Eigenvalue Degeneracy as a Possible ``Mathematical Mechanism'' for Phase Transitions
1970; American Institute of Physics; Volume: 41; Issue: 3 Linguagem: Inglês
10.1063/1.1658912
ISSN1520-8850
AutoresH. Eugene Stanley, M. Blume, Koichiro Matsuno, S. Milošević,
Tópico(s)Opinion Dynamics and Social Influence
ResumoSome years ago Ashkin and Lamb noted that the phase transition in the two-dimensional Ising model with nearest-neighbor interaction was characterized mathematically by an asymptotic degeneracy of the largest eigenvalue of a linear operator. More recently Kac and Thompson showed this eigenvalue degeneracy also characterized the phase transition in the Kac model (with weak, long-range forces), suggesting that this ``mathematical mechanism'' is not restricted to systems with short-range forces. However both the Kac model and the Ising model consider the ``spins'' to be one-dimensional unit vectors assuming only the discrete values +1 and −1. We are therefore led to consider the nature of the phase transition in one of the few exactly soluble models in which the spins can assume an entire continuum of orientations-the closed linear chain of classical spins of arbitrary spin dimensionality interacting isotropically through the Hamiltonian Hν=− ∑ i=1N Ji,i+1Si(ν)·Si+1(ν),where the exchange constants Ji,i+1 are arbitrary numbers. We find for all spin dimensionalities ν that the two-spin correlation function may be expressed as ρN(ν) (γ) = (λ1(ν)/λ0(ν))r, where λ0(ν) and λ1(ν) are the largest and next-largest eigenvalues of a certain linear operator. Thus the onset of longrange order, limr→∞ρN(ν) (γ)≠0, is characterized by the degeneracy of λ0(ν) and λ1(ν). This onset of long-range order occurs at a critical temperature Tc(ν)=0, for all values of spin dimensionality ν.
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