Characterization of Lee-Yang polynomials. (English) Zbl 1251.30005

Author’s abstract: “The Lee-Yang circle theorem describes complex polynomials of degree \(n\) in \(z\) with all their zeros on the unit circle \(|z|=1\). These polynomials are obtained by taking \(z_1=\cdots=z_n=z\) in certain multiaffine polynomials \(\Psi(z_1,\dots,z_n)\) which we call Lee-Yang polynomials (they do not vanish when \(|z_1|,\dots,|z_n| < 1\) or \(|z_1|,\dots,|z_n| > 1\)). We characterize the Lee-Yang polynomials \(\Psi\) in \(n+1\) variables in terms of polynomials \(\Phi\) in \(n\) variables (those such that \(\Phi(z_1,\dots,z_n)\not= 0\) when \(|z_1|,\dots,|z_n| < 1\)). This characterization gives us a good understanding of Lee-Yang polynomials and allows us to exhibit some new examples. In the physical situation where the \(\Psi\) are temperature dependent partition functions, we find that those \(\Psi\) which are Lee-Yang polynomials for all temperatures are precisely the polynomials with pair interactions originally considered by T. D. Lee and C. N. Yang [Phys. Rev., II. Ser. 87, 410–419 (1952; Zbl 0048.43401)].”
See also the author’s paper [“Extension of the Lee-Yang circle theorem”, Phys. Rev. Lett. 26, 303–304 (1971; doi:10.1103/PhysRevLett.26.870)].


30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
26C10 Real polynomials: location of zeros
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics


Zbl 0048.43401
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