The commensurate - disordered phase transition in 2 D classical ATNNI model studied by DMRG

The classical two-dimensional anisotropic triangular nearest-neighbor Ising (ATNNI) model is studied by the density matrix renormalization group (DMRG) technique when periodic boundary conditions are imposed. Applying the finite-size scaling to the DMRG results a commensurate-disordered (C-D) phase transition line as well as temperature and magnetic critical exponents are calculated. We conclude that the (C-D) phase transition in the ATNNI model belongs to the same universality class as the ordered-disordered phase transition of the Ising model. Analysis of semi-finite systems of small size in one or more directions has been used as a powerful tool in extracting of critical properties of two-dimensional classical models and corresponding one-dimensional quantum models. Although finite or 1D systems themselves do not display any critical behavior, it is, however, possible to extract critical parameter values as well as critical exponents. Temperature, ordering magnetic field, and finite-size deviations from criticality are all described by the same set of the critical exponents [1]. This paper is focused on infinite strips of finite width where the relevant numerical data are obtained from the transfer matrix methods, in particular, the Density Matrix Renormalization Group (DMRG) method. In 1992 the DMRG technique has been invented by S. R. White [2] in real space for one– dimensional (1D) quantum spin Hamiltonians. Three years later T. Nishino [3] applied this numerical technique to classical spin 2D models that is based on the renormalization group transformation for the transfer matrix for the open boundary conditions. DMRG treatment of 2D classical systems exceeds the classical Monte Carlo approach in accuracy, speed, and size of the systems [4]. Recently, we have modified the DMRG method for the 2D classical models, imposing periodic boundary conditions (PBC) on strip boundaries, and found a relation that helped to determine an optimal strip width L opt in order to obtain correct values of critical temperature and exponents [5] using the finite-size scaling (FSS). We have obtained results of very high accuracy exceeding the DMRG method with standard open boundary conditions. Our method does not require any extrapolation analysis of the data. The use of DMRG for 2D classical models may follow one of two different approaches: (i) DMRG method is applied to strips of finite width and from two largest transfer-matrix eigenvalues or the free energy estimated with high precision, the critical properties of the system are calculated by the FSS analysis (here, we use this approach).