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Yang-Lee edge singularities from experimental high field magnetization data

Christian Binek

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

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.

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Density of Yang-Lee zeros for the Ising ferromagnet

Seung‐Yeon Kim

The densities of Yang-Lee zeros for the Ising ferromagnet on the L x L square lattice are evaluated from the exact grand partition functions (L=3 approximately 16). The properties of the density of Yang-Lee zeros are discussed as a function of temperature T and system size L. The three different classes of phase transitions for the Ising ferromagnet--first-order phase transition, second-order phase transition, and Yang-Lee edge singularity--are clearly distinguished by estimating the magnetic scaling exponent yh from the densities of zeros for finite-size systems. The divergence of the density of zeros at Yang-Lee edge in high temperatures (Yang-Lee edge singularity), which has been detected only by the series expansion until now for the square-lattice Ising ferromagnet, is obtained from the finite-size data. The identification of the orders of phase transitions in small systems is also discussed using the density of Yang-Lee zeros.

arXiv:cond-mat/0607257•Physical Review E•Jul 26, 2006•26 citations

Probing Yang–Lee edge singularity by central spin decoherence

Bo-Bo Wei

Yang-Lee edge singularities are the branch point of free energy on the complex plane of physical parameters and have been shown to be the simplest universality class of phase transitions.However, Yang-Lee edge singularities have not been regarded as experimentally observable since they occur at complex physical parameters which are unphysical.A recent discovery regarding the relation between partition functions and probe spin coherence makes it experimentally feasible to access the complex plane of physical parameters.However, how to extract the critical point and the critical exponent of Yang-Lee edge singularities in many-body systems, which occurs only at the thermodynamic limit, has still been elusive.Here we show that the quantum coherence of a probe spin coupled to finite-size Ising-type spin systems presents universal scaling behavior near the Yang-Lee edge singularity.The finite-size scaling behavior of the quantum coherence of the probe spin predicts that one can extract the critical point and the critical exponent of the Yang-Lee edge singularity of an Ising-type spin system in the thermodynamic limit from the spin coherence measurement of the probe spin coupled to finite-size Ising-type spin systems.This finding provides a practical approach to studying the nature of Yang-Lee edge singularities of many-body systems.In 1952, Yang and Lee [1, 2] initiated the examination of phase transitions by studying the partition function zeros, termed Lee-Yang zeros, on the complex plane of a magnetic field h (or fugacity z for fluids).They found that for a system above its critical temperature T c , the partition function must be nonzero throughout some neighborhood of the real axis of the magnetic field within a gap determined by two edges at  ( ) h T YL .Therefore the free energy is an analytic function of the real magnetic field.On the other hand, below the critical temperature T c , the Lee-Yang zeros will come arbitrarily to the real axis as the thermodynamic limit is taken, destroying the analyticity of the free energy in h for those real fields at which the Lee-Yang zeros accumulate.Kortman and Griffiths [3] pointed out that the edges of Lee-Yang zeros of a many-body system in the thermodynamic limit are singularity points, termed Yang-Lee edge singularities [4].Fisher [4] proposed that the Yang-Lee edge singularity could be considered as a new second order phase transition point with associated critical exponents.Renormalization group analyses [4] show that the Yang-Lee edge singularity is associated with an j i 3 theory and the crossover dimensionality of the Yang-Lee edge singularity is d c = 6.Later, the investigation of the Yang-Lee edge singularity has been extended to many other models, such as the classical nvector model [5], the quantum Heisenberg model [5], the spherical model [6], the quantum one-dimensional (1D) transverse Ising model [7], the hierarchical model [8], branched polymers [9], the directed-site animalsenumeration problem [10], Ising models on fractal lattices [11], Ising systems with correlated disorder [12], fluid models with repulsive-core interactions [13], the antiferromagnetic Ising model [14] etc.A comprehensive review of the study of Yang-Lee edge singularities can be found in [15, 16].Recently, Kibble-Zurek scaling was shown to appear at the Yang-Lee edge singularity [17], which supports that the Yang-Lee edge singularity is similar to conventional second order phase transitions.Yang-Lee edge singularity occurs in the ferromagnetic Ising models above its critical temperatures in a purely imaginary magnetic field  h i (figure 1).For  > ( ) h h T YL , the partition function acquires zeros, which becomes dense on the line with > ( ) Ih h T YL .The density of Lee-Yang zeros  ( ) g T h , in the thermodynamic limit behaves as  ~ -s ( ) | ( )| g T h h h T , YL when   ( ) h h T YL from above, with σ being the universal critical

arXiv:1611.08074•New Journal of Physics•Aug 8, 2017•24 citations

Yang-Lee Edge Singularity Of The Infinite-Range Ising Model

Seung‐Yeon Kim

The Ising ferromagnet, consisting of magnetic spins, is the simplest system showing phase transitions and critical phenomena at finite temperatures. The Ising ferromagnet has played a central role in our understanding of phase transitions and critical phenomena. Also, the Ising ferromagnet explains the gas-liquid phase transitions accurately. In particular, the Ising ferromagnet in a nonzero magnetic field has been one of the most intriguing and outstanding unsolved problems. We study analytically the partition function zeros in the complex magnetic-field plane and the Yang-Lee edge singularity of the infinite-range Ising ferromagnet in an external magnetic field. In addition, we compare the Yang-Lee edge singularity of the infinite-range Ising ferromagnet with that of the square-lattice Ising ferromagnet in an external magnetic field.

