Andrej Gendiar

Two-Dimensional Tensor Product Variational Formulation

We propose a numerical self-consistent method for 3D classical lattice models, which optimizes the variational state written as two-dimensional product of tensors. The variational partition function is calculated by the corner transfer matrix renormalization group (CTMRG), which is a variant of the density matrix renormalization group (DMRG). Numerical efficiency of the method is observed via its application to the 3D Ising model.

Corner Transfer Matrix Renormalization Group Method Applied to the Ising Model on the Hyperbolic Plane

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 p...

Product Wave Function Renormalization Group: Construction from the Matrix Product Point of View

We present a construction of a matrix product state (MPS) that approximates the largest-eigenvalue eigenvector of a transfer matrix T of a two-dimensional classical lattice model. A state vector created from the upper or the lower half of a finite size cluster approximates the largest-eigenvalue eigenvector. Decomposition of this state vector into the MPS gives a way of extending the MPS recursively. The extension process is a special case of the product wave function renormalization group (PWFRG) method, that accelerates the numerical calculation of the infinite system density matrix renormalization group (DMRG) method. As a result, we successfully give the physical interpretation of the PWFRG method, and obtain its appropriate initial condition.

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 (5,4)-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.

Ising model on a hyperbolic lattice studied by the corner transfer matrix renormalization group method

We study a two-dimensional ferromagnetic Ising model on a series of regular lattices, which are represented as a 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 the 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 toward the limit p → ∞, where the system coincides with the Ising model on the Bethe lattice.

A Classical Background for the Wave Function Prediction in the Infinite System Density Matrix Renormalization Group Method

We report a physical background of the wave function prediction in the infinite system density matrix renormalization group (DMRG) method, from the view point of two-dimensional vertex model, a typical lattice model in statistical mechanics. Singular value decomposition applied to rectangular corner transfer matrices naturally draws matrix product representation for the maximal eigenvector of the row-to-row transfer matrix. The wave function prediction can be expressed as the insertion of an approximate half-column transfer matrix. This insertion process is in accordance with the scheme proposed by McCulloch recently.

Thermodynamic model of social influence on two-dimensional square lattice: Case for two features

We propose a thermodynamic multi-state spin model in order to describe equilibrial behavior of a society. Our model is inspired by the Axelrod model used in social network studies. In the framework of the statistical mechanics language, we analyze phase transitions of our model, in which the spin interaction J is interpreted as a mutual communication among individuals forming a society. The thermal fluctuations introduce a noise T into the communication, which suppresses long-range correlations. Below a certain phase transition point Tt, large-scale clusters of the individuals, who share a specific dominant property, are formed. The measure of the cluster sizes is an order parameter after spontaneous symmetry breaking. By means of the Corner transfer matrix renormalization group algorithm, we treat our model in the thermodynamic limit and classify the phase transitions with respect to inherent degrees of freedom. Each individual is chosen to possess two independent features f=2 and each feature can assume one of q traits (e.g. interests). Hence, each individual is described by q2 degrees of freedom. A single first-order phase transition is detected in our model if q>2, whereas two distinct continuous phase transitions are found if q=2 only. Evaluating the free energy, order parameters, specific heat, and the entanglement von Neumann entropy, we classify the phase transitions Tt(q) in detail. The permanent existence of the ordered phase (the large-scale cluster formation with a non-zero order parameter) is conjectured below a non-zero transition point Tt(q)≈0.5 in the asymptotic regime q→∞.

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.

Phase transition of clock models on a 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 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.

Modulation of Local Magnetization in Two-Dimensional Axial-Next-Nearest-Neighbor Ising Model

The axial next-nearest-neighbor Ising model is studied in two dimensions at finite temperature using the density matrix renormalization group. The model exhibits phase transition of the second-order between the antiphase in low temperature and the modulated phase in high temperature. Observing the domain wall free energy, we confirm that the modulation period in high-temperature side is well explained by the free-fermion picture.