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Study of the Potts Model on the Honeycomb and Triangular Lattices: Low-Temperature Series and Partition Function Zeros PDF

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Study of the Potts Model on the Honeycomb and Triangular 8 Lattices: Low-Temperature Series and Partition Function Zeros 9 9 1 Heiko Feldmann(a)∗, Anthony J. Guttmann(b)†, Iwan Jensen(b)‡, Robert Shrock(a)§, Shan-Ho n Tsai(a)∗∗ a J (a) Institute for Theoretical Physics 8 2 State University of New York ] h Stony Brook, N. Y. 11794-3840, USA c e m (b) Department of Mathematics and Statistics - t The University of Melbourne a t s . Parkville, Vic 3052, Australia t a m - d n We present and analyze low-temperature series and complex-temperature o c partition function zeros for the q-state Potts model with q = 4 on the honey- [ 1 comb lattice and q = 3,4 on the triangular lattice. A discussion is given as v 5 to how the locations of the singularities obtained from the series analysis cor- 0 3 1 relate with the complex-temperature phase boundary. Extending our earlier 0 8 work, we include a similar discussion for the Potts model with q = 3 on the 9 / t honeycomb lattice and with q = 3,4 on the kagom´e lattice. a m 05.20.-y, 64.60.C, 75.10.H - d n o c : v ∗email: [email protected] i X †email: [email protected] r a ‡email: [email protected] §email: [email protected] ∗∗email: [email protected] 1 I. INTRODUCTION The 2D q–state Potts models [1,2] for various q have been of interest as examples of different universality classes for phase transitions and, for q = 3,4, as models for the adsorp- tion of gases on certain substrates. Unlike the q = 2 (Ising) case, however, for q 3, the ≥ free energy has never been calculated in closed form for arbitrary temperature. It is thus of continuing value to obtain further information about the 2D Potts model. It has long been recognized that a very powerful method for doing this is via the calculation and analysis of series expansions for thermodynamic quantities such as the specific heat, magnetization, and susceptibility [3]. For q = 2,3, and 4, the respective 2D q-state Potts ferromagnets have con- tinuous phase transitions, and the critical singularities and associated exponents are known exactly [2,4,5]. Recently, two of us have calculated and analyzed long low-temperature se- ries expansions for the Potts model with q = 3 on the honeycomb lattice and for the Potts model with q = 3 and q = 4 on the kagom´e lattice [6]. These have been used to make very precise estimates of the respective critical points, to confirm a formula for the honeycomb lattice and to strengthen a previous refutation of an old conjecture for the kagom´e lattice. The other three authors have recently used a relation between complex-temperature (CT) properties of the Potts model on a given lattice and physical properties of the Potts anti- ferromagnet (AF) on the dual lattice to rule out other conjectures [7] and have calculated complex-temperature zeros of the partition function for these three cases of q and lattice type [8]. The study of properties of spin models with the magnetic field and temperature generalized to complex values was pioneered by Yang and Lee [9] for the magnetic field and Fisher for the temperature [10]. Some of the earliest work on CT properties of spin models dealt with zeros of the partition function [10–12]. Another major reason for early interest in CT properties of spin models was the fact that unphysical, CT singularities complicated the analysis of low-temperature series expansions to get information about the location and critical exponents of the physical phase transition [13]. HereweshallpresentaunifiedstudyofthePottsmodelonthehoneycomblatticeforq = 4 andonthetriangularlatticeforq = 3andq = 4. Foreachq valueandlatticetype, ourresults include (i) long, low-temperature series for the specific heat, spontaneous magnetization, and initial susceptibility derived using the finite-lattice method [14,15], extended by noting the structure of the correction terms [16]; (ii) a calculation of the complex-temperature zeros and, from these, an inference about the CT phase boundary; and (iii) a discussion of how the positions of the physical and unphysical singularities extracted from the series analysis correlate with the CT phase boundary. Since both the critical exponents and the location of the paramagnetic-to-ferromagnetic phase