7 One-dimensional q-state Potts model with multi-site 1 0 interactions 2 n a Lo¨ıc Turban J 1 Groupe de Physique Statistique, Institut Jean Lamour, Universit´e de Lorraine, 3 CNRS (UMR 7198), Vandœuvre l`es Nancy Cedex, F-54506,France ] E-mail: [email protected] h c e Abstract. A one-dimensional (1D) q-state Potts model with N sites, m-site m interaction K in a field H is studied for arbitrary values of m. Exact results for - t the partition function and the two-point correlation function are obtained at H = 0. a t The system in a field is shown to be self-dual. Using a change of Potts variables, it is s . mapped onto a standard 2D Potts model, with first-neighbour interactions K and H, t a onacylinderwithhelicalboundaryconditions(BC).The2DsystemhasalengthN/m m and a transverse size m. Thus the Potts chain with multi-site interactions is expected - to develop a 2D critical singularity along the self-duality line, (eqK −1)(eqH −1)=q, d n when N/m→∞ and m→∞. o c [ Keywords: Potts model, multi-site interactions, self-duality, helical boundary 1 conditions v 8 5 0 9 Submitted to: J. Phys. A: Math. Theor. 0 . 1 1. Introduction 0 7 1 The standard Potts model is a lattice statistical model with pair interactions between : v q-state variables attached to neighbouring sites [1,2]. Multi-site Potts models can be i X constructed by extending to an arbitrary number of states existing multispin Ising r a models for which q = 2. In this way, a self-dual three-site Potts model on the triangular lattice was introduced by Enting [3,4], which corresponds to the Baxter-Wu model [5,6] when q = 2. Similarly, a 2D self-dual Potts model with m-site interactions in one direction and n-site interactions in the other [7–9] follows from the Ising version with n = 1 [10]. Multi-site interactions may be generated from two-site interactions in a position- space renormalisation group transformation and thus have to be included in the initial Hamiltonian. In this way Schick andGriffiths have introduced a three-state Potts model on the triangular lattice with two- and three-site interactions [11]. For any value of q it can been reformulated as a standard q-state Potts model with two-site interactions on a Potts model with multi-site interactions 2 3-12 lattice [12]. When the three-site interactions are restricted to up-pointing triangles, the model is self-dual [13,14] and related to a 20-vertex model [13,15]. Extending the results of Fortuin and Kasteleyn [16] for pair interactions, a random-cluster representation for Potts models with multi-site interactions has been introduced [17–19] and exploited in Monte Carlo simulations [20]. Multi-site interactions enter naturally when the site percolation process is formulated as a Potts model in the limit q → 1 [21–25]. Various Potts multi-site interactions have also been used to model conformational transitions in polypeptide chains [26–29]. With s = 0,1,...,q − 1 denoting a q-state Potts variable attached to site j, a j multi-site interaction can take one of the following forms m−1 m−1 (a) −K δ , (b) −Kδ s , (1.1) sj,sj+1 q j+l ! j=1 l=0 Y X where δn,n′ is the standard Kronecker delta and δq(n) is a Kronecker delta modulo