Arbitrary-width structure of the Potts partition-function-zero accumulation locus on self-dual square-lattice strips (the Chang–Shrock conjecture)

The $Q$-state Potts model partition function $Z(G,Q,v)$ (a Tutte-polynomial specialization) was computed exactly by Chang and Shrock on infinite-length self-dual cyclic square-lattice strips restricted to the self-dual manifold $Q=v^2$, for transverse widths $L_y\le 4$. Their exact zero-accumulation loci display a systematic pattern — real-axis crossings at the odd Beraha values $Q_{2\ell+1}=4\cos^2(\pi/(2\ell+1))$, a semi-infinite segment $4\le Q\le\infty$ present for odd $L_y$ and absent for the even widths computed, division of each plane into at least $2L_y$ regions with a fixed assignment of the dominant transfer-matrix eigenvalue to each region, confluence of all arcs at $Q=4$, and isolated zeros near the even Beraha values $Q_{2\ell}$ — which they conjectured to hold for arbitrary width $L_y$. The problem asks for a proof or disproof of that conjecture and for a determination of whether the locus crosses the real $Q$ axis outside $0\le Q\le 4$ for sufficiently wide strips. The strongest later work (Jacobsen–Saleur's boundary Temperley–Lieb analysis) derives, at physics level of rigor, the real-axis crossing positions at every finite width but none of the global structure; no proof or disproof of the conjecture for arbitrary width is known, so the problem is open.

Background

The $Q$-state Potts model assigns to each vertex of a graph $G$ one of $Q$ spin states and couples neighboring spins with Boltzmann weight $v=e^{K}-1$, where $K$ is the dimensionless exchange. In the Fortuin–Kasteleyn cluster representation (an expansion over spanning subgraphs $G'=(V,E')$ of $G$, each weighted by $Q^{k(G')}\,v^{e(G')}$, with $k(G')$ the number of connected components and $e(G')$ the number of occupied edges) the partition function is a polynomial in $Q$ and $v$, equivalent to a specialization of the Tutte polynomial of $G$; at $v=-1$ it reduces to the chromatic polynomial counting proper $Q$-colorings of $G$. Because $Z$ is a polynomial, its zeros can lie off the real parameter axes, and the loci on which zeros accumulate in the thermodynamic limit (complex-temperature zeros, the complex-temperature analogue of the Lee–Yang zeros, which are the zeros of the Ising partition function in the complex magnetic field) encode the analytic structure of the free energy — in two dimensions, the complex-temperature phase diagram.

For a strip of a lattice, i.e. a graph of fixed transverse width $L_y$ and length $L_x$ with $L_x\to\infty$, the partition function is computable exactly by transfer matrices and takes the form $Z=\sum_j c_j(Q)\,\lambda_j(Q,v)^{L_x}$, a finite sum of eigenvalue powers with polynomial amplitudes $c_j$. The Beraha–Kahane–Weiss theorem then states that the zeros accumulate, as $L_x\to\infty$, exactly on the locus where two or more dominant (largest-modulus) terms are equal in magnitude, possibly supplemented by isolated points where a dominant amplitude $c_j$ vanishes. The accumulation locus $\mathcal{B}$ can therefore be determined exactly, without approximation, provided one can control which eigenvalues dominate where in the complex plane.

A distinguished submanifold for the square lattice is the self-dual manifold $Q=v^2$. Potts duality maps $v\to Q/v$, so this manifold is invariant under duality, just as the square lattice itself is self-dual. Chang and Shrock studied strips that preserve this property: the CDBC strip $G_{\rm CDBC}[L_y\times L_x]$ (self-dual cyclic boundary conditions) is an infinite-length strip with periodic longitudinal boundary conditions plus one extra external vertex connected to every vertex on one of the two sides running along the length (so the external vertex has degree $L_x$), which makes the graph self-dual (its vertex and face counts are equal and its average degree is 4, as on the square lattice). Restricting to $Q=v^2$, they computed the exact accumulation loci $\mathcal{B}_v$ and $\mathcal{B}_Q$ in the complex $v$ and $Q$ planes for widths $1\le L_y\le 4$. Their transfer-matrix analysis organizes the partition function as

$Z\big(G_{\rm CDBC}[L_y\times L_x],Q,v\big)=Q\,v^{L_xL_y}\sum_{d=1}^{L_y+1}\sum_j\bar\kappa^{(d)}(Q)\,\big(\bar\lambda_{L_y,d,j}\big)^{L_x},$

where the eigenvalues $\bar\lambda_{L_y,d,j}=v^{-L_y}\lambda_{L_y,d,j}$ are labeled by a sector number $d\in\{1,\ldots,L_y+1\}$ and an index $j$ within that sector, the amplitudes $\bar\kappa^{(d)}(Q)$ are $L_x$-independent polynomials in $Q$, and the single eigenvalue of the last sector is identically $\bar\lambda_{L_y,L_y+1}=1$; $\bar\lambda_{L_y,d,\max}$ denotes the eigenvalue of largest magnitude within sector $d$, with which member $j$ of the sector attains the maximum depending on the point of the complex plane under consideration. Already at $L_y=4$ there are 126 such eigenvalues, roots of high-degree polynomials with no closed form.

