The 2x2 hard-square lattice gas on the square lattice (equivalently the 2-NN exclusion model, in which each particle excludes its nearest and next-nearest neighbors) undergoes a continuous transition from a disordered fluid to a fourfold-degenerate columnar phase at a critical activity z_c of about 97.5 (critical density rho_c of about 0.932) that is known only from Monte Carlo simulation; unlike the hard hexagon model it has no exact solution. Mandal, Nath and Rajesh (J. Stat. Mech. 2017) estimated z_c = 105.35 by computing, within an uncontrolled pairwise approximation, the interfacial tension between two differently ordered columnar phases and locating the activity where it vanishes. The open problem is to turn that estimate into a controlled determination: quantify or bound the error of the pairwise scheme (including its defect-activity and overhang-counting choices) and validate the scheme on related columnar-ordering models. No exact solution and no error control for the pairwise scheme are known.

Background

The model considered here is the 2x2 hard-square lattice gas: each particle occupies a 2x2 block of sites of the square lattice, particles interact only through excluded volume (blocks may touch but not overlap), and every particle carries an activity z (the fugacity weighting each configuration in the grand-canonical partition function). It is equivalent to the 2-NN exclusion model, in which occupying a site forbids occupation of its eight nearest and next-nearest-neighbor sites. At high activity the model breaks translational symmetry into a columnar (smectic-like, i.e. layered) phase: the lower-left corners ('heads') of the squares preferentially occupy either the even or the odd rows, or the even or the odd columns, giving four symmetry-related ordered phases. Monte Carlo studies locate the continuous disordered-to-columnar transition at critical activity zc≈97.5z_c\approx97.5 and critical density ρc≈0.932\rho_c\approx0.932 (density = covered area fraction), with critical behavior in the Ashkin-Teller universality class, a family of critical points realizable as two coupled Ising models that interpolates between Ising and four-state Potts criticality (Fernandes, Arenzon and Levin 2007; Feng, Blöte and Nienhuis 2011; Ramola, Damle and Dhar 2015). Unlike the hard hexagon model, which Baxter solved exactly in 1980, the 2x2 hard-square model is not exactly solvable, and its critical parameters are known only numerically.

Analytic estimates of zcz_c have a long and mostly disappointing history. High-activity perturbation expansions (Ramola and Dhar 2012: zc=14.86z_c=14.86 at second order), Flory-type mean-field theory, and density functional theory underestimate zcz_c by nearly an order of magnitude. Estimates based on the interfacial tension have been more successful. The interfacial tension σ\sigma between two differently ordered columnar phases is defined through the ratio of partition functions with and without an interface, e−σNy=∑IZ(I)/Z(0)e^{-\sigma N_y}=\sum_{\mathcal{I}}Z^{(\mathcal{I})}/Z^{(0)} for a lattice of width NyN_y; near a continuous transition the tension between ordered domains must vanish, so the activity at which a computed σ(z)\sigma(z) vanishes estimates zcz_c (Baxter and Pearce computed exactly such a tension for the hard hexagon model in 1982). Slotte (1983), using the mapping of this model to an Ising model with first- and second-neighbor interactions, obtained zc=135.63z_c=135.63. Nath, Dhar and Rajesh (2016) developed the defect-overhang expansion: defects are squares of the wrong type inside an ordered domain, and overhangs are excursions of the interface that fold back on themselves. For the 2x2 case the zero-defect, one-defect, and height-one-overhang levels give zc=48.25z_c=48.25, 52.4952.49, and 54.8754.87 respectively, while the two-defect calculation appeared intractable.

The source paper, Mandal, Nath and Rajesh (2017), introduced a pairwise approximation analogous to the pairwise closures of liquid-state theory: the partition function of each domain is factorized into a product of partition functions of 'tracks', strips of four consecutive lattice rows with boundary conditions fixed by the interface, each computed exactly by transfer matrices. This allows multiple defects and effective overhangs (defects adjacent to the interface are counted as overhangs), but the interface is constrained to be a directed walk (monotone, with no upward steps), and the activity of defects is set equal to the particle activity, zD=zz_D=z. The vanishing of the computed tension gives zc=105.35z_c=105.35 and ρc=0.947\rho_c=0.947, in reasonable agreement with Monte Carlo; however, a variant of the counting rules that excludes overhangs gives zc=43.28z_c=43.28, less than half the value. The authors conclude from an entropy-balance argument - the entropy of the system with an interface decreases on excluding overhangs while that of the system without an interface is unchanged - that the presence of overhangs is important for the calculation of the interfacial tension, but this argument has not been turned into a controlled justification that distinguishes the two counting rules. Crucially, the scheme is uncontrolled: it has no small parameter, no error estimate, and no proof that successive refinements - wider tracks (longer-range pair correlations), explicit clusters of more than two defects, or non-directed interfaces - converge toward the true critical parameters.