q. When K > 0 the ground state is q-times degenerate in the first case (the standard one) whereas the degeneracy depends on m and is given by qm−1 in the second case. As an example, when q = m = 3 the degenerate ground states are the following ones: 000 000 012 210 (a) 111 , (b) 111 120 021 (1.2)    222  222 201 102 In the present work we generalize for q-state Potts variables some results recently   obtained for the 1D Ising model with multispin interactions [30]. The Hamiltonian of the q-state Potts chain takes the following form m−1 −βH [{s}] = K qδ s −1 +H [qδ (s )−1] , β = (k T)−1. (1.3) N q j+l q j B " ! # j l=0 j X X X where m > 1. We assume ferromagnetic interactions K ≥ 0 and H ≥ 0, too. The Kronecker delta modulo q is given by: q−1 1 2iπks 1 when s = 0 (mod q) δ (s) = exp = . (1.4) q q q ( 0 otherwise k=0 (cid:18) (cid:19) X Introducing the Potts spins [31,32] 2iπs j σ = exp , (1.5) j q (cid:18) (cid:19) the Hamiltonian in (1.3) can be rewritten as †: q−1 m−1 q−1 −βH [{σ}] = K σk +H σk. (1.6) N j+l j j k=1 l=0 j k=1 XX Y XX † One may also express the Potts interaction using clock angular variables (see appendix A). Potts model with multi-site interactions 3 t t j N s j+3p=N j t N−2 t t j N−1 s j+3p=N−1 j t N t t j N−2 s j+3p=N−2 j t N−1 Figure 1. t-variables entering into the expression (2.3) of s for m = 3 and different j valuesofthedistanceN−j fromtheendofthechain. Thet-variables,definedin(2.2), are the sums of m s-variables (circles) with the convention s = 0 when i > N. The i second lines are subtracted from the first so that only s is remaining. j When q = 2, σ = ±1, k = 1 and the Ising multispin Hamiltonian studied in [30] is j recovered, which a posteriori justifies the choice of interaction (b) in (1.3). The zero-field partition function of the Potts chain with m-site interaction K is obtained for free BC in section 2 and for periodic BC in section 3. The periodic BC result allowsadeterminationoftheeigenvalues ofTm whereTisthesite-to-sitetransfer- matrix. The two-site correlation function is calculated in section 4. In section 5 the system with periodic BC is shown to be self-dual when the external field H is turned on. In section 6 the system with free BC is mapped onto a standard 2D Potts model with first-neighbour interactions K and H, length N/m and transverse size m. The mapping of 1D Potts models with m-site and n-site interactions is discussed in section 7. The conclusion in section 8 is followed by 4 appendices. 2. Zero-field partition function for free BC With free BC the zero-field Hamiltonian of a chain with N Potts spins, with m-site interaction K, takes the following form N−m+1 m−1 (f) −βH [{s}] = K qδ s −1 (2.1) N q j+l " ! # j=1 l=0 X X when written interms of the Potts variables s . Let us introduce the new Potts variables j t = 0,...,q−1 defined as j m−1 t = s (mod q), j = 1,...,N , (2.2) j j+l l=0 X Potts model with multi-site interactions 4 with the convention s = 0 when i > N in (2.2). Note that the relationship between old i and new variables is one-to-one with the inverse transformation given by (see figure 1): p s = (t −t ) (mod q), j +pm = N −l, l = 0,...,m−1. (2.3) j rm+j rm+j+1 r=0 X Using (2.2) in (2.1) one obtains a system of N −m+1 non-interacting Potts spins in a field K with N−m+1 (f) −βH [{t}] = K [qδ (t )−1]. (2.4) N q j j=1 X The canonical partition function is easily obtained and reads: N−m+1 N ZN(f) = Tr{t}e−βH(Nf)[{t}] = Trtj eK[qδq(tj)−1] Trtj 1 j=1 j=N−m+2 Y Y = qm−1 e(q−1)K +(q −1)e−K N−m+1 . (2.5) Note that although only N(cid:2) − m + 1 new vari(cid:3)ables enter into the expression of the transformed Hamiltonian (2.4), one has to trace over the N Potts variables t in (2.5). j When q = 2 the Ising result (equation (2.6) in [30]) is recovered. The free energy can be decomposed as follows (f) (f) F = −k T lnZ = Nf +F (m), (2.6) N B N b s where the bulk free energy per site f = −k T ln e(q−1)K +(q −1)e−K , (2.7) b B does not depend on m wher(cid:2)eas the surface contri(cid:3)bution exp[(q −1)K]+(q −1)exp(−K) F (m) = (m−1)k T ln , (2.8) s B q (cid:26) (cid:27) is m-dependent. 3. Zero-field partition function for periodic BC Let us now evaluate the partition function for a periodic chain with N sites and m > 1. To simplify the discussion we consider only the case where N is a multiple of m. Then the Hamiltonian takes the following form N=pm m−1 (p) −βH [{s}] = K qδ s −1 , (3.1) N=pm q j+l " ! # j=1 l=0 X X with s = s . Making use of the change of variables (2.2), it can be rewritten as: N+j j N=pm (p) −βH [{t}] = K [qδ (t )−1]. (3.2) N=pm q j j=1 X Potts model with multi-site interactions 5 With periodic BC the correspondence between {s} and {t} Potts configurations is no longer one-to-one and the new variables have to satisfy a set of m − 1 constraints [30,33–35]. There are several {s} configurations leading to the same {t}. One of these configurations, {s′}, is obtained by changing s into j s′ = s +∆ (mod q), j = 1,...,N . (3.3) j j j where the shifts ∆ = 0,...,q−1 have to satisfy some constraint. Let us first consider j t = m−1s (mod q). One can freely choose the first m−1 shifts (qm−1 choices) 1 l=0 l+1 and t keeps its value when ∆ is such that m−1∆ = 0 (mod q). Since t and t 1P m l=0 l+1 j j+1 have the shifts ∆ (l = 1,...,m−1 in common, the value of ∆ leaving t invariant is j+l P j j equal to the value of ∆ leaving t invariant. When N = pm a periodic repetition j+m j+1 with period m of the first m shifts acting on {s} leaves {t} invariant. Thus there are qm−1 Potts configurations {s′} leading to the same {t} ‡. When {s} is a ground-state configuration, qm−1 gives the ground-state degeneracy. In the following we shall make use of the Potts spin variables: m−1 2iπt 2iπt τ = exp j = σ , τ∗ = exp − j = τq−1, τ τ∗ = τq = 1. (3.4) j q j+l j q j j j j (cid:18) (cid:19) l=0 (cid:18) (cid:19) Y According to (1.4) one has: q−1 2iπkt Tr τk = exp j = qδ (k). (3.5) τj j q q tXj=0 (cid:18) (cid:19) For later use, note that the Boltzmann factor N=pm e−βH(Np)[{t}] = e−K + e(q−1)K −e−K δq(tj) , (3.6) j=1 Y (cid:2) (cid:0) (cid:1) (cid:3) can be rewritten as N=pm eqK −1 q−1 e−βH(Np)[{τ}] = e−NK 1+ τk q j " # j=1 k=0 Y X e−K N N=pm q−1 = eqK +q −1)+ eqK−1 τk , (3.7) q j " # (cid:18) (cid:19) j=1 k=1 Y (cid:0) (cid:1)X using (1.4) and (3.4), Let us consider the product of Potts spins p−1 P = τ τ∗ , i = 1,...,m−1. (3.8) i