The exact loci for $L_y\le 4$ show a strikingly regular pattern. In the interval $-2\le v\le -1$ the locus crosses the negative real $v$ axis at $v=-2$ and at $v_{2\ell+1}=-2\cos(\pi/(2\ell+1))$ for $1\le\ell\le L_y$; equivalently, in the interval $1\le Q\le 4$, $\mathcal{B}_Q$ crosses the positive real axis at the odd Beraha values $Q_{2\ell+1}=4\cos^2(\pi/(2\ell+1))$ for $0\le\ell\le L_y$, where $Q_1=4$ (the Beraha numbers $4\cos^2(\pi/r)$ have played a special role in the theory of chromatic polynomials since Beraha, Kahane and Weiss). For odd $L_y$ the locus also contains the semi-infinite real segments $-\infty\le v\le -2$ and $4\le Q\le\infty$; the even widths computed show no such segment. All curves of the locus meet at $v=-2$ (equivalently $Q=4$), where every transfer-matrix eigenvalue has unit modulus. Isolated zeros off the locus occur at $Q=0$ (from the explicit prefactor above, a zero proven by the authors for all widths) and near the even Beraha values $Q_{2\ell}$, $1\le\ell\le L_y+1$, where the amplitude of the dominant eigenvalue vanishes. Finally, the locus divides each plane into at least $2L_y$ regions, with a systematic rule for which eigenvalue dominates in each region. Chang and Shrock elevated these observations to a conjecture for arbitrary width (their Section V) and flagged one further issue: their properties 'do not preclude' real-axis crossings outside $0\le Q\le 4$ for sufficiently wide strips. (The noncompactness of $\mathcal{B}_v$ and $\mathcal{B}_Q$ — arcs running to complex infinity, i.e. passage through $1/v=0$ and $1/Q=0$ — is asserted by the authors generally on the restricted manifold, as a result rather than as an open question.)

The mechanism behind part of this pattern is understood. Saleur's exact study of the two-dimensional antiferromagnetic Potts model showed that at Beraha values, transfer-matrix eigenvalues in adjacent sector numbers become equal in magnitude, reflecting the structure of the Temperley–Lieb algebra — the algebra generated by operators $U_i$ obeying $U_i^2=\sqrt{Q}\,U_i$, $U_iU_{i\pm1}U_i=U_i$, and $U_iU_j=U_jU_i$ for $|i-j|\ge 2$, whose representations merge when $q=e^{i\pi/r}$ is a root of unity. Jacobsen and Saleur's later analysis of the boundary chromatic polynomial — the Potts partition function on an annulus whose outer-rim spins see $Q_s$ colors instead of $Q$; the CDBC self-dual strip corresponds to $Q_s=Q-1$ with edge weight $v_e=\pm\sqrt{Q}$ — showed via the representation theory of the boundary Temperley–Lieb (blob) algebra (the boundary variant of the Temperley–Lieb algebra augmented by a marker operator acting on one rim) that dominant eigenvalues in the $L$- and $(L+2)$-leg sectors (loop-model sectors with $L$ non-contracted strands threading the annulus) cross exactly at the positions observed by Chang and Shrock for every finite width, with the number of crossings cut off by the width through $L\le 2W$ and the whole pattern terminating at $Q=4$, the end of the Berker–Kadanoff (massless antiferromagnetic) phase. This explains, at physics level of rigor, the real-axis crossing positions but not the global complex-arc structure, the region count, the dominance assignment away from the real axis, or the odd/even parity of the semi-infinite segment.

