Thermodynamic-limit analytic structure of the non-integrable hard-square lattice gas: zero locus, equimodular gaps, the $z=-1$ conjectures, and a discriminant criterion for integrability

The hard-square model, a lattice gas with nearest-neighbor exclusion on the square lattice, is the canonical non-integrable counterpart of Baxter's exactly solved hard-hexagon model. A 2014 study by Assis, Jacobsen, Jensen, Maillard and McCoy compared the two models through partition-function zeros and transfer-matrix eigenvalues on tori, cylinders and free strips, up to $40\times 40$ sites and 30-site transfer matrices, and found systematic differences: hard-square zeros fill two-dimensional regions rather than lying on curves and appear to converge to a wedge near the critical activity $z_c$; the equimodular curves of the transfer-matrix eigenvalues have gaps, matched by 'glitches' in the density of zeros on the negative real axis; for strip widths $12\le L_h\le 28$ some equimodular curves cross the negative axis to the right of $z=-1$; and the discriminants of the hard-hexagon transfer-matrix characteristic polynomials contain square factors that are absent for hard squares. The paper posed six open questions, none of which has been answered: the fate of the area-filling zeros as the system size $L\to\infty$; the measure and free-energy consequences of the gaps and glitches; whether the right-of-(-1) crossings persist and move rightward; whether discriminant factorization can be proven to characterize integrability; proofs of three conjectures on the generating functions of the partition functions at $z=-1$; and whether hard squares possess additional singularities or natural boundaries beyond the renormalization-group critical points $z_c$ and $z_d$.

Background

The hard-square model is a lattice gas on the square lattice: each site is either occupied by at most one particle or empty, nearest-neighbor sites may not both be occupied (nearest-neighbor exclusion), and a configuration with $n$ particles carries weight $z^n$, where the complex number $z$ is the fugacity (also called activity). The grand partition function $Z_L(z)$, the sum of these weights over all allowed configurations, is equivalently the independence polynomial of the $L\times L$ grid graph. In the thermodynamic limit the free energy per site is expected to be analytic in $z$ except at singular points. Long series expansions (up to 92 terms) locate a transition on the positive real axis at $z_c\approx 3.796$, from a disordered fluid to a fourfold sublattice-ordered solid, in the Ising universality class, and a transition on the negative real axis at $z_d\approx -0.119$ whose leading exponents match those of the Lee-Yang edge singularity, the endpoint of the line of zeros of the ferromagnetic Ising partition function.

The triangular-lattice analogue, the hard-hexagon model, was solved exactly by Baxter in 1980: the free energy is explicitly known and has no singular points other than $z_c^{\rm hh}=(11+5\sqrt5)/2\approx 11.090$, $z_d^{\rm hh}=(11-5\sqrt5)/2\approx -0.090$, and infinity, and the solution rests on an infinite-dimensional algebra: the Bethe ansatz, an exact technique for diagonalizing transfer matrices, and corner transfer matrices, a related exact representation of the partition function built from quadrant decompositions. Hard squares admit no such algebra and are the standard example of a non-integrable lattice gas; since both models arise as the infinite-magnetic-field limit of the Ising model on the square and triangular lattices respectively, the pair is the cleanest available laboratory for asking what integrability contributes to analytic structure. The question has force because in the exactly solved two-dimensional Ising model at zero field the susceptibility was found to have a dense set of singularities coinciding with the accumulation locus of the partition-function zeros, indicating a natural boundary, i.e. a curve across which no analytic continuation is possible - a phenomenon of analysis with no counterpart in the exactly solved free energy.

