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Computational Methods in Science and Technology

Algorithm for calculation of the maximum eigenvalue of transfer matrix for the linear S = 1/2 Heisenberg model

M. Bieliński, G. Kamieniarz

Published Jan 1, 1997• 0 citations
Publisher

Abstract

The paper presents the algorithm and description of the meigval procedure written in FORTRAN 90 language which was designed for calculation of the maximum eigenvalue of transfer matrix for one-dimensional S = 1/2 Heisenberg model.The procedure is available at the Poznań Supercomputing and Networking Centre on a Cray J916 supercomputer in the numerical procedure library /lib/mestat.a.

DOI
10.12921/cmst.1997.03.01.83-86

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•Sep 25, 2016•0 citations
The paper presents the algorithm and description of the meigval procedure written in FORTRAN 90 language which was designed for calculation of the maximum eigenvalue of transfer matrix for one-dimensional S = 1/2 Heisenberg model.The procedure is available at the Poznań Supercomputing and Networking Centre on a Cray J916 supercomputer in the numerical procedure library /lib/mestat.a.
The paper presents the algorithm for calculation of the maximum eigenvalue of the transfer matrix for one-dimensional S=1/2 Heisenberg model with nearest and next-nearest neighbours interactions.This value permits a calculation of the partition function and free energy of the quantum system.This algorithm is implemented in the mevnnn procedure written in FORTRAN 90.The procedure is available at the Poznań Supercomputing and Networking Center on Cray T3E supercomputer in the library /usr/local/lib/libms.a
A new fast algorithm for calculating a few maximum (or minimum) eigenvalues and the corresponding eigenvectors of large N × N Hermitian matrices is presented. The method is based on a molecular dynamics algorithm for N coupled harmonic oscillators. The time step for iteration is chosen so that only the normal mode with the maximum eigenvalues grows exponentially. Other eigenvalues and eigenvectors are obtained one by one from the largest eigenvalue by repeating the process in subspaces orthogonal to the previous modes. The characteristics of the algorithm lie in the simplicity, speed (CPU time α N2), and memory efficiency (O(N) besides the matrix). The effectiveness of the algorithm is illustrated by calculation of the groundstate and first-excited state energies of the Heisenberg model for an antiferromagnetic chain with N up to 16384.
The one-dimensional S=1 quantum spin system has been mapped onto the equivalent two-dimensional Ising model on the basis of the equivalence theorem of Suzuki. The classical representation of the partition function has been expressed in terms of the maximum eigenvalue of the transfer matrix and evaluated by means of the numerically exact transfer-matrix method. The physical meaning of two kinds of correlation functions in RSD and CBD has been made clear in terms of the concept of the maximum eigenstate. Calculations have been made for both ferromagnetic and antiferromagnetic Heisenberg chains with several single-site anisotropy values, both positive and negative. Good agreements are found with some existing numerical results.
In order to calculate 3-dimensional Ising model, we develop a method to build a much smaller transfer matrix containing the largest eigenvalue from the original 2 N 1 N 2 ×2 N 1 N 2 matrix V 1 . Firstly, the transfer matrix V 1 is written as the linear combination of several basic vectors. Secondly, we divide the basic vectors into several subgroups. The multiplication of a basic vector and V 1 can be written as the linear combination of basic vectors from the same subgroup. Finally, we use a new transfer matrix V 2 to describe the relationship between basic vectors of the same subgroup. V 2 is much smaller than the original transfer matrix and contains the largest eigenvalue of V 1 . We use this method to calculate the specific heat per atom C pa and the magnetic momentum per atom M pa . The results show that there exists a pair of temperature and magnetic field intensity where the specific heat gets to its maximum value. When N 1 N 2 increases, the maximum value of specific heat becomes larger.