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Physical review. B, Condensed matter

Yang-Lee edge singularities at high temperatures

Douglas A. Kurtze, Michael Ellis Fisher

Published Oct 1, 1979• 121 citations
Publisher

Abstract

The density of Lee-Yang zeros, in the thermodynamic limit, for classical $n$-vector models and for the quantum Heisenberg model is studied in the asymptotic high-temperature limit. It is shown that the high-temperature series expansions for these models reduce, in this limit, to the corresponding low-density expansions for the monomer-dimer problem with negative dimer activity. If the density of zeros, $g({h}^{\ensuremath{'}\ensuremath{'}})$, on the imaginary axis of the complex reduced-magnetic-field plane, $h=\frac{H}{{k}_{B}T}={h}^{\ensuremath{'}}+i{h}^{\ensuremath{'}\ensuremath{'}}$, has an algebraic singularity at the edge of the gap in the zero distribution, $g({h}^{\ensuremath{'}\ensuremath{'}})\ensuremath{\sim}{[|{h}^{\ensuremath{'}\ensuremath{'}}|\ensuremath{-}{h}_{0}(T)]}^{\ensuremath{\sigma}}$, then $\ensuremath{\sigma}$ is independent of $n$ in this limit. Analyzing dimer density series on various lattices by means of the ratio test, Dlog Pad\'e, the recursion-relation method, and inhomogeneous differential approximants, we obtain the estimates $\ensuremath{\sigma}=\ensuremath{-}0.163\ifmmode\pm\else\textpm\fi{}3$ for $d=2$ dimensions and $\ensuremath{\sigma}=0.086\ifmmode\pm\else\textpm\fi{}15$ for $d=3$.

DOI
10.1103/physrevb.20.2785

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Yang-Lee zeros of certain antiferromagnetic models

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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.

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The density of Lee-Yang zeros, in the thermodynamic limit, for classical nnn-vector models and for the quantum Heisenberg model is studied in the asymptotic high-temperature limit. It is shown that the high-temperature series expansions for these models reduce, in this limit, to the corresponding low-density expansions for the monomer-dimer problem with negative dimer activity. If the density of zeros, g(h\ensuremath′\ensuremath′)g({h}^{\ensuremath{'}\ensuremath{'}})g(h\ensuremath′\ensuremath′), on the imaginary axis of the complex reduced-magnetic-field plane, h=HkBT=h\ensuremath′+ih\ensuremath′\ensuremath′h=\frac{H}{{k}_{B}T}={h}^{\ensuremath{'}}+i{h}^{\ensuremath{'}\ensuremath{'}}h=kB​TH​=h\ensuremath′+ih\ensuremath′\ensuremath′, has an algebraic singularity at the edge of the gap in the zero distribution, g(h\ensuremath′\ensuremath′)\ensuremath∼[∣h\ensuremath′\ensuremath′∣\ensuremath−h0(T)]\ensuremathσg({h}^{\ensuremath{'}\ensuremath{'}})\ensuremath{\sim}{[|{h}^{\ensuremath{'}\ensuremath{'}}|\ensuremath{-}{h}_{0}(T)]}^{\ensuremath{\sigma}}g(h\ensuremath′\ensuremath′)\ensuremath∼[∣h\ensuremath′\ensuremath′∣\ensuremath−h0​(T)]\ensuremathσ, then \ensuremathσ\ensuremath{\sigma}\ensuremathσ is independent of nnn in this limit. Analyzing dimer density series on various lattices by means of the ratio test, Dlog Pad'e, the recursion-relation method, and inhomogeneous differential approximants, we obtain the estimates \ensuremathσ=\ensuremath−0.163\ifmmode±\else\textpm\fi3\ensuremath{\sigma}=\ensuremath{-}0.163\ifmmode\pm\else\textpm\fi{}3\ensuremathσ=\ensuremath−0.163\ifmmode±\else\textpm\fi3 for d=2d=2d=2 dimensions and \ensuremathσ=0.086\ifmmode±\else\textpm\fi15\ensuremath{\sigma}=0.086\ifmmode\pm\else\textpm\fi{}15\ensuremathσ=0.086\ifmmode±\else\textpm\fi15 for d=3d=3d=3.
The edge of the gap in the distribution of Yang-Lee zeros at H=iH0(T)H=i{H}_{0}(T)H=iH0​(T) on the imaginary magnetic field axis in ferromagnets above Tc{T}_{c}Tc​ is essentially a critical point. In terms of the edge exponents $\ensuremath{\delta}$ and $\ensuremath{\eta}$, the density of zeros obeys $\mathcal{G}({H}^{\ensuremath{'}\ensuremath{'}})\ensuremath{\sim}{[{H}^{\ensuremath{'}\ensuremath{'}}\ensuremath{-}{H}_{0}(T)]}^{\ensuremath{\sigma}}$, with $\ensuremath{\sigma}=\frac{1}{\ensuremath{\delta}}=\frac{(d\ensuremath{-}2+\ensuremath{\eta})}{({d}_{2}\ensuremath{-}\ensuremath{\eta})}$. Classical behavior ($\ensuremath{\sigma}=\frac{1}{2}$) occurs for $d>{d}^{\ifmmode\times\else\texttimes\fi{}}=6$. The appropriate field-theoretic renormalization group entails a $w{\ensuremath{\phi}}^{3}$ coupling and, with $\ensuremath{\epsilon}=6\ensuremath{-}d>~0$, yields $\ensuremath{\eta}\ensuremath{\approx}\ensuremath{-}\frac{\ensuremath{\epsilon}}{9}$ for all $n<\ensuremath{\infty}$. This correlates well with refined series estimates for d=2d=2d=2 and d=3d=3d=3 and with exact results for d=1d=1d=1 ($\ensuremath{\eta}=\ensuremath{-}1$).
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.
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.
The distribution of Yang-Lee zeros in the ferromagnetic Ising model in both two and three dimensions is studied on the complex field plane directly in the thermodynamic limit via the tensor network methods. The partition function is represented as a contraction of a tensor network and is efficiently evaluated with an iterative tensor renormalization scheme. The free-energy density and the magnetization are computed on the complex field plane. Via the discontinuity of the magnetization, the density of the Yang-Lee zeros is obtained to lie on the unit circle, consistent with the Lee-Yang circle theorem. Distinct features are observed at different temperatures---below, above, and at the critical temperature. Application of the tensor network approach is also made to the qqq-state Potts models in both two and three dimensions and a previous debate on whether, in the thermodynamic limit, the Yang-Lee zeros lie on a unit circle for q>2q>2q>2 is resolved: they clearly do not lie on a unit circle except at the zero temperature. For the Potts models (q=3,4,5,6)(q=3,4,5,6)(q=3,4,5,6) investigated in two dimensions, as the temperature is lowered the radius of the zeros at a fixed angle from the real axis shrinks exponentially towards unity with the inverse temperature.
The quantum one-dimensional transverse Ising model with an imaginary longitudinal field is studied at T=0T=0T=0 using a real-space renormalization-group method. Evaluation is made of the real part of the ground-state longitudinal magnetization which is related to the Yang-Lee distribution of zeros of the partition function for the two-dimensional classical Ising model. The edge exponents are found as $\ensuremath{\sigma}=\ensuremath{-}0.25\ifmmode\pm\else\textpm\fi{}0.05$ and $\ensuremath{\eta}=\ensuremath{-}1.0\ifmmode\pm\else\textpm\fi{}0.02$, in agreement with series expansion estimates.
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.