3 research outputs found
Phase transition of clock models on hyperbolic lattice studied by corner transfer matrix renormalization group method
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 the fixed boundary conditions
are imposed, for the cases N>=3 up to N=30. The model with N=3, which is
equivalent to the 3-state Potts model on the hyperbolic lattice, exhibits the
first order phase transition. A mean-field like phase transition of the second
order is observed for the cases N>=4. When N>=5 we observe the Schottky type
specific heat below the transition temperature, where its peak hight at low
temperatures scales as N^{-2}. From these facts we conclude that the phase
transition of classical XY-model deep inside the hyperbolic lattices is not of
the Berezinskii-Kosterlitz-Thouless type.Comment: REVTeX style, 4 pages, 6 figures, submitted to Phys. Rev.
Phase Transition of the Ising model on a Hyperbolic Lattice
The matrix product structure is considered on a regular lattice in the
hyperbolic plane. The phase transition of the Ising model is observed on the
hyperbolic lattice by means of the corner-transfer-matrix
renormalization group (CTMRG) method. Calculated correlation length is always
finite even at the transition temperature, where mean-field like behavior is
observed. The entanglement entropy is also always finite.Comment: 4 pages, 3 figure
Periodic boundary conditions on the pseudosphere
We provide a framework to build periodic boundary conditions on the
pseudosphere (or hyperbolic plane), the infinite two-dimensional Riemannian
space of constant negative curvature. Starting from the common case of periodic
boundary conditions in the Euclidean plane, we introduce all the needed
mathematical notions and sketch a classification of periodic boundary
conditions on the hyperbolic plane. We stress the possible applications in
statistical mechanics for studying the bulk behavior of physical systems and we
illustrate how to implement such periodic boundary conditions in two examples,
the dynamics of particles on the pseudosphere and the study of classical spins
on hyperbolic lattices.Comment: 30 pages, minor corrections, accepted to J. Phys.