1 α +1 γ α  = exp  bρ − 1 v − α  , (2 . 42) do satisfy:  v + α  n =  v − α  n = 1 α = 1 , ..., p − 1 . (2 . 43) hence b oth expr essions (2.41) and (2.42) b ecome n − 1 order p olynomials in matrices v + α and v − α corresp ondin gly . n=2: F or n = 2 the fu rther job is extremely f acilitat ed due to th e fact that A b ecomes then of the form A = p Y k =1 P k Q k (2 . 44) where P ’s and Q ’s are some kno wn linear com binations of ord inary γ ma- trices and similar holds for B + , B − matrices from (2.35). The matrices V + , V − from (2.35) h a v e also simple form and thus T r M q disen tangles for n = 2 to b e th e su m only four s ummands of the Pfaffian t yp e i.e. T r ( P 1 P 2 ...P s ). n > 2: Unfortunately this disen tanglemen t is no m ore p ossible for n > 2 as the p olynomial W (see (2.12)) no more is representa ble un iquely as the p ro du ct of linear com b inations of γ ’s and neither is B . Hence the m ulti-sum b ecomes more complicated. Nev er theless it is ob vious that T r M q problem reduces to calculation of T r ( γ i 1 γ i 2 ...γ i s ) f or some collect ions of γ ’s. n=2: In the case of Ising mo del, the four arising Pfaffians con trib ute to the p ar- tition fu nction to giv e [22]: Z = 2 pq − 1 ( p,q Q k ,l =1 h c h2 a ′ c h2 b ′ − sh2 a ′ cos π q (2 l + 1) − sh2 b ′ cos π p (2 k + 1) i 1 2 + + p,q Q k ,l =1 h c h2 a ′ c h2 b ′ − sh2 a ′ cos π q (2 l + 1) − sh2 b ′ cos 2 π k p i 1 2 + + p,q Q k ,l =1 h c h2 a ′ c h2 b ′ − sh2 a ′ cos 2 π l q − sh2 b ′ cos π p (2 k + 1) i 1 2 + − σ p,q Q k ,l =1 h c h2 a ′ c h2 b ′ − sh2 a ′ cos 2 π l q − sh2 b ′ cos 2 π k p i 1 2 ) (2 . 45) 10 where σ denotes the s ign of T − T c and a ′ = 2 a , b ′ = 2 b . Both the square ro ot and the σ -sign hav e app eared here b ecause of the use of P f 2 = det relation. n > 2: Again, for n > 2, as we shall see in the follo win g section, although the generalizat ion of the Pfaffian is p ossible to t he case of arb itrary n , it s r elation to an y v aluable generalizati on of determinan t do es not to b e v alid as the arising sign um lik e function no mo re is an epimorph ism of S k on to Z n , exce pt for n = 2 of cour se. ( S k - the symmetric group of k -element al p ermutat ions). Ho wev er, one m a y write M q as the p olynomial in γ ’s and then use the general form ula from the f ollo wing section. This represen tatio n of M q in terms of γ i 1 ...γ i s pro du cts is giv en in th e Ap- p end ix. As a result we hav e the follo wing structure of the complete partitio n function for the Z n v ector Potts mo dels: M q = 1 n 2 n − 1 X j 1 ,j 2 =0 X ~ Π ∈ Γ m G  ~ Π  b Ω  ~ Π; j 1 , j 2  , (2 . 46) where G ( ... ) are kno w n functions of parameters a and b , (see the Ap- p end ix: (A.15)) and T r b Ω = ω i or T r = 0 , where i = i  ~ Π , j 1 , j 2  ∈ Z ′ n = { 0 , 1 , ..., n − 1 } . The dep endence of i on its indices is easy to b e derive d using the most general, appropriate formula f or T r ( γ i 1 γ i 2 ...γ i s ) s upplied by the next section. I I I. T race form ula for an y elemen t of C ( n ) 2 p In th is sectio n the explicit form ula for trace of an y eleme n t of C ( n ) 2 p algebra is delive red. The v ery form ula is crucial for getting the complete partition f unction for P otts mo dels and hence (see [2] p . 