The Yang–Lee edge singularity in spin models on connected and non-connected rings

Abstract

Renormalization group arguments based on a ıφ3 field theory lead us to expect a certain universal behavior for the density of partition function zeros in spin models with short-range interaction. Such universality has been tested analytically and numerically in different d = 1 and higher dimensional spin models. In d = 1, one finds usually the critical exponent σ = −1/2. Recently, we have shown in the d = 1 Blume–Emery–Griffiths (BEG) model on a periodic static lattice (one ring) that a new critical behavior with σ = −2/3 can arise if we have a triple degeneracy of the transfer matrix eigenvalues. Here we define the d = 1 BEG model on a dynamic lattice consisting of connected and non-connected rings (non-periodic lattice) and check numerically that also in this case we have mostly σ = −1/2 while the new value σ = −2/3 can arise under the same conditions of the static lattice (triple degeneracy) which is a strong check of universality of the new value of σ. We also show that although such conditions are necessary, they are not sufficient to guarantee the new critical behavior.

Classical phase transitions in a one-dimensional short-range spin model

“…Three eigenvalues are equal by modulus exactly at the point where flipping occurs. Such a behaviour signals existence of a point with unusual Lee-Yang edge singularity exponent [56].…”

Unusual Yang-Lee edge singularity in the one-dimensional axial-next-to-nearest-neighbor Ising model

“…However, in the works [18,19,20] one has found another critical behavior (σ = −2/3). The models investigated in [18,19,20] have three-state per site and only nearest-neighbor interaction. Here we have shown that σ = −2/3 also appears in the one-dimensional spin-1/2 ANNNI model which contains a next-to-nearestneighbor interaction and only two states per site.…”

“…Even for one-dimensional spin models there are no analytic expressions for the Yang-Lee zeros in general. In order to save computer time, instead of using the analytic solution for Z N given in (4) in terms of the transfer matrix eigenvalues or in terms of the trace of powers of the transfer matrix as in Tr T N , we use an alternative 3 exact expression derived in [20] for any spin model which can be solved via a finite transfer matrix. Namely, since λ i , i = 1, 2, 3, 4 are solutions of the secular equation P 4 (λ) ≡ λ 4 − a 3 λ 3 + a 2 λ 2 − a 1 λ + a 0 = 0, we have shown, formula (11) of [20], that (4) can be identified with Z N = −N ln g 4 P 4 (1/g) g N = −N ln 1 − a 3 g + a 2 g 2 − a 1 g 3 + a 0 g 4…”

“…[13,14,15,16,17]. However, in the works [18,19,20] one has found another critical behavior (σ = −2/3). The models investigated in [18,19,20] have three-state per site and only nearest-neighbor interaction.…”

“…The alternative formula (37) has a diagrammatic interpretation as a connected Feynman diagram of a zero-dimensional Gaussian field theory[20].…”