Spin models with an eigenvalue of small multiplicity
The author makes a thorough study of spin models with modulus 1 having an eigenvalue with multiplicity at most two. Main results are as follows: First, he remarks that the multiplicity of an eigenvalue \(D\theta^{- 1}\) of a spin model \(S= (X, w)\), where \(D\) denotes the loop variable, coincides with the number \(m(\theta)\) of \(x\) in \(X\) such that \(w(b, x)= \theta\). Second, he shows: (I) If \(m(\theta)= 1\), then \(\theta^4= 1\). Moreover, if \(\theta^2= -1\), then \(S= S_1\times S_2\), where \(S_1\) is the Ising model. (II) Assume that \(m(\theta)= 2\) and \(\theta\neq 1\). Two elements \(a\) and \(b\) of \(X\) are called adjacent if \(w(a, b)= \theta\), and so the concept of a cycle is naturally introduced. Now let \((a_0, a_1,\dots, a_{k- 1})\) be a cycle of length \(k\neq 4\) such that \(a_0, a_1,\dots, a_{k- 1}\) are distinct, \(w(a_i, a_{i+ 1})= \theta\) for \(i= 0, 1,\dots, k- 2\) and \(w(a_0, a_{k- 1})= \theta\). Then \(w(a_i, a_j)= \theta^{(i- j)^2}\) for all \(i\) and \(j\). (III) Assume that \(m(\theta)= 2\) and \(\theta\neq 1\). Assume that \(S\) has a cycle of length \(k\neq 4\). Then one of the following holds: (i) \(k\) is a power of an odd prime and \(\theta\) is a primitive \(k\)th root of unity. (ii) \(k= 2^f\) with \(f\geq 3\) and \(\theta\) is a primitive \(2k\)th root of unity. (iii) \(k= 2p^f\) where \(p\) is an odd prime and \(\theta\) is a primitive \(k\)th or \(2k\)th root of unity. (IV) Assume that \(m(\theta)= 2\) and \(\theta\neq 1\). Assume that \(S\) contains a cycle of length \(k\neq 4\) and \(S\) does not contain a cycle of length 4. Further, assume that \(\theta^k\neq 1\) when \(k\equiv 2\pmod 4\). Then \(S= S_1\times S_2\), where \(S_1\) is a cyclic model with \(k\) elements.
scientific article; zbMATH DE number 802360