Rigorous truncation-error foundations for corner transfer matrix computations of the hard-square lattice gas

In 1980 Baxter, Enting and Tsang introduced the corner transfer matrix (CTM) computation of the hard-square lattice gas — the square-lattice gas with nearest-neighbor exclusion — generating the 24-term high-density series for the sublattice order parameter and estimating the critical activity $z_c = 3.7962 \pm 0.0001$. Their infinite-dimensional CTM equations were solved only approximately, by truncating to the $n$ numerically dominant eigenvalues, with accuracy rules validated empirically; and their series yielded a specific-heat exponent $\alpha = 0.09 \pm 0.05$, in apparent conflict with the Ising value $\alpha = 0$, so the paper left the universality class and the full exponent set unsettled. The physics questions have since been decisively resolved — the transition is in the two-dimensional Ising universality class, confirmed numerically at the $10^{-11}$ level — but the methodological question posed by the truncation itself has never been answered: the CTM equations are exact only in the infinite-dimensional limit, and the method's own authors document that finite truncations are certified only by observed convergence (Baxter 1999) and that the accuracy and complexity claims remain unproved (Chan 2012). There is still no rigorous convergence theory and no explicit error bound for finite-truncation CTM approximations of this nonintegrable model, and the rules certifying which series coefficients and free-energy digits a given truncation delivers remain empirically calibrated. The problem posed here is to supply that missing foundation: prove convergence of $n$-eigenvalued truncated CTM solutions with explicit, computable error bounds for the partition function per site, density, and order parameter of the hard-square lattice gas, and prove or refute the practical series-accuracy rule.

Background

The hard-square lattice gas is the simplest model of particles with excluded volume on a lattice: each site of the square lattice is either empty or occupied by one particle, particles carry an activity (fugacity) $z$, and two nearest-neighbor sites may not both be occupied. At low activity the system is a disordered fluid; above a critical activity $z_c \approx 3.79625$ particles condense preferentially onto one of the two checkerboard sublattices, and the transition is continuous. The natural order parameter is the sublattice density difference $\rho_2 - \rho_1$. Particles can be assigned different activities $z_1$ and $z_2$ on the two sublattices (a staggered activity, the analogue of a magnetic field); its conjugate response is the staggered susceptibility, the quantity a two-variable ($z_1$, $z_2$) series analysis would probe, but no such series for the plain model was published.

Series analysis of this model goes back to Gaunt and Fisher (1965), who obtained $\beta = 0.125 \pm 0.005$. Baxter, Enting and Tsang (1980) attacked the model with corner transfer matrices (CTMs). A CTM is Baxter's device for computing the partition function of a two-dimensional model as the trace of a product of four matrices, one per quadrant of the lattice; each corner matrix $A(a)$ — indexed by the occupation $a \in \{0,1\}$ of a boundary site — and its associated edge matrix $F(a,b)$ satisfy self-consistent equations that are infinite dimensional in the thermodynamic limit. Truncating these matrices to their $n$ numerically dominant eigenvalues turns the equations into a finite fixed-point problem and allows exact generation of power series in $z$ (or $z^{-1}$ in the ordered phase); a discarded-eigenvalue accuracy rule predicts how many leading coefficients are exact. With 24 terms of the high-density series for $\rho_2 - \rho_1$, Baxter, Enting and Tsang estimated $z_c = 3.7962 \pm 0.0001$; their series gave a specific-heat exponent $\alpha = 0.09 \pm 0.05$ (as recounted by Guo and Blöte 2002), seemingly at odds with the Ising value $\alpha = 0$, although the authors noted in proof that Ising exponents were not excluded. The truncation accuracy rules were validated only empirically — the CTM eigenvalues of this model are not known in closed form — as the method's own authors later documented explicitly (Baxter 1999; Chan 2012).

When next-nearest-neighbor (diagonal) interactions are added, the model acquires integrable manifolds — loci in parameter space on which the model becomes exactly solvable: the hard-square gas with diagonal interactions is equivalent to special cases of Baxter's exactly solved hard-hexagon family (Baxter 1980; Baxter–Pearce 1982, 1983; Huse 1983; Pearce 1985 even solved the magnetic, i.e. staggered-activity, version on those manifolds; Dhar 1982 connected it to directed-site animals). The plain nearest-neighbor-exclusion gas, however, is nonintegrable and remains exactly unsolved (Assis et al 2014), so every CTM quantity for it rests on numerics.

The critical-behavior questions left open by the 1980 paper were settled over the following two decades. Rácz (1980) obtained $\nu = 0.999 \pm 0.001$. Kamieniarz and Blöte (1993) and Todo and Suzuki (1996) found Ising scaling dimensions to $10^{-6}$ and $10^{-7}$; Guo and Blöte (2002), using transfer matrices up to width 34 in extended precision, verified the full Ising universal-parameter set — including the correction-to-scaling spectrum — to margins of order $10^{-11}$, determined $\rho_c = 0.3677429990410(3)$, and extracted the magnetic exponent from derivatives with respect to the staggered field — the response the staggered susceptibility measures. Chan (2012), regenerating and extending the low-density series to 92 terms with the CTM renormalization group (CTMRG, the modern iterative descendant of the CTM truncation), found $\alpha = 0.0020(17)$. Monte Carlo work likewise treats the Ising universality class as settled (Fernandes, Arenzon and Levin 2007).

