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Physical review. B, Condensed matter

Structure factors associated with the melting of a (31) ordered phase on a centered-rectangular lattice gas: Effective scaling in a three-state chiral-clock-like model

Norman C. Bartelt, T. L. Einstein, Lyle D. Roelofs

Published Apr 1, 1987• 23 citations
Publisher

Abstract

By computing the structure factor through the melting transition of a commensurate (3\ifmmode\times\else\texttimes\fi{}1) phase in a version of Baxter's generalized hard hexagon model, we simulate what might be observed in diffraction experiments (especially those using low-energy electrons) on chemisorption systems which have similar Landau-Ginzburg-Wilson Hamiltonians. In the commensurate and disordered phases we observe that the shift of the peak of the critical scattering from the commensurate position is roughly proportional to the inverse correlation length in the reduced temperature range 0.015--0.15. The proportionality constant appears to depend on the position of the transition on the phase boundary (i.e., on the chemical potential). We see no sign of an incommensurate floating phase between the disordered and commensurate phases. Effective critical exponents are consistent with those expected from the three-state Potts model. To the precision of this calculation (a few percent), the structure factor scales over approximately 10% of the Brillouin-zone area.

DOI
10.1103/physrevb.35.4812

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Journal of Physics A Mathematical and General•Dec 7, 1992•4 citations
By computing the structure factor through the melting transition of a commensurate (3\ifmmode\times\else\texttimes\fi{}1) phase in a version of Baxter's generalized hard hexagon model, we simulate what might be observed in diffraction experiments (especially those using low-energy electrons) on chemisorption systems which have similar Landau-Ginzburg-Wilson Hamiltonians. In the commensurate and disordered phases we observe that the shift of the peak of the critical scattering from the commensurate position is roughly proportional to the inverse correlation length in the reduced temperature range 0.015--0.15. The proportionality constant appears to depend on the position of the transition on the phase boundary (i.e., on the chemical potential). We see no sign of an incommensurate floating phase between the disordered and commensurate phases. Effective critical exponents are consistent with those expected from the three-state Potts model. To the precision of this calculation (a few percent), the structure factor scales over approximately 10% of the Brillouin-zone area.
We investigate the classical chiral Ashkin-Teller model on a square lattice with the corner transfer matrix renormalization group algorithm. We show that the melting of the period-4 phase in the presence of a chiral perturbation takes different forms depending on the coefficient of the four-spin term in the Ashkin-Teller model. Close to the clock limit of two decoupled Ising models, the system undergoes a two-step commensurate-incommensurate transition as soon as the chirality is introduced, with an intermediate critical floating phase bounded by a Kosterlitz-Thouless transition at high temperature and a Pokrovsky-Talapov transition at low temperature. By contrast, close to the four-states Potts model, we argue for the existence of a unique commensurate-incommensurate transition that belongs to the chiral universality class, and for the presence of a Lifshitz point where the ordered, disordered, and floating phases meet. Finally, we map the whole phase diagram, which turns out to be in qualitative agreement with the 40-year-old prediction by Huse and Fisher.
The scaling behaviour of the hard square lattice gas with diagonal interactions is examined on the basis of Baxter's (1980, 1981, 1982) recent exact solution. It is demonstrated that all the corrections to scaling found may be accounted for in terms of a single irrelevant scaling field that scales simply as a length. The lattice cut-off is proposed as the source of these corrections, although other sources cannot be ruled out. The melting of the 3*1 commensurate ordered phase in this model is examined in a more general context in which the line of Potts critical points found by Baxter appears to be a line of multicritical points on which the effective uniaxially chiral symmetry breaking field vanishes.
We present a transfer-matrix scaling study of a hard-square lattice-gas model with an exclusion shell extending up to second-nearest neighbors, and with a uniaxial finite repulsion active at third-nearest-neighbor separations. The phase diagram, as a function of temperature and chemical potential, shows three distinct regions bounded by two merging critical lines. At low temperature the system undergoes a transition to an ordered phase whose c(4\ifmmode\times\else\texttimes\fi{}2) structure is characterized---within this calculation---by the asymptotic degeneracy of the four largest eigenvalues of the transfer matrix. At high temperature this phase is reached from the disordered state via an intermediate, degenerate phase which, on average, exhibits long-range order in one dimension only. These two transition regimes are found to be separated by a multicritical point which is likely to arise from the competition between interaction anisotropy and thermal-disordering effects. Evidence of anisotropic critical behavior is found in the correlation lengths exponent ratios, ${\ensuremath{\nu}}_{?}$/${\ensuremath{\nu}}_{\ensuremath{\perp}}$, which appear to vary continuously along both transition lines. However, the accuracy of the predictions we make for the thermodynamics of the model is still limited by sensitive finite-size effects which are presumably emphasized by the relatively extended range of the interactions.
Using the corner-transfer matrix renormalization group approach, we revisit the three-state chiral Potts model on the square lattice, a model proposed in the eighties to describe commensurate-incommensurate transitions at surfaces, and with direct relevance to recent experiments on chains of Rydberg atoms. This model was suggested by Huse and Fisher to have a chiral transition in the vicinity of the Potts point, a possibility that turned out to be very difficult to definitely establish or refute numerically. Our results confirm that the transition changes character at a Lifshitz point that separates a line of Pokrosky-Talapov transitions far enough from the Potts point from a line of direct continuous order-disorder transition close to it. Thanks to the accuracy of the numerical results, we have been able to base the analysis entirely on effective exponents to deal with the crossovers that have hampered previous numerical investigations. The emerging picture is that of a new universality class with exponents that do not change between the Potts point and the Lifshitz point, and that appear to be consistent with those of a self-dual version of the model, namely correlation lengths exponents νx=2/3 in the direction of the asymmetry and νy=1 perpendicular to it, an incommensurability exponent β¯=2/3, a specific heat exponent that keeps the value α=1/3 of the three-state Potts model, and a dynamical exponent z=3/2. These results are in excellent agreement with experimental results obtained on reconstructed surfaces in the nineties, and shed light on recent Kibble-Zurek experiments on the period-3 phase of chains of Rydberg atoms.
The helical Potts model, a generalization of the qqq-state Potts model relevant to the commensurate-incommensurate transition, is studied on the square lattice. Close to zero temperature this model can be mapped onto the six-vertex Baxter model in direct fields. This mapping indicates the presence of an incommensurate phase, characterized by algebraic decay of correlations with an exponent $\ensuremath{\eta}\ensuremath{\rightarrow}\frac{3}{{q}^{2}}$ as $T\ensuremath{\rightarrow}0$. A Lifshitz-like point occurs when the ferromagnetic, modulated, and disordered phases meet.
The authors have performed extensive Monte Carlo simulations of two-dimensional layered Ising lattice gases with (m-1)-neighbour exclusion in the x direction and pair attraction in the y direction. They studied the specific heat of the models in systems with different rectangular shapes and the spin-spin correlation function in the x direction near to the critical point. For the m=3 model both the thermal and magnetic exponents were estimated to be close to the known values of the three-state Potts model. Estimates for the critical exponents of the m=4 model are consistent with a commensurate-incommensurate transition; however they find no independent evidence for the existence of a floating phase.