454) for solving sev eral ma jor problems of statistical physics b eing unsolv ed till n o w s ince many y ears. The p roblem of explicit trace formula for M q ∈ C ( n ) 2 p , is decisiv e in calculatio n of Z function fo r those mo dels on the la ttice in which the transfer matrix is an element of C ( n ) 2 p . W e pro ceed no w to deriv atio n of the v ery formula. Note! By d efinition, in this section T r map is normalized i.e. T r I = 1. The deriv ation has the form of a sequence of lemmas. 11 Lemma 1. Let k 6 = n mod n , k ∈ N ; then T r ( γ i 1 ...γ i k ) = 0 . ♦ Pro of: The same as for u sual Clifford algebras. Use the matrix U d efined b y (2.18). ♦ Lemma 2. T r ( γ i 1 γ i 2 ...γ i k ) 6 = 0 iff there exists p ermutation δ ∈ S k n , suc h that i σ ( 1) = i σ ( 2) = ... = i σ ( n ) , i σ ( n +1) = ... = i σ ( 2 n ) , ... , i σ ( k n − n +1) = ... = i σ ( k n ) . ♦ Pro of: Th e p ro of follo ws from obs erv ation that due to (A.1) if no n -tuple of the same γ ’s exists then T r ( ... ) = 0. Oth er steps of the p ro of are redu ced to this first one. ♦ It is therefore trivial to n ote, bu t imp ortan t to realize, th at: Lemma 3. T r ( γ i 1 ...γ i k ) = 0 or l ∈ Z n - the multiplica tiv e cyclic group of n -th ro ots of un it y . ♦ In Lemma 3 k is again an arbitrary int eger while in all p receding lemmas, and in the follo wing, i 1 , i 2 , ..., i k run from 1 to num b er of generators of the giv en algebra. T his num b er w as c hosen to b e ev en, ho w ev er note [15] that the ”o dd case” problem is reduced to this v ery one due to the prop erties of generalized Clifford algebra r epresen tations. The m a jor prob lem now is to d etermine this v alue ”0 or l ” for arbitrary set of indices i 1 , i 2 , ..., i k . In order to do that define a signum lik e function K (un fortunately it is an epimorphism only for n = 2) - as follo ws: Def: K : S p → Z n ; Θ σ ( 1) Θ σ ( 2) ... Θ σ ( p ) = K ( σ ) Θ 1 Θ 2 ... Θ p , where Θ’s satisfy (A.1) except for the condition γ n i = 1, whic h is now re- placed by Θ 2 i = 1. ♦ This defin ition b eing adapted, it is no w not v ery difficult to prov e: Lemma 4. 12 T r  γ i 1 ...γ i pn  = K (Σ) K ( σ ) , for a) i σ ( 1) = ... = i σ ( n ) , ... , i σ ( pn − n +1) = ... = i σ ( pn ) and b) i e σ ( n ) < i e σ (2 n ) < ... < i e σ ( pn ) , where e σ ≡ Σ ◦ σ , while Σ is a p erm utation of the elemen ts { n, 2 n, ..., pn } . (The group of Σ’s is n aturally iden tified w ith an appr opriate subgroup of S pn ). ♦ Pro of: The pro of relies on observ ation th at these are only differen t n - tuples whic h are ”rigidly” shifted ones trough the others, i.e. th ere is no p ermutatio n w ithin any giv en n -tuple. ♦ The generalization of Lemma 4 to the arbitrary case of some of the n -tup les b eing equal - is straigh tforw ard. (T he necessary change of conditions a) and b) is ob v ious). This in mind and from other lemmas w e finally get: Theorem: T r  γ i 1 ...