What has never been supplied is the rigorous foundation for the CTM truncation itself. The accuracy rules remain empirical calibrations: Chan (2012) determined how many series coefficients a given truncation size yields by comparing successive truncations; Okunishi, Hieida and Akutsu (1999) established the eigenvalue distribution of CTMs only as an asymptotic numerical observation; Baxter (1999) used the CTM to evaluate the hard-square partition function per site at $z = 1$ to 43 decimal places with no provable error bound; and the only rigorous CTM-derived statements known — the Collatz–Wielandt-certified upper bounds on the hard-square growth constant (the exponential growth rate of the number of independent sets on the lattice) by Chan and Rechnitzer (2018), obtained by feeding a CTM-constructed trial vector into the Collatz–Wielandt formula, which bounds the largest eigenvalue of a nonnegative matrix from above and below through such positive trial vectors — bound a different object and give no error control for the truncated CTM equations. Meanwhile the method family has grown far beyond hard squares (CTMRG and tensor-network contraction are standard tools), all without a truncation-error theory for nonintegrable models. Rigorous correlation-decay results for hard-core models are known only at activities well below $z_c$, so a full theory up to criticality is expected to require new ideas.

Problem Statement

For the hard-square lattice gas on the square lattice (nearest-neighbor exclusion, particle activity $z$), formulated through Baxter's corner-transfer-matrix equations — the self-consistent equations for the corner matrices $A(a)$ and edge matrices $F(a,b)$, indexed by boundary occupation numbers. In the disordered phase these are the translation-invariant CTM equations (Baxter 1978; written in the form used here in Baxter 1999, eq. (8), and Chan 2012, eqs. (1)–(2), with $\kappa = \eta/\xi$), whose infinite-dimensional solution determines the partition function per site $\kappa(z)$ and the density $\rho(z)$. In the ordered (high-density) phase the relevant object is the symmetry-broken, sublattice-resolved CTM system used by Baxter, Enting and Tsang (1980) to generate the $z^{-1}$ series for the sublattice densities $\rho_1$, $\rho_2$ and the order parameter $\rho_2 - \rho_1$; its written form is given in that paper (and in Baxter 1978). In both settings the approximation is the finite truncation introduced by Baxter, Enting and Tsang (1980) that retains the $n$ dominant eigenvalues of the corner matrices. Establish a rigorous foundation for these finite-truncation approximations:

(a) Convergence. For the direct CTM scheme of Baxter, Enting and Tsang (1980), including its symmetry-broken ordered-phase variant, prove (or disprove) that as $n \to \infty$ the $n$-eigenvalued truncated solutions converge to the infinite-dimensional CTM solution, in a norm sufficient to control $\kappa(z)$, $\rho(z)$, and the order parameter $\rho_2 - \rho_1$, at least for activities away from the critical point, in both the disordered (low-$z$) and ordered (high-$z$, $z^{-1}$-expansion) regimes, and characterize how the achievable error grows as $z$ approaches the critical activity $z_c \approx 3.7962$.

(b) Explicit error bounds. Provide a-priori and/or computable a-posteriori bounds on the truncation error of $\kappa(z)$, $\rho(z)$, and the order parameter $\rho_2 - \rho_1$ in terms of $n$ and the retained and discarded eigenvalue data, stating explicitly the range of activities over which the bounds hold, whether their constants are pointwise in $z$ or uniform on subcritical intervals, and whether the discarded-eigenvalue input is an a-priori bound or a quantity computed from the truncated system. In the symmetry-broken ordered phase the corresponding bounds are those on the sublattice densities $\rho_1$ and $\rho_2$.

(c) Series-accuracy rule. In the CTMRG setting in which it is documented (Chan 2012), prove (or refute) the empirical rule that if the largest discarded corner-matrix eigenvalue begins contributing at order $z^{a}$ in the low-activity expansion (here $a$ is that leading series order, not the boundary-occupation index $a$ of $A(a)$), then the truncated computation reproduces the coefficients of $\kappa(z)$ exactly through order $z^{4a-1}$ — a rule at present validated only by comparing successive truncations — stating explicitly which contraction variant is certified: the expression for $\kappa$ in terms of the truncated $A$, $F$, versus the practical variant that substitutes the iteratively updated $A_l$, $F_l$, whose term counts are known only empirically.

The question concerns the nonintegrable nearest-neighbor-exclusion model, whose CTM spectra are not known in closed form; results for integrable relatives (hard hexagons, or the model with diagonal interactions on its integrable manifolds) do not answer it. Items (a) and (b) concern the direct CTM scheme of Baxter, Enting and Tsang (1980), including its symmetry-broken ordered-phase variant; item (c) concerns the CTMRG setting in which the discarded-eigenvalue rule is documented. Since CTMRG (Nishino and Okunishi 1996) postdates and differs from the direct scheme, and has not reproduced its ordered-phase series (Chan 2012), a result established for one scheme does not by itself complete a deliverable formulated for the other. Partial results — for example rigorous bounds valid below a correlation-decay threshold strictly smaller than $z_c$ — are acceptable contributions but must state their range of validity explicitly; a convergence or error-bound result valid on any interval $[0, z_0)$ with $0 < z_0 \le z_c$ resolves the corresponding item over that interval, provided it also names $z_0$ and states that the behavior as $z \to z_c$ is left open.

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