The instrument of the comparison is the finite-size transfer matrix. For a strip of width $L_h$ with periodic transverse boundary conditions (cylindrical transfer matrix $T^{C}$) or open transverse boundary conditions (free-free transfer matrix $T^{F}$), the partition function is $Z=\operatorname{Tr} T^{L_v}$, and the eigenvalues $\lambda_j(z)$ of $T(z;L_h)$ control the zeros: as the length $L_v$ grows, zeros of $Z$ appear where the eigenvalues of largest modulus conspire. The equimodular curves are the loci in the complex $z$-plane on which two or more of the largest-modulus eigenvalues have equal modulus; zeros of the partition function concentrate near them, and the density of zeros on the negative real segment matches the derivative of the phase difference between the leading eigenvalues. Hard-core lattice gases always have a line segment of zeros on the negative real axis for finite systems. The discriminant of the transfer-matrix characteristic polynomial vanishes where that polynomial has a repeated root, i.e. where two eigenvalues coincide and the transfer matrix in general fails to be diagonalizable, so its factorization properties constrain how eigenvalues can exchange dominance.

The 2014 comparison of Assis, Jacobsen, Jensen, Maillard and McCoy produced the following picture. Hard-hexagon zeros lie on curves converging to $z_c^{\rm hh}$ and the equimodular curves are gap-free; hard-square zeros instead fill two-dimensional regions and, near $z_c$, appear to converge to a wedge, with qualitatively different behavior in the three lattice-size families $L=6n$, $6n+2$ and $6n+4$ (a 'mod 6' effect). On the negative axis the hard-square zero density exhibits glitches, sharp localized depletions, at locations matching gaps in the equimodular curves, with no hard-hexagon analogue. For $12\le L_h\le 28$ some equimodular curves of $T^{C}$ and $T^{F}$ cross the negative real axis to the right of $z=-1$. And the discriminants of the hard-hexagon characteristic polynomials contain square factors, i.e. multiplicity-two roots, for every width studied, which excludes gaps in the equimodular curves and singularities of the maximal eigenvalue on the negative axis, whereas hard squares show no such factorization.

Hard squares also have a special algebraic property at $z=-1$: all eigenvalues of the cylindrical transfer matrix have unit modulus there. Fendley, Schoutens and van Eerten conjectured from the connection with supersymmetric lattice fermions that these eigenvalues are roots of unity, numbers $\omega$ with $\omega^m=1$; Jonsson proved this via a bijection with rhombus tilings, and Baxter computed factorized characteristic polynomials. Building on exact data up to width 29 (cylindrical) and 20 (free), the 2014 paper conjectured further structure for the partition functions at $z=-1$. Here the generating function of a sequence of partition-function values is the power series $\sum_m a_m x^m$ in the strip length, and a direction is called periodic when the boundary condition in that direction is periodic, possibly with a twist. Conjecture 1, that all generating functions along periodic directions are repeating, that is, eventually periodic sequences of integer coefficients; Conjecture 2, that they are logarithmic derivatives of products $\prod_j(1-x^{p_j})^{m_j}$ with integer $p_j$ and $m_j$; and Conjecture 3, that for the torus and cylinder they equal the negative logarithmic derivative of the transfer-matrix characteristic polynomial; together with conjectured least-common-multiple formulas for the periods of these repeating sequences. None of these conjectures is proven, and the paper states that the relation of the equimodularity of all eigenvalues at $z=-1$ to the analytic behavior of the physical free energy is completely unknown.

Problem Statement

Determine the thermodynamic-limit analytic structure of the non-integrable hard-square lattice gas in the complex fugacity plane, and decide whether the contrasts observed against the integrable hard-hexagon model can be converted into proven statements about integrability. Assis, Jacobsen, Jensen, Maillard and McCoy (2014) pose this objective through six linked questions, all of which remain open:

(1) Fate of the area-filling zeros. Finite-size hard-square partition-function zeros fill bounded two-dimensional regions of the $z$-plane and, near $z_c\approx 3.796$, appear to converge to a wedge in families depending on $L \bmod 6$. Determine whether, as $L\to\infty$, the zeros remain area-filling, collapse onto one-dimensional curves, or accumulate on a wedge or other finer structure.

(2) Measure of the gaps and glitches. Determine whether the gaps in the equimodular curves (the loci where two or more of the largest-modulus transfer-matrix eigenvalues have equal modulus) and the corresponding glitches - sharp localized depletions - in the density of zeros on the negative real axis shrink to a set of measure zero (possibly dense), remain of positive measure, or give rise to new non-analytic structures in the thermodynamic free energy and density.

