Summary
Let us consider a one-dimensionalS=1/2 Ising lattice with spin variables {s j/j ∈Z} and interacting via a long-ranged pair potential possessing translational invariance and defined by an inverse-power lawW ik =S j Φ jk S k , Φ jk =f(|j−k|),f(n) =qɛn −p. Here ε is a positive energy to be used as a scaling factor andq=±1 determines the ferromagnetic or antiferromagnetic character of the interaction; in contrast to the ferromagnetic case, the antiferromagnetic one was seldom studied. Continuing along the lines of our previous work (Nuovo Cimento B,83, 188 (1984)), we carried out Monte Carlo calculations for the caseq=+1,p=2, and found results suggesting a Thouless transition at the reduced temperaturek B T c/ε=0.21875±0.00625, and also noticed a remarkable qualitative similarity between the two potential models. In the absence of more stringent rigorous results, we conjecture that antiferromagnetic potential models defined by 1≤p≤2 can support phase transitions to ordered states.
Riassunto
Si consideri un reticolo di Ising unidimensionale conS=1/2, variabili di spin {S j/j∈Z} ed interazioni attraverso un potenziale a coppia che possieda invarianza traslazionale e sia definito da una potenza inversa della distanzaW ik =S j Φ jk S k , Φ jk =f(|j−k|),f(n) =qɛn −p. Qui ε è un'energia positiva, usata come fattore di scala, eq=±1 determina il carattere ferromagnetico od antiferromagnetico dell'interazione; a differenza dal caso ferromagnetico, quello antiferromagnetico è stato raramente studiato. Continuando lungo le linee del nostro precedente lavoro (vediNuovo Cimento B,83, 188 (1984)), abbiamo eseguito calcoli di simulazione Monte Carlo per il casoq=+1,p=2, trovando risultati che suggeriscono una transizione alla Thouless alla temperatura ridottak B T c/ε=0.21875±0.00625; is osserva inoltre una notevole somiglianza qualitativa tra i due modelli di potenziale. In assenza di risultati rigorosi piú stringenti, congetturiamo che modelli di potenziale antiferromagnetici definiti da 1≤p≤2 possano produrre transizioni di fase a stati ordinati.
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Romano, S. Computer simulation of a long-ranged Ising antiferromagnet in one dimension.. Nuov Cim B 89, 1–13 (1985). https://doi.org/10.1007/BF02728500
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DOI: https://doi.org/10.1007/BF02728500