Density of Yang-Lee zeros for the Ising ferromagnet

Yang-Lee zeros of certain antiferromagnetic models

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.

Condensation of Lee-Yang zeros in scalar field theory

We show that, at the critical temperature, there is a class of Lee-Yang zeros of the partition function in a general scalar field theory, which location scales with the size of the system with a characteristic exponent expressed in terms of the isothermal critical exponent δ. In the thermodynamic limit the zeros belonging to this class condense to the critical point ζ=1 on the real axis in the complex fugacity plane while the complementary set of zeros (with Reζ<1) covers the unit circle. Although the aforementioned class degenerates to a single point for an infinite system, when the size is finite it contributes significantly to the partition function and reflects the self-similar structure (fractal geometry, scaling laws) of the critical system. This property opens up the perspective to formulate finite-size scaling theory in effective QCD, near the chiral critical point, in terms of the location of Lee-Yang zeros.

Exact partition function zeros of the Wako-Saitô-Muñoz-Eaton β hairpin model

I compute exact partition function zeros of β hairpins, using both analytic and numerical methods, extending previous work [J. Lee, Phys. Rev. Lett. 110, 248101 (2013)] where only a restricted class of hairpins was considered. The zeros of β hairpins with an odd number of peptide bonds are computed and the difference of the distribution of zeros from those for an even number of peptide bonds is explained in terms of additional entropy of liberating the extra bond at the turn region. Upon the introduction of a hydrophobic core in the central region of the hairpin, the zeros are distributed uniformly on two concentric circles corresponding to the hydrophobic collapse and the transition to the fully folded conformation. One of the circles dissolves as the core moves toward the turn or the tip region, which is explained in terms of the similarity of the intermediate state with the folded or unfolded states. The exact partition function zeros for a hairpin with a more complex structure of native contacts, the 16 C-terminal residues of streptococcal protein G B1, are numerically computed and their loci are closely approximated by concentric circles.

Exact partition function zeros of the Wako-Saitô-Muñoz-Eaton protein model

I compute exact partition function zeros of the Wako-Saitô-Muñoz-Eaton model for various secondary structural elements and for two proteins, 1BBL and 1I6C, by using both analytic and numerical methods. Two-state and barrierless downhill folding transitions can be distinguished by a gap in the distribution of zeros at the positive real axis.

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

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.

Critical behavior at edge singularities in one-dimensional spin models

In ferromagnetic spin models above the critical temperature (T>T_{cr}) the partition function zeros accumulate at complex values of the magnetic field (H_{E}) with a universal behavior for the density of zeros rho(H) approximately mid R:H-H_{E}mid R:;{sigma} . The critical exponent sigma is believed to be universal at each space dimension and it is related to the magnetic scaling exponent y_{h} via sigma=(d-y_{h})y_{h} . In two dimensions we have y_{h}=125(sigma=-16) while y_{h}=2(sigma=-12) 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(sigma=-23) can emerge if we have a triple degeneracy of the transfer matrix eigenvalues.

Partition function zeros of the one-dimensional Blume-Capel model in transfer matrix formalism

Zeros of the partition function of the one-dimensional ferromagnetic and antiferromagnetic Blume-Capel models have been studied by using the transfer matrix method in the thermodynamic limit and for finite size chains. The equation for the distribution of zeros of the partition function in the thermodynamic limit is derived. The distribution of the Yang-Lee and Fisher zeros are studied for a variety of values of the parameters of the model. Densities of the Yang-Lee and Fisher zeros are investigated and a singular behavior of the corresponding densities of zeros at the edge points is shown. The edge singularity exponents are calculated analytically for the densities of the Yang-Lee and Fisher zeros. It is found that for both cases edge singularity exponents are universal and equal to sigma = -1/2 .