Stiffness from Disorder in Triangular-Lattice Ising Thin Films

We study the triangular lattice Ising model with a finite number of vertically stacked layers and demonstrate a low temperature reentrance of two Berezinskii-Kosterlitz-Thouless transitions, which results in an extended disordered regime down to $T=0$. Numerical results are complemented with the derivation of an effective low-temperature dimer theory. Contrary to order by disorder, we present a new scenario for fluctuation-induced ordering in frustrated spin systems. While short-range spin-spin correlations are enhanced by fluctuations, quasi-long-range ordering is precluded at low enough temperatures by proliferation of topological defects.

Figure 1

(a) Mapping to a dimer state (each arrow stands for the majority spin of a chain). (b) Physical topological defects corresponding to height dislocations $\mathrm{\Delta }h=\pm 2$. (c) Example of a monomer, which has no physical realization in the spin model. (d) Renormalization-group flow diagram near the lowest-$T$ BKT transition ($g=1$). The trajectory schematically shows the bare coupling of the spin model with $N_z>N_{c1}$, while the $y_v=0$ axis corresponds to the close-packed dimer model. (e) Example of a vortex of the spin operator $\psi ={\sigma }_1^z+{\sigma }_2^z{e}^{2\pi i/3}+{\sigma }_3^z{e}^{4\pi i/3}$ associated with an isolated topological defect (note that three parallel spins correspond to $|\psi |=0$, as expected for a vortex core).

Figure 2

(a) Phase diagram of the QIM (4) (data are MC results from Ref. [17]) and the trajectories of quasi-2D classical systems ($J_z/J=0.5$; $N_z=2$, 6, and 24). The phase boundaries are guides to the eye. (b) Phase diagram of the classical model compared with the QC mapping.

Figure 3

(a) $T$ dependence of the critical exponent $\eta$. (b) $T$ dependence of $C_6$. (c)–(e) $R=G(L/2,0)/G(L/4,0)$ for $N_z=24$, 6, and 2. The shaded regions in (a) and the insets of (c) and (d) indicate the BKT phase.

Figure 4

Comparison of exponents in the low-$T$ regime. The shaded region corresponds to the BKT phase (the horizontal lines indicate $\eta =1/4$ and $\eta =1/9$). The dashed (solid) lines are the results of simulating the dimer model with interactions up to first (second) order in $w$ (error bars are smaller than the line width). The points are the results of MC simulations of the spin model [Fig. 3].