Problem Statement

The problem, as left open by Mandal, Nath and Rajesh (2017), is to convert the uncontrolled interfacial-tension estimate of the disordered-to-columnar transition of the 2x2 hard-square (2-NN exclusion) lattice gas on the square lattice into a controlled determination of the critical parameters zcz_c and ρc\rho_c. Concretely, for the model defined by the grand-canonical partition function of non-overlapping 2×22\times2 squares with activity zz:

  1. Quantify or rigorously bound the error of the pairwise interfacial-tension estimate zc=105.35z_c=105.35, ρc=0.947\rho_c=0.947 - in particular the effect of the choice zD=zz_D=z for the defect activity and of the microscopic rules by which defects are counted as overhangs. The variant that excludes overhangs gives zc=43.28z_c=43.28, a spread of more than a factor of two; the source argues on entropy-balance grounds that overhangs must be included, but no controlled justification currently distinguishes the two counting rules, and the reliability of the estimate cannot at present be judged without calibrating against the Monte Carlo answer.

  2. Determine whether the same scheme transfers to other lattice gases that order columnarly, taking as the concrete test cases mixtures of hard squares and dimers and systems of d×2d\times2 hard rectangles - that is, whether it yields reliable estimates of their disordered-columnar phase boundaries, where 'reliable' denotes agreement with the available Monte Carlo phase boundaries. These phase boundaries have so far been estimated only with the same uncontrolled machinery.

References

  1. Dipanjan Mandal, Trisha Nath, R. Rajesh, 'Estimating the critical parameters of the hard square lattice gas model', Journal of Statistical Mechanics (2017) 043201. DOI: 10.1088/1742-5468/aa650e, arXiv:1702.02332, https://doi.org/10.1088/1742-5468/aa650e
  2. Trisha Nath, Deepak Dhar, R. Rajesh, 'Stability of columnar order in assemblies of hard rectangles or squares', EPL 114 (2016) 10003. DOI: 10.1209/0295-5075/114/10003, arXiv:1601.02198, https://doi.org/10.1209/0295-5075/114/10003
  3. R. J. Baxter, 'Hard hexagons: exact solution', Journal of Physics A: Mathematical and General 13 (1980) L61-L70. DOI: 10.1088/0305-4470/13/3/007, https://doi.org/10.1088/0305-4470/13/3/007
  4. R. J. Baxter, P. A. Pearce, 'Hard hexagons: interfacial tension and correlation length', Journal of Physics A: Mathematical and General 15 (1982) 897-910. DOI: 10.1088/0305-4470/15/3/027, https://doi.org/10.1088/0305-4470/15/3/027
  5. P. A. Slotte, 'Phase diagram of the square-lattice Ising model with first- and second-neighbour interactions', Journal of Physics C: Solid State Physics 16 (1983) 2935-2951. DOI: 10.1088/0022-3719/16/15/015, https://doi.org/10.1088/0022-3719/16/15/015
  6. Heitor C. Marques Fernandes, Jeferson J. Arenzon, Yan Levin, 'Monte Carlo simulations of two-dimensional hard core lattice gases', The Journal of Chemical Physics 126 (2007) 114508. DOI: 10.1063/1.2539141, https://doi.org/10.1063/1.2539141
  7. Xiaomei Feng, H. W. J. Blöte, B. Nienhuis, 'Lattice gas with nearest- and next-to-nearest-neighbor exclusion', Physical Review E 83 (2011) 061153. DOI: 10.1103/PhysRevE.83.061153, https://doi.org/10.1103/PhysRevE.83.061153
  8. K. Ramola, K. Damle, D. Dhar, 'Columnar Order and Ashkin-Teller Criticality in Mixtures of Hard Squares and Dimers', Physical Review Letters 114 (2015) 190601. DOI: 10.1103/PhysRevLett.114.190601, https://doi.org/10.1103/PhysRevLett.114.190601
  9. K. Ramola, D. Dhar, 'High-activity perturbation expansion for the hard square lattice gas', Physical Review E 86 (2012) 031135. DOI: 10.1103/PhysRevE.86.031135, https://doi.org/10.1103/PhysRevE.86.031135
  10. Dipanjan Mandal, R. Rajesh, 'Columnar-disorder phase boundary in a mixture of hard squares and dimers', Physical Review E 96 (2017) 012140. DOI: 10.1103/PhysRevE.96.012140, arXiv:1704.08052, https://doi.org/10.1103/PhysRevE.96.012140
  11. Nathann T. Rodrigues, Tiago J. Oliveira, 'Husimi-lattice solutions and the coherent-anomaly-method analysis for hard-square lattice gases', Physical Review E 103 (2021) 032153. DOI: 10.1103/PhysRevE.103.032153, arXiv:2103.00965, https://doi.org/10.1103/PhysRevE.103.032153
  12. A. A. A. Jaleel, J. E. Thomas, D. Mandal, Sumedha, R. Rajesh, 'Rejection-free cluster Wang-Landau algorithm for hard-core lattice gases', Physical Review E 104 (2021) 045310. DOI: 10.1103/PhysRevE.104.045310, arXiv:2108.01402, https://doi.org/10.1103/PhysRevE.104.045310
  13. J. Kundu, R. Rajesh, 'Asymptotic behavior of the isotropic-nematic and nematic-columnar phase boundaries for the system of hard rectangles on a square lattice', Physical Review E 91 (2015) 012105. DOI: 10.1103/PhysRevE.91.012105, https://doi.org/10.1103/PhysRevE.91.012105