Vertical Density Matrix Algorithm: A Higher-Dimensional Numerical Renormalization Scheme based on the Tensor Product State Ansatz

摘要

We present a new algorithm to calculate the thermodynamic quantities ofthree-dimensional (3D) classical statistical systems, based on the ideas of thetensor product state and the density matrix renormalization group. We representthe maximum-eigenvalue eigenstate of the transfer matrix as the product oflocal tensors which are iteratively optimized by the use of the ``verticaldensity matrix'' formed by cutting the system along the transfer direction.This algorithm, which we call vertical density matrix algorithm (VDMA), issuccessfully applied to the 3D Ising model. Using the Suzuki-Trottertransformation, we can also apply the VDMA to two-dimensional (2D) quantumsystems, which we demonstrate for the 2D transverse field Ising model.