Phase transitions and crtical exponents in the six-vertex model on kagome lattices

TL;DR

This study analyzes the six-vertex model on the kagome lattice using directed-loop Monte Carlo to map both magnetization- and vortex-driven phase diagrams. It identifies four vortex-lattice phases with distinct winding-number patterns and demonstrates two BKT-type melting transitions, along with a nonuniversal FM–PM critical line where the thermal exponent $y_t$ varies with vertex weights; notably, a new transition between VL-H and VL-T occurs with $y_t=1.340(3)$. The work also establishes an Ising-type universality at $a=0.3$ and shows the absence of fractional vortices in this setup, enriching the understanding of ice-like physics on complex lattices. The results have potential relevance for experimental kagome ice realizations (e.g., colloids, nanomagnets) and motivate future extensions to more general vertex models and advanced simulation techniques.

Abstract

Inspired by the experimental realization of direct kagome spin ice [Yue et al., Nat. Nanotechnol. 19, 1101 (2024)], the theoretical six-vertex model on the kagome lattice is systematically simulated using the directed loop Monte Carlo method. Four distinct vortex lattice phases are identified: (i) antiferromagnetic leg states and vortex lattice order on both triangular and honeycomb faces, with a winding number $k=1$. (ii) ferromagnetic leg states and vortex lattice order on both types of faces, with $k=-2$ on the honeycomb faces and $k=1$ on the triangular faces. (iii) paramagnetic leg states and vortex lattice order on the triangular faces with $k=1$; and (iv) paramagnetic leg states and vortex lattice order on the honeycomb faces with $k=1$. As for ferromagnetic to different types of paramagnetic phase, besides the Ising universality class with $y_t=1$, varying critical exponents have also been found with different values of vertex weights. The transition between the third type and fourth type of vortex lattice phases occurs with the new exponent $y_t=1.340(3)$. The third and fourth types of the vortex lattice phase to the vortex disorder phase are found to be of the Berezinskii-Kosterlitz-Thouless type. These findings contribute to the search for and understanding of ice on complex lattices.

Figures (19)

• Figure 1: The colloidal particle-based ice (a) employs elongated optical traps to confine colloidal particles (b) in an ASI configuration on a Kagome lattice.
• Figure 2: Configuration of 6V model. (a1)-(c2) show six types of vertices and their leg configurations; (d1)-(d3) show three types of vertex positions and their leg labels; (e1) shows a possible water ice configuration;(e2) shows $2\times2$ vertex configurations. The unit cells have three vertices and an upright triangular shape $\mathlarger{\triangle}$.
• Figure 3: Two types of loop close methods. If one type of loop closure is missing, it will result in longer loops, causing the code to slow down. Similar to the process in quantum Monte Carlo simulations, a linked list is also constructed. For instance, leg 3 and leg 8 are interlinked with each other.
• Figure 4: Equation groups of the directed loop probabilities $a_{i,j}$ (a) before sorting of the vertex weight $W_i, i=2, 3, 6$ and (b) after sorting.
• Figure 5: The vortex configurations on the triangular and hexagonal faces are marked with the winding number $k$ in red numbers beside them, and the rotation direction is indicated by colored arrows. (a)-(d) $k=1$, (e)-(f) $k=1$, (g)-(h) $k=-2$.