Macroscopic Degeneracy and Emergent Frustration in a Honeycomb Lattice Magnet

Using a hybrid method based on fermionic diagonalization and classical Monte Carlo techniques, we investigate the interplay between itinerant and localized spins, with competing double- and superexchange interactions, on a honeycomb lattice. For moderate superexchange, a geometrically frustrated triangular lattice of hexagons forms spontaneously. For slightly larger superexchange a dimerized ground state is stable that has macroscopic degeneracy. The presence of these states on a nonfrustrated honeycomb lattice highlights novel phenomena in this itinerant electron system: emergent geometrical frustration and degeneracy related to a symmetry intermediate between local and global.

Figure 1

Schematic view of (a) the honeycomb lattice and (b) the brick-wall lattice having the same topology.

Figure 2

(a)–(d) DOS at low, intermediate, and high temperatures for different values of $J_{\mathrm{AFM}}$ ($\gamma =0.04$). In (a) the $T=0.001$ curve shows the DOS of free fermions on a honeycomb lattice in the thermodynamic limit. In (d) the $T=0.001$ curve represents a gapped insulating phase, the seemingly finite DOS at $E_F$ being a broadening effect. Except for the FM phase, all ground states are gapped.

Figure 3

(a) Snapshot from MC simulations supplemented by optimization routines showing an emergent triangular lattice (black circles) formed by FM hexagons at $J_{\mathrm{AFM}}=0.14$. Spins within each hexagon are almost FM, a small canting angle between groups of three is illustrated by shading. The colored spins illustrate the $2\pi /3$-angle order of the Yafet-Kittel state. Schematic view of (b),(c) two dimer states related by a translational symmetry (see text) and (d) a canted dimer state.

Figure 4

(a) $T\mathrm{\text{−}}{J}_{\mathrm{AFM}}$ phase diagram at half filling obtained by MC simulations for a $12\times 12$ lattice. (b) The energy of the various states: “alm. FM” refers to a spiral with the longest wavelength supported by the lattice, converging to FM in the thermodynamic limit. Similar finite-size effects are reported in doped 1D and 2D lattices [28]. “Hex” denotes the emergent Yafet-Kittel order between hexagons depicted in Fig. 3a, the energy was optimized with respect to the canting angle within the hexagons. “Dimers” and “C. Dim” are the highly degenerate FM and canted dimer states, and “AFM” denotes perfectly AFM order. The black crosses are energies obtained by unbiased MC calculations and a subsequent energy optimization.