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Eigenvalue Degeneracy and Existence and Non‐existence of a Phase Transition in One Dimension
Author(s) -
Thompson Colin J.
Publication year - 1969
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1969484299
Subject(s) - eigenvalues and eigenvectors , mathematics , degeneracy (biology) , phase transition , dimension (graph theory) , ising model , perturbation (astronomy) , mathematical physics , order (exchange) , combinatorics , physics , statistical physics , condensed matter physics , quantum mechanics , biology , bioinformatics , finance , economics
We consider a one dimensional Ising chain with interaction potential J ( k ) such that J ( k ) = 0 when k > n . By a perturbation argument we show that long range order exists at sufficiently low temperatures if and only ifThis is consistent with Dyson's recent theorems and in addition predicts that when J ( k ) = k −2 there is no long range order.