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Physica A Statistical Mechanics and its Applications

Transfer-matrix renormalization group method for general Markov random fields

Yukun Huang, Sung‐Nien Yu

Published Dec 2, 2010• 4 citations
Publisher
DOI
10.1016/j.physa.2010.11.024

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We seek numerical calculations of partition functions of general Markov random fields (MRFs) by using the biorthonormal transfer-matrix renormalization-group (BTMRG) method. The BTMRG is a modification of the conventional TMRG, a variant of the density-matrix renormalization-group (DMRG), which automatically truncates the Hilbert space so that the properties of large systems can be precisely calculated while the dimensions of the renormalized transfer matrix remain constant. We apply the BTMRG to the decimation of the fundamental transfer matrix for general MRFs. Four binary 2nd-order MRFs are selected for numerical simulations. Results of simulations show that all four models exhibit the phase transition phenomenon. This work shows that our BTMRG method is superior to the conventional TMRG in accuracy, computational speed, and in the possibility of treating a much larger system.
This chapter introduces two kinds of RG methods for solving the leading eigenvalue and eigenvectors of a transfer matrix: TMRG (transfer matrix renormalization group) and CTMRG (corner transfer matrix renormalization group). These methods are developed to study the thermodynamic properties of two-dimensional classical statistical models. Furthermore, in the framework of MPS, the fixed-point equations of these methods are derived, and the steps for efficiently solving these equations are outlined.