transition are known exactly for these models, 1 we shall focus mainly on getting new information on complex-temperature properties from the series and CT zeros. Using the results of Refs. [6–8], we shall also discuss subject (iii) for the Potts model with q = 3 model on the honeycomb lattice and with q = 3 and 4 on the kagom´e lattice. It is useful to perform a unified analysis of this type because, aside from well-understood exceptions [17], the physical and complex-temperature singularities of the thermodynamic functions lie on the continuous locus of points which serves as the B boundariesofthecomplex-temperature phases [19]; consequently, anapproximateknowledge (or exact knowledge, if available) of where this boundary lies is of considerable help in checking which CT singularities that one extracts from a series analysis are trustworthy and which are not. This will be discussed further below. Note that low-temperature series on the honeycomb lattice correspond to high-temperature series on the triangular lattice, and vice versa. II. MODEL The (isotropic, nearest-neighbor) q-state Potts model at temperature T on a lattice Λ is defined by the partition function Z = Xe−βH (2.1) {σn} with the Hamiltonian H = J X(1−δσnσn′)+H X(1−δ0 σn) (2.2) hnn′i n where σ = 0,...,q 1 are Z -valued variables on each site n Λ, β = (k T)−1, and nn′ n q B − ∈ h i denotes pairs of nearest-neighbor sites. The symmetry group of the Potts Hamiltonian is the symmetric group on q objects, S . We use the notation K = βJ, q a = z−1 = eK (2.3) eK 1 x = − . (2.4) √q and µ = e−βH (2.5) 2 (The variable z was denoted u in Ref. [6].) The (reduced) free energy per site is denoted as f = βF = lim N−1lnZ, where N denotes the number of sites in the lattice. − Ns→∞ s s There are actually q types of external fields which one may define, favoring the respective values σ = 0,..,q 1; it suffices for our purposes to include only one. The order parameter n − (magnetization) is defined to be qM 1 m = − (2.6) q 1 − where M = σ = lim ∂f/∂h. With this definition, m = 0 in the S -symmetric, dis- h→0 q h i ordered phase, and m = 1 in the limit of saturated ferromagnetic (FM) long-range order. Finally, the (reduced, initial) susceptibility is denoted as χ¯ = β−1χ = ∂m/∂h . We con- h=0 | sider the zero-field model, H = 0, unless otherwise stated. For J > 0 and the dimensionality of interest here, d = 2, the q-state Potts model has a phase transition from the symmetric, high-temperature paramagnetic (PM) phase to a low-temperature phase involving sponta- neous breaking of the S symmetry and onset of ferromagnetic (FM) long-range order. This q transition is continuous for 2 q 4 and first-order for q 5. As noted above, the model ≤ ≤ ≥ has the property of duality [1,2,21,22], which relates the partition function on a lattice Λ with temperature parameter x to another on the dual lattice with temperature parameter 1 a+q 1 x (x) = , i.e. a (a) = − . (2.7) d d ≡ D x ≡ D a 1 − Other exact results include formulas for the PM-FM transition temperature on the square, triangular, and honeycomb lattices [1,22], and calculations of the free energy at the phase transition temperature, and of the related latent heat for q 5 [23]. The case J < 0, i.e., ≥ the Potts antiferromagnet (AF), has also been of interest because of its connection with graph colorings. Depending on the type of lattice and the value of q, the antiferromagnetic model may have a low-temperature phase with AFM long-range order. Alternatively, it may not have any finite-temperature PM-AFM phase transition but instead may exhibit nonzero ground state entropy. For the Potts model on the honeycomb lattice, the well-known q = 2 (Ising) case [24,25] falls into the former category, while the model with q 3 falls into the ≥ latter category [26,27] with nonzero ground state entropy [27–30]. Reviews of the model include Refs. [2,31]. For the q-state Potts model, from duality and a star-triangle relation, together with a plausible assumption of a single transition, one can derive algebraic equations that yield the PM-FM critical points for the honeycomb (hc) and triangular (t) lattices [22]. The equation for the honeycomb lattice is x3 3x √q = 0 , i.e., a3 3a2 3(q 1)a q2 +3q 1 = 0 (honeycomb) (2.8) − − − − − − − 3 and, asfollows fromeq. (2.7), thecorresponding formulafor thetriangularlatticeisobtained by the replacement x 1/x: → √qx3 +3x2 1 = 0 , i.e., a3 3a+2 q = 0 (triangular) . (2.9) − − − It