Magnon crystallization in the kagome lattice antiferromagnet

Loop-gas description of the localized-magnon states on the kagome lattice with open boundary conditions

“…We are thus faced with characterizing the loop gas that corresponds to the linearly independent states among equation (3.1). Previously, we have observed relations [41] for which we currently have no geometric interpretation. Thus, at this point we rather go back to the wave functions equation (3.1) and examine the linear relations between them.…”

“…These include three cases consisting of N hex = n × n hexagons and N = 3 n 2 + 4 n − 1 sites, and two hexagonal-shaped samples containing N hex = 7 and 19 hexagons (N = 30 and 72 spins, respectively). As in previous work [41] for periodic boundary conditions, we started by enumerating all allowed n -loop configurations { i }. We then make use of the fact that, apart from a global 2In the case of periodic boundary conditions, there are actually linear relations between the hard-hexagon states on the torus, but from the point of view of counting, this deficit is compensated by the state with ω − ( ì 0) = ω 0 , see also figure 1 [36,41].…”

“…Recently, we have investigated the full loop-gas description of the localized-magnon states on finite lattices with N ≤ 72 sites [41]. Following the general philosophy that periodic boundary conditions reduce finite-size effects by eliminating explicit boundaries, we have employed periodic boundary conditions in that investigation.…”

“…There are several issues associated with these winding loops. Firstly, they lead to an enormous number of geometrically allowed configurations before taking linear relations into account [41]. Secondly, on a torus a nested configuration of two loops can actually be expressed as a linear combination of hard-hexagon and winding configurations, as is evidenced by the counting of ground states in the two-magnon sector [36,41].…”

“…Firstly, they lead to an enormous number of geometrically allowed configurations before taking linear relations into account [41]. Secondly, on a torus a nested configuration of two loops can actually be expressed as a linear combination of hard-hexagon and winding configurations, as is evidenced by the counting of ground states in the two-magnon sector [36,41]. Thirdly, one finds that the number of linearly independent loop configurations does not describe all ground states of the spin-1/2 kagome Heisenberg antiferromagnet on a torus [41] which raises the question whether the loop gas does yield a complete description of the ground-state manifold of the spin-1/2 kagome Heisenberg antiferromagnet around the saturation field in the thermodynamic limit.…”