(3) Crossings right of $z=-1$. For $12\le L_h\le 28$ equimodular curves of the cylindrical and free-free transfer matrices $T^{C}(z;L_h)$ and $T^{F}(z;L_h)$ cross the negative real axis to the right of $z=-1$, sometimes more than one curve per width. Determine whether this persists for $L_h>28$ and, if so, whether the rightmost crossing moves to the right in the limit - which would force a re-evaluation of the role of negative-axis zeros in analytic continuation.

(4) Discriminant factorization as an integrability criterion. The discriminants of the hard-hexagon transfer-matrix characteristic polynomials contain square factors (multiplicity-two roots), which excludes gaps in the equimodular curves and singularities of the maximal eigenvalue on the negative axis; hard squares show no such factorization. Prove or disprove that this factorization - which the source suggests should follow in the integrable case by extending reflection-equation (boundary Yang-Baxter) methods, the algebraic framework behind exact solutions with boundaries - guarantees the absence of equimodular-curve gaps for integrable models, and address the bolder converse: whether the absence of such factorization permits gaps in general.

(5) The $z=-1$ conjectures. Prove or refute, for the generating functions (power series in the strip-length variable $x$) of the hard-square partition functions at $z=-1$ in directions with periodic boundary conditions, including twisted periodic boundaries, the following four claims, each with independent standing: Conjecture 1, that these generating functions are repeating, i.e. eventually periodic sequences of integer coefficients; Conjecture 2, that they are logarithmic derivatives of products $\prod_j(1-x^{p_j})^{m_j}$ with integer $p_j$ and $m_j$; Conjecture 3, that the torus and cylinder generating functions equal the negative logarithmic derivative of the characteristic polynomial of $T^{C}$ and $T^{F}$ at $z=-1$ respectively; and the fourth claim, the conjectured least-common-multiple period structure of these sequences. Conjecture 2 is the stronger statement and implies Conjecture 1, so the two do not stand or fall together: a proof of Conjecture 2 settles Conjecture 1 as well, and a refutation of Conjecture 1 refutes Conjecture 2 as well, whereas the converse implications do not hold. Settling any one of the four claims - by proof or refutation - is an independently reportable result, and a full resolution of (5) requires all four to be settled.

(6) Analytic structure beyond the renormalization-group predictions. Determine whether the more complicated finite-size approach of hard squares to the critical activities $z_c\approx 3.796$ and $z_d\approx -0.119$ implies additional singularities or a different analytic structure for the thermodynamic free energy - for example dense natural boundaries - beyond the isolated regular critical points with finitely many algebraic and logarithmic singularities predicted by the renormalization group and conformal field theory.

Questions (1), (2), (3) and (6) concern the zero and singularity structure; question (5) concerns the algebraic mechanism at $z=-1$ that underlies the crossings of (3) and the criterion of (4); question (4) is the proposed synthesis into a characterization of integrability. Full completion requires each question to be settled - by proof, disproof, or a validated determination of the limiting behavior; rigorous or conclusive partial resolutions of individual questions constitute progress on the shared objective.

