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Existence of a phase-transition in a one-dimensional Ising ferromagnet. (English) Zbl 1306.47082
Summary: Existence of a phase-transition is proved for an infinite linear chain of spins \(\mu_j=\pm 1\), with an interaction energy \[ H=-\sum J(i-j)\mu_i\mu_j, \] where \(J(n)\) is positive and monotone decreasing, and the sums \(\sum J(n)\) and \(\sum(\log\log n)\) \([n^3 J(n)]^{-1}\) both converge. In particular, as conjectured by M. Kac and C. J. Thompson [J. Math. Phys. 10, 1373–1386 (1969; Zbl 1306.82004)], a transition exists for \(J(n)=n^{-\alpha}\) when \(1<\alpha<2\). A possible extension of these results to Heisenberg ferromagnets is discussed.

47N50 Applications of operator theory in the physical sciences
82B26 Phase transitions (general) in equilibrium statistical mechanics
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82D40 Statistical mechanics of magnetic materials
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