World Academy of Science, Engineering and Technology, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering•Jan 1, 2015•0 citations

Griffiths singularities in random magnets: Results for a soluble model

Alan J. Bray, Deng Huifang

A soluble, but nontrivial, model of a dilute Ising ferromagnet is studied, with infinite-range interactions but finite average connectivity c. The density of (Yang-Lee) zeros of the partition function in the complex z=exp(-2H) plane (where H is the external magnetic field in units of the temperature) is calculated explicitly in the high-temperature phase for large but finite c and small \ensuremath{\Vert}H\ensuremath{\Vert}. The density of zeros on the unit circle H=i\ensuremath{\theta} has the form \ensuremath{\rho}(\ensuremath{\theta})\ensuremath{\sim}exp{-[cf(K)/\ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert}]ln/B (1/\ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert})} for \ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert}\ensuremath{\rightarrow}0. The function f(K) (K=J/T) vanishes at the critical coupling ${K}_{C}$(c). Heuristic arguments are given for the form of \ensuremath{\rho}(\ensuremath{\theta}) expected for systems with short-range interactions.

Physical review. B, Condensed matter•Oct 1, 1989•53 citations

Yang-Lee Zeros of the Antiferromagnetic Ising Model

Seung‐Yeon Kim

There exists the famous circle theorem on the Yang-Lee zeros of the ferromagnetic Ising model. However, the Yang-Lee zeros of the antiferromagnetic Ising model are much less well understood than those of the ferromagnetic model. The precise distribution of the Yang-Lee zeros of the antiferromagnetic Ising model only with nearest-neighbor interaction J on LxL square lattices is determined as a function of temperature a=e(2betaJ) (J<0), and its relation to the phase transitions is investigated. In the thermodynamic limit (L-->infinity), the distribution of the Yang-Lee zeros of the antiferromagnetic Ising model cuts the positive real axis in the complex x=e(-2betaH) plane, resulting in the critical magnetic field +/-H(c)(a), where H(c)>0 below the critical temperature a(c)=square root of 2-1. The results suggest that the value of the scaling exponent y(h) is 1 along the critical line for a<a(c).

Physical Review Letters•Sep 23, 2004•68 citations

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

Denis Dalmazi, Fernanda Lopes Sá

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.