will be useful to have the explicit solutions for the cases studied here. For q = 4 on the honeycomb lattice, eq. (2.8) reduces to (a 5)(a + 1)2 = 0, yielding the PM-FM critical − point a = z−1 = 5 (2.10) hc,PM−FM,q=4 hc,PM−FM,q=4 together with a double root at the complex-temperature value a = z−1 = 1 (2.11) hc,2,q=4 hc,2,q=4 − For q = 3 on the triangular lattice, eq. (2.9) has the solutions a = a = cos(2π/9)+√3sin(2π/9) = 1.879385... (2.12) t,1,q=3 t,PM−FM,q=3 i.e., z = 0.5320889..., t,PM−FM,q=3 a = cos(2π/9) √3sin(2π/9) = 0.347296... (2.13) t,2,q=3 − − and a = 2cos(2π/9) = 1.532089... (2.14) t,3,q=3 − − For q = 4, eq. (2.9) reduces to (a 2)(a + 1)2 = 0, so that the physical PM-FM critical − point is given by a = a = 2 (2.15) t,1,q=4 t,PM−FM,q=4 and there is a double root at the CT value a = 1 . (2.16) t,2,q=4 − III. SERIES EXPANSIONS The low-temperature series expansion is based on perturbations from the fully aligned groundstateandisexpressed intermsofthelow-temperaturevariablez andthefieldvariable 4 y = 1 µ. Details of the methods can be found in Ref. [6], so here it suffices to say that − in order to derive series in z for the specific heat, magnetization and susceptibility one need only calculate the expansion in y to second order, i.e., Z = Z (z)+yZ (z)+y2Z (z), (3.1) 0 1 2 where Z (z) is a series in z formed by collecting all terms in the expansion of Z containing k factors of yk. We use the finite-lattice method [15] to approximate the infinite-lattice parti- tion function Z by a product of partition functions Z on finite (m n) lattices, with each m,n × Z calculated by transfer matrix techniques. As explained in Ref. [6], this leads to a series m,n in z correct to order w (m 2)+m 1, where w is the maximal number of sites contained s s − − within the largest width w of the rectangles, and m is the number of nearest neighbors of each site. The implementation of the algorithm on the honeycomb lattice [6] has w = 2w s and m = 3. The triangular lattice is represented as a square lattice with additional inter- actions along one of the diagonals, and in this case w = w and m = 6. In addition, we s make use of a recent extension procedure discussed in Ref. [16], which allows us to calculate additional series terms. The extension procedure forthe 4-statePottsmodel onthe honeycomb latticeis thesame as for the 3-state model [6]. For a given width the expansion is correct to order 2w+2, and we calculated the series up to w = 12. Next we look at the integer sequences d (w) obtained s by taking the difference between the expansions obtained from successive widths w, Zw+1(z) Zw(z) = z2w+3Xds(w)zs. (3.2) − s≥0 In this case the formulae for the correction terms are simply given by polynomials of order 2s+ k. We managed to find formulae for the first 4 correction terms, which enabled us to calculate the series for the specific heat C, magnetization m, and susceptibility χ¯ to order 30. The resulting series for m, χ¯, and the (reduced) specific heat C¯ = C/(k K2) are given B in Table 1. The extension procedure for the triangular lattice is essentially the same as for the hon- eycomb lattice. The only difference is that the order of the polynomials is s+k. For a given width the expansion is correct to order 4w + 5, and we calculated the series up to w = 14 for q = 3 and up to w = 12 for q = 4. We found formulae for the first 7 or 8 correction terms in the case q = 3 and the first 6 or 7 correction terms for q = 4. The series were thus derived to order 69 (60) for the specific heat and magnetization and to order 68 (59) for the susceptibility in the case q = 3 (q = 4). The resulting series for m, χ¯, and the (reduced) specific heat C¯ = C/(k K2) are given in Tables 2 and 3. B 5 IV. ANALYSIS OF SERIES A. Honeycomb Lattice, q = 4 We have analyzed the series using dlog Pad´e approximants (PA’s) and differential ap- proximants (DA’s); for a general review of these methods, see Ref. [3]. We first comment on the physical PM-FM phase transition. The series yield a value for the critical point in excellent agreement with the known value z = 1/5. For example, the differen- hc,PM−FM,q=4 tial approximants of the type [L/M ,M ] with L = 1 and L = 2 to the specific heat series 0 1 yield z = 0.19993(4) and 0.19991(5), while those for the magnetization yield hc,PM−FM,q=4 0.19999(3) and 0.20005(7), respectively, with similarly good agreement for other values of L andfortheapproximantstothesusceptibility. Concerningthecriticalexponents