Tutte-Beraha crossings of the partition-function-zero loci of the restricted Potts model on lattice strips (Chang-Shrock conjecture)

Chang and Shrock (2006) computed exactly the accumulation loci of partition-function zeros of the $Q$-state Potts model on infinite-length lattice strips of widths $L_y$ up to 5 (square, triangular, and honeycomb lattices, with several boundary conditions), with $Q$ and the temperature-like variable $v$ restricted to the ferromagnetic critical curve of the corresponding two-dimensional lattice. For the cyclic (and equivalently Moebius) strips they found that the equimodular locus in the $Q$ plane, ${\cal B}_Q$, crosses the real axis at the Tutte-Beraha numbers $Q_{2\ell}=4\cos^2(\pi/(2\ell))$, $\ell=1,\dots,L_y$, besides a universal crossing at $Q=4$, and they conjectured that this pattern holds for strips of arbitrary width, without proof. The problem posed here is the rigorous resolution of that conjecture - a general proof or an explicit counterexample - including the eigenvalue-dominance argument that places the crossings on the locus itself. No proof or refutation of the conjecture is known in the literature to date.

Background

The $Q$-state Potts model assigns to each vertex $i$ of a graph $G$ a spin $\sigma_i\in\{1,\dots,Q\}$ and has partition function $Z(G,Q,v)=\sum_{\{\sigma\}}\exp(K\sum_{\langle ij\rangle}\delta_{\sigma_i\sigma_j})$, where $K=\beta J$ is the reduced coupling and $v=e^K-1$ is the temperature-like variable. By the Fortuin-Kasteleyn cluster expansion, $Z(G,Q,v)=\sum_{G'\subseteq G}Q^{k(G')}v^{e(G')}$ (with $k$ and $e$ the numbers of connected components and edges of the subgraph $G'$), so both $Q$ and $v$ may be continued to complex values; in this form $Z$ is equivalent to the Tutte polynomial of graph theory. On the square, triangular, and honeycomb lattices, the ferromagnetic phase transition of the two-dimensional (2D) thermodynamic limit lies on the curves $Q=v^2$ (square), $Q=v^2(v+3)$ (triangular), and $Q^2+3Qv-v^3=0$ (honeycomb). The 'restricted' Potts model of Chang and Shrock constrains $Q$ and $v$ to satisfy the corresponding curve, leaving a single complex variable: zeros of $Z$ are studied in the complex $v$ plane and mapped to the complex $Q$ plane.

For a strip of transverse width $L_y$ and length $L_x=m$, with free, cyclic (periodic in the length direction $x$), cylindrical (periodic in the transverse direction $y$), toroidal (periodic in both directions), Moebius (cyclic with a reversal of the transverse coordinate), or Klein-bottle (toroidal with such a reversal) boundary conditions, $Z$ has the transfer-matrix form $Z=\sum_{d=0}^{L_y}c^{(d)}\sum_j(\lambda_{\Lambda,L_y,d,j})^m$, where the eigenvalues $\lambda$ fall into sectors labeled $d=0,\dots,L_y$ with coefficients $c^{(d)}=U_{2d}(\sqrt{Q}/2)$ given by Chebyshev polynomials of the second kind. In the limit $m\to\infty$ the zeros accumulate on curves ${\cal B}$ determined by the equality of the moduli of the two dominant eigenvalues (the 'equimodular curves'); ${\cal B}_v$ and ${\cal B}_Q$ denote these loci in the $v$ and $Q$ planes. The central algebraic actors are the Tutte-Beraha numbers $Q_r=4\cos^2(\pi/r)$, which arise as accumulation points of chromatic-polynomial zeros (Beraha-Kahane-Weiss) and as the values at which the Temperley-Lieb algebra - the algebra underlying the Potts transfer matrix - becomes reducible (Saleur). For cyclic square-lattice strips, Saleur further proved that at $Q=Q_{2\ell}$ the maximal-magnitude eigenvalues of sectors $\ell$ and $\ell-1$ are equal in modulus. It has not been possible to solve exactly for the free energy $f(\Lambda,Q,v)$ at arbitrary $Q$ and $v$ on any lattice of dimension $d\ge2$ - the only exact solution for arbitrary $v$ is the $d=2$ Ising case $Q=2$ - which is why exactly solvable infinite-length strips serve as a laboratory for the analytic structure of $Z$.

