Critical behavior at edge singularities in one-dimensional spin models

Abstract

In ferromagnetic spin models above the critical temperature ͑T Ͼ T cr ͒ the partition function zeros accumulate at complex values of the magnetic field ͑H E ͒ with a universal behavior for the density of zeros ͑H͒ ϳ͉H − H E ͉ . The critical exponent is believed to be universal at each space dimension and it is related to the magnetic scaling exponent y h via = ͑d − y h ͒ / y h . In two dimensions we have y h =12/ 5 ͑ =−1/ 6͒ while y h =2 ͑ =−1/ 2͒ in d = 1. For the one-dimensional Blume-Capel and Blume-Emery-Griffiths models we show here, for different temperatures, that a value y h =3 ͑ =−2/ 3͒ can emerge if we have a triple degeneracy of the transfer matrix eigenvalues.

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

“…The log-log fits in figures 5(a) and 5(b) confirm the FSS relations (19) and (35). They furnish the estimates y E h = 2.9960 (using the real pat of the zeros) and y ρ h = 2.9866.…”

“…A similar 1 expression has been derived before in [17] and analogous formulae for the onedimensional spin-1 Blume-Capel and Blume-Emery-Griffiths models have appeared in [16] and [19] respectively 2 . We interpret (18) as a cubic equation for A = A(ϕ) such that when we plug it back in (5) we get two eigenvalues with the same absolute value according to (10).…”

“…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.…”

“…[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.…”

“…Our results support the universality of σ = −2/3. As in [19,20], the triple degeneracy of the transfer matrix (TM) eigenvalues is necessary to evade the well known result σ = −1/2. Such condition requires a fine-tuning of the couplings of the model which explains why the authors of [17] have only found σ = −1/2 for the same model treated here.…”