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The one-dimensional Potts model with long-range interactions: a renormalization group approach

Sergio A Cannas, Aglaé C N de Magalhães

1997
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Journal of Physics A: Mathematical and General
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The one-dimensional q-state Potts model with ferromagnetic pair interactions which decay with the distance r as 1/r α is considered. We calculate, through a real-space renormalization group technique using Kadanoff blocks of length b, the critical temperature T c (b, q, α) and the correlation length critical exponent ν(b, q, α) as a function of α for different values of q. Some of the very few known rigorous results for general q are reproduced by our approach. Several asymptotic behaviours are
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... derived analytically for q = 2, 3 in the b → ∞ limit. We also obtain extrapolated critical temperatures (b = ∞) for arbitrary values of α > 1 and for q = 2, 3, 4, which we believe approximate the exact ones well, except in the region near α = 2. Furthermore, the use of another extrapolation procedure suitable only in the vicinity of α = 2 led us to conjecture that the exact critical temperature T c (q, α = 2) is the same for any value of q. We also verify that T c (q, α → 1) ∝ (α − 1) −1 ∀q, which is consistent with a recent conjecture of Tsallis. where to each site, i, we associate a Potts variable, σ i , which can assume q integer values (σ i = 1, 2, . . . , q), r ij is the distance (in crystal units) between sites i and j (i.e. r ij = i − j = 1, 2, 3 · · ·), J > 0 is the ferromagnetic coupling constant between nearest neighbours, δ(σ i , σ j ) is the Kronecker delta function, and the sum (i,j ) runs over all distinct pairs of sites of a one-dimensional lattice of N sites. The α → ∞ limit corresponds to the first-neighbour model, while the α = 0 limit corresponds to the infinite-range ferromagnet which, after a rescaling J → J /N, yields basically the mean-field approach. This model, in its plain formulation (α → ∞ of equation (1) ) or in a more general one with many-body interactions, is at the heart of a complex network of relations between geometrical and/or thermal statistical models, such as for example various types of percolation, vertex models, generalized resistor and diode network problems, classical spin models, etc (see [15] and references therein). On the other hand, the one-dimensional Potts model with LR interactions has definitely not been studied so much. In particular, very few rigorous results for general q are known. Let us summarize some of the most relevant results to date: (i) this model exhibits LR order at finite temperatures [16] T T c (q, α) for 1 < α 2; for α → 1 the critical temperature diverges and for α 1 the thermodynamic limit is not defined and the system becomes non-extensive; (ii) for α > 2 (SR interactions) it has no phase transition at finite temperatures [16] for all q 1, more precisely, T c = 0; (iii) it has been proved that for α = 2 the order parameter is discontinuous at T = T c = 0 for any q [16]; (iv) for q = 1 the percolation threshold satisfies 1/p c 2ζ(α) for 1 < α 2, where ζ(α) is the Riemann Zeta function [17] . All the following additional results correspond to the q = 2 case, which is, up to now, the best one studied: (v) for 1 < α < 1.5 the critical exponents are classical [18]; (vi) the region 1.5 < α < 2 shows non-trivial critical exponents, which are not known exactly. Approximate results in the latter region were obtained by different methods such as (among others): series expansions [19], finite-range scaling approximations [20], coherent anomaly method [21], real-space renormalization group [22], -expansions [3, 6], around α = 2 where the critical behaviour is of an essential singularity type [23] , and α = 1.5. Some approximate results for the critical temperature and the correlation length critical exponent ν were obtained for a wide range of values of q using finite-range-scaling calculations [24] . The α = 2 (i.e. the 1/r 2 potential) case is of particular interest because for q = 2 it can be mapped onto the spin-1 2 Kondo problem [25] (which is related to recent developments in high temperature superconductivity [26] ) and for a general value of q > 2 it may be related to higher spin generalizations of the Kondo problem [23] . In order to calculate the critical temperature and the critical exponent ν of the q-state LR Potts model in the extensive region 1 < α 2 we use a real-space renormalization group (RG) method, the cumulant method of Niemeijer and van Leeuwen [27], based on a construction of Kadanoff blocks using the majority rule. Although the convergence of the cumulant method for a fixed block size can become questionable in some cases (for a discussion on the advantages and disadvantages of the method see, for example [28, 29] ), a

doi:10.1088/0305-4470/30/10/014
fatcat:4wbufeojora7bcl3falhc4vcnu