γ i pn  = P ′ σ ∈ S pn P ~ p P Σ ∈ S ~ p K ( Σ) K ( σ ) × δ  i e σ (1) , ..., i e σ ( p 1 n )  × × δ  i e σ ( p 1 n +1) , ..., i e σ ([ p 1 + p 2 ] n )  × ... × δ  i e σ ( pn − p l n +1) , ..., i e σ ( pn )  , with the notation to follo w. ♦ Notation: ~ p = ( p 1 , p 2 , ..., p l ), p i ≥ 1, l P i =1 p i = p , e σ = Σ ◦ σ , and S ~ p is a sub group of S pn isomorphic (for example!) to the group of all blo c k matrices obtained via p ermutati ons of ”blo c k columns” of the matrix      I p 1 n I p 2 n . . . I p l n      , where I k is the ( k × k ) unit matrix. δ - her e den otes the multi-indexed Kronec k er delta i.e. it assigns zero unless all its argumen ts are equal and in this v ery case δ ( ... ) = 1. The sum Σ ′ is m ean t to tak e into accoun t only those p er m utations that do satisfy the conditions: a) σ (1) < σ (2) < ... < σ ( p 1 n ), ... , σ ( pn − p l n + 1) < ... < σ ( pn ), and b) σ ( 1) < σ ( p 1 n + 1) < ... < σ ( pn − p l n + 1). Commen ts: 1) F or the case of n = 2 th e theorem giv es us the Pfaffian of the pro duct γ i 1 , ..., γ i p 2 , as in the case, (and only! for n = 2) K (Σ) = 1 and we are left, as a result with only Σ ′ sum, while Kronec ker deltas b ecome fu nctions of the same n um b er of indices i j . 2) Th e theorem solv es our p roblem of T r M q , as any elemen t of generalized Clifford alge bra is a p olynomial in γ ’s satisfying (A.1), including M q ∈ C ( n ) 2 p . 13 IV. Final c ommen ts for the P art 1 of the presen tation W e hav e carried out our t wen tieth cen tury inv estigation for the Z n v ector P otts mod el kno wn also und er the n ame of planar Pot ts mo del. The similar inv estigation of the other P otts mo dels, i.e. standard P otts mo dels with t wo-site int eraction [23] and multi site in teractions as w ell, is b eing no w carried out. Ho wev er, it is to b e noted here that the mo del considered in [1 4] p ossesses transfer matrix M = B A , where matrix A is a particular case of the one defined b y (2.19) while B , though also expressed b y Z k matrices defined in section I I , has a different p olynomial (in these op erators) in th e exp oten tial. As for t he m ultisite in teractions, the al gebras t o b e u sed are the universal generalized Clifford algebras, in tro duced in [7]. Needless to say that these are standard Po tts mo dels wh ic h are of more int erest b ecause of their relation to a n um b er of outstand ing problems in lattic e statistics [2,23 ]. T o this end let u s expr ess our suggestion th at the m o dels of lattice statistics could b e adapted (thanks to sp ecific interpretation) to the domain of ur ban economics wh ic h , u sing th e n otion of entrop y and inform ation int ro du ces as a m atter of fact a kind of thermo dyn amics [13]. V. An outline of the second planned part con ten t The con tent of the pr eceding chapters-exce pt for th e algorithm f or calc ula- tion of the complete partition fun ction as presen ted ab ov e - was already pu b- lished in t wen ty fir st cen tury [25],[26]. T his esp ecially concerns the Clifford algebra tec hnique omn ipresent here and p lanned to play similar leading role in the next part of our review. T he con tent of [26 ] from 2001 indicates the idea and a wa y ho w to use generalized Clifford algebra for chiral Po tts m o dels on the plane. The incessan tly gro win g area of app lications of C lifford alge - bras and naturalness of their use in formulating pr oblems f or d irect calcula- tion enti tles on e to call them Clifford num b ers. The generalized “unive rsal” Clifford n um b ers a re here in tr o duced via k-ubic form Q k replacing quadratic one in familiar construction of an approp riate ideal of tensor alge