rm+i rm+i+1 r=0 Y Making use of m−1 τ τ∗ = σ σ σ∗ σ∗ = σ σ∗ (3.9) rm+i rm+i+1 rm+i rm+i+l rm+i+l (r+1)m+i rm+i (r+1)m+i ! l=1 Y ‡ Note that the initial configuration, {s}, corresponding to ∆ =0 ∀j, is taken into account. j Potts model with multi-site interactions 6 and taking into account the periodic BC, one obtains the constraints p−1 P = σ σ∗ = 1, i = 1,...,m−1, (3.10) i rm+i (r+1)m+i r=0 Y to be satisfied by the τ-configurations in (3.8). When m > 2 other constraints can be constructed, for instance from τ τ∗ , but these are automatically satisfied since rm+i rm+i+2 they can be written as products of the fundamental ones: τ τ∗ τ τ∗ . rm+i rm+i+1 rm+i+1 rm+i+2 1 Thus with the new Potts spin variables, taking the constraints into account, the | {z } partition function is given by: m−1 ZN(p=)pm = qm−1Tr{τ}e−βH(Np)[{τ}] δPi,1. (3.11) i=1 Y To go further we need an explicit expression for the Kronecker delta, δ . Consider the Pi,1 geometric series q−1 1−Xq f(X) = Xk = , (3.12) 1−X k=0 X it vanishes when X is a qth root of unity other than 1 and is equal to q when X = 1. Since P in (3.8) is a qth root of unity, the constraint can be written as (cf. (1.4)) i q−1 q−1 p−1 1 1 δ = Pk = τk τq−k , (3.13) Pi,1 q i q rm+i rm+i+1 k=0 k=0 r=0 X XY where (3.4) has been used. The partition function in (3.11) now takes the following form: m−1 q−1 p−1 ZN(p=)pm = Tr{τ}e−βH(Np)[{τ}] 1+ Pik , Pik = τrkm+iτrqm−+ki+1. (3.14) ! i=1 k=1 r=0 Y X Y The first product has the following expansion: m−1 q−1 1+ Pk = 1+ Pk + PkPk′ + PkPk′Pk′′ +···+ Pq−1.(3.15) i i i i′ i i′ i′′ i i=1 k=1 ! i,k i<i′,k,k′ i<i′<i′′,k,k′,k′′ i Y X X X X Y The expression of Pk in (3.14) is periodic with period m. There are two consecutive i Potts spins contributing to the product for each period and the sum of their exponents vanishes modulo q. Besides 1 the expansion (3.15) generates terms containing from l = 2 to m spins for each period with m possible spatial configurations {α }. These l l spatial configurations are labelled by the l spin exponents, each varying from 1 to q−1 (cid:0) (cid:1) with a sum which remains vanishing modulo q in the products, due to the Potts spins properties (3.4). As shown in appendix B, for l spins the number ν of allowed exponent l distributions is given by: 1 ν = (q −1)l +(−1)l(q−1) . (3.16) l q (cid:2) (cid:3) Potts model with multi-site interactions 7 Combining these results, the expansion can be written as (m) m−1 q−1 m l νl 1+ Pk = 1+ Ξβl , (3.17) i αl ! Yi=1 Xk=1 Xl=2 αXl=1βXl=1 where Ξβl is a product for each period of l Potts spins in configuration α with an αl l exponent distribution β . l The partition function in (3.14) splits in two parts: (m) m l νl ZN(p=)pm = Tr{τ}e−βH(Np)[{τ}]+ Tr{τ}e−βH(Np)[{τ}]Ξβαll . (3.18) A Xl=2 αXl=1βXl=1 B In A, according to (3.5), each of the pm factors in (3.7) contributes to the trace by: | {z } | {z } q−1 e−K Tr eqK +q −1+ eqK −1 τk = e−K eqK +q −1 . (3.19) q τj j " # k=1 (cid:0) (cid:1)X (cid:0) (cid:1) In B, for each period, Ξβl contains l supplementary spin terms of the form τk′ with αl j k′ = 1,2,...,q − 1. Thus the trace involves p(m − l) factors given by (3.19) and pl factors of the form q−1 e−K Tr eqK+q −1 τk′+ eqK−1 τk+k′ = e−K eqK−1 , (3.20) q τj j j " # k=1 (cid:0) (cid:1) (cid:0) (cid:1)X (cid:0) (cid:1) where the non-vanishing contribution comes from the term q − k′ in the sum over k according to (3.5). Collecting the different contributions to the partition function, we finally obtain m m Z(p) = ν e(q−1)K+(q−1)e−K p(m−l) e(q−1)K−e−K pl N=pm l l l=0 (cid:18) (cid:19) X (cid:2) (cid:3) (cid:2) (cid:3) m m eqK −1 pl = e(q−1)K+(q−1)e−K N 1+ ν , (3.21) l l eqK+q−1 " # l=2 (cid:18) (cid:19) (cid:18) (cid:19) (cid:2) (cid:3) X where ν , given by (3.16), is such that ν = 1 and ν = 0. For q = 2 l 0 1 1 when l is even ν = (3.22) l ( 0 when l is odd and the Ising result, equation (3.13) in [30], is recovered. Let T be the transfer matrix from |σ σ ...σ i to |σ σ ...σ i. As j j+1 j+m−2 j+1 j+2 j+m−1 discussed in appendix C, its mth power is real and symmetric. The real eigenvalues of Tm, ω , and their degeneracy, g , can be deduced from the expression (3.21) of the l l partition function (see (C.6)). Potts model with multi-site interactions 8 4. Zero-field correlation function In this section the zero-field correlation function is obtained for free BC and m > 1. The correlations between the Potts variables at sites i and i′ are evaluated by taking the thermal average of the following expression: qδq(si −si′)−1 . (4.1) (q−1) It is equal to one when the two sites are in the same state and has a vanishing average in a fully disordered system. Making use of (1.4) and (1.5) the numerator in (4.1) can be expressed in terms of Potts spins as: q−1 q−1 qδq(si −si′)−1 = exp 2iπk(sqi −si′) = σikσi∗′k. (4.2) k=1 (cid:20) (cid:21) k=1 X X Let us first suppose that i′ = i+rm. Taking into account (3.9) one may write r−1 σiσi∗+rm = τr′m+iτr∗′m+i+1, (4.3) r′=0 Y and the correlation function takes the following form: G(f)(i,i+rm) = qδq(si −si+rm)−1 = Tr e−βH(Nf)[{τ}] q−1 r−1 τk τ∗k . (4.4) N (q−1) {τ} (q−1)Z(f) r′m+i r′m+i+1 (cid:28) (cid:29) N k=1r′=0 X Y Following the same steps that led to (3.7), the Boltzmann factor for free BC can be written as e(q−1)K +(q −1)e−K N−m+1 N−m+1 eqK −1 q−1 e−βH(Nf)[{τ}] = 1+ τk′ q eqK +q −1 j (cid:20) (cid:21) j=1 " k′=1 # Y X Z(f) N−m+1 eqK −1 q−1 = N 1+ τk′ (4.5) qN eqK +q −1 j j=1 " k′=1 # Y X (f) where the expression of Z in (2.5) has been used §. Inserting this expression in (4.4), N one obtains: q−1 r−1 1 G(f)(i,i+rm) = Tr τk τ∗k × N qN(q −1) {τ} r′m+i r′m+i+1 k=1 r′=0 X Y N−m+1 eqK −1 q−1 × 1+ τk′ , (4.6) eqK +q −1 j " # j=1 k′=1 Y X The trace over {τ} contains r factors with j = r′m+i of the form q−1 eqK −1 eqK −1 Tr τk + τk+k′ = q , (4.7) τr′m+i r′m+i eqK +q −1 r′m+i eqK +q −1 " k′=1 # X § Takingthe traceoverthe N Pottsspinsin(4.5),allthe termsintheproductinvolvingτ vanishand j the trace over 1 gives qN. Potts model with multi-site interactions 9 theonlynon-vanishingcontributioncomingfromthesecondtermfork′ = q−k according to (3.5). The same result is obtained for the r factors with j = r′m+i+1 and k′ = k. The trace over the remaining N − 2r Potts spins contributes a factor qN−2r, the sum over k gives q −1, so that, finally: eqK −1 2r rm (f) G (i,i+rm) = = exp − . (4.8) N eqK +q −1 ξ (cid:20) (cid:21) (cid:18) (cid:19) As