atthistran- sition, the value q = 4 is the borderline between the interval 2 q 4 where this transition ≤ ≤ is second-order and the interval q > 4 where it is first order. Related to this, the q = 4 2D Potts model has the special feature that the thermodynamic functions have strong confluent logarithmic corrections to their usual algebraic scaling forms [4] at the PM-FM transition (on any lattice). For example, the singularities in the specific heat and magnetization are C t −2/3( ln t )−1 for t 0, where t = (T T )/T , and M ( t)1/12( ln t )−1/8 sing c c sing ∼ | | − | | → − ∼ − − | | for t 0−. Consequently, simple fits of the series to an algebraic singularity without this → confluent logarithmic correction are not expected to agree well with the known singularities. Indeed, this was the general experience in early series work, and the same is found for the longer series here. As an illustration, a naive fit to a simple algebraic singularity for the spe- cific heat would yield the value α′ 0.5 rather than the known value α′ = 2/3. Since these ∼ confluent singularities may also affect singularities at complex-temperature points, it could be useful in future work, as was noted earlier for the square-lattice model [32], to carry out a more sophisticated analysis of the series including these confluent singularities. However, because our primary focus here is on obtaining new information on complex-temperature properties rather than reproducing exactly known results for the critical exponents of the physical PM-FM singularity, and because it is not known if the confluent logarithmic cor- rections do affect the CT singularities, we have not tried to include such logarithmic factors in fits to the CT singularities. Proceeding to CT singularities, we find evidence for one on the negative real z axis at z = 0.33(1) , i.e a = 3.0(1) (4.1) hc,ℓ,q=4 hc,ℓ,q=4 − − Here the subscript ℓ stands for “leftmost” singularity on the negative real axis. We shall present below, as an application of the mapping discussed in Ref. [7], an analytic derivation 6 of the exact value a = 3. Clearly, the value extracted from the series analysis is in hc,ℓ,q=4 − excellent agreement with the exact determination. By the mapping of Ref. [7], it follows that the singularity in the specific heat at this point a , as approached from larger negative hc,ℓ,q=4 a, i.e. smaller negative z, is the same as the singularity in the specific heat of the q = 4 Potts antiferromagnet on the triangular lattice at the T = 0 critical point as approached from finite temperature. We also find evidence from the series analyses for at least one complex conjugate (c. c.) pair of singularities. One such pair is observed at z , z∗ = 0.02(2) 0.38(1)i (4.2) hc,cc1,q=4 hc,cc1,q=4 ± The central values correspond to a , a∗ = 0.14 2.6i. As we shall show later, hc,cc1,q=4 hc,cc1,q=4 ± this pair of singularities is consistent with lying on the complex-temperature phase boundary . B B. Triangular Lattice, q = 3 The series yield values for the PM-FM critical point in excellent agreement with the ex- actly known expression, eq. (2.12). For example, the first-order DA’s of the form [L/M ,M ] 0 1 with L = 1 for the free energy yield z = 0.532095(85), in complete agreement, to t,PM−FM,q=3 within the uncertainty, with the known value given by eq. (2.12). For reference, the thermal and field exponents for the 2D q = 3 Potts model are y = 6/5 and y = 28/15, so that the t h critical exponents for the specific heat, magnetization, and susceptibility are α = α′ = 1/3, β = 1/9, and γ = γ′ = 13/9 = 1.444... [2,5]. The above approximants yield the exponent α′ = 0.331(27), again in agreement with the known value. Similar statements apply to the magnetization and susceptibility. Concerning complex-temperature singularities, the series for m and χ¯ indicate a singu- larity on the negative real axis, at z 0.71 and z 0.65. If we assume that t,−,q=3 t,−,q=3 ≃ − ≃ − this is, as it should be, the same singularity, and average the positions, we get z z = 0.68(5) (4.3) t,−,q=3 t,ℓ,q=3 ≡ − or equivalently, a = 1.47(11) (4.4) t,ℓ,q=3 − where the numbers in parentheses are our estimates of the theoretical uncertainties. We observe thatour numerical determination ineq. (4.4) isconsistent, to within theuncertainty, 7 with being equal to the value given by the root in eq. (2.14), whence our use of the symbol a in eq. (4.4). t,ℓ,q=3 We find a complex-conjugate