Integrability vs non-integrability: Hard hexagons and hard squares compared

In this paper we compare the integrable hard hexagon model with the non-integrable hard squares model by means of partition function roots and transfer matrix eigenvalues. We consider partition functions for toroidal, cylindrical, and free-free boundary conditions up to sizes $40\times40$ and transfer matrices up to 30 sites. For all boundary conditions the hard squares roots are seen to lie in a bounded area of the complex fugacity plane along with the universal hard core line segment on the negative real fugacity axis. The density of roots on this line segment matches the derivative of the phase difference between the eigenvalues of largest (and equal) moduli and exhibits much greater structure than the corresponding density of hard hexagons. We also study the special point $z=-1$ of hard squares where all eigenvalues have unit modulus, and we give several conjectures for the value at $z=-1$ of the partition functions.Comment: 46 page

Hard hexagon partition function for complex fugacity

We study the analyticity of the partition function of the hard hexagon model in the complex fugacity plane by computing zeros and transfer matrix eigenvalues for large finite size systems. We find that the partition function per site computed by Baxter in the thermodynamic limit for positive real values of the fugacity is not sufficient to describe the analyticity in the full complex fugacity plane. We also obtain a new algebraic equation for the low density partition function per site.Comment: 49 pages, IoP styles files, lots of figures (png mostly) so using PDFLaTeX. Some minor changes added to version 2 in response to referee report

On rational approximation of algebraic functions

We construct a new scheme of approximation of any multivalued algebraic function $f(z)$ by a sequence $\{r_{n}(z)\}_{n\in \mathbb{N}}$ of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by $f(z)$. Compared to the usual Pad\'e approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Pad\'e Conjecture and Nuttall's Conjecture for the sequence $\{r_{n}(z)\}_{n\in \mathbb{N}}$ in the complement \mathbb{CP}^1\setminus \D_{f}, where \D_{f} is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family $\{r_{n}(z)\}_{n\in \mathbb{N}}$. As an application we settle the so-called 3-conjecture of Egecioglu {\em et al} dealing with a 4-term recursion related to a polynomial Riemann Hypothesis.Comment: 25 pages, 8 figures, LaTeX2e, revised version to appear in Advances in Mathematic

Complex-temperature phase diagram of Potts and RSOS models

We study the phase diagram of Q-state Potts models, for Q=4 cos^2(PI/p) a Beraha number (p>2 integer), in the complex-temperature plane. The models are defined on L x N strips of the square or triangular lattice, with boundary conditions on the Potts spins that are periodic in the longitudinal (N) direction and free or fixed in the transverse (L) direction. The relevant partition functions can then be computed as sums over partition functions of an A\_{p-1} type RSOS model, thus making contact with the theory of quantum groups. We compute the accumulation sets, as N -> infinity, of partition function zeros for p=4,5,6,infinity and L=2,3,4 and study selected features for p>6 and/or L>4. This information enables us to formulate several conjectures about the thermodynamic limit, L -> infinity, of these accumulation sets. The resulting phase diagrams are quite different from those of the generic case (irrational p). For free transverse boundary conditions, the partition function zeros are found to be dense in large parts of the complex plane, even for the Ising model (p=4). We show how this feature is modified by taking fixed transverse boundary conditions.Comment: 60 pages, 16 figures, 2 table

Algebraic methods for chromatic polynomials

The chromatic polynomials of certain families of graphs can be calculated by a transfer matrix method. The transfer matrix commutes with an action of the symmetric group on the colours. Using representation theory, it is shown that the matrix is equivalent to a block-diagonal matrix. The multiplicities and the sizes of the blocks are obtained. Using a repeated inclusion-exclusion argument the entries of the blocks can be calculated. In particular, from one of the inclusion-exclusion arguments it follows that the transfer matrix can be written as a linear combination of operators which, in certain cases, form an algebra. The eigenvalues of the blocks can be inferred from this structure. The form of the chromatic polynomials permits the use of a theorem by Beraha, Kahane and Weiss to determine the limiting behaviour of the roots. The theorem says that, apart from some isolated points, the roots approach certain curves in the complex plane. Some improvements have been made in the methods of calculating these curves. Many examples are discussed in detail. In particular the chromatic polynomials of the family of the so-called generalized dodecahedra and four similar families of cubic graphs are obtained, and the limiting behaviour of their roots is discussed

Phase diagram of the chromatic polynomial on a torus

We study the zero-temperature partition function of the Potts antiferromagnet (i.e., the chromatic polynomial) on a torus using a transfer-matrix approach. We consider square- and triangular-lattice strips with fixed width L, arbitrary length N, and fully periodic boundary conditions. On the mathematical side, we obtain exact expressions for the chromatic polynomial of widths L=5,6,7 for the square and triangular lattices. On the physical side, we obtain the exact ``phase diagrams'' for these strips of width L and infinite length, and from these results we extract useful information about the infinite-volume phase diagram of this model: in particular, the number and position of the different phases.Comment: 72 pages (LaTeX2e). Includes tex file, three sty files, and 26 Postscript figures. Also included are Mathematica files transfer6_sq.m and transfer6_tri.m. Final version to appear in Nucl. Phys.