Melting scenario of the period-3 phase of the hard-square model with diagonal interactions: Lifshitz point location, anisotropic critical exponents, and floating-phase boundaries

The hard-square lattice gas with couplings along the two diagonal directions of the square lattice hosts a period-3 solid that melts either directly in the Huse-Fisher chiral universality class near Baxter's integrable three-state Potts point, or in two steps through an extremely narrow floating phase bounded by a Pokrovsky-Talapov transition on the solid side and a Kosterlitz-Thouless transition on the fluid side. A 2022 corner-transfer-matrix renormalization group study pinned down the gross phase diagram and the conditional exponent set $\bar{\beta}=2/3$, $\alpha=1/3$, $z_{\mathrm{dyn}}=\nu_{x+y}/\nu_{x-y}=3/2$, but could not locate the Lifshitz point where the chiral regime gives way to the two-step regime, could not access the commensurate-direction correlation-length exponent $\nu_{x+y}$ (CTMRG accesses only lattice-axis correlations, which yield $\nu_{x-y}$, and $z_{\mathrm{dyn}}=3/2$ follows only if hyperscaling is assumed), and could not resolve the floating-phase boundaries, in particular $T_{KT}$. The problem of determining the boundary between direct chiral and two-step melting, measuring $\nu_{x+y}$ and $\nu_{x-y}$ independently of hyperscaling, and fixing the floating-phase boundaries in this paradigmatic two-dimensional classical model therefore remains open.

Background

Adsorbed monolayers and, more recently, chains of Rydberg atoms realize commensurate-incommensurate (C-IC) transitions: the melting of a solid commensurate with an underlying lattice into a phase with incommensurate modulation. The hard-square model is a paradigmatic classical lattice gas for this problem. Each site of the square lattice carries $n\in\{0,1\}$ (empty or filled), a hard-core rule forbids two nearest neighbours from being filled simultaneously, and the grand-canonical weight of a configuration is $z^{\sum n_{i}}$ times the Boltzmann factor of the diagonal-interaction Hamiltonian

$$\beta H=-M\sum_{x,y}n_{x,y}n_{x+1,y+1}-L\sum_{x,y}n_{x,y}n_{x+1,y-1},$$

where $z=e^{\mu}$ is the activity (fugacity) and $M,L$ are dimensionless couplings along the two diagonal directions. For $M=L=0$ the pure hard-square lattice gas has an Ising-like ordering transition at $z\simeq 3.7962$ into a period-2 (sublattice) solid. Diagonal interactions enrich this picture with a period-3 ($3\times 1$) solid at intermediate activities, and Baxter's exact solution of an integrable manifold of this model shows that on that manifold the period-3 solid melts at a three-state Potts point while the period-2 melting is Ising-tricritical; Huse's 1983 multicritical-scaling analysis of exactly this lattice gas is the origin of the chiral-melting question. Away from the integrable Potts point the model is not integrable, and the Huse-Fisher domain-wall theory of C-IC melting with unequal wall energies leaves two generic scenarios for melting a period-$p$ solid with $p=3$: (a) a direct continuous transition in a non-conformal chiral universality class, characterized by the product of the incommensurate wave-vector shift $q-q_{0}$ with the correlation length tending to a strictly positive constant, or (b) a two-step melting through a floating phase, a critical incommensurate phase with algebraically decaying correlations, entered from the solid side through a Pokrovsky-Talapov (PT) transition with exponents $(\nu_{x+y},\nu_{x-y},\bar{\beta})=(1,1/2,1/2)$ and left through a Kosterlitz-Thouless (KT) transition with exponentially diverging correlation length. Here $\bar{\beta}$ is defined by $q-q_{0}\sim t^{\bar{\beta}}$ with $t$ the distance to criticality, and $\nu_{x+y}$, $\nu_{x-y}$ are the correlation-length exponents along the two diagonal directions, the natural directions of the domain walls. The point where the floating phase collapses onto the direct chiral transition line is a Lifshitz point; it is distinct from the disorder (Lifshitz) line inside the disordered phase across which the dominant wave-vector leaves commensurarity, an object that is separately locatable.