bra. On e of the epimorp hic images of u niv ersal algebras k − C n ≡ T ( V ) /I ( Q k ) is the algebra C l n ( k ) w ith n generators and th ese are the algebras to b e used here. Because generalized Clifford algebras C l n ( k ) p ossess inheren t Z k grading - this prop ert y mak es th em an efficient apparatus to d eal with spin lattice systems. This efficiency is illustrated in [26] by d eriv ation of t wo m a jor observ ations. Firstly , the partition functions for vect or and planar P otts mo dels and other mo del with Z n in v arian t Hamiltonian are p olynomials in generalized hyp erb olic fun ctions of the n -th order. Secondly , the pr oblem of algorithmic calculation of the partition function for an y v ector Pot ts mod el as treated h ere is reduced to the calculation of traces of pr o ducts of the gen- erators of the generalized C lifford algebra. Finally the expression f or suc h 14 traces for arbitrary colle ction of generator matrices is deriv ed in [26]. Since the same 2001 y ear, du e to the a uthors of [27] we kno w the form of the k -state P otts mo del p artition f unction (equiv alen t to the T utte p olynomial) for a lattice strip of fixed width and arbitrary length. F rom 2005 year Alan D. Sok al 54 pages review [28] aimed for mathemati- cians to o one m a y learn th at ”‘ the multiv ariate T u tte p olynomial (known to physic ists as the Potts-model partition function) can b e defined on an arbitrary finite graph G, or more generall y on an arbitrary matroid M, and enco des muc h imp ortan t com binatorial information ab out the graph”’. Alan D. Sok al discusses ther e ”‘some questions concerning the complex zeros of the m u ltiv ariate T u tte p olynomial, along with their ph ysical in ter- pretations in statistica l mechanics (in connection with the Y ang–Lee ap- proac h to p hase transitions) and electrical circuit th eory .”’ Q uite numerous op en p roblems are also p osed in [28]. F or m an y references see b oth [27] and [28]. Coming bac k for a wh ile to generalized Clifford algebra we mark their b eing just only mentio ned in the abstract of Baxter pap er [29] in whic h: ”‘The partition function of the N-state su p erintegrable c hiral P otts mo d el is ob- tained exactly and explici tly (if not completely rigorously) for a finite lat tice with particular b oundary cond itions”’. ”‘ The asso ciated Hamilto nian has a very simple form, su ggesting that ma y b e a more d irect algebraic metho d (p erhaps a generalized C lifford algebra) f or obtaining its eigen v alues.”’ Bax- ter - including his famous b o ok [198 2 Exactly Solv ed Mo dels in S tatistica l Mec han ics] comes bac k several time to the idea of Clifford or Clifford-lik e algebras’ p oten tial imp ortance for the still n ot solve d problem of complete partitions fun ction obtained in a wa y I sing mo del wa s splved w ith help of Clifford algebras p rop erties b eing used in a natural and elegan t w a y . Man- ageable in a und erstandable w a y . Let us quote after Baxter from [30]:”’ ”This is rather in triguing - w e are in m u c h the same p osition with the c h i- ral P otts mo del as we w ere in 1951, when Professor Y ang en tered the field of statistica l mec h anics by calculating MO for the Ising mo del. So on the o ccasion of his 70th bir thda