Critical behavior of the Ising model is investigated at the center of large scale finite size systems, where the lattice is represented as the tiling of pentagons. The system is on the hyperbolic plane, and the recursive structure of the lattice makes it possible to apply the corner transfer matrix renormalization group method. From the calculated nearest neighbor spin correlation function and the spontaneous magnetization, it is concluded that the phase transition of this model is mean-field like. One parameter deformation of the corner Hamiltonian on the hyperbolic plane is discussed.
We consider the thermal phase transition from a paramagnetic to a stripe-antiferromagnetic phase in the frustrated two-dimensional square-lattice Ising model with competing interactions
J1<0 (nearest neighbor, ferromagnetic) and
J2>0 (second neighbor, antiferromagnetic). The striped phase breaks a
Z4 symmetry and is stabilized at low temperatures for
g=J2/∣J1∣>1/2. Despite the simplicity of the model, it has proved difficult to precisely determine the order and the universality class of the phase transitions. This was done convincingly only recently by Jin et al. [Phys. Rev. Lett. 108, 045702 (2012)]. Here, we further elucidate the nature of these transitions and their anomalies by employing a combination of cluster mean-field theory, Monte Carlo simulations, and transfer-matrix calculations. The
J1-
J2 model has a line of very weak first-order phase transitions in the whole region
1/2<g<g∗, where ${g}^{*}=0.67\ifmmode\pm\else\textpm\fi{}0.01$. Thereafter, the transitions from
g=g∗ to $g\ensuremath{\rightarrow}\ensuremath{\infty}$ are continuous and can be fully mapped, using universality arguments, to the critical line of the well-known Ashkin-Teller model from its four-state Potts point to the decoupled Ising limit. We also comment on the pseudo-first-order behavior at the Potts point and its neighborhood in the Ashkin-Teller model on finite lattices, which in turn leads to the appearance of similar effects in the vicinity of the multicritical point
g∗ in the
J1-
J2 model. The continuous transitions near
g∗ can therefore be mistaken for first-order transitions, and this realization was the key to understanding the paramagnetic-striped transition for the full range of
g>1/2. Most of our conclusions are based on Monte Carlo calculations, while the cluster mean-field and transfer-matrix results provide useful methodological benchmarks for weakly first-order behaviors and Ashkin-Teller criticality.
We study two-dimensional ferromagnetic Ising model on a series of regular lattices, which are represented as the tessellation of polygons with p>=5 sides, such as pentagons (p=5), hexagons (p=6), etc. Such lattices are on hyperbolic planes, which have constant negative scalar curvatures. We calculate critical temperatures and scaling exponents by use of the corner transfer matrix renormalization group method. As a result, the mean-field like phase transition is observed for all the cases p>=5. Convergence of the calculated transition temperatures with respect to p is investigated towards the limit p->infinity, where the system coincides with the Ising model on the Bethe lattice.
Two-dimensional ferromagnetic N -state clock models are studied on a hyperbolic lattice represented by tessellation of pentagons. The lattice lies on the hyperbolic plane with a constant negative scalar curvature. We observe the spontaneous magnetization, the internal energy, and the specific heat at the center of sufficiently large systems, where fixed boundary conditions are imposed, for the cases N>or=3 up to N=30 . The model with N=3 , which is equivalent to the three-state Potts model on the hyperbolic lattice, exhibits a first-order phase transition. A mean-field-like phase transition of second order is observed for the cases N>or=4 . When N>or=5 we observe a Schottky-type specific heat below the transition temperature, where its peak height at low temperatures scales as N(-2) . From these facts we conclude that the phase transition of the classical XY model deep inside hyperbolic lattices is not of the Berezinskii-Kosterlitz-Thouless type.
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.
We obtain phase diagrams of a spin-3/2 Ising model on a square lattice by using a corner transfer matrix renormalization group method. We find that three phases appear in the phase diagrams; they are the ferromagnetic(FM) phase, the paramagnetic(PM) phase and a ferrimagnetic(FI) phase. It turns out that a tricritical point appears in a region of space spanned by parameters of coupling constants and the temperature. We estimate values of critical exponent ν and α at a transition temperature between the FM phase and the PM phase, at that between the FI phase and the FM phase, and also at that between the FI phase and the PM phase, by using the finite size scaling analysis.