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Investigation of the Maier–Saupe–Zwanzig Model in the Apollonian Network

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Abstract

We analyze the thermodynamic behavior of a discrete version of the Maier–Saupe model for the nematic phase transitions in liquid crystals in the Apollonian network. This simple model, which we call Maier–Saupe–Zwanzig (MSZ) model, has been investigated in a variety of situations and geometric substrates. In terms of a single energy parameter, it has been shown to account for the well-known transition between uniaxially ordered and disordered phases in nematic liquid crystals. We consider a special Apollonian lattice, and use an exact transfer matrix approach, to investigate the occurrence of these phase transitions. We obtain numerical precise results for the free energy and its derivatives, as well as for the correlation lengths, which come from the iterations of recursion relations for the transfer matrix elements of successive generations of the lattice. Results are compared with similar findings on a diamond hierarchical lattice. In contrast with the diamond lattice, and in agreement with similar analyses for the Ising model, we find some special thermodynamic features, including peculiar behavior of the correlations functions, but no singular behavior of the free energy.

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A Supplementary Material file is available as indicated in [42].

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Funding

 This work was supported by the following Brazilian agencies and institutions: the National Council for Scientific and Technological Development (CNPq) through grants 465259/2014-6 (APV and SRS), 422561/2018-5 and 304257/2019-2 (RFSA); the Coordination for the Improvement of Higher Education Personnel (CAPES), the National Institute of Science and Technology Complex Fluids (INCT-FCx), the National Institute of Science and Technology for Complex Systems (INCT-SC), and the São Paulo Research Foundation (FAPESP – 2014/50983-3).

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Authors and Affiliations

  1. Instituto de Física, Universidade Federal da Bahia, Salvador, 40170-115, Bahia, Brazil

    Cícero T. G. dos Santos & Roberto F. S. Andrade

  2. Instituto Federal do Sertão Pernambucano, Petrolina, 56302-100, Pernambuco, Brazil

    Cícero T. G. dos Santos

  3. Instituto de Fisica, Universidade de Sao Paulo, Rua do Matao, 1371, Sao Paulo, 05508-090, Sao Paulo, Brazil

    André P. Vieira & Silvio R. Salinas

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  1. Cícero T. G. dos Santos
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Correspondence to Cícero T. G. dos Santos.

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This article is part of the Topical Collection on From Equilibrium Statistical Physics to Complex Systems: A Special Issue in Honor of Silvio R. Salinas

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T. G. dos Santos, C., P. Vieira, A., R. Salinas, S. et al. Investigation of the Maier–Saupe–Zwanzig Model in the Apollonian Network. Braz J Phys 53, 98 (2023). https://doi.org/10.1007/s13538-023-01297-7

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