y we are able to present him n ot on ly with this meeting in honor of h is great contributions to theoretical ph ysics, b ut also with an outstanding problem worth y of his mettle. Plu s a c h ange, c’est la m ˆ eme c h ose.”’Here and there P . P . Martin’s b o ok [31] and his many subsequ en t pap ers in tw entie th as w ell in t w en t y first cen tury [ 32] are in evitable source of id eas and i nspiration. And here no w comes the 2005 y ear. 2005 w as declared b y some authors to b e the ma jor breakthrough for the chiral Po tts mo del. T his concerns an outstanding problem of the order parameters. See Baxter again [33] and [34]. T here in [33] and then in [34] Baxter d eals again with th e problem of the order parameter in the c hiral Potts mo del. He r ecalls that an elegan t conjecture f or this was m ade in 1983 and that it h as since b een su ccessfully tested against series expansions, bu t as far as the author of [34] is a w are 15 there is as y et no pro of of the conjecture. 2005. Again Pr ofessor Baxter. Here is an abstr act of his Annual Confer- ence 2005 of th e Au stralian Mathematical Societ y p ublic talk . : entitled Lat- tice mo dels in statistical mec hanics: t he chiral P otts mo del :”‘There are a f ew lattice mo d els of inte racting systems th at can b e solv ed exactly , in the sense th at one can calculate the fr ee energy in the thermo dyn amic limit of a large sys tem. Th e in teresting ones are mostly tw o-dimensional, suc h as the Ising mo del and the six and eigh t-v ertex mo dels. A comparitiv ely recen t addition to the list is the c hiral Pot ts mo d el. T his is more difficult mathematicall y than its pr edecessors. While its free energy was calc ulated in 1988, u n til no w there has only b een a conjecture (a v ery elegan t one) for the order parameters, i.e. the sp onta neous magnetizations. This conjecture has now b een ve rified, and in this talk I sh all discuss the d ifficulties encoun- tered and th e metho d us ed. ”’ The solution from 1988 he is referr ing to apparen tly refers to pap ers such as [35 ], [36], [37] , [38] an d others later, see [39]. By the w a y? - the authors of [39 ] implicitly indicate Generalized Clif- ford Algebras - in a fo otnote (3) r eferring there to Morris pap ers from 1967 and 1968 [qu oted in all Kwasniewski pap ers on sub ject]. The 69 pages, 30 figures reviw [39] w as written in honor of Onsager’s ninetieth birthda y ,also in order to p resen t ”‘some exact results in th e c hiral Potts mo dels and to translate these results into language more trans paren t to physicist s”’. This is more or less what th e second part is planned to b e ab out. By no means it sholud include review of n umerous cotribu tions of Professor F.Y. W u including n ot only P otts mo d els [see: h ttp://www.ph ysics.neu.edu/Departmen t/Vt wo/fa cult y/wu.../wupubu p dated81803.h tm but such fascinating pap ers as [40] of sp ecifically p ersonal in terest of the au- thors [http:// ii.u wb.edu.pl/akk/publ1.htm]. Pa p ers published in Adv an ces in Applied Clifford Alge bras suc h as [42] and the pap ers b y the authors are to b e in cluded in the second part of this review to o. App endix 1. C ( n ) 2 p generalized Clifford algebra is defined [15] to b e generated by γ 1 , ..., γ 2 p matrices satisfying: