Chromatic-count conjectures and the infinite-width limit of partition-function zeros for Potts models on honeycomb-lattice strips

The exact study of the q-state Potts model on infinite-length strips of the honeycomb lattice by Chang and Shrock (arXiv:cond-mat/0703014) left three linked issues open, which later literature has not closed. First, the identities counting the distinct transfer-matrix eigenvalues of the chromatic polynomial on cylindrical (periodic-transverse, free-longitudinal) honeycomb strips of arbitrary even width $L_y$ --- stated there as Conjectures 2.1 and 2.2, eqs. (2.14)-(2.15) --- remain unproven, even though the corresponding counts for the full Potts partition function were proven in the same paper and for the square and triangular lattices in companion work. Second, the $L_y\to\infty$ limit of the continuous accumulation loci $\mathcal B$ of partition-function zeros, in both the $q$ plane and the $v$ plane, is not established: finite-width data suggest convergence of envelope curves and crossings to the two-dimensional honeycomb criticality roots of $v^3-3qv-q^2=0$ (e.g. $q_c(-1)=(3+\sqrt5)/2$), but the limiting locus and the mechanism of convergence (arc-endpoint pinching versus motion of real-axis crossing points versus bubble formation) are undetermined. Third, the thermodynamic and combinatorial meaning of formal criticality at noninteger $q$ --- where the Fortuin-Kasteleyn cluster representation can yield unphysical partition-function values --- is unsettled.

Background

The $q$-state Potts model assigns to each vertex $i$ of a graph $G=(V,E)$ a spin $\sigma_i$ taking one of $q$ values and couples nearest neighbours through the Hamiltonian $H=-J\sum_{\langle ij\rangle}\delta_{\sigma_i,\sigma_j}$. Its partition function admits the Fortuin-Kasteleyn (FK) cluster representation $Z(G,q,v)=\sum_{G'\subseteq G} q^{k(G')}v^{|E(G')|}$, where the sum runs over spanning subgraphs $G'$ of $G$, $k(G')$ is the number of connected components, $|E(G')|$ the number of occupied edges, and $v=e^{\beta J}-1$ is a temperature-like variable. This representation defines $Z$ for arbitrary --- in particular noninteger and complex --- $q$ and $v$. At the antiferromagnetic zero-temperature point $v=-1$, $Z(G,q,-1)$ equals $P(G,q)$, the chromatic polynomial, which counts proper $q$-colourings of $G$ for positive integer $q$.

A lattice strip of width $L_y$ and arbitrarily great length is quasi-one-dimensional: its infinite-length free energy is analytic in temperature, yet its structure encodes two-dimensional physics increasingly well as $L_y$ grows. With free longitudinal boundary conditions and either free (FF) or periodic transverse boundary conditions (PF, called cylindrical), the transfer matrix $T$ acts on FK connectivity states --- set partitions of $\{1,\ldots,L_y\}$, restricted to noncrossing partitions (counted by the Catalan numbers $C_{L_y}$) in the FF case --- and yields $Z(G,q,v)=\sum_j c_j(\lambda_j)^m$, where $m$ is the number of repeated slices and the $\lambda_j$ are eigenvalues of $T$. A central structural question is the exact number $N_{Z,G,BC,L_y}$ of distinct eigenvalues as a function of $L_y$, and its chromatic counterpart $N_{P,G,BC,L_y}$, which is smaller at $v=-1$ because additional eigenvalue degeneracies occur there that are not induced by lattice symmetries. Cylindrical honeycomb strips exist only for even $L_y$ and possess a length-two rotational symmetry in addition to reflection symmetry; Chang and Shrock proved the full-Potts counts $N_{Z,hc,PF,L_y}$ exactly (their Theorem 2.3, via Lemmas 2.1-2.2, using Mobius-function and Euler-totient bookkeeping of rotational periodicity together with a type-I/type-II classification of reflection-symmetric partitions), but for the chromatic counts they could only state conjectures. The relevant companion quantities come from cyclic (free-transverse, periodic-longitudinal) honeycomb strips, whose chromatic eigenvalues decompose into angular-momentum ('level') sectors $d$; $n_P(hc,L_y,0)$ denotes the count in the zero-momentum sector and $N_{P,hc,FP,d}$ the total chromatic count at width $d$, both tabulated in earlier work of the same authors.

In the infinite-length limit at fixed width, the zeros of $Z$ in the complex $q$ plane (fixed $v$) or complex $v$ plane (fixed $q$) accumulate on curves and arcs forming a continuous locus $\mathcal B$ --- the finite-width analogue of the complex-temperature singularities of the two-dimensional model, studied for the square and triangular lattices by Salas and Sokal and by Jacobsen, Salas and Sokal in their series 'Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models' (parts I-VI, 2001-2011). On the infinite honeycomb lattice the Potts criticality condition is exactly known, $v^3-3qv-q^2=0$; at $v=-1$ its roots are $q=(3+\sqrt5)/2$ and $q=(3-\sqrt5)/2$, and at the Beraha values $q_r=4\cos^2(\pi/r)$ (root-of-unity-related values with a distinguished role in chromatic-polynomial theory) the roots admit trigonometric closed forms. What is not known exactly for the honeycomb lattice is the full two-dimensional accumulation locus of chromatic zeros --- in sharp contrast to the triangular lattice, where Baxter's exact solution of the chromatic polynomial of large triangular lattices provides it, and which is what allowed the triangular-lattice strip program to be anchored to an exact target. Indeed, the chromatic (zero-temperature antiferromagnetic Potts) free energy of the honeycomb lattice is not exactly known for any $q\ge3$ (the $q=2$ Ising case is exactly solvable), so strip-based determination is the only current route.

