Corner Transfer Matrix Renormalization Group Method Applied to the Ising Model on the Hyperbolic Plane

Critical behavior of the Ising model is investigated at the center of large scale finite size systems, where the lattice is represented as the tiling of pentagons. The system is on the hyperbolic plane, and the recursive structure of the lattice makes it possible to apply the corner transfer matrix renormalization group method. From the calculated nearest neighbor spin correlation function and the spontaneous magnetization, it is concluded that the phase transition of this model is mean-field like. One parameter deformation of the corner Hamiltonian on the hyperbolic plane is discussed.Comment: 4 pages, 5 figure

Magnetic phase diagram of the S=1/2 antiferromagnetic zigzag spin chain in the strongly frustrated region: cusp and plateau

We determine the magnetic phase diagram of the antiferromagnetic(AF) 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.Comment: 4 pages, 8 figures, revised version, to appear in J.Phys.Soc.Jp

Wilson-like real-space renormalization group and low-energy effective spectrum of the XXZ chain in the critical regime

We present a novel real-space renormalization group(RG) for the one-dimensional XXZ model in the critical regime, reconsidering the role of the cut-off parameter in Wilson's RG for the Kondo impurity problem. We then demonstrate the RG calculation for the XXZ chain with the free boundary. Comparing the hierarchical structure of the obtained low-energy spectrum with the Bethe ansatz result, we find that the proper scaling dimension is reproduced as a fixed point of the RG transformation.Comment: 4 pages, 6 figures, typos corrected, final versio

Hyperbolic Deformation Applied to S = 1 Spin Chains - Scaling Relation in Excitation Energy -

We investigate excitation energies of hyperbolically deformed S = 1 spin chains, which are specified by the local energy scale f_j^{~} = \cosh j \lambda, where j is the lattice index and \lambda is the deformation parameter. The elementary excitation is well described by a quasiparticle hopping model, which is also expressed in the form of hyperbolic deformation. It is possible to estimate the excitation gap \Delta in the uniform limit \lambda \rightarrow 0, by means of a finite size scaling with respect to the system size N and the deformation parameter \lambda.Comment: 5 pages, 4 figure

Product Wave Function Renormalization Group: construction from the matrix product point of view

We present a construction of a matrix product state (MPS) that approximates the largest-eigenvalue eigenvector of a transfer matrix T, for the purpose of rapidly performing the infinite system density matrix renormalization group (DMRG) method applied to two-dimensional classical lattice models. We use the fact that the largest-eigenvalue eigenvector of T can be approximated by a state vector created from the upper or lower half of a finite size cluster. Decomposition of the obtained state vector into the MPS gives a way of extending the MPS, at the system size increment process in the infinite system DMRG algorithm. As a result, we successfully give the physical interpretation of the product wave function renormalization group (PWFRG) method, and obtain its appropriate initial condition.Comment: 8 pages, 8 figure

Hyperbolic Deformation on Quantum Lattice Hamiltonians

A group of non-uniform quantum lattice Hamiltonians in one dimension is introduced, which is related to the hyperbolic $1 + 1$-dimensional space. The Hamiltonians contain only nearest neighbor interactions whose strength is proportional to $\cosh j \lambda$, where $j$ is the lattice index and where $\lambda \ge 0$ is a deformation parameter. In the limit $\lambda \to 0$ the Hamiltonians become uniform. Spacial translation of the deformed Hamiltonians is induced by the corner Hamiltonians. As a simple example, we investigate the ground state of the deformed $S = 1/2$ Heisenberg spin chain by use of the density matrix renormalization group (DMRG) method. It is shown that the ground state is dimerized when $\lambda$ is finite. Spin correlation function show exponential decay, and the boundary effect decreases with increasing $\lambda$.Comment: 5 pages, 4 figure