γ i γ j = ω γ j γ i , i < j , γ n i = 1 , i, j = 1 , 2 , ..., 2 p . ( A. 1) The very algebra h as - up to equiv alence - only one ir reducible an d f aithful represent ation, and its generators can b e repr esen ted as tensor pro ducts of generalized P auli matrices: σ 1 = ( δ i +1 ,j ) , σ 2 =  ω i δ i +1 ,j  , σ 3 =  ω i δ i,j  , ( A. 2) where i, j ∈ Z ′ n = { 0 , 1 , ..., n − 1 } - the additiv e cyclic grou p. One easily c h ec ks, that { σ i } 3 1 do satisfy (A.1) f or n b eing o dd . 16 Let I denotes since now the unit ( n × n ) matrix and let γ 1 = σ 3 ⊗ I ⊗ I ⊗ ... ⊗ I ⊗ I , γ 2 = σ 1 ⊗ σ 3 ⊗ I ⊗ ... ⊗ I ⊗ I , . . . γ p = σ 1 ⊗ σ 1 ⊗ σ 1 ⊗ ... ⊗ σ 1 ⊗ σ 3 , ¯ γ 1 = σ 2 ⊗ I ⊗ I ⊗ ... ⊗ I ⊗ I , ¯ γ 2 = σ 1 ⊗ σ 2 ⊗ I ⊗ ... ⊗ I ⊗ I , . . . ¯ γ p = σ 1 ⊗ σ 1 ⊗ σ 1 ⊗ ... ⊗ σ 1 ⊗ σ 2 , ( A. 3) then { γ i , ¯ γ j , i, j = 1 , ..., p } do satisfy (A.1) with ω replaced by ω − 1 , hence (A.3) are generato rs of the algebra isomorphic to C ( n ) 2 p (isomorphism is giv en b y σ 1 ↔ σ 3 in (A.3)) (This very (A.3) repr esen tation wa s chosen for tec hnical reason - we get, for example, in calculations of section I I, the matrix U without co efficient s etc.). It is also to b e n oted that for n b eing o dd σ 3 = σ n − 1 1 σ 2 . ( A. 4) The case of n b eing ev en leads to similar represent ation with σ 1 un- c hanged b ut σ 2 and σ 3 no w equal to: σ 2 =  ξ i δ i +1 ,j  , σ 3 = ξ σ n − 1 1 σ 2 , ( A. 5) where ξ is a primitive 2 n -th r o ot of unit y suc h that ξ 2 = ω . (A.3) th en with these appropr iate for case n = 2 ν generalized P auli matrices, pro vides us with the same t yp e representa tion of C ( n ) 2 p as the one for the ca se n = 2 ν + 1. One th en easily pro v es that σ n − 1 3 σ 2 = ω σ 1 , σ n − 1 2 σ 1 = σ − 1 3 for n = 2 ν + 1 ( A. 6) and σ n − 1 3 σ 2 = ξ − 1 ω σ 1 , σ n − 1 2 σ 1 = ξ − 1 σ − 1 3 for n = 2 ν . ( A. 7) 2. In this part of the App endix we deriv e one u seful formula, necessary for section I I. Let x b e any elemen t of an asso ciativ e, finite d imensional algebra with unit y 1 . 17 Then exp { x } = n − 1 P i =0 f i ( x ) , where f i ( x ) = ∞ P k =0 x nk + i ( nk + i )! , i = 0 , ..., n − 1 . ( A. 8) W e no w express these f i ’s in terms expotentia ls. F or that to do it is su fficien t to note that f i ( ω x ) = ω i f i ( x ) , i = 0 , 1 , ..., n − 1 . ( A. 9) The (A.9) rev eals the Z n symmetry prop erties of these generalized ”cosh” functions and we get from this set of relations f i ( x ) = 1 n n − 1 X k =0 ω − k i exp n ω k x o , i = 0 , ..., n − 1 . ( A. 10) 3. F or consideratio ns of the section I I we n eed the follo win g Lemma Let U b e as x ab o v e and in addition let U n = 1 . Then V defined as follo ws V = 1 n n − 1 P i =0 U i , has the prop erty : V n = V . ♦ ( A. 11) Pro of: F or the pr o of, ju st n ote that for some a i ’s V n = n − 1 X i =0 a i U i , and b oth sides of this iden tit y equation m ust b e symmetric in U i monomials. One conclud es therefore that a i = a j , i, j = 0 , 1 , ..., n − 1, h ence - coun ting the num b er of all arising su mmands - one arr iv es at the conclusion of the Lemma. 4. Here, the pro of of the (2.3 0) form ula follo ws. Let k = l + r , 0 < r ≤ n − 1. Th en V + k V + l = 1 n 2 n − 1 X i 1 =0 n − 1 X i 2 =0 ω − r i 1  ω − 1 U  i 1 + i 2 . In tro duce no w new su mmation ind ices: i 1 and i = i 1 + i 2 . 