Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1012.4454 · Physical Review E

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

Denis Dalmazi, Fernanda Lopes Sá

Published Nov 5, 2010• 5 citations
arXivHTMLPDF

Abstract

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 σ=-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 σ=-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 σ=-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.

DOI
10.1103/physreve.82.051108

Related papers

Generalized partition function zeros of 1D spin models and their critical behavior at edge singularities

Denis Dalmazi, Fernanda Lopes Sá

Here we study the partition function zeros of the one-dimensional Blume–Emery–Griffiths model close to their edge singularities. The model contains four couplings ( H , J , Δ, K ) including the magnetic field H and the Ising coupling J . We assume that only one of the three couplings ( J , Δ, K ) is complex and the magnetic field is real. The generalized zeros z i tend to form continuous curves on the complex z -plane in the thermodynamic limit. The linear density at the edges z E diverges usually with ρ( z ) ∼ | z − z E | σ and σ = −1/2. However, as in the case of complex magnetic fields (Yang–Lee edge singularity), if we have a triple degeneracy of the transfer matrix eigenvalues a new critical behavior with σ = −2/3 can appear as we prove here explicitly for the cases where either Δ or K is complex. Our proof applies for a general three-state spin model with short-range interactions. The Fisher zeros (complex J ) are more involved; in practice, we have not been able to find an explicit example with σ = −2/3 as far as the other couplings ( H , Δ, K ) are kept as real numbers. Our results are supported by numerical computations of zeros. We show that it is absolutely necessary to have a non-vanishing magnetic field for a new critical behavior. The appearance of σ = −2/3 at the edge closest to the positive real axis indicates its possible relevance for tricritical phenomena in higher-dimensional spin models.

Journal of Physics A Mathematical and Theoretical•May 28, 2010•5 citations

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

Xianzhi Wang, Jai Sam Kim

It is shown that for a one-dimensional lattice system in a purely imaginary magnetic field, if the interaction is finite range, the nature of the Yang-Lee edge singularity is universal, independent of the spin and interaction strengths. The edge singularity corresponds to the twofold degeneracy of the largest eigenvalues of the transfer matrix. For the Blume-Emery-Griffiths ferromagnet, the tricritical point and the edge pseudosingularity may exist. The tricritical point corresponds to the triple degeneracy of the eigenvalues. The edge pseudosingularity corresponds to the twofold degeneracy of the nonlargest eigenvalues.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Oct 1, 1998•15 citations

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

Denis Dalmazi, Fernanda Lopes Sá

Renormalization group arguments based on a ıφ 3 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.

Journal of Physics A Mathematical and Theoretical•Nov 3, 2008•6 citations

Critical behavior at edge singularities in one-dimensional spin models

Denis Dalmazi, Fernanda Lopes Sá

In ferromagnetic spin models above the critical temperature $(T>{T}_{\mathrm{cr}})$ the partition function zeros accumulate at complex values of the magnetic field $({H}_{E})$ with a universal behavior for the density of zeros $\ensuremath{\rho}(H)\ensuremath{\sim}{\ensuremath{\mid}H\ensuremath{-}{H}_{E}\ensuremath{\mid}}^{\ensuremath{\sigma}}$. The critical exponent $\ensuremath{\sigma}$ is believed to be universal at each space dimension and it is related to the magnetic scaling exponent ${y}_{h}$ via $\ensuremath{\sigma}=(d\ensuremath{-}{y}_{h})∕{y}_{h}$. In two dimensions we have ${y}_{h}=12∕5\phantom{\rule{0.3em}{0ex}}(\ensuremath{\sigma}=\ensuremath{-}1∕6)$ while ${y}_{h}=2\phantom{\rule{0.3em}{0ex}}(\ensuremath{\sigma}=\ensuremath{-}1∕2)$ in $d=1$. For the one-dimensional Blume-Capel and Blume-Emery-Griffiths models we show here, for different temperatures, that a value ${y}_{h}=3\phantom{\rule{0.3em}{0ex}}(\ensuremath{\sigma}=\ensuremath{-}2∕3)$ can emerge if we have a triple degeneracy of the transfer matrix eigenvalues.

arXiv:0809.4510•Physical Review E•Sep 30, 2008•6 citations

Yang-Lee Edge Singularity of the Ising Model on a Honeycomb Lattice in an External Magnetic Field

Seung‐Yeon Kim

The Ising model in an external magnetic field is one of the most outstanding and intriguing unsolved problems. One of the important properties of the Ising model in an external magnetic field is the Yang-Lee edge singularity. The zeros in the complex magnetic-field plane of the grand partition function of the Ising model on a honeycomb lattice in an external magnetic field are exactly calculated by using the exact integer values for the density of states on L × 2L honeycomb lattices (up to L = 14). The critical line of the Yang-Lee edge singularity of the Ising model on a honeycomb lattice in an external magnetic field is accurately investigated from its Yang-Lee zeros.