References

1. M. Assis, J. L. Jacobsen, I. Jensen, J.-M. Maillard, B. M. McCoy, 'Integrability vs non-integrability: Hard hexagons and hard squares compared', J. Phys. A: Math. Theor. 47, 445001 (2014). DOI: 10.1088/1751-8113/47/44/445001; arXiv:1406.5566. https://arxiv.org/abs/1406.5566
2. M. Assis, J. L. Jacobsen, I. Jensen, J.-M. Maillard, B. M. McCoy, 'The hard hexagon partition function for complex fugacity', J. Phys. A: Math. Theor. 46, 445202 (2013). DOI: 10.1088/1751-8113/46/44/445202; arXiv:1306.6389. https://arxiv.org/abs/1306.6389
3. R. J. Baxter, 'Hard hexagons: exact solution', J. Phys. A: Math. Gen. 13, L61-L70 (1980). DOI: 10.1088/0305-4470/13/3/007. https://doi.org/10.1088/0305-4470/13/3/007
4. R. J. Baxter, 'Hard Squares for z = -1', Annals of Combinatorics 15, 185-195 (2011). DOI: 10.1007/s00026-011-0089-2; arXiv:0709.4324. https://arxiv.org/abs/0709.4324
5. P. Fendley, K. Schoutens, H. van Eerten, 'Hard squares with negative activity', J. Phys. A: Math. Gen. 38, 315-322 (2005). DOI: 10.1088/0305-4470/38/2/002. https://doi.org/10.1088/0305-4470/38/2/002
6. J. Jonsson, 'Hard Squares with Negative Activity and Rhombus Tilings of the Plane', Electron. J. Combin. 13(1), #R67 (2006). DOI: 10.37236/1093. https://doi.org/10.37236/1093
7. I. Jauslin, J. L. Lebowitz, 'High-Fugacity Expansion, Lee-Yang Zeros, and Order-Disorder Transitions in Hard-Core Lattice Systems', Commun. Math. Phys. 364, 655-682 (2018). DOI: 10.1007/s00220-018-3269-7; arXiv:1708.01912. https://arxiv.org/abs/1708.01912
8. M. Assis, J. L. Jacobsen, I. Jensen, J.-M. Maillard, B. M. McCoy, 'Analyticity of the Ising susceptibility: an interpretation', J. Phys. A: Math. Theor. 50, 365203 (2017). DOI: 10.1088/1751-8121/aa8128; arXiv:1705.02541. https://arxiv.org/abs/1705.02541
9. J. L. Jacobsen, J. Salas, C. R. Scullard, 'Phase diagram of the triangular-lattice Potts antiferromagnet', J. Phys. A: Math. Theor. 50, 345002 (2017). DOI: 10.1088/1751-8121/aa778f; arXiv:1702.02006. https://arxiv.org/abs/1702.02006
10. T. Aslyamov, I. Akhatov, 'Zeros of partition functions in the NPT-ensemble', Phys. Rev. E 100, 052118 (2019). DOI: 10.1103/PhysRevE.100.052118; arXiv:1811.03371. https://arxiv.org/abs/1811.03371
11. D. Dhar, R. Rajesh, A. Kumar, 'Phase transitions in systems of particles with only hard-core interactions', EPL 143, 61001 (2023). DOI: 10.1209/0295-5075/acf6dc; arXiv:2307.05597. https://arxiv.org/abs/2307.05597
12. X. Lyu, 'Lattice and PT symmetries in tensor-network renormalization group: Case study of a hard-square lattice gas model', Phys. Rev. B 114, 034111 (2026). DOI: 10.1103/vl23-xsyr; arXiv:2603.25492. https://arxiv.org/abs/2603.25492
13. S. Sarma, 'A numerical study of the zeros of the grand partition function of k-mers on strips of width k', J. Phys. A: Math. Theor. 58, 165001 (2025). DOI: 10.1088/1751-8121/adc9e8; arXiv:2409.07744. https://arxiv.org/abs/2409.07744
14. A. Haldar, O. Tavakol, H. Ma, T. Scaffidi, 'Hidden Critical Points in the Two-Dimensional O(n>2) Model: Exact Numerical Study of a Complex Conformal Field Theory', Phys. Rev. Lett. 131, 131601 (2023). DOI: 10.1103/PhysRevLett.131.131601; arXiv:2303.02171. https://arxiv.org/abs/2303.02171
15. M. V. Tamm, M. Dudka, N. Pospelov, G. Oshanin, S. Nechaev, 'From steady-state TASEP model with open boundaries to 1D Ising model at negative fugacity', J. Stat. Mech. 2022, 033201 (2022). DOI: 10.1088/1742-5468/ac52a5; arXiv:2112.04377. https://arxiv.org/abs/2112.04377
16. P. A. Pearce, 'Hard hexagons, hard squares and hard mathematics', J. Phys. A: Math. Theor. 49, 411003 (2016). DOI: 10.1088/1751-8113/49/41/411003. https://doi.org/10.1088/1751-8113/49/41/411003