Numerical calculation of the restricted density of states for the three-dimensional Ising model

J. Engels, L. Fromme, M. Seniuch

Engels, Jürgen, Fromme, L, Seniuch, M

Henk W. J. Blöte, G. Kamieniarz

Rick J. Creswick

The numerical transfer matrix for the partition function of discrete lattice models is generalized to allow the calculation of the density of states \ensuremath{\Omega}(E), and the restricted density of states \ensuremath{\Omega}(E,M). Given \ensuremath{\Omega}(E,M) the partition function is expressed as a polynomial in the variables x=${\mathit{e}}^{\mathrm{\ensuremath{\beta}}\mathit{h}}$ and y=${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}}$. These algorithms are illustrated with calculations for the Ising model on finite square lattices. The zeros of the partition function are examined in both the complex x and y planes. Finite size scaling analysis of the zeros leads to very accurate estimates for the critical temperature and critical exponents.

Nobuya Maeshima, Yasuhiro Hieida, Yasuhiro Akutsu +2 more

We present a new algorithm to calculate the thermodynamic quantities of three-dimensional (3D) classical statistical systems, based on the ideas of the tensor product state and the density matrix renormalization group. We represent the maximum-eigenvalue eigenstate of the transfer matrix as the product of local tensors that are iteratively optimized by the use of the "vertical density matrix" formed by cutting the system along the transfer direction. This algorithm, which we call vertical density matrix algorithm (VDMA), is successfully applied to the 3D Ising model. Using the Suzuki-Trotter transformation, we can also apply the VDMA to 2D quantum systems, which we demonstrate for the 2D transverse field Ising model.

Kurt Binder