Problem Statement

For the $Q$-state Potts model on infinite-length CDBC (self-dual cyclic boundary conditions) strips of the square lattice, $G_{\rm CDBC}[L_y\times L_x]$ with $L_x\to\infty$ (periodic longitudinal boundary conditions plus one external vertex joined to all $L_x$ vertices of one longitudinal side, making the strip graph self-dual), restricted to the self-dual manifold $Q=v^2$:

Notation: on this manifold the partition function has the exact form $Z\big(G_{\rm CDBC}[L_y\times L_x],Q,v\big)=Q\,v^{L_xL_y}\sum_{d=1}^{L_y+1}\sum_j\bar\kappa^{(d)}(Q)\,\big(\bar\lambda_{L_y,d,j}\big)^{L_x}$, where the transfer-matrix eigenvalues $\bar\lambda_{L_y,d,j}=v^{-L_y}\lambda_{L_y,d,j}$ are labeled by the sector number $d\in\{1,\ldots,L_y+1\}$ and an index $j$ running over the eigenvalues of that sector, the amplitudes $\bar\kappa^{(d)}(Q)$ are $L_x$-independent polynomials in $Q$, and the single eigenvalue of sector $d=L_y+1$ is identically $\bar\lambda_{L_y,L_y+1}=1$. Throughout, $\bar\lambda_{L_y,d,\max}$ denotes the eigenvalue of largest magnitude within sector $d$; which member $j$ of the sector attains this maximum depends on the point of the complex plane under consideration.

(a) Prove or disprove the Chang–Shrock conjecture that, for every width $L_y\ge 1$, the accumulation loci $\mathcal{B}_v$ and $\mathcal{B}_Q$ of partition-function zeros in the complex $v$ and $Q$ planes have the following structure:

• in the interval $-2\le v\le -1$, $\mathcal{B}_v$ crosses the negative real axis at $v=-2$ and at $v_{2\ell+1}=-2\cos\big(\pi/(2\ell+1)\big)$ for $1\le\ell\le L_y$, and, in the interval $1\le Q\le 4$, $\mathcal{B}_Q$ crosses the positive real axis at the odd Beraha values $Q_{2\ell+1}=4\cos^2\big(\pi/(2\ell+1)\big)$ for $0\le\ell\le L_y$, where $Q_1=4$;
• the semi-infinite real segments $-\infty\le v\le -2$ and $4\le Q\le\infty$ belong to the locus for odd $L_y$ (no such segment was found for the even widths computed);
• all curves of $\mathcal{B}_v$ meet at $v=-2$ (equivalently, all curves of $\mathcal{B}_Q$ meet at $Q=4$), where every transfer-matrix eigenvalue has unit modulus;
• $\mathcal{B}_v$ divides the $v$ plane into at least $2L_y$ regions and $\mathcal{B}_Q$ divides the $Q$ plane into at least $2L_y$ regions, with the dominant eigenvalue assigned systematically across those regions: one member of sector $d=1$ (denoted $\bar\lambda_{L_y,1,\max}$ there) dominates in the regions extending to complex infinity, a different member of sector $d=1$ dominates in the region containing the interval $4\le Q\le\infty$ (for odd $L_y$, two members of sector $d=1$ are degenerate in magnitude on this interval, giving rise to its line segment on the locus), $\bar\lambda_{L_y,k+1,\max}$ dominates in the region containing the interval $Q_{2k+1}\le Q\le Q_{2k+3}$ for $1\le k\le L_y-1$ (this range is empty for $L_y=1$), and $\bar\lambda_{L_y,L_y+1}=1$ dominates in the region containing the interval $Q_{2L_y+1}\le Q\le Q_1=4$;
• besides the locus, the partition function has an isolated zero at $Q=0$ (proven by the authors for all widths) and, for each $1\le\ell\le L_y+1$, zeros close to the even Beraha value $Q=Q_{2\ell}$, which become isolated accumulation points at these values as $L_x\to\infty$ through the vanishing there of the amplitude of the dominant eigenvalue.

(b) Determine whether, for sufficiently large $L_y$, $\mathcal{B}_Q$ also crosses the real $Q$ axis at points outside the interval $0\le Q\le 4$ (i.e. with $Q<0$ or $Q>4$), for both parities of $L_y$; the only part of this range already asserted in (a) is the odd-$L_y$ semi-infinite segment $4\le Q\le\infty$.

These two items form one coherent objective: a complete structure theorem for the zero-accumulation locus of the self-dual Potts strip at arbitrary width, of which item (a) is the core and item (b) the authors' explicitly flagged open question about the same locus.

References

1. Shu-Chiuan Chang and Robert Shrock, 'Partition Function Zeros of a Restricted Potts Model on Self-Dual Strips of the Square Lattice', International Journal of Modern Physics B 21, 1755–1773 (2007). DOI: 10.1142/S021797920703703x; arXiv:cond-mat/0602178, https://arxiv.org/abs/cond-mat/0602178
2. Shu-Chiuan Chang and Robert Shrock, 'Partition Function Zeros of a Restricted Potts Model on Lattice Strips and Effects of Boundary Conditions', Journal of Physics A: Mathematical and General 39, 10277–10295 (2006). DOI: 10.1088/0305-4470/39/33/002; arXiv:cond-mat/0602440, https://arxiv.org/abs/cond-mat/0602440
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