pair of singularities at z , z∗ = 0.0209(1) 0.531(1)i (4.5) t,e,q=3 t,e,q=3 ± From our analysis of the respective series, we infer the following values of singular exponents at the points (4.5): (α′, β, γ′) = (1.19(1), 0.18(1), 1.17(1)) (4.6) zt,e,q=3 − The central values in eq. (4.5) correspond to a , a∗ = 0.0740 1.88i (4.7) t,e,q=3 t,e,q=3 ± From the CT zeros to be discussed below, we see clearly that the c. c. members of this pair are endpoints of arcs of CT zeros of Z, corresponding to continuous arcs of sin- gularities of the free energy in the thermodynamic limit. (This type of correspondence with endpoints on is indicated by the subscripts e here and in other cases below.) B In passing, we observe that the exponent values in eq. (4.6) are not too different from the respective exponents obtained from the series analysis of Ref. [33] the singularities u = 0.301939(5) 0.3787735(5)i in the 2D spin-1 Ising model on the square lattice, s − ± namely [33] (α′, β, γ′) = (1.1693(3), 0.1690(2), 1.1692(2)). The c.c. pair of points u s − is analogous to the pair in eq. (4.7) because the members of this pair were shown [34] to be endpoints of arcs of CT zeros protruding into the complex-temperature extension of the FM phase of the spin-1 square-lattice Ising model. We also observe that the values of both these sets of exponents are consistent with the equality α′ = γ′. However, we already know that such an equality, even if it were to hold for these cases, is not a general result for singular exponents at endpoints of arcs of a CT boundary protruding into the complex- B temperature extension of the FM phases for a spin model. A counterexample is provided by the (isotropic, spin 1/2)Ising model on thetriangular lattice. In this case, one candetermine the complex-temperature phase diagram exactly, and consists of the union of the unit cir- B cle u+1/3 = 2/3 and the semi-infinite line segment u 1/3 [35], where u = z2. | | −∞ ≤ ≤ − Thus, in this case there is an exactly known analogue to the arc endpoints, viz., the endpoint at u = 1/3 (where the subscript e denotes “endpoint”) of the line segment protruding into e − the complex-temperature extension of the FM phase. An analysis of low-temperature series [36] had earlier yielded the inference that γ′ = 5/4, while exact results [35] yielded α′ = 1 e e (and β = 1/8), so that α′ = γ′. e − e 6 e 8 We find a second c. c. pair at zt,e′,q=3,zt∗,e′,q=3 = −0.515(3)±0.322(3)i (4.8) with exponents (α′, β, γ′) = (1.2(1), 0.25(10), 1.2(1)). The central values in eq. (4.8) − correspond to at,e′,q=3,a∗t,e′,q=3 = −1.40±0.873i (4.9) This pair is consistent with lying on the CT phase boundary, as will be discussed below. It should be noted that we would not expect the low-temperature series to be sensitive to the thirdroot,a ,ofeq. (2.9),sincethisrootismaskedbythenearersingularitya (that t,2,q=3 t,3,q=3 is, z = 0.652704... is closer to the origin in the z plane than z = 2.879385..). t,3,q=3 t,2,q=3 − − C. Triangular Lattice, q = 4 For the physical PM-FM critical point of the q = 4 Potts model on the triangular lattice, the discussion that we gave above for the honeycomb lattice applies; that is to say, the position of the physical singularity is well approximated, but the critical exponents are not, due to the presence of confluent logarithms. For complex-temperature properties, we first note that the series do not give a firm indication of a singularity on the negative real axis. We find a complex-conjugate pair of singularities at z , z∗ = 0.0304(2) 0.498(2)i (4.10) t,e,q=4 t,e,q=4 ± We have also studied the exponents at this pair of singularities. If one assumes that there are no strong confluent singularities present, such as the logarithms that are present at the physical critical point, then from our series analysis we extract the following values, with their quoted uncertainties: (α′, β, γ′) = (1.18(2), 0.17(2), 1.20(2)) (4.11) zt,e,q=4 − However, we caution that it is not known whether strong confluent singularities are present at the points (4.10), and if they are, then the values in eq. (4.11) would have a lower degree of reliability. The central values in eq. (4.10) correspond to a , a∗ = 0.122 2.00i (4.12) t,e,q=4 t,e,q=4 ± As in the q = 3 case, from the CT zeros to be presented below, we see clearly that the c. c. members of this pair of singularities are endpoints of arcs of CT zeros of Z. 9

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