Journal of the Korean Physical Society•Aug 1, 2020•4 citations

Yang-Lee edge singularities from experimental high field magnetization data

Christian Binek

The isothermal magnetization m(H) of the metamagnet FeCl2 is measured in axial magnetic fields 0≤µ0Ha≤12 T at temperatures 34≤T≤53 K above the Néel temperature, where the system is essentially a two-dimensional Ising ferromagnet. The analysis of the data indicates experimental accessibility of the critical exponent µ of the Yang-Lee edge singularities. They manifest themselves in divergences of the density functions g(θ), which quantify the distribution of the zeros of the partition function on the Lee-Yang unit circle in the complex plane. In accordance with the hypothesis of universality, a critical exponent close to the theoretical prediction for the two-dimensional Ising ferromagnet, µ = -1/6, is found.

•Jan 1, 2001•0 citations
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 σ=-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 σ=-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 σ=-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.
Here we study the partition function zeros of the one-dimensional Blume–Emery–Griffiths model close to their edge singularities. The model contains four couplings ( H , J , Δ, K ) including the magnetic field H and the Ising coupling J . We assume that only one of the three couplings ( J , Δ, K ) is complex and the magnetic field is real. The generalized zeros z i tend to form continuous curves on the complex z -plane in the thermodynamic limit. The linear density at the edges z E diverges usually with ρ( z ) ∼ | z − z E | σ and σ = −1/2. However, as in the case of complex magnetic fields (Yang–Lee edge singularity), if we have a triple degeneracy of the transfer matrix eigenvalues a new critical behavior with σ = −2/3 can appear as we prove here explicitly for the cases where either Δ or K is complex. Our proof applies for a general three-state spin model with short-range interactions. The Fisher zeros (complex J ) are more involved; in practice, we have not been able to find an explicit example with σ = −2/3 as far as the other couplings ( H , Δ, K ) are kept as real numbers. Our results are supported by numerical computations of zeros. We show that it is absolutely necessary to have a non-vanishing magnetic field for a new critical behavior. The appearance of σ = −2/3 at the edge closest to the positive real axis indicates its possible relevance for tricritical phenomena in higher-dimensional spin models.
It is shown that for a one-dimensional lattice system in a purely imaginary magnetic field, if the interaction is finite range, the nature of the Yang-Lee edge singularity is universal, independent of the spin and interaction strengths. The edge singularity corresponds to the twofold degeneracy of the largest eigenvalues of the transfer matrix. For the Blume-Emery-Griffiths ferromagnet, the tricritical point and the edge pseudosingularity may exist. The tricritical point corresponds to the triple degeneracy of the eigenvalues. The edge pseudosingularity corresponds to the twofold degeneracy of the nonlargest eigenvalues.
Renormalization group arguments based on a ıφ 3 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.
In ferromagnetic spin models above the critical temperature (T>Tcr)(T>{T}_{\mathrm{cr}})(T>Tcr​) the partition function zeros accumulate at complex values of the magnetic field (HE)({H}_{E})(HE​) with a universal behavior for the density of zeros $\ensuremath{\rho}(H)\ensuremath{\sim}{\ensuremath{\mid}H\ensuremath{-}{H}_{E}\ensuremath{\mid}}^{\ensuremath{\sigma}}$. The critical exponent $\ensuremath{\sigma}$ is believed to be universal at each space dimension and it is related to the magnetic scaling exponent yh{y}_{h}yh​ via $\ensuremath{\sigma}=(d\ensuremath{-}{y}_{h})∕{y}_{h}$. In two dimensions we have ${y}_{h}=12∕5\phantom{\rule{0.3em}{0ex}}(\ensuremath{\sigma}=\ensuremath{-}1∕6)$ while ${y}_{h}=2\phantom{\rule{0.3em}{0ex}}(\ensuremath{\sigma}=\ensuremath{-}1∕2)$ in d=1d=1d=1. For the one-dimensional Blume-Capel and Blume-Emery-Griffiths models we show here, for different temperatures, that a value ${y}_{h}=3\phantom{\rule{0.3em}{0ex}}(\ensuremath{\sigma}=\ensuremath{-}2∕3)$ can emerge if we have a triple degeneracy of the transfer matrix eigenvalues.
The Ising model in an external magnetic field is one of the most outstanding and intriguing unsolved problems. One of the important properties of the Ising model in an external magnetic field is the Yang-Lee edge singularity. The zeros in the complex magnetic-field plane of the grand partition function of the Ising model on a honeycomb lattice in an external magnetic field are exactly calculated by using the exact integer values for the density of states on L × 2L honeycomb lattices (up to L = 14). The critical line of the Yang-Lee edge singularity of the Ising model on a honeycomb lattice in an external magnetic field is accurately investigated from its Yang-Lee zeros.
The isothermal magnetization m(H) of the metamagnet FeCl2 is measured in axial magnetic fields 0≤µ0Ha≤12 T at temperatures 34≤T≤53 K above the Néel temperature, where the system is essentially a two-dimensional Ising ferromagnet. The analysis of the data indicates experimental accessibility of the critical exponent µ of the Yang-Lee edge singularities. They manifest themselves in divergences of the density functions g(θ), which quantify the distribution of the zeros of the partition function on the Lee-Yang unit circle in the complex plane. In accordance with the hypothesis of universality, a critical exponent close to the theoretical prediction for the two-dimensional Ising ferromagnet, µ = -1/6, is found.