Yang-Lee edge singularity of a one-dimensional Ising ferromagnet with arbitrary spin

Wang, Xianzhi; Kim, Jai Sam

doi:10.1103/physreve.58.4174

Cited by 16 publications

(29 citation statements)

References 23 publications

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“…In this case the linear density was already known exactly [2] furnishing σ = −1/2. Numerical works have confirmed σ = −1/2 in several one-dimensional spin models [13,14,15,16] including, see [17], the same model discussed here. An early exception is a special type of three-state Potts model [18].…”

Section: Introductionsupporting

confidence: 67%

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

Dalmazi

2010

Phys. Rev. E

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We show here for the one-dimensional spin-1/2 axial-next-to-nearest-neighbor Ising model in an external magnetic field that the linear density of Yang-Lee zeros may diverge with critical exponent =−2/ 3 at the Yang-Lee edge singularity. The necessary condition for this unusual behavior is the triple degeneracy of the transfer-matrix eigenvalues. If this condition is absent we have the usual value =−1/ 2. Analogous results have been found in the literature in the spin-1 Blume-Emery-Griffths model and in the three-state Potts model in a magnetic field with two complex components. Our results support the universality of =−2/ 3 which might be a one-dimensional footprint of a tricritical version of the Yang-Lee edge singularity possibly present also in higher-dimensional spin models.

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Section: Introductionsupporting

confidence: 67%

“…The universality of σ is known for a long time [6,7,12] and checked explicitly in D = 1 in several models, see e.g. [13,14,15,16,17]. However, in the works [18,19,20] one has found another critical behavior (σ = −2/3).…”

Section: Resultsmentioning

confidence: 99%

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

Dalmazi

2010

Phys. Rev. E

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“…Manipulating (13) and (14) we further eliminate λ 1 and find an equation for A(ϕ): a 2 0 (1 + 2 cos ϕ) 3 + 4 cos 2 ϕ 2 a 3 1 + a 0 a 3 2 −a 2 1 a 2 2 −2 (1 + 2 cos ϕ) (2 + cos ϕ) a 0 a 1 a 2 = 0 (15) which is the same expression obtained in [13] for the Blume-Capel model (b = 1). At this point, in principle, the Yang-Lee zeros are determined by (15), once we check (11). After taking the continuum limit in the corresponding solution u(ϕ k ) one could derive the density of zeros and obtain the exponent σ.…”

Section: General Setup and Analytic Resultsmentioning

confidence: 99%

“…It is not feasible to substitute them in the solutions of the cubic equation (6), which are a bit complicated too, and then check the condition (11) for arbitrary values of the parameters b, c, x. Fortunately, in order to compute the exponent σ or y h we only need to study the vicinity of the YLES. The YLES is located at A(ϕ = 0) since at ϕ = 0 the two, presumably, largest eigenvalues of the transfer matrix coincide (λ 1 = λ 2 ) which guarantees, see the original works [20], [21] and [11] for a recent work, the existence of phase transition. Therefore, only the behavior of the solutions of the equation (15) about ϕ = 0 is required, as already noticed in [13], remembering of course that (11) must hold.…”

Section: General Setup and Analytic Resultsmentioning

confidence: 99%

Critical behavior at edge singularities in one-dimensional spin models

Dalmazi

Sá

2008

Phys. Rev. E

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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.

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“…As we have already mentioned, there are some other models, for which the Lee-Yang theorem does not hold [42][43][44][45][46][47][48][49][50][51][52][53]. Unlike the above examples, the problem we have considered in this paper concerns ferromagnetic Ising model, and the Lee-Yang theorem was proven [55] to hold for any Ising-like model with ferromagnetic interaction, see also [56].…”

Section: Discussionmentioning

confidence: 99%

Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks

Krasnytska

Berche

Holovatch

et al. 2016

J. Phys. A: Math. Theor.

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We analyze the partition function of the Ising model on graphs of two different types: complete graphs, wherein all nodes are mutually linked and annealed scale-free networks for which the degree distribution decays as P (k) ∼ k −λ . We are interested in zeros of the partition function in the cases of complex temperature or complex external field (Fisher and Lee-Yang zeros respectively). For the model on an annealed scale-free network, we find an integral representation for the partition function which, in the case λ > 5, reproduces the zeros for the Ising model on a complete graph. For 3 < λ < 5 we derive the λ-dependent angle at which the Fisher zeros impact onto the real temperature axis. This, in turn, gives access to the λ-dependent universal values of the critical exponents and critical amplitudes ratios. Our analysis of the Lee-Yang zeros reveals a difference in their behaviour for the Ising model on a complete graph and on an annealed scale-free network when 3 < λ < 5. Whereas in the former case the zeros are purely imaginary, they have a non zero real part in latter case, so that the celebrated Lee-Yang circle theorem is violated.

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Yang-Lee edge singularity of a one-dimensional Ising ferromagnet with arbitrary spin

Cited by 16 publications

References 23 publications

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

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

Critical behavior at edge singularities in one-dimensional spin models

Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks

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