We propose a new fast numerical renormalization group method, the corner transfer matrix renormalization group (CTMRG) method, which is based on a unified scheme of Baxter's corner transfer matrix method and White's density matrix renormalization group method. The key point is that a product of four corner transfer matrices coincides with the density matrix. We formulate the CTMRG method as a renormalization of 2D classical models.

We report a real-space renormalization group (RSRG) algorithm, which is formulated through Baxter's corner transfer matrix (CTM), for two-dimensional (d = 2) classical lattice models. The new method performs the renormalization group transformation according to White's density matrix algorithm, so that variational free energies are minimized within a restricted degree of freedom m. As a consequence of the renormalization, spin variables on each corner of CTM are replaced by a m-state block spin variable. It is shown that the thermodynamic functions and critical exponents of the q = 2, 3 Potts models can be precisely evaluated by use of the renormalization group method.Comment: 20 pages, 10 ps figures, JPSJ style files are include

We determine the magnetic phase diagram of the S = 1/2 antiferromagnetic zigzag spin chain in the strongly frustrated region, using the density matrix renormalization group method. We find the magnetization plateau at 1/3 of the full moment accompanying the spontaneous symmetry breaking of the translation, the cusp singularities above and/or below the plateau, and the even-odd effect in the magnetization curve. We also discuss the formation mechanisms of the plateau and cusps briefly.KEYWORDS: frustration, cusp, 1/3 plateau, even-odd effect For the purpose of clarifying the role of the frustration in low-dimensional quantum spin systems, the S = 1/2 antiferromagnetic zigzag spin chain has been attracting considerable attention, since it minimally contains the frustrating interaction without loss of the translational invariance.1-4) The Hamiltonian of the model is given bywhere S is the S = 1/2 spin operator, H is the magnetic field, and J 1 and J 2 denote the nearest and next nearest neighbor couplings, respectively. We introduce the notation α = J 2 /J 1 for simplicity. Recently, the zigzag chain was realized as SrCuO 2 , 5) Cu(ampy)Br 2 , 6)(N 2 H 5 )CuCl 3 7) and F 2 PIMNH. 8)The Hamiltonian (1) has a very simple form, but captures a variety of behaviors induced by the frustration. In particular, the zigzag chain in a magnetic field has been studied actively [9][10][11][12][13] and it has been clarified that cusp singularities appear near the saturation field and/or in the low field region for α ≤ 0.6, in accordance with the frustration-driven shape change of the dispersion curve of the elementary excitations.12-14) However, the magnetic phase diagram including the strongly frustrated region (α > 0.6) has not been acquired yet. Here we remark that the magnetization curve of α = 0.6 is still quite different from the one in the α → ∞ limit (see Fig.2 (a)). Thus, as α is increased beyond α = 0.6, the intrinsic structural change of the magnetization curve can be expected particularly around α ≃ 1, where the most significant competition is achieved.In this paper we address the magnetization process of the zigzag chain in the strongly frustrated region(α > 0.6), using the density matrix renormalization group (DMRG) method.15) We then find that the obtained phase diagram exhibits rich physics, as is shown in Fig.1. For 0.56 < ∼ α < ∼ 1.25, the magnetization plateau appears at 1/3 of the full moment, accompanying the spontaneous breaking of the translational symmetry with the period three. Moreover, the cusp singularities in the magnetization curve show quite interesting behavior; as α is increased, the high field cusp merges into the 1/3 plateau at α ≃ 0.82. Also the low field cusp merges the 1/3 plateau at α ≃ 0.7, but it appears again when α > 0.7. In addition, we find an interesting even-odd effect in the magnetization curve for α > 0.7. In analyzing the phase diagram, an important feature of the zigzag chain is that the Hamiltonian (1) interpolates between the single Heisenberg chain(J 2 = 0) and the double Hei...

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

( ) SynopsisThe authors propose a fast numerical renormalization group method -the product wave function renormalization group (PWFRG) methodfor 1D quantum lattice models and 2D classical ones. A variational wave function, which is expressed by a matrix product, is improved through a self-consistent calculation. The new method has the same fixed point as the density matrix renormalization group method.-to appear in J. Phys. Soc. Jpn. -1

We apply a recently developed numerical renormalization group, the cornertransfer-matrix renormalization group (CTMRG), to 2D classical lattice models at their critical temperatures. It is shown that the combination of CTMRG and the finite-size scaling analysis gives two independent critical exponents.

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 generalize the corner transfer matrix renormalization group, which consists of White's density matrix algorithm and Baxter's method of the corner transfer matrix, to three dimensional (3D) classical models. The renormalization group transformation is obtained through the diagonalization of density matrices for a cubic cluster. A trial application for 3D Ising model with m=2 is shown as the simplest case.Comment: 15 pages, Latex(JPSJ style files are included), 8 ps figures, submitted to J. Phys. Soc. Jpn., some references are correcte