18 Then we ha v e V + k V + l = 1 n 2 n − 1 X i 1 =0 ω − r i 1 n − 1 X i =0  ω − 1 U  i = 0 , b ecause th e s ummation ov er i 1 giv es zero. The p ro of for other ( + , − ), ( − , +), ( − , − ) cases is the same. 5. In this part of the App end ix w e derive the multisum structure of the complete p artition fun ction Z , for the Z n -v ecto r P otts model with any n ≥ 2. Since now on w e shall u se the follo win g abbreviations: Γ k ≡ Z ′ n ⊗ Z ′ n ⊗ ... ⊗ Z ′ n , ( k summand s) , where Z ′ n = { 0 , 1 , ..., n − 1 } , and ω n 2 − 1 2 ≡ ρ ( n ) ≡ ρ . Recall also u + k , u − k ; k ∈ Z ′ n and v + α , v − α ; α = 1 , ..., p − 1, defined in section I I. Then w e ha ve, according to (A.10): B + k = n − 1 P l 1 =0 f l 1 ( bρ )  u + k  l 1 p − 1 Q α =1 n − 1 P s =0 f s ( bρ ) ( v + α ) s , B − k = n − 1 P l 2 =0 f l 2  bρ − 1   u − k  l 2 p − 1 Q α ′ =1 n − 1 P t =0 f t  bρ − 1  ( v − α ) t . ( A. 12) In ord er to manage with abun dance of indices w e introdu ce a further , delib erate notatio n. Notation: ~ L ∈ Γ 2 , ~ T , ~ S ∈ Γ p − 1 i.e. ~ L ≡ ( l 1 , l 2 ) w here l 1 , l 2 ∈ Z ′ n etc. W e also defin e: F  b ; k , ~ L, ~ S , ~ T  ≡ ω k ( l 1 + l 2 ) ρ l 2 − l 1 f l 1 ( bρ ) f l 2  bρ − 1  p − 1 Q i =1 f s i ( bρ ) ρ − s i × p − 1 Q j =1 f t j  bρ − 1  ρ t j , b Ξ  ~ L, ~ S , ~ T  ≡  ¯ γ n − 1 p γ 1  l 1  γ n − 1 1 ¯ γ p  l 2 p − 1 Q i =1  ¯ γ n − 1 i γ i +1  s i p − 1 Q j =1  γ n − 1 j +1 ¯ γ j  t j and B k ≡ B + k B − k . ♦ 19 Hence we ma y write B k = X ~ L, ~ S , ~ T F  b ; k , ~ L, ~ S , ~ T  b Ξ  ~ L, ~ S , ~ T  ( A. 13) where ~ L ∈ Γ 2 , ~ S , ~ T ∈ Γ p − 1 . It is imp ortant to recall n o w th at: M q = n − 1 X k =0 [ B k , A ] q V + k V − k and [ B k , A ] 6 = 0 . W e in tro duce therefore again an approp riate notation. Notation: Λ  a ; ~ I  ≡ p Q r =1 λ i r ( a ) ω − i r , and ˆ Γ  ~ I  = p Q r =1  γ n − 1 r ¯ γ r  i r , while (see (2.21)) U = p Y r =1 γ n − 1 r ¯ γ r , (recall also (2.22) ). ♦ This b eing adapted w e m a y wr ite: A = X ~ I ∈ Γ p Λ  a ; ~ I  ˆ Γ  ~ I  ( A. 14) In v estigat ion of the multisum structur e of M q matrix even tuates in rather transparent form of it, if one, (for the last time!) introdu ces an o v erall index, or rather ”m u ltiindex” for all reapp earing su mmations. Notation: ~ Π ≡  k , ~ L 1 , ..., ~ L q , ~ S 1 , ..., ~ S q , ~ T 1 , ..., ~ T q , ~ I 1 , ..., ~ I q  , G  ~ Π  = q Q r =1 F  b ; k , ~ L r , ~ S r , ~ T r  Λ  a ; ~ I r  , ˆ Ω  ~ Π , j 1 , j 2  =  q Q r =1 b Ξ  ~ L r , ~ S r , ~ T r  ˆ Γ  ~ I r   U j 1 − j 2 ω − k ( j 1 + j 2 ) . ♦ All this together taken int o accoun t leads to M q = 1 n 2 n − 1 X j 1 ,j 2 =0 X ~ Π ∈ Γ m G  ~ Π  ˆ Ω  ~ Π , j 1 , j 2  ( A. 15) 20 where ~ Π ∈ Γ m ; m = 3 pq + 1. References [1] Y u. A. Bashilo v and S. V. Pokro vsky; Comm un. Math Phys. 76 (1980 ), 129-1 41 [2] R. J. Baxter F. R. S., Exactly Solve d Mo dels in Statistic al Me chanics ; Acad. Pr ess, Lond on, 1982 [3] W. Greub, M ultiline ar Algebr a ; 2nd E dition Sp ringer-V erlag New Y ork (1978 ) [4] K. Huang, Statistic al Me c hanics ; John Wiley and Sons, Inc. New Y ork (1963 ) [5] B. Kaufman; P h ys. Rev. 76 (1949), 1232 [6] J. B. Kogut; Reviews of Mo dern Ph ys. 51 (1979 ), 659 [7] A. K. Kw a ´ s niewski; J. Math. 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