Two subtleties sharpen the honeycomb strip questions. First, at special points such as $q=0$ and $q=1$ the limits of infinite length and $q$ approaching the special point do not commute, so the locus obtained depends on the order of limits. Second, for negative $v$ and noninteger $q$ the FK sum can yield negative or complex values of $Z$, so the Potts antiferromagnet is physically defined only for integer $q$; crossings of $\mathcal B$ at noninteger values such as $q_c(-1)=(3+\sqrt5)/2$ are 'formal' algebraic statements whose thermodynamic or combinatorial meaning requires separate analysis. Chromatic zeros and asymptotic loci for honeycomb strips with cyclic longitudinal boundary conditions had been computed earlier up to width 5 by Chang and Shrock, and the 2007 source paper extended the exact transfer-matrix treatment to cylindrical strips, proved the full-Potts counting theorems, tabulated chromatic counts through $L_y=12$, plotted zero loci for $L_y\le5$ (free) and $L_y=4,6$ (cylindrical), and analysed internal energy and specific heat on infinite-length strips --- leaving open the three questions stated below.

Problem Statement

The common objective is to complete the exact structural theory of the Potts partition function on infinite-length honeycomb-lattice strips --- the one regular planar lattice for which neither the chromatic strip counts nor the infinite-width zero locus is settled --- by resolving the following three linked questions left open by Chang and Shrock (arXiv:cond-mat/0703014).

(1) Prove, or refute, the conjectured exact eigenvalue-counting identities for the chromatic polynomial on cylindrical (periodic-transverse, free-longitudinal) honeycomb strips of arbitrary even width $L_y$. With $N'_{P,hc,PF,L_y}$ denoting the number of distinct transfer-matrix eigenvalues after reduction by the length-two rotational symmetry of the cylindrical strip, and $N_{P,hc,PF,L_y}$ the smaller number obtained by further reduction by reflection symmetry, $n_P(hc,L_y,0)$ the count in the zero-momentum (level $d=0$) sector of cyclic honeycomb strips, $N_{P,hc,FP,d}$ the total chromatic eigenvalue count for cyclic honeycomb strips of width $d$, and $\phi$ the Euler totient function, the identities read

$$\frac{L_y}{2}\,N'_{P,hc,PF,L_y}-n_P(hc,L_y,0)=\sum_{\substack{d\mid L_y,\; d\ \mathrm{even}\\ 1\le d<L_y}}\phi\!\left(\frac{L_y}{d}\right)\,N_{P,hc,FP,d}$$

and

$$2N_{P,hc,PF,L_y}-N'_{P,hc,PF,L_y}=\begin{cases}\tfrac{1}{2}\,N_{P,hc,FP,L_y/2}, & L_y/2\ \mathrm{odd},\ L_y/2>1,\\ \tfrac{1}{2}\left[N_{P,hc,FP,L_y/2}+N_{P,hc,FP,(L_y-2)/2}\right], & L_y/2\ \mathrm{even}.\end{cases}$$

Together they should yield a closed formula for $N_{P,hc,PF,L_y}$ valid for arbitrary even $L_y$ (currently the identities are consistent with all published tabulated values, which extend through $L_y=12$; as in the source conjectures, the case split of the second identity covers odd $L_y/2>1$ and even $L_y/2$, while the smallest width $L_y=2$, i.e. $L_y/2=1$, is tabulated separately in the source rather than falling under a branch of that identity).

(2) Establish the $L_y\to\infty$ limit of the continuous accumulation loci $\mathcal B$ of partition-function zeros, in the $q$ plane for fixed $v$ and in the $v$ plane for fixed $q$: determine the limiting locus --- in particular whether it crosses the real $q$ axis at the values given by the two-dimensional honeycomb criticality condition $v^3-3qv-q^2=0$, e.g. $q_c(-1)=(3+\sqrt5)/2$ at $v=-1$, and in the $v$ plane whether the leftmost arcs cross at the complex-temperature singularities such as the value $v\simeq-3.7645$ for $q=3$ (obtained by duality, $v_{hc}=3/v_{tri}$, from the triangular-lattice paramagnetic-antiferromagnetic singularity, and not a root of the cubic) --- and establish the mechanism by which the finite-width loci approach it: rightward motion of the points where $\mathcal B$ crosses the real axis, pinching of complex-conjugate arc endpoints, formation of bubble regions, or a combination.

(3) Settle the interpretation of the formal noninteger-$q$ criticality: how algebraic crossings of $\mathcal B$ at noninteger $q$ --- at which the Fortuin-Kasteleyn cluster representation can yield unphysical (negative or complex) partition-function values --- are related to thermodynamic or combinatorial quantities on honeycomb strips and in their $L_y\to\infty$ limit.

These three questions are inseparable parts of one objective: the counting identities (1) are the structural backbone that makes the strip spectra, and hence the loci $\mathcal B$, exactly computable at arbitrary width; the limit (2) is the target those computations approach; and (3) is the interpretive question attaching to precisely the antiferromagnetic region where the decisive crossings lie. Extending the exact transfer-matrix expressions beyond $L_y=7$ (free) and $L_y=6$ (cylindrical), and to other longitudinal boundary conditions such as cyclic or toroidal, is an auxiliary computational task in service of the same objective rather than an independent completion criterion.

References

1. Shu-Chiuan Chang and Robert Shrock, 'Exact Potts Model Partition Functions for Strips of the Honeycomb Lattice', Journal of Statistical Physics 130, 1011-1024 (2008). DOI: 10.1007/s10955-007-9461-3, arXiv:cond-mat/0703014, https://arxiv.org/abs/cond-mat/0703014
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