Local and global magnetization on the Sierpinski carpet.

The phase transition of the classical Ising model on the Sierpinski carpet, which has the fractal dimension log_{3}^{}8≈1.8927, is studied by an adapted variant of the higher-order tensor renormalization group method. The second-order phase transition is observed at the critical temperature T_{c}^{}≈1.478. Position dependence of local functions is studied through impurity tensors inserted at different locations on the fractal lattice. The critical exponent ß associated with the local magnetization varies by two orders of magnitude, depending on lattice locations, whereas T_{c}^{} is not affected. Furthermore, we employ automatic differentiation to accurately and efficiently compute the average spontaneous magnetization per site as a first derivative of free energy with respect to the external field, yielding the global critical exponent of ß≈0.135.

Corner transfer matrix renormalization group analysis of the two-dimensional dodecahedron model.

We investigate the phase transition of the dodecahedron model on the square lattice. The model is a discrete analog of the classical Heisenberg model, which has continuous O(3) symmetry. In order to treat the large on-site degree of freedom q=20, we develop a massively parallelized numerical algorithm for the corner transfer matrix renormalization group method, incorporating EigenExa, the high-performance parallelized eigensolver. The scaling analysis with respect to the cutoff dimension reveals that there is a second-order phase transition at T_{c}^{}=0.4398(8) with the critical exponents ν=2.88(8) and ß=0.21(1). The central charge of the system is estimated as c=1.99(6).

Finite-m scaling analysis of Berezinskii-Kosterlitz-Thouless phase transitions and entanglement spectrum for the six-state clock model.

We investigate the Berezinskii-Kosterlitz-Thouless transitions for the square-lattice six-state clock model with the corner-transfer matrix renormalization group (CTMRG). Scaling analyses for effective correlation length, magnetization, and entanglement entropy with respect to the cutoff dimension m at the fixed point of the CTMRG provide transition temperatures consistent with a variety of recent numerical studies. We also reveal that the fixed-point spectrum of the corner-transfer matrix in the critical intermediate phase of the six-state clock model is characterized by the scaling dimension consistent with the c=1 boundary conformal field theory associated with the effective Z_{6} dual sine-Gordon model.

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 ν=1.62±0.02 and ß=0.12±0.01. We also extract the central charge from the classical analog of entanglement entropy as c=1.90±0.02, which cannot be explained by the minimal series of conformal field theory.

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.

Phase transition of the Ising model on a fractal lattice.

The phase transition of the Ising model is investigated on a planar lattice that has a fractal structure. On the lattice, the number of bonds that cross the border of a finite area is doubled when the linear size of the area is extended by a factor of 4. The free energy and the spontaneous magnetization of the system are obtained by means of the higher-order tensor renormalization group method. The system exhibits the order-disorder phase transition, where the critical indices are different from those of the square-lattice Ising model. An exponential decay is observed in the density-matrix spectrum even at the critical point. It is possible to interpret that the system is less entangled because of the fractal geometry.

Mean-field universality class induced by weak hyperbolic curvatures.

Order-disorder phase transition of the ferromagnetic Ising model is investigated on a series of two-dimensional lattices that have negative Gaussian curvatures. Exceptional lattice sites of coordination number seven are distributed on the triangular lattice, where the typical distance between the nearest exceptional sites is proportional to an integer parameter n. Thus, the corresponding curvature is asymptotically proportional to -n(-2). Spontaneous magnetization and specific heat are calculated by means of the corner transfer matrix renormalization group method. For all the finite n cases, we observe the mean-field-like phase transition. It is confirmed that the entanglement entropy at the transition temperature is linear in (c/6)ln n, where c = 1/2 is the central charge of the Ising model. The fact agrees with the presence of the typical length scale n being proportional to the curvature radius.

Weak correlation effects in the Ising model on triangular-tiled hyperbolic lattices.

The Ising model is studied on a series of hyperbolic two-dimensional lattices which are formed by tessellation of triangles on negatively curved surfaces. In order to treat the hyperbolic lattices, we propose a generalization of the corner transfer matrix renormalization group method using a recursive construction of asymmetric transfer matrices. Studying the phase transition, the mean-field universality is captured by means of a precise analysis of thermodynamic functions. The correlation functions and the density-matrix spectra always decay exponentially even at the transition point, whereas power-law behavior characterizes criticality on the Euclidean flat geometry. We confirm the absence of a finite correlation length in the limit of infinite negative Gaussian curvature.