[Paper Review] Sublattice extraordinary-log phase and new special point of the antiferromagnetic Potts model

Zhang Li-ru, Chengxiang Ding+2 more|arXiv (Cornell University)|Jan 21, 2023

Physics of Superconductivity and Magnetism4 citations

TL;DR

This study investigates surface criticality in a 3D antiferromagnetic Potts model with emergent O(2) symmetry, showing that antiferromagnetic next-nearest-neighbor (NNN) surface interactions drive a transition from the extraordinary-log phase to an ordinary phase, and further induce a novel sublattice extraordinary-log phase dominated by sublattice order. The special point between the ordinary phase and this new phase belongs to a previously unreported universality class, distinct from the well-known XY model special transition.

ABSTRACT

We study the surface criticality of a three-dimensional classical antiferromagnetic Potts model, whose bulk critical behaviors belongs to the XY model because of emergent O(2) symmetry. We find that the surface antiferromagnetic next-nearest neighboring interactions can drive the extraordinary-log phase to the ordinary phase, the transition between the two phases belongs to the universality class of the well-known special transition of the XY model. Further strengthening the surface next-nearest neighboring interactions, the extraordinary-log phase reappears, but the main critical behaviors are dominated on the sublattices of the model; the special point between the ordinary phase and the sublattice extraordinary-log phase belongs to a new universality class.

Motivation & Objective

• To explore how surface next-nearest-neighbor (NNN) antiferromagnetic interactions influence surface critical behavior in a 3D antiferromagnetic Potts model with emergent O(2) symmetry.
• To investigate whether tuning surface NNN interactions can stabilize a new type of extraordinary-log phase dominated by sublattice order.
• To determine the universality class of the special transition between the ordinary phase and the newly identified sublattice extraordinary-log phase.
• To clarify the role of symmetry—particularly O(2) and Z2 lattice symmetries—in shaping surface criticality beyond standard XY model behavior.

Proposed method

• Numerical study using large-scale Monte Carlo simulations with system sizes up to L = 128 on a simple cubic lattice.
• Employing a combination of Metropolis local updates and geometric clustering algorithms to efficiently sample critical configurations.
• Mapping discrete Potts spins to continuous O(2) vectors via θi = 2πσi/3 to access O(2) symmetry and critical scaling.
• Analyzing surface squared magnetization ms1², surface magnetic susceptibility χs1, and parallel correlation function C∥(L/2) to extract critical exponents.
• Computing surface structure factor F(k) in momentum space to identify sublattice order patterns and confirm strip-order correlations.
• Applying logarithmic scaling ansatz m² ∝ [ln(L/L₀)]⁻q to extract decay exponents and compare with known universality classes.

Figure 1: Surface phase diagram of the antiferromagnetic Potts model ( 1 )with $T=T_{c}^{\rm bulk}=1.22603$ and $J_{\rm s}=5$ .

Experimental results

Research questions

• RQ1 Can antiferromagnetic NNN surface interactions drive the surface from the extraordinary-log phase to an ordinary phase in the antiferromagnetic Potts model?
• RQ2 Does further strengthening NNN interactions lead to a new type of extraordinary-log phase dominated by sublattice order?
• RQ3 What is the universality class of the special transition between the ordinary phase and the sublattice extraordinary-log phase?
• RQ4 How do the O(2) spin symmetry and Z2 lattice symmetry jointly influence the critical behavior of the new surface phase?
• RQ5 Is the critical decay exponent q of the sublattice magnetization consistent with known universality classes, or does it indicate a new one?

Key findings

• Antiferromagnetic NNN surface interactions drive the system from the extraordinary-log phase to an ordinary phase, with correlation functions showing the same power-law exponent as the XY model’s ordinary phase but decaying faster at short distances.
• Further increasing NNN interaction strength stabilizes a new sublattice extraordinary-log phase, where the main critical behavior is dominated by sublattice order, as confirmed by the surface structure factor showing enhanced peaks at (0, ±π) and (±π, 0).
• The decay exponent q for the parallel correlation function C∥(L/2) and the sublattice magnetization m²_s1A is q = 0.59(3), matching the XY model’s ordinary phase exponent.
• The squared magnetization m²_s1 follows a different logarithmic scaling with exponent q₁ = 1.9(2), indicating distinct critical behavior from the main correlation function.
• The special transition between the ordinary phase and the sublattice extraordinary-log phase belongs to a new universality class, distinct from the well-known special transition of the XY model.
• The surface structure factor F(k) confirms the presence of a finite-size strip order, consistent with sublattice magnetization dominance and supporting the emergence of a new critical phase.

Figure 2: Surface critical behavior of the antiferromagnetic Potts model ( 1 ) at $T=T_{c}^{\rm bulk}=1.22603$ , $J_{s}=5$ : (a) surface susceptibility $\chi_{\rm s1}$ , (b) surface squared magnetization $m^{2}_{\rm s1}$ , (c) surface structure factor $F_{1}$ , and (d) surface specific heat $C_{\rm