Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets

Using a specially designed Monte Carlo algorithm with directed loops, we investigate the triangular lattice Ising antiferromagnet with coupling beyond the nearest neighbors. We show that the first-order transition from the stripe state to the paramagnet can be split, giving rise to an intermediate nematic phase in which algebraic correlations coexist with a broken symmetry. Furthermore, we demonstrate the emergence of several properties of a more topological nature such as fractional edge excitations in the stripe state, the proliferation of double domain walls in the nematic phase, and the Kasteleyn transition between them. Experimental implications are briefly discussed.

Figure 1

Representative phase diagrams of the TLIAF with $J_1/T\to \infty$, determined by MC simulation. All phase boundaries were determined from the winding number. (a) In the presence of $J_2$ and $J_3$ interactions there are 3 possible phases, and all transitions are first order. (Inset) Illustration of first to fifth neighbors of the central site and bond labeling. (b) In the presence of a $J_5$ interaction an intermediate nematic state is revealed. Blue (red) dots show the position of a second order Kasteleyn transition calculated analytically [14, 17] (with MC calculations). (c) Stripe phase. (d) Nematic phase with fluctuating double domain walls (shown in black). (e) Zig-zag phase.

Figure 2

Winding number sectors of the TLIAF. (a) An Ising configuration within the nearest-neighbor ground state manifold can be mapped onto a dimer covering of the dual honeycomb lattice. The dimer crossings of the reference lines are used to calculate the winding number $W=(W_1,W_2)$. (b) The allowed winding number sectors, illustrated for $L=12$. The $W=(0,0)$ sector (circled in blue) has a macroscopic degeneracy, while the sectors containing stripe (zig-zag) states have a two (four)-fold degeneracy.

Figure 3

Representative plots of the structure factor in the 4 phases (see Fig. 1). (a) In the stripe phase there are a set of Bragg peaks, and similarly (b) in the zig-zag phase. (c) The nematic phase is characterized by pairs of peaks, with power-law singularities, and the peak splitting is proportional to the density of double domain walls. (d) The critical paramagnet also has peaks with power-law singularities, but at different positions compared to the nematic state.

Figure 4

Fractional edge excitations in the low-temperature stripe state. While the stripe state is fluctuationless in the bulk, defect triangles (red) can be created at open boundaries at an energy cost of $2J_2-8J_3+4J_4$. The snapshot shown is taken from a Monte Carlo simulation of the $J_1-J_2$ model performed on a cylinder at $T=1.5J_2$ (for comparison $T_1=6.39J_2$).

Figure 5

Temperature dependence of the winding number with $J_2=1$, $J_3=0.4$ and variable $J_5$. Monte Carlo simulations on an $L=96$ cluster are used to measure $W_{\max }=\max (|W_1|,|W_2|,|W_2-W_1|)$, and errors are typically smaller than the point size. For $J_5=0$ (black) there is a direct first order phase transition from the low-temperature stripe state ($W_{\max }=L$) to the high-temperature critical paramagnet ($W_{\max }\to 0$). For all other values of $J_5$ there is a second order phase transition at $T_{\mathrm{DDW}}=2.29$ between the stripe state and a nematic state (variable $W_{\max }$), followed by a first order transition to the critical paramagnet at higher $T$.