{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T05:02:35Z","timestamp":1775538155707,"version":"3.50.1"},"reference-count":35,"publisher":"AIP Publishing","issue":"1","content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2007,1,1]]},"abstract":"<jats:p>We consider the Potts model with competing two-step interactions and spin values 1, 2, 3 on a semi-infinite Cayley tree of order 2. We describe ground states and verify the Peierls condition for the model. Using a contour argument we show the existence of three different Gibbs measures associated with translation invariant ground states.<\/jats:p>","DOI":"10.1063\/1.2408398","type":"journal-article","created":{"date-parts":[[2007,1,17]],"date-time":"2007-01-17T02:18:25Z","timestamp":1169000305000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":13,"title":["On contour arguments for the three state Potts model with competing interactions on a semi-infinite Cayley tree"],"prefix":"10.1063","volume":"48","author":[{"given":"Farrukh","family":"Mukhamedov","sequence":"first","affiliation":[{"name":"Universidade de Aveiro Departamento de Fisica, , Campus Universit\u00e1rio de Santiago, 3810-193 Aveiro, Portugal"}]},{"given":"Utkir","family":"Rozikov","sequence":"additional","affiliation":[{"name":"Institute of Mathematics , 29, F. Hodjaev Strasse, 700125, Tashkent, Uzbekistan"}]},{"given":"Jos\u00e9 F. F.","family":"Mendes","sequence":"additional","affiliation":[{"name":"Universidade de Aveiro Departamento de Fisica, , Campus Universit\u00e1rio de Santiago, 3810-193 Aveiro, Portugal"}]}],"member":"317","published-online":{"date-parts":[[2007,1,16]]},"reference":[{"key":"2023063020571418400_c1","volume-title":"Exactly Solved Models in Statistical Mechanics","year":"1982"},{"key":"2023063020571418400_c2","doi-asserted-by":"publisher","first-page":"5297","DOI":"10.1103\/PhysRevB.21.5297","article-title":"Ising model with solitons, phasons, and the devil\u2019s staircase","volume":"21","year":"1980","journal-title":"Phys. Rev. B"},{"key":"2023063020571418400_c3","first-page":"345","article-title":"Rigidity of the critical phases on a Cayley tree","volume":"3","year":"2001","journal-title":"Mosc. Math. J."},{"key":"2023063020571418400_c4","year":"2004"},{"key":"2023063020571418400_c5","doi-asserted-by":"publisher","first-page":"722","DOI":"10.1063\/1.2008132","article-title":"Ordering in two-dimensional Ising models with competing interactions","volume":"31","year":"2005","journal-title":"Low Temp. Phys."},{"key":"2023063020571418400_c6","doi-asserted-by":"publisher","first-page":"340","DOI":"10.1103\/PhysRev.124.340","article-title":"Symmetry of Excitons in Cu2O","volume":"124","year":"1961","journal-title":"Phys. Rev."},{"key":"2023063020571418400_c7","doi-asserted-by":"publisher","first-page":"939","DOI":"10.1007\/BF01055710","article-title":"Ground states of VBS models on Cayley trees","volume":"66","year":"1992","journal-title":"J. Stat. Phys."},{"key":"2023063020571418400_c8","doi-asserted-by":"publisher","first-page":"1125","DOI":"10.1007\/BF01086840","article-title":"On pure phases of the three-state ferromagnetic Potts model on the second-order Bethe lattice","volume":"85","year":"1990","journal-title":"Theor. Math. Phys."},{"key":"2023063020571418400_c9","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1007\/BF02634202","article-title":"A description of periodic extremal Gibbs measures of some lattice models on the Cayley tree","volume":"111","year":"1997","journal-title":"Theor. Math. Phys."},{"key":"2023063020571418400_c10","volume-title":"Gibbs Measures and Phase Transitions","year":"1988"},{"key":"2023063020571418400_c11","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1007\/BF01609493","article-title":"Peierls condition and the number of ground states","volume":"61","year":"1978","journal-title":"Commun. Math. Phys."},{"key":"2023063020571418400_c12","doi-asserted-by":"publisher","first-page":"912","DOI":"10.1007\/BF01039015","article-title":"Structure of ground states in three-dimensional Ising model with tree-step interaction","volume":"33","year":"1977","journal-title":"Theor. Math. Phys."},{"key":"2023063020571418400_c13","doi-asserted-by":"publisher","first-page":"577","DOI":"10.1007\/BF01017186","article-title":"Phase diagram of the Ising model on a Cayley tree in the presence of competing interactions and magnetic field","volume":"40","year":"1985","journal-title":"J. Stat. Phys."},{"key":"2023063020571418400_c14","volume-title":"Introduction to Mathematical Statistical Physics","year":"2000"},{"key":"2023063020571418400_c15","doi-asserted-by":"publisher","first-page":"80","DOI":"10.1016\/0375-9601(94)90121-X","article-title":"A new criterion for the location of phase transitions for spin systems on recursive lattices","volume":"188","year":"1994","journal-title":"Phys. Lett. A"},{"key":"2023063020571418400_c16","doi-asserted-by":"publisher","first-page":"017103","DOI":"10.1103\/PhysRevE.67.017103","article-title":"Critical temperature of the Potts models on the kagome lattice","volume":"67","year":"2003","journal-title":"Phys. Rev. E"},{"key":"2023063020571418400_c17","first-page":"P10011","article-title":"How to compute loop correction to Bethe Approximation","volume-title":"J. Stat. Mech.: Theory Exp.","year":"2005"},{"key":"2023063020571418400_c18","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1023\/B:JOSS.0000012509.10642.83","article-title":"On Gibbs measures of models with competing ternary and binary interactions and corresponding von Neumann algebras","volume":"114","year":"2004","journal-title":"J. Stat. Phys."