arXiv:1012.4454•Physical Review E•Nov 5, 2010•5 citations
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.
The densities of Yang-Lee zeros for the Ising ferromagnet on the L x L square lattice are evaluated from the exact grand partition functions (L=3 approximately 16). The properties of the density of Yang-Lee zeros are discussed as a function of temperature T and system size L. The three different classes of phase transitions for the Ising ferromagnet--first-order phase transition, second-order phase transition, and Yang-Lee edge singularity--are clearly distinguished by estimating the magnetic scaling exponent yh from the densities of zeros for finite-size systems. The divergence of the density of zeros at Yang-Lee edge in high temperatures (Yang-Lee edge singularity), which has been detected only by the series expansion until now for the square-lattice Ising ferromagnet, is obtained from the finite-size data. The identification of the orders of phase transitions in small systems is also discussed using the density of Yang-Lee zeros.
Yang-Lee edge singularities are the branch point of free energy on the complex plane of physical parameters and have been shown to be the simplest universality class of phase transitions.However, Yang-Lee edge singularities have not been regarded as experimentally observable since they occur at complex physical parameters which are unphysical.A recent discovery regarding the relation between partition functions and probe spin coherence makes it experimentally feasible to access the complex plane of physical parameters.However, how to extract the critical point and the critical exponent of Yang-Lee edge singularities in many-body systems, which occurs only at the thermodynamic limit, has still been elusive.Here we show that the quantum coherence of a probe spin coupled to finite-size Ising-type spin systems presents universal scaling behavior near the Yang-Lee edge singularity.The finite-size scaling behavior of the quantum coherence of the probe spin predicts that one can extract the critical point and the critical exponent of the Yang-Lee edge singularity of an Ising-type spin system in the thermodynamic limit from the spin coherence measurement of the probe spin coupled to finite-size Ising-type spin systems.This finding provides a practical approach to studying the nature of Yang-Lee edge singularities of many-body systems.In 1952, Yang and Lee [1, 2] initiated the examination of phase transitions by studying the partition function zeros, termed Lee-Yang zeros, on the complex plane of a magnetic field h (or fugacity z for fluids).They found that for a system above its critical temperature T c , the partition function must be nonzero throughout some neighborhood of the real axis of the magnetic field within a gap determined by two edges at  ( ) h T YL .Therefore the free energy is an analytic function of the real magnetic field.On the other hand, below the critical temperature T c , the Lee-Yang zeros will come arbitrarily to the real axis as the thermodynamic limit is taken, destroying the analyticity of the free energy in h for those real fields at which the Lee-Yang zeros accumulate.Kortman and Griffiths [3] pointed out that the edges of Lee-Yang zeros of a many-body system in the thermodynamic limit are singularity points, termed Yang-Lee edge singularities [4].Fisher [4] proposed that the Yang-Lee edge singularity could be considered as a new second order phase transition point with associated critical exponents.Renormalization group analyses [4] show that the Yang-Lee edge singularity is associated with an j i 3 theory and the crossover dimensionality of the Yang-Lee edge singularity is d c = 6.Later, the investigation of the Yang-Lee edge singularity has been extended to many other models, such as the classical nvector model [5], the quantum Heisenberg model [5], the spherical model [6], the quantum one-dimensional (1D) transverse Ising model [7], the hierarchical model [8], branched polymers [9], the directed-site animalsenumeration problem [10], Ising models on fractal lattices [11], Ising systems with correlated disorder [12], fluid models with repulsive-core interactions [13], the antiferromagnetic Ising model [14] etc.A comprehensive review of the study of Yang-Lee edge singularities can be found in [15, 16].Recently, Kibble-Zurek scaling was shown to appear at the Yang-Lee edge singularity [17], which supports that the Yang-Lee edge singularity is similar to conventional second order phase transitions.Yang-Lee edge singularity occurs in the ferromagnetic Ising models above its critical temperatures in a purely imaginary magnetic field  h i (figure 1).For  > ( ) h h T YL , the partition function acquires zeros, which becomes dense on the line with > ( ) Ih h T YL .The density of Lee-Yang zeros  ( ) g T h , in the thermodynamic limit behaves as  ~ -s ( ) | ( )| g T h h h T , YL when   ( ) h h T YL from above, with σ being the universal critical
The Ising ferromagnet, consisting of magnetic spins, is the simplest system showing phase transitions and critical phenomena at finite temperatures. The Ising ferromagnet has played a central role in our understanding of phase transitions and critical phenomena. Also, the Ising ferromagnet explains the gas-liquid phase transitions accurately. In particular, the Ising ferromagnet in a nonzero magnetic field has been one of the most intriguing and outstanding unsolved problems. We study analytically the partition function zeros in the complex magnetic-field plane and the Yang-Lee edge singularity of the infinite-range Ising ferromagnet in an external magnetic field. In addition, we compare the Yang-Lee edge singularity of the infinite-range Ising ferromagnet with that of the square-lattice Ising ferromagnet in an external magnetic field.
A soluble, but nontrivial, model of a dilute Ising ferromagnet is studied, with infinite-range interactions but finite average connectivity c. The density of (Yang-Lee) zeros of the partition function in the complex z=exp(-2H) plane (where H is the external magnetic field in units of the temperature) is calculated explicitly in the high-temperature phase for large but finite c and small \ensuremath{\Vert}H\ensuremath{\Vert}. The density of zeros on the unit circle H=i\ensuremath{\theta} has the form \ensuremath{\rho}(\ensuremath{\theta})\ensuremath{\sim}exp{-[cf(K)/\ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert}]ln/B (1/\ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert})} for \ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert}\ensuremath{\rightarrow}0. The function f(K) (K=J/T) vanishes at the critical coupling KC{K}_{C}KC​(c). Heuristic arguments are given for the form of \ensuremath{\rho}(\ensuremath{\theta}) expected for systems with short-range interactions.
There exists the famous circle theorem on the Yang-Lee zeros of the ferromagnetic Ising model. However, the Yang-Lee zeros of the antiferromagnetic Ising model are much less well understood than those of the ferromagnetic model. The precise distribution of the Yang-Lee zeros of the antiferromagnetic Ising model only with nearest-neighbor interaction J on LxL square lattices is determined as a function of temperature a=e(2betaJ) (J<0), and its relation to the phase transitions is investigated. In the thermodynamic limit (L-->infinity), the distribution of the Yang-Lee zeros of the antiferromagnetic Ising model cuts the positive real axis in the complex x=e(-2betaH) plane, resulting in the critical magnetic field +/-H(c)(a), where H(c)>0 below the critical temperature a(c)=square root of 2-1. The results suggest that the value of the scaling exponent y(h) is 1 along the critical line for a<a(c).
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.