Kitaev model in regular hyperbolic tilings

TL;DR

This work analyzes the Kitaev model on regular hyperbolic tilings ${\{p,3\}}$ by evaluating two colorings (Kekulé for $p$ even and Kitaev for $p\equiv 0\pmod{3}$) and mapping to Majorana fermions with a $\mathbb{Z}_2$ gauge structure. Combining exact diagonalization, a ground-state flux conjecture, and analytic isolated-dimer limits, the authors characterize phase diagrams that interpolate between the Euclidean honeycomb and the trivalent Bethe lattice as $p$ grows. They derive low-energy effective Hamiltonians in the isolated-dimer limit, confirming toric-code phases with calculable vison gaps, and obtain exact Bethe-lattice phase boundaries by a Green’s-function approach, clarifying how loops and flux sectors shape the topological phases. The results illuminate how lattice curvature and coloring determine gapless regions and topological order, offering a unified view that connects hyperbolic, Euclidean, and Bethe-lattice Kitaev physics and suggesting directions for exploring non-Abelian anyons and random colorings.

Abstract

We study the Kitaev model on regular hyperbolic trivalent tilings. Depending on the length $p$ of the elementary polygons, we examine two distinct tri-colorings of the tiling. Using a recent conjecture on the ground-state flux sector, we compute the phase diagram via exact diagonalizations and derive analytical expressions for the effective Hamiltonians in the isolated-dimer limit which are valid for all values of $p$. Our results interpolate between the Euclidean honeycomb lattice and the trivalent Bethe lattice ($p=\infty$) for which we derive the exact solution of the phase boundaries.

Figures (10)

• Figure 1: A piece of the $\{8,3\}$ tiling in the Poincaré disk conformal representation with Kekulé coloring, in which each polygon is only made up of two colors. The red, green, and blue links are associated with $J_x$, $J_y$, and $J_z$ couplings, respectively.
• Figure 2: A piece of the $\{9,3\}$ tiling in the Poincaré disk conformal representation with Kitaev coloring, in which each polygon is a periodic arrangement of the three colors red, green, and blue). The color code is the same as in Fig. \ref{['fig:disk_8']}.
• Figure 3: A piece of the $\{\infty,3\}$ tiling (trivalent Bethe lattice) in the Poincaré disk conformal representation, for which the three-edge coloring is irrelevant. The color code is the same as in Fig. \ref{['fig:disk_8']}.
• Figure 4: The fermion gap $\Delta_{\rm f}$ for the Kekulé coloring is shown as a function of $J_z$ along the line $J_x=J_y$, with the constraint $J_x+J_y+J_z=1$, which fixes the energy unit. The inset shows a zoom around $J_z=1/3$. The results were obtained from ED of ad hoc clusters Conder06 with a gauge configuration realizing the ground-state flux sector [see Eq. \ref{['eq:fluxmin']}], except for $p=\infty$ where the exact solution discussed in Sec. \ref{['sec:Bethe']} was used. When $J_z=1$ ($J_x=J_y=0$), the isolated-dimer limit is recovered and $\Delta_{\rm f}=2$ (see Sec. \ref{['sec:dimer']}). For $J_z=0$, the isolated $p$-gon limit is obtained, and the gap is given by Eq. \ref{['eq:gap']}. In the Bethe lattice ($p=\infty$), one has $\Delta_{\rm f}=0$ for $J_z\in [0,\sqrt{2}-1$] (see Sec. \ref{['sec:Bethe']}).
• Figure 5: The fermion gap $\Delta_{\rm f}$ for the Kitaev coloring is shown as a function of $J_z$ along the line $J_x=J_y$, with the constraint $J_x+J_y+J_z=1$ (see also the caption for Fig. \ref{['fig:gap_Ke']}). Inset: A zoom around $J_z=1/3$. Unlike the Kekulé coloring, the phase diagram with the Kitaev coloring quickly converges to that of the Bethe lattice as $p$ increases.