},{"key":"2023063020571418400_c19","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1007\/s10955-004-2056-3","article-title":"On Gibbs measures of models with competing ternary and binary interactions and corresponding von Neumann algebras II","volume":"119","year":"2005","journal-title":"J. Stat. Phys."},{"key":"2023063020571418400_c20","doi-asserted-by":"publisher","first-page":"L39","DOI":"10.1088\/0305-4470\/15\/1\/008","article-title":"Three-state square lattice Potts antiferromagnet","volume":"15","year":"1982","journal-title":"J. Phys. A"},{"key":"2023063020571418400_c21","first-page":"L02003","article-title":"Loop expansion around the Bethe-Peierls approximation for lattice models","volume-title":"J. Stat. Mech.: Theory Exp.","year":"2006"},{"key":"2023063020571418400_c22","doi-asserted-by":"publisher","first-page":"300","DOI":"10.1016\/0378-4371(85)90094-9","article-title":"Critical behaviour in three-state Potts antiferromagnets on a Bethe lattice","volume":"131","year":"1985","journal-title":"Physica A"},{"key":"2023063020571418400_c23","doi-asserted-by":"publisher","first-page":"811","DOI":"10.1088\/0305-4470\/16\/4\/018","article-title":"Potts model on Bethe lattices I. General results","volume":"16","year":"1983","journal-title":"J. Phys. A"},{"key":"2023063020571418400_c24","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1017\/S0305004100027419","article-title":"Some generalized order-disorder transformations","volume":"48","year":"1952","journal-title":"Proc. Cambridge Philos. Soc."},{"key":"2023063020571418400_c25","volume-title":"Gibbs States on Countable Sets","year":"1974"},{"key":"2023063020571418400_c26","doi-asserted-by":"publisher","first-page":"1185","DOI":"10.1007\/BF01040127","article-title":"Phase diagrams of classical lattice systems. I","volume":"25","year":"1975","journal-title":"Theor. Math. Phys."},{"key":"2023063020571418400_c27","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1007\/BF01038255","article-title":"Phase diagrams of classical lattice systems. II","volume":"26","year":"1976","journal-title":"Theor. Math. Phys."},{"key":"2023063020571418400_c28","doi-asserted-by":"crossref","first-page":"929","DOI":"10.1007\/BF02634109","article-title":"Partition structures of the group representation of the Cayley tree into cosets by finite-index normal subgroups and their applications to the description of periodic Gibbs distributions","volume":"112","year":"1997","journal-title":"Theor. Math. Phys."},{"key":"2023063020571418400_c29","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1007\/s11005-004-5117-2","article-title":"On q\u2212 component models on Cayley tree: Contour method","volume":"71","year":"2005","journal-title":"Lett. Math. Phys."},{"key":"2023063020571418400_c30","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1007\/s10955-005-8029-3","article-title":"A constructive description of ground states and Gibbs measures for Ising model with two-step interactions on Cayley tree","volume":"122","year":"2006","journal-title":"J. Stat. Phys."},{"key":"2023063020571418400_c31","volume-title":"Theory of Phase Transitions: Rigorous Results","year":"1982"},{"key":"2023063020571418400_c32","doi-asserted-by":"crossref","first-page":"2187","DOI":"10.1103\/PhysRevE.52.2187","article-title":"Field behavior of an Ising model with competing interactions on the Bethe lattice","volume":"52","year":"1995","journal-title":"Phys. Rev. B"},{"key":"2023063020571418400_c33","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1103\/PhysRevLett.54.163","article-title":"Strange attractor in the Ising model with competing interactions on the Cayley tree","volume":"54","year":"1985","journal-title":"Phys. Rev. Lett."},{"key":"2023063020571418400_c34","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1103\/RevModPhys.54.235","article-title":"The Potts model","volume":"54","year":"1982","journal-title":"Rev. Mod. Phys."},{"key":"2023063020571418400_c35","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1007\/BF01293605","article-title":"Modulated phase of an Ising system with competing interactions on a Cayley tree","volume":"43","year":"1981","journal-title":"Z. Phys. B: Condens. Matter"}],"container-title":["Journal of Mathematical Physics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/jmp\/article-pdf\/doi\/10.1063\/1.2408398\/14066781\/013301_1_online.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/jmp\/article-pdf\/doi\/10.1063\/1.2408398\/14066781\/013301_1_online.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,30]],"date-time":"2023-06-30T20:59:14Z","timestamp":1688158754000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/jmp\/article\/48\/1\/013301\/290437\/On-contour-arguments-for-the-three-state-Potts"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,1,1]]},"references-count":35,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2007,1,1]]}},"URL":"https:\/\/doi.org\/10.1063\/1.2408398","relation":{},"ISSN":["0022-2488","1089-7658"],"issn-type":[{"value":"0022-2488","type":"print"},{"value":"1089-7658","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2007,1]]},"published":{"date-parts":[[2007,1,1]]},"article-number":"013301"}}