The one-dimensional quantum analog, hard-core bosons with nearest-neighbour blockade and next-nearest-neighbour repulsion realized by Rydberg-atom chains, has been studied intensively with the density-matrix renormalization group (DMRG), an algorithm for one-dimensional quantum systems: its period-3 phase also melts either chirally near a three-state Potts point or through a narrow floating phase, with floating widths below $10^{-2}$ resolved on both sides of the Potts point. The two-dimensional classical model is the subject of the source study. Nyckees and Mila (2022) contracted its partition-function tensor network with the corner-transfer-matrix renormalization group (CTMRG), an iterative contraction algorithm that builds Baxter's corner transfer matrices at finite bond dimension $\chi$, and mapped the phase diagram in two planes: the plane where the two diagonal couplings are equal in magnitude and opposite in sign (previously studied by Monte Carlo, which found $\bar{\beta}\simeq 0.8$), and the plane with fixed ferromagnetic coupling $L=2$ varying $z$ and $M$, which is close to the one-dimensional quantum model. They established direct chiral melting near the Potts point with $\bar{\beta}\simeq 2/3$ and $\alpha\simeq 1/3$, the two-step PT+KT scenario with a floating phase too narrow to locate its KT boundary $T_{KT}$ at high activity, an Ising transition inside the period-2 phase (becoming first-order beyond a tricritical Ising point in the $L=2$ plane, in agreement with Baxter's prediction), and the disorder line of the fluid. Three limitations of the method define the open problem. First, standard CTMRG yields transfer matrices along the lattice axes only, so correlation lengths along the diagonal (commensurate and incommensurate) directions cannot be measured directly. The lattice-axis correlation length gives $\nu=\min(\nu_{x-y},\nu_{x+y})$, expected to equal $\nu_{x-y}$, so $\nu_{x-y}$ and the wave-vector exponent $\bar{\beta}$ are recoverable from the CTMRG data and the Huse-Fisher relation $\nu_{x-y}=\bar{\beta}$ can be tested rather than assumed; what remains inaccessible is $\nu_{x+y}$, and hence the dynamical exponent $z_{\mathrm{dyn}}=\nu_{x+y}/\nu_{x-y}$. Second, the quoted exponent set is conditional: taking the Huse-Fisher relation $\nu_{x-y}=\bar{\beta}=2/3$ and assuming the hyperscaling relation $\nu_{x+y}+\nu_{x-y}=2-\alpha$ with $\alpha$ keeping its Potts value $1/3$ along the chiral line gives $\nu_{x+y}=1$ and hence the dynamical exponent $z_{\mathrm{dyn}}=\nu_{x+y}/\nu_{x-y}=3/2$, but neither hyperscaling nor the constancy of $\alpha$ has been independently tested. Third, the floating phase is so narrow that its boundaries, and hence the position of the Lifshitz point bounding the chiral regime, remain undetermined; at $z=0.31$, on the low-activity side of the Potts point, the reduced value $\bar{\beta}=0.59$ is equally consistent with a crossover from the PT value $1/2$, so the very existence of a floating phase on that side is unestablished.

Problem Statement

The scientific objective is to complete the characterization of the commensurate melting of the period-3 solid of the two-dimensional classical hard-square model with diagonal interactions, defined by the hard-core nearest-neighbour exclusion, the activity $z$, and

$$\beta H=-M\sum_{x,y}n_{x,y}n_{x+1,y+1}-L\sum_{x,y}n_{x,y}n_{x+1,y-1}.$$

In the parameter planes already mapped by CTMRG (opposite diagonal couplings of equal magnitude, and fixed $L=2$ varying $z$ and $M$), the open question named by the source study is:

1. Regime boundary: locate precisely the Lifshitz point(s) at which the direct chiral (Huse-Fisher) melting of the period-3 phase gives way to the two-step melting through a floating phase, equivalently the point where the floating phase collapses onto the chiral transition line, on both sides of Baxter's integrable three-state Potts point; this settles whether a floating phase also exists on the low-activity side, where the measured $\bar{\beta}=0.59$ at $z=0.31$ could equally be a crossover from the Pokrovsky-Talapov value $1/2$.

Two related characterizations, left open by the same study, frame and support this question without forming additional mandatory deliverables:

2. Anisotropic critical behavior: measure the correlation-length exponents $\nu_{x+y}$ and $\nu_{x-y}$ of the chiral transition, with $\nu_{x-y}$ obtained from the lattice-axis correlation length so that the Huse-Fisher relation $\nu_{x-y}=\bar{\beta}$ can be tested rather than assumed and $\nu_{x+y}$ from correlations along the commensurate diagonal inaccessible to standard CTMRG, and use them to test rather than assume the hyperscaling relation $\nu_{x+y}+\nu_{x-y}=2-\alpha$; this confirms or refutes the conditional dynamical exponent $z_{\mathrm{dyn}}=\nu_{x+y}/\nu_{x-y}=3/2$ and the constancy of $\alpha=1/3$ along the chiral line.

3. Floating-phase boundaries: determine the temperatures $T_{PT}$ and $T_{KT}$ bounding the floating phase and its width as functions of the activity wherever a floating phase exists, with controlled finite-size or finite-bond-dimension extrapolation and quantified error bars; these boundaries are the means by which the Lifshitz point of item 1 is tracked.

The core open question is item 1: the source study names the precise location of the Lifshitz point that separates the chiral transition from the floating phase as its only remaining open issue. Item 3 is instrumental to it, since the Lifshitz point is where the splitting between $T_{PT}$ and $T_{KT}$ shrinks to zero, and item 2 is the critical characterization of the chiral regime whose extent item 1 fixes; neither is an additional mandatory deliverable beyond the regime boundary itself. Any sound method is acceptable, for example transfer matrices oriented along the diagonals, Monte Carlo with directional correlation analysis, rotated or higher-$\chi$ tensor-network geometries, DMRG on the quantum model within its exact scaling-limit mapping, or a combination, provided the model and observable conventions match those of the source study and the discrimination between direct and two-step melting rests on the standard criterion, the behaviour of the product $\xi\,(q-q_{0})$ or an equivalent asymptotic diagnostic; a controlled effective-exponent crossover analysis qualifies as such a diagnostic insofar as it extrapolates the effective exponents toward the asymptotic limit, rather than reading the melting scenario off effective exponents at finite distance from criticality.

References

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