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Phase-space networks and connectivity of the kagome antiferromagnet

Brandon B. Le, Seung-Hun Lee, Gia-Wei Chern

TL;DR

The paper investigates the phase-space geometry of the coplanar ground-state manifold in the kagome Heisenberg antiferromagnet by constructing phase-space networks where nodes are coplanar configurations and edges correspond to weathervane-loop transitions. It contrasts the full network (all loop updates) with a short-loop subnetwork restricted to elementary six-spin moves, revealing a Gaussian connectivity distribution with $k^* \sim L^{2}$ and fractal-phase-space structure ($d_B \approx 3.15$) for the short-loop case, while including longer loops introduces nonlocal shortcuts and destroys fractality. Spectrally, short-loop networks exhibit Gaussian densities, whereas full networks show suppressed edge weight due to competing loop scales. These results connect microscopic energetic constraints to global network geometry, offering insight into slow dynamics, trapping, and ergodicity breaking in constrained frustrated magnets and providing a general framework for constraint-induced phase-space structure.

Abstract

We study the coplanar ground-state manifold of the kagome Heisenberg antiferromagnet using a phase-space network representation, in which nodes correspond to coplanar ground states and edges represent transitions generated by weathervane loop rotations. In the coplanar manifold, each configuration can be mapped to a three-coloring problem on the dual honeycomb lattice, where a weathervane mode corresponds to a closed loop of two alternating colors. By comparing networks that include all weathervane loops with networks restricted to elementary six-spin loops, we examine how energetic constraints shape phase-space structure. We find that connectivity distributions are sharply peaked in large systems, while restrictions to short loops reduce typical connectivity. Spectral properties further distinguish the two cases, with short-loop networks exhibiting Gaussian spectra and full networks displaying non-Gaussian features associated with correlated loop updates. Finally, a box-counting analysis reveals distinct fractal properties of the two networks, demonstrating how energetic constraints control the global geometry of configuration space. These results show that the hierarchy of weathervane loop rotations provides a direct link between microscopic constraints and emergent phase-space geometry in a frustrated magnet.

Phase-space networks and connectivity of the kagome antiferromagnet

TL;DR

The paper investigates the phase-space geometry of the coplanar ground-state manifold in the kagome Heisenberg antiferromagnet by constructing phase-space networks where nodes are coplanar configurations and edges correspond to weathervane-loop transitions. It contrasts the full network (all loop updates) with a short-loop subnetwork restricted to elementary six-spin moves, revealing a Gaussian connectivity distribution with and fractal-phase-space structure () for the short-loop case, while including longer loops introduces nonlocal shortcuts and destroys fractality. Spectrally, short-loop networks exhibit Gaussian densities, whereas full networks show suppressed edge weight due to competing loop scales. These results connect microscopic energetic constraints to global network geometry, offering insight into slow dynamics, trapping, and ergodicity breaking in constrained frustrated magnets and providing a general framework for constraint-induced phase-space structure.

Abstract

We study the coplanar ground-state manifold of the kagome Heisenberg antiferromagnet using a phase-space network representation, in which nodes correspond to coplanar ground states and edges represent transitions generated by weathervane loop rotations. In the coplanar manifold, each configuration can be mapped to a three-coloring problem on the dual honeycomb lattice, where a weathervane mode corresponds to a closed loop of two alternating colors. By comparing networks that include all weathervane loops with networks restricted to elementary six-spin loops, we examine how energetic constraints shape phase-space structure. We find that connectivity distributions are sharply peaked in large systems, while restrictions to short loops reduce typical connectivity. Spectral properties further distinguish the two cases, with short-loop networks exhibiting Gaussian spectra and full networks displaying non-Gaussian features associated with correlated loop updates. Finally, a box-counting analysis reveals distinct fractal properties of the two networks, demonstrating how energetic constraints control the global geometry of configuration space. These results show that the hierarchy of weathervane loop rotations provides a direct link between microscopic constraints and emergent phase-space geometry in a frustrated magnet.
Paper Structure (8 sections, 6 equations, 8 figures)

This paper contains 8 sections, 6 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Kagome Phase-Space Networks
  3. Phase-Space Connectivity
  4. Exact Connectivity Distributions
  5. Connectivity Distributions of Large Networks
  6. Spectral Densities
  7. Fractal Properties
  8. Conclusions

Figures (8)

  • Figure 1: Three equivalent representations of a random coplanar ground state of a $6\times 6$ kagome lattice with periodic boundary conditions. (a) A coplanar spin configuration on the kagome lattice, with arrows indicating spin directions separated by $120^\circ$. (b) The same state represented as an antiferromagnetic three-state Potts configuration on the kagome lattice, where each site is assigned one of three labels $\{R, G, B\}$ corresponding to the coplanar spin orientations. (c) The associated three-coloring problem on the dual honeycomb lattice: each kagome site maps to an edge of the honeycomb lattice, and the Potts label of that site determines the color of the corresponding dual edge.
  • Figure 2: Examples of coplanar ground states of a $6\times 6$ kagome lattice with periodic boundary conditions, shown in the three-state Potts representation where each color corresponds to one of the three spin orientations separated by $120^\circ$. (a) A $\sqrt{3}\times\sqrt{3}$ state. (b) A $q=0$ state. (c) A representative disordered coplanar configuration, with an elementary six-spin weathervane loop (purple) and a longer weathervane loop (orange) highlighted.
  • Figure 3: (a) Full coplanar ground-state phase-space network and (b) short-loop network for a $6\times 3$ kagome lattice. Each node represents a coplanar ground-state configuration, and each edge represents a weathervane rotation. The color of an edge corresponds to its associated barrier energy, and the size of a node is indicative of its degree.
  • Figure 4: Exact connectivity distributions for (a) a $6\times 3$ kagome lattice and (b) a $9\times 3$ kagome lattice. The number of nodes in the full (blue) and short-loop (green) network is plotted for each degree value.
  • Figure 5: Connectivity distributions for the full (blue) and short-loop (green) networks of (a) $6\times 6$, (b) $15\times 15$, (c) $30\times 30$, and (d) $45\times 45$ kagome lattices. Each data point represents the probability $P(k)$ of choosing a random configuration from the phase-space network that contains $k$ flippable weathervane loops. The data is fit with a Gaussian curve.
  • ...and 3 more figures