Critical behavior of the two-dimensional icosahedron model

In the context of a discrete analog of the classical Heisenberg model, we investigate the critical behavior of the icosahedron model, where the interaction energy is defined as the inner product of neighboring vector spins of unit length pointing to the vertices of the icosahedron. The effective correlation length and magnetization of the model are calculated by means of the corner-transfer-matrix renormalization group (CTMRG) method. A scaling analysis with respect to the cutoff dimension $m$ in CTMRG reveals a second-order phase transition characterized by the exponents $\nu =1.62\pm 0.02$ and $\beta =0.12\pm 0.01$. We also extract the central charge from the classical analog of entanglement entropy as $c=1.90\pm 0.02$, which cannot be explained by the minimal series of conformal field theory.

Figure 1

(a) Numbering of the vertices of the icosahedron. (b) Local Boltzmann weight in Eq. (2) defined for a “black” plaquette, and its tensor representation.

Figure 2

Icosahedron model on the diagonal lattice, where $W$ on each black plaquette represents the local Boltzmann weight of Eq. (2). The partition function can be represented by a tensor network on the square lattice. The dashed lines show the division of the system into quadrants corresponding to CTMs.

Figure 3

Temperature dependence of magnetization $M$ for several $m$. Inset: Magnified view in the region $0.54\le T\le 0.59$.

Figure 4

Probability $P^{\left(s\right)}$ of finding ${\mathit{v}}^{\left(s\right)}$ at the center of the system with $m=500$. Plus marks and green circles denote $P^{\left(1\right)}$ and $P^{\left(12\right)}$, respectively. Blue squares denote $P^{\left(s\right)}$ for $s=2$, 3, 4, 5, and 6, where these probabilities are the same. Red crosses denote $P^{\left(s\right)}$ for $s=7$, 8, 9, 10, and 11.

Figure 5

Finite-$m$ scalings for (a) correlation length in Eq. (6), (b) magnetization $M$ in Eq. (4), and (c) entanglement entropy in Eq. (10).