A systematic way to find and construct exact finite dimensional matrix product stationary states

We explain how to construct matrix product stationary states which are composed of finite-dimensional matrices. Our construction explained in this article was first presented in a part of [Hieida and Sasamoto:J. Phys. A: Math. Gen. 37 (2004) 9873] for general models. In this article, we give more details on the treatment than in the above-mentioned reference, for one-dimensional asymmetric simple exclusion process(ASEP).Comment: This article will appear in the proceedings of "Workshop on Matrix Product State Formulation and Density Matrix Renormalization Group Simulations (MPS&DMRG)" to be published by World Scientifi

深層学習を用いた感情モデルの構築：感情メカニズムの理解へ向けた構成的アプローチ

電気通信大学201

Universal Asymptotic Eigenvalue Distribution of Density Matrices and the Corner Transfer Matrices in the Thermodynamic Limit

We study the asymptotic behavior of the eigenvalue distribution of the Baxter's corner transfer matrix (CTM) and the density matrix (DM) in the White's density-matrix renormalization group (DMRG), for one-dimensional quantum and two-dimensional classical statistical systems. We utilize the relationship ${\rm DM}={\rm CTM}^4$ which holds for non-critical systems in the thermodynamic limit. Using the known diagonal form of CTM, we derive exact asymptotic form of the DM eigenvalue distribution for the integrable $S=1/2$ XXZ chain (and its related integrable models) in the massive regime. The result is then recast into a ``universal'' form without model-specific quantities, which leads to $\omega_{m}\sim \exp[-{\rm const.}(\log m)^2]$ for $m$-th DM eigenvalue at larg $m$. We perform numerical renormalization group calculations (using the corner-transfer-matrix RG and the product-wavefunction RG) for non-integrable models, verifying the ``universal asymptotic form'' for them. Our results strongly suggest the universality of the asymptotic eigenvalue distribution of DM and CTM for a wide class of systems.Comment: 4 pages, RevTeX, 4 ps figure

Numerical Renormalization Approach to Two-Dimensional Quantum Antiferromagnets with Valence-Bond-Solid Type Ground State

We study the ground-state properties of the two-dimensional quantum spin systems having the valence-bond-solid (VBS) type ground states. The ``product-of-tensors'' form of the ground-state wavefunction of the system is utilized to associate it with an equivalent classical lattice statistical model which can be treated by the transfer-matrix method. For diagonalization of the transfer matrix, we employ the product-wavefunction renormalization group method which is a variant of the density-matrix renormalization group method. We obtain the correlation length and the sublattice magnetization accurately. For the anisotropically ``deformed'' S=3/2 VBS model on the honeycomb lattice, we find that the correlation length as a function of the deformation parameter behaves very much alike as that in the S=3/2 VBS chain.Comment: 9 pages and 11 non-embedded figures, REVTex, submitted to New Journal of Physic

Fluctuation in e-mail sizes weakens power-law correlations in e-mail flow

Power-law correlations have been observed in packet flow over the Internet. The possible origin of these correlations includes demand for Internet services. We observe the demand for e-mail services in an organization, and analyze correlations in the flow and the sequence of send requests using a Detrended Fluctuation Analysis (DFA). The correlation in the flow is found to be weaker than that in the send requests. Four types of artificial flow are constructed to investigate the effects of fluctuations in e-mail sizes. As a result, we find that the correlation in the flow originates from that in the sequence of send requests. The strength of the power-law correlation decreases as a function of the ratio of the standard deviation of e-mail sizes to their average.Comment: 8 pages, 6 figures, EPJB accepte

Two-Dimensional Tensor Product Variational Formulation

We propose a numerical self-consistent method for 3D classical lattice models, which optimizes the variational state written as two-dimensional product of tensors. The variational partition function is calculated by the corner transfer matrix renormalization group (CTMRG), which is a variant of the density matrix renormalization group (DMRG). Numerical efficiency of the method is observed via its application to the 3D Ising model.Comment: 9 pages, 4 figures, submitted to Prog. Theor. Phy

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

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 which 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 two-dimensional (2D) quantum systems, which we demonstrate for the 2D transverse field Ising model.Comment: Unnecessary files are removed. 8 pages, 7 figures, submitted to Phys.Rev.

Self-Consistent Tensor Product Variational Approximation for 3D Classical Models

We propose a numerical variational method for three-dimensional (3D) classical lattice models. We construct the variational state as a product of local tensors, and improve it by use of the corner transfer matrix renormalization group (CTMRG), which is a variant of the density matrix renormalization group (DMRG) applied to 2D classical systems. Numerical efficiency of this approximation is investigated through trial applications to the 3D Ising model and the 3D 3-state Potts model.Comment: 12 pages, 6 figure