Phase diagram of a truncated tetrahedral model

Phase diagram of a discrete counterpart of the classical Heisenberg model, the truncated tetrahedral model, is analyzed on the square lattice, when the interaction is ferromagnetic. Each spin is represented by a unit vector that can point to one of the 12 vertices of the truncated tetrahedron, which is a continuous interpolation between the tetrahedron and the octahedron. Phase diagram of the model is determined by means of the statistical analog of the entanglement entropy, which is numerically calculated by the corner transfer matrix renormalization group method. The obtained phase diagram consists of four different phases, which are separated by five transition lines. In the parameter region, where the octahedral anisotropy is dominant, a weak first-order phase transition is observed.

Figure 1

Truncated tetrahedron (shown in the middle, parametrized by $t=0.5$) is depicted as the interpolation between the octahedron (on the left for $t=0$) and the tetrahedron (on the right for $t=1$).

Figure 2

Temperature dependence of the classical analog of the entanglement entropy $S_v^{}$ for $t=0.2$, $t=0.3$, and $t=0.4$. The vertical dotted lines denote phase boundaries, and each phase is labeled by the Roman number.

Figure 3

Phase diagram of the TTM with respect to the parameter $t$ and the temperature $T$. The circles denote the second-order phase transition. The phase boundaries shown by triangles are identified as the first-order ones.

Figure 4

Effective exponent ${\nu }_{\mathrm{eff}}^{}\left(L\right)$ in Eq. (10) calculated at the I–II phase boundary when $t=0.1$ (squares), and that at the I–III boundary when $t=0.35$ (circles).

Figure 5

Effective exponent ${\eta }_{\mathrm{eff}}^{}\left(L\right)$ in Eq. (12) calculated at the I–II phase boundary when $t=0.1$ (squares), and that at the I–III boundary when $t=0.35$ (circles).

Figure 6

Free energy per site $f_t^{}\left(T\right)$ with respect to the parameter $t$ at fixed temperature $T=0.5$ under the fixed (triangles) and the free (circles) boundary conditions.

Figure 7

Temperature dependence of the free energy per site $f_t^{}\left(T\right)$ in Eq. (4) when $t=0$, where the TTM coincides with the octahedral model. The triangles and the circles correspond to $f_t^{}\left(T\right)$ calculated under fixed and free boundary condition, respectively.

Figure 8

Free energy per site $f_{t=0.2}^{}\left(T\right)$ around the II–IV phase boundary calculated for both fixed and free boundary conditions; the inset shows their difference.