expected, this expression can be rewritten in terms of transfer matrix eigenvalues (C.6) as (ω /ω )r. The correlation length, given by 2 0 m eqK +q −1 −1 ξ = ln , (4.9) 2 eqK −1 (cid:20) (cid:18) (cid:19)(cid:21) diverges at the zero-temperature critical point when K → ∞. Let us now consider the case where i′ −i is not a multiple of m. Using the Potts spin variables (1.5) and (3.4) the inverse transformation in (2.3) translates into: p σi = τr′m+iτr∗′m+i+1, i+l = N −pm, l = 0,...,m−1.(4.10) r′=0 Y In the same way let p−r σi∗′ = τr∗′m+i′τr′m+i′+1, i′+l′ = N−pm+rm, l′ = 0,...,m−1,(4.11) r′=0 Y with, in both cases, τ = 1 when j > N. Since i′−i = rm+l−l′, we need l 6= l′. In the j product σ σ∗, the last factor contributed by σ is either τ∗ or τ when l = 0 whereas i i′ i N−l+1 N for σj∗′ it is either τN−l′+1 or τN∗ when l′ = 0. Thus these factors cannot all disappear in the product when l 6= l′. At least one of them leads to a vanishing trace over {τ} in the correlation function since the product over j in the Boltzmann factor (4.5) ends at N −m+1. It follows that: G(f)(i,i′) = 0, i′ −i 6= rm. (4.12) N When m = 2 and q = 2 this argument no longer applies. With m = 2 the τ and the τ∗ j j always appear twice inthe product σ σ∗ for values ofj ≥ i′. Accordingly, the correlation i i′ function does not vanish since τ2 = τ∗2 = 1 when q = 2. The difference between q = 2 j j andq > 2 when m = 2 canbe understoodby looking atthe behaviour ofthecorrelations in the ground state. For q = 2 there are 2 degenerate ground states which, using Potts variables, are given by 00000... and 11111... so that h2δq(si − si′) − 1i = 1. When q = 3, for example, there are 3 degenerate ground states, 00000..., 12121... and 21212..., leading to h3δq(si −si′)−1i = 0 when i′ −i is odd. 5. Self-duality under external field In this section standard methods [4,7,36] are used to show that the Potts chain with multi-site interactions and periodic BC is self-dual under external field. Potts model with multi-site interactions 10 s~ a s~ i+1 i+3/2 s s s s s s s i i+1 i+2 i i+1 i+2 i+3 s~ s~ s~ s~ s~ s~ s~ i−1 i i+1 i−3/2 i−1/2 i+1/2 i+3/2 s s i i m=3 m=4 Figure2. PositionofthedualPottsvariables(squares)enteringinthedefinitions(5.6) of u and v relative to the original ones (circles) for odd and even values of m. i i According to (1.3), the partition function is given by: N m−1 Z(p)(K,H) = e−N(K+H)Tr exp qKδ s exp[qHδ (s )] . (5.1) N {s} q j+l q j " !# j=1 l=0 Y X Introducing the auxiliary function C(X,x) = eqX −1+qδ (x), (5.2) q one obtains the identity: q−1 1 2iπxy eqXδq(y) = 1+ eqX −1 δ (y) = C(X,x)exp . (5.3) q q q x=0 (cid:18) (cid:19) (cid:0) (cid:1) X Thus the partition function can be rewritten as: e−N(K+H) N q−1 q−1 (p) Z (K,H) = Tr C(K,u )C(H,v ) N qN {s} j j Yj=1uXj=0vXj=0 m−1 1 2iπ × exp v s +u s . (5.4) j j j j+l q q " !# l=0 X Regrouping the factors containing s in the last exponential and reordering the sums, i one obtains e−N(K+H) N N 1 q−1 2iπs w (p) i i Z (K,H) = Tr C(K,u )C(H,v ) exp N qN {u,v} j j q q Yj=1 Yi=1 sXi=0 (cid:18) (cid:19) e−N(K+H) N N = Tr C(K,u )C(H,v ) δ (w ), (5.5) qN {u,v} j j q i j=1 i=1 Y Y where w stands for v + m−1u . i i l=0 i−l P