In this program Chang and Shrock determined ${\cal B}_v$ and ${\cal B}_Q$ exactly for strips of the square, triangular, and honeycomb lattices with widths $L_y\le5$ and various boundary conditions. Two systematic findings emerged. First, for every geometry studied that has a nontrivial continuous locus, ${\cal B}_v$ passes through $v=-2$ and ${\cal B}_Q$ through $Q=4$, the value at the upper end of the interval $0\le Q\le4$ where the 2D ferromagnetic Potts transition is second order. Second, for all cyclic (and, equivalently, Moebius) strips studied, ${\cal B}_Q$ also crosses the real axis at $Q=Q_{2\ell}$ for $\ell=1,\dots,L_y$. For cyclic self-dual strips of the square lattice, treated in their companion paper, the crossings instead occur at the interleaved odd values $Q_{2\ell+1}$: there the real $Q$ axis is divided into intervals whose endpoints are these crossing points, a different transfer-matrix eigenvalue dominates in each interval, and consecutive dominant eigenvalues are equimodular at the endpoints, while isolated zeros of $Z$ near certain Beraha values (for example near $Q=Q_4=2$) were traced to the vanishing, at these values, of that paper's sector coefficients $\kappa^{(d)}=Q\bar\kappa^{(d)}=\prod_{k=1}^{d}(Q-s_{d,k})$ with $s_{d,k}=4\cos^2(\pi k/(2d))$. For the cyclic strips of the main paper, by contrast, the sector coefficients $c^{(d)}=U_{2d}(\sqrt{Q}/2)$ have zeros $Q=4\cos^2(\pi k/(2d+1))$, $k=1,\dots,2d$, none of which equals an even-indexed Beraha number $Q_{2\ell}$, so no sector coefficient vanishes at $Q_{2\ell}$, and the crossings there must come instead from degeneracies in the magnitudes of the transfer-matrix eigenvalues. The crossing pattern was verified for every width computed ($L_y\le5$), but a general proof was lacking for all three lattices, and the authors recorded the crossing property as a conjecture for arbitrary width. They also observed that for some geometries (for example the $L_y=3$ toroidal square strip and cyclic self-dual square strips) the loci are noncompact, extending to complex infinity, while for others they are compact.

Problem Statement

For the restricted Potts model ($Q$ and $v$ constrained to the two-dimensional ferromagnetic critical curve of the respective lattice) on infinite-length strips ($L_x\to\infty$) of the square, triangular, and honeycomb lattices, resolve the Chang-Shrock conjecture: prove or refute that, for cyclic (and equivalently Moebius, since ${\cal B}$ is identical for these two boundary conditions) strips of arbitrary transverse width $L_y$ of each of the three lattices, the locus ${\cal B}_Q$ crosses the real $Q$ axis at the Tutte-Beraha numbers

$Q \;=\; Q_{2\ell} \;=\; 4\cos^2\!\left(\frac{\pi}{2\ell}\right), \qquad \ell = 1,\dots,L_y,$

in addition to the universal crossing at $Q=4$. The index range $\ell=1,\dots,L_y$ is that of the source; at the endpoint $\ell=1$, i.e. $Q_2=0$, the source records the crossing as a point where ${\cal B}_Q$ has a cusp. The conjecture makes no exclusivity claim: the loci may also cross the real axis elsewhere (for example on the negative real axis for $L_y\ge3$). An affirmative answer must establish dominance: at each $Q_{2\ell}$, the equimodular pair of transfer-matrix eigenvalues must be shown to dominate all sectors and all eigenvalues, so that the point lies on the accumulation locus itself; mere equimodularity of some eigenvalue pair, as in the known square-lattice adjacent-sector result of Saleur, does not suffice. A negative answer is an explicit counterexample: a lattice, boundary condition, width $L_y$, and index $\ell\le L_y$ for which the crossing fails (since the crossing has been verified computationally for all cyclic and Moebius strips with $L_y\le5$, such a counterexample must lie beyond the source's computed cases). The compact-versus-noncompact behavior of the loci observed in the source program (for example the noncompact ${\cal B}$ of the $L_y=3$ toroidal square strip and of the cyclic self-dual square strips) is background context rather than part of the completion boundary: the sources report noncompactness only as an observation and pose no classification question. The broader question of how the strip loci converge, as $L_y\to\infty$, to the locus of the two-dimensional thermodynamic limit (for example, whether the $Q=4$ crossing persists) is the program's motivation but is not part of the completion boundary, since the sources pose no determinate criterion for it.

References

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