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

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.

We study the zero-temperature magnetization process (M −H curve) of one-dimensional quantum antiferromagnets using a variant of the density-matrix renormalization group method. For both the S = 1/2 zig-zag spin ladder and the S = 1 bilinear-biquadratic chain, we find clear cusp-type singularities in the middle-field region of the M − H curve. These singularities are successfully explained in terms of the double-minimum shape of the energy dispersion of the low-lying excitations. For the S = 1/2 zig-zag spin ladder, we find that the cusp formation accompanies the Fermi-liquid to non-Fermi-liquid transition.PACS numbers: 75.10. Jm, 75.40.Cx, 75.30.Cr Low-dimensional antiferromagnetic (AF) quantum spin systems with various spin magnitude S and various spatial structures, have been a field of active researches, both experimentally and theoretically. In particular, the magnetization process (M − H curve, M : magnetization, H: magnetic field) of AF spin chain at low-temperatures have recently drawn much attention, because it exhibits various phase-transition-like behaviors: e.g., the critical phenomena ∆M ∼ √ H − H c associated with the gapped excitation (excitation gap ∝ H c ) [1][2][3] or with the saturated magnetization (at the saturation field H s ), [2][3][4][5][6][7] magnetization plateau, [8] and, the first-order transition.[9] These are field-induced phase transitions of the ground state, which reflect non-trivial energy-level structure at zero field.What we consider in this Letter is another type of singularity which has not been discussed so much: the cusp singularity at H = H cusp in the middle-field region (H c < H cusp < H s ). Existence this type of singularity was first demonstrated by Parkinson [10] for the integrable Uimin-Lai-Sutherland (or, SU(3)) chain, [11] and has also been known for some other integrable spin chains, [12][13][14][15] and ladders. [16,17] Derivation of the magnetization cusp for each model, however, relies on the model's integrability in an essential manner, restricting the Hamiltonian to be of somewhat unrealistic form. Hence, whether or not this type of singularity can be found in realistic systems is a highly non-trivial question. In the present Letter, we make the first systematic numerical study of the "middle-field cusp singularity" (MFCS, for short) in the zero-temperature M − H curve for non-integrable systems, through which we give a positive answer for the above question. Namely, we show that the S = 1/2 zig-zag spin ladder and the S = 1 bilinear-biquadratic chain actually exhibit MFCS in the M − H curve.The numerical method we employ is the productwavefunction renormalization group (PWFRG), [18] which is a variant of the density-matrix renormalization group (DMRG).[19] Efficiency of the PWFRG in calculation of the M − H curve has been demonstrated in previous studies; [3,20] the PWFRG will allow us to obtain the M − H curve in the thermodynamic limit with enough accuracy to detect "weak" (non-divergent) singularity like the MFCS.Consider the S = 1/2 zig-zag spin la...

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.