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First-order transitions in large-q one-dimensional Potts model with power-law decaying interactions

Zvonko Glumac, Katarina Uzelac

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

We have performed exact analytical calculations of the cluster probability distribution and interface free energy in the finite q-state Potts chains with interactions decaying with distance r as r^(1+sigma). By presenting the partition function in form of the polynomial in q, the advantage of using the Hamiltonian configurations compared with graph or connectivity approach was examined. Scaling analysis of the probability performed for large q (q >= 16) yields the first-order phase transition for 0 < sigma < 1.

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We present a large- q expansion of the 2d q -states Potts model free energies up to order 9 in 1/√ q . Its analysis leads us to an ansatz which, in the first-order region, incorporates properties inferred from the known critical regime at q = 4, and predicts, for q > 4, the n -th energy cumulant scales as the power (3 n /2 - 2) of the correlation length. The parameter-free energy distributions reproduce accurately, without reference to any interface effect, the numerical data obtained in a simulation for q = 10 with lattices of linear dimensions up to L = 50. The pure-phase specific heats are predicted to be much larger, at q ⩽ 10, than the values extracted from the current finite-size scaling analysis of the extrema. Implications for safe numerical determinations of interface tensions are discussed.
We study phase transitions of the ferromagnetic qqq-state Potts chain with random nearest-neighbor couplings having a variance ${\mathrm{\ensuremath{\Delta}}}^{2}$ and with homogeneous long-range interactions, which decay with distance as a power ${r}^{\ensuremath{-}(1+\ensuremath{\sigma})}, \ensuremath{\sigma}>0$. In the large-qqq limit the free-energy of random samples of length $L\ensuremath{\le}2048$ is calculated exactly by a combinatorial optimization algorithm. The phase transition stays first order for $\ensuremath{\sigma}<{\ensuremath{\sigma}}_{c}(\mathrm{\ensuremath{\Delta}})\ensuremath{\le}0.5$, while the correlation length becomes divergent at the transition point for ${\ensuremath{\sigma}}_{c}(\mathrm{\ensuremath{\Delta}})<\ensuremath{\sigma}<1$. In the latter regime the average magnetization is continuous for small enough $\mathrm{\ensuremath{\Delta}}$, but for larger $\mathrm{\ensuremath{\Delta}}$---according to the numerical results---it becomes discontinuous at the transition point, thus the phase transition is expected of mixed order.
Relying on the fractal character of the largest clusters at criticality, we employ a finite-size scaling analysis to obtain an accurate phase-diagram of the percolation transition in chains with bond concentration decaying as a power-law on the form 1/ r(1+sigma) . For the particular case of sigma=1, no percolation transition is observed to occur at a finite dilution, in contrast with the finite temperature Kosterlitz-Thouless transition exhibited in Ising and Potts chains with inverse square-law couplings. The fractal dimension of the critical percolation cluster is found to follow distinct dependencies on the decay exponent being numerically fitted by d(f) =0.35+4sigma/5 for 0<sigma<1/2 and d(f) = (1+sigma) /2 for 1/2<sigma<1 . We also compute average mass ratios of the two largest clusters at criticality.
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