Corner Transfer Matrix Renormalization Group Method

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. PACS codes: 05.50.+q, 02.70.-c, 75.10.Hk.The renormalization group is one of the basic concepts in physics [1, 2]. Real space representation of the renormalization group — the real space renormalization group — has been applied to various lattice models [3]. Recently, White established a numerical renormalization algorithm, which is referred as ‘density matrix renormalization group (DMRG) method ’ [4, 5]. The method has been applied to various one-dimensional (1D) quantum lattice models [5, 6, 7], because it is possible to treat large scale systems with relatively small numerical calculation. Although DMRG method was originally proposed as a renormalization procedure for 1D quantum systems, the method has an implicit relation with 2D classical models. Östlund and Rommer analyzed the thermodynamic limit of the DMRG method [8], and pointed out that the method is a mapping from 1D quantum lattice models to effective classical lattice models. They show that the orthogonal matrix, which represents the block-spin transformation, plays a role of a transfer matrix of the classical model. What is the classical model, then? Roughly speaking, the model corresponds to a 2D square-lattice model, which is obtained through the Trotter decomposition [9, 10] of the operator exp(−β ˆ H) [11]; the row-to-row transfer matrix corresponds to the imaginary time shift operator exp(−∆β ˆ H); the transfer matrix discussed by Östlund and Rommer is a renormalized column-to-column transfer matrix. The relation between the DMRG method and 2D classical systems leads a new view point. We find that the density matrix is expressed as a product of Baxter’s corner transfer matrices (CTMs) [12, 13, 14]. Moreover, the DMRG method and