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arXiv:2309.14562 · Physical review. E

Yang-Lee zeros of certain antiferromagnetic models

Muhammad Sedik, Junaid Majeed Bhat, Abhishek Dhar, B. Sriram Shastry

Published Jul 11, 2024• 2 citations

Abstract

We revisit the somewhat less studied problem of Yang-Lee zeros of the Ising antiferromagnet. For this purpose, we study two models, the nearest-neighbor model on a square lattice and the more tractable mean-field model corresponding to infinite-ranged coupling between all sites. In the high-temperature limit, we show that the logarithm of the Yang-Lee zeros can be written as a series in half odd integer powers of the inverse temperature, k, with the leading term ∼k^{1/2}. This result is true in any dimension and for arbitrary lattices. We also show that the coefficients of the expansion satisfy simple identities (akin to sum rules) for the nearest-neighbor case. These identities are verified numerically by computing the exact partition function for a two-dimensional square lattice of size 16×16. For the mean-field model, we write down the partition function (termed the mean-field polynomials) for the ferromagnetic (FM) and antiferromagnetic (AFM) cases and derive from them the mean-field equations. We analytically show that at high temperatures the zeros of the AFM mean-field polynomial scale as ∼k^{1/2} as well. Using a simple numerical method, we find the roots lie on certain curves (the root curves), in the thermodynamic limit for the mean-field polynomials for the AFM case as well as for the FM one. Our results show a new root curve that was not found earlier. Our results also clearly illustrate the phase transition expected for the FM and AFM cases, in the language of Yang-Lee zeros. Moreover, for the AFM case, we observe that the root curves separate two distinct phases of zero and nonzero complex staggered magnetization, and thus depict a complex phase boundary.

DOI: 10.1103/physreve.110.014117

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We revisit the somewhat less studied problem of Yang-Lee zeros of the Ising antiferromagnet. For this purpose, we study two models, the nearest-neighbor model on a square lattice and the more tractable mean-field model corresponding to infinite-ranged coupling between all sites. In the high-temperature limit, we show that the logarithm of the Yang-Lee zeros can be written as a series in half odd integer powers of the inverse temperature, k, with the leading term ∼k^{1/2}. This result is true in any dimension and for arbitrary lattices. We also show that the coefficients of the expansion satisfy simple identities (akin to sum rules) for the nearest-neighbor case. These identities are verified numerically by computing the exact partition function for a two-dimensional square lattice of size 16×16. For the mean-field model, we write down the partition function (termed the mean-field polynomials) for the ferromagnetic (FM) and antiferromagnetic (AFM) cases and derive from them the mean-field equations. We analytically show that at high temperatures the zeros of the AFM mean-field polynomial scale as ∼k^{1/2} as well. Using a simple numerical method, we find the roots lie on certain curves (the root curves), in the thermodynamic limit for the mean-field polynomials for the AFM case as well as for the FM one. Our results show a new root curve that was not found earlier. Our results also clearly illustrate the phase transition expected for the FM and AFM cases, in the language of Yang-Lee zeros. Moreover, for the AFM case, we observe that the root curves separate two distinct phases of zero and nonzero complex staggered magnetization, and thus depict a complex phase boundary.

We study Yang-Lee zeros as they approach the thermodynamic limit of the two-dimensional (2D) nearest-neighbor antiferromagnetic Ising model on square and triangular lattices. We employ the numerical linked cluster expansion (NLCE) equipped with exact enumeration (EE) of the partition function to estimate the Laplacian of the free energy, which is proportional to the zeros density. Using a modified NLCE, where the expansion can be carried directly on the Yang-Lee zeros of the involved clusters, we estimate the density of Yang-Lee zeros in the thermodynamic limit. NLCE gives significantly more zeros than EE in the complex field plane providing more insights on how the root curves look in the thermodynamic limit. For the square lattice at $T\ensuremath{\ll}{T}_{c}$, the results suggest that two vertical lines at $\ifmmode\pm\else\textpm\fi{}{h}_{c}(T)$ in the complex field plane (i.e., two concentric circles in the complex fugacity plane) are the thermodynamic root curves. A similar picture is found for the triangular lattice for phase transitions at large values of magnetic field while further study is needed for phase transitions at smaller values of magnetic field. The convergence of the NLCE and (EE) calculations of the partition function to the thermodynamic limit is studied in both lattices and the temperature-field phase diagram is obtained from Yang-Lee zeros using both methods. This NLCE-based approach will facilitate the study of different types of phase transitions using Yang-Lee zeros in future research.

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).

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.

To study the relations between Yang-Lee's theory of the phase transitions and the distribution of the zeros of the partition functions in the antiferromagnetic case, the partition functions of finite (4×4 and 4×6) Ising model with the nearest and the next nearest neighbor interactions are calculated. The distributions of the zeros in the fugacity z plane of the partition function for various combinations of the values of J / k T - J '/ k T or tanh ( J /2 k T )-tanh ( J '/2 k T ) are investigated. The patterns of the distribution of the zeros show several types corresponding to ferromagnetic state, antiferromagnetic state and superantiferromagnetic state. In the antiferromagnetic ordered state the locus of the zeros is found to be nearly two concentric circles and to cross the positive real axis at two points z c and 1/ z c when the next neighbor ferromagnetic interaction is introduced. This means that our system shows the phase transitions at finite critical fields H c and - H c . The locus of zeros of Fisher's model of super-exchange antiferromagnet is also shown to cross the positive real axis at z c and 1/ z c below the Néel temperature.

The partition functions of the two- and three-dimensional finite Ising models in the presence of a magnetic field have been calculated with use of a high-speed computer. Periodic boundary conditions were imposed. All the numbers of configurations for the 3×3×3, 5×5, and 4×6 lattices are tabulated. The distribution of zeros of the partition functions in the complex magnetic-field plane has been obtained both for the ferro- and antiferromagnetic cases. The zeros in the ferromagnetic cases are distributed on the unit circle according to the Yang-Lee theorem, and the dependence on temperature of the distribution functions has been studied. In the antiferromagnetic case most of zeros are on the negative real axis, but a certain number of complex roots appear in the left half of the plane. The magnetization and the magnetic susceptibility have been calculated as functions of temperature.

We study the complex-temperature phase diagram of the square-lattice Ising model for nonzero external magnetic field H, i.e., for 0\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}\ensuremath{\infty}, where \ensuremath{\mu}=${\mathit{e}}^{\mathrm{\ensuremath{-}}2\mathrm{\ensuremath{\beta}}\mathit{H}}$. We also carry out a similar analysis for -\ensuremath{\infty}\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}0. The results for the interval -1\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}1 provide a way of continuously connecting the two known exact solutions of this model, viz., at \ensuremath{\mu}=1 (Onsager, Yang) and \ensuremath{\mu}=-1 (Lee and Yang). Our methods include numerical calculations of complex-temperature zeros of the partition function and an analysis of low-temperature series expansions. For real nonzero H, the inner branch of a lima\ifmmode \mbox{\c{c}}\else \c{c}\fi{}on bounding the FM phase breaks and forms two complex-conjugate arcs. We study the singularities and associated exponents of thermodynamic functions at the endpoints of these arcs. For \ensuremath{\mu}0, there are two line segments of singularities on the negative and positive u axis, and we carry out a similar study of the behavior at the inner endpoints of these arcs, which constitute the nearest singularities to the origin in this case. Finally, we also determine the exact complex-temperature phase diagrams at \ensuremath{\mu}=-1 on the honeycomb and triangular lattices and discuss the relation between these and the corresponding zero-field phase diagrams. \textcopyright{} 1996 The American Physical Society.