The density matrix renormalization group (DMRG) method is applied to the interaction round a face (IRF) model. When the transfer matrix is asymmetric, singular-value decomposition of the density matrix is required. A trial numerical calculation is performed on the square lattice Ising model, which is a special case of the IRF model.
Given a Hamiltonian with a continuous symmetry one can generally factorize that symmetry and consider the dynamics on invariant Hilbert Spaces. In Statistical Mechanics this procedure is known as the vertex-IRF map, and in certain cases, like rotational invariant Hamiltonians, can be implemented via group theoretical techniques. Using this map we translate the DMRG method, which applies to 1d vertex Hamiltonians, into a formulation adequate to study IRF Hamiltonians. The advantage of the IRF formulation of the DMRG method ( we name it IRF-DMRG), is that the dimensions of the Hilbert Spaces involved in numerical computations are smaller than in the vertex-DMRG, since the degeneracy due to the symmetry has been eliminated. The IRF-DMRG admits a natural and geometric formulation in terms of the paths or string algebras used in Exactly Integrable Systems and Conformal Field Theory. We illustrate the IRF-DMRG method with the study of the SOS model which corresponds to the spin 1/2 Heisenberg chain and the RSOS models with Coxeter diagram of type A, which correspond to the quantum group invariant XXZ chain.
An interaction-round-a-face density-matrix renormalization-group (IRF-DMRG) method is developed for higher integer spin chain models which are rotational invariant. The expressions of the IRF weights associated with the nearest-neighbor spin-S interaction ${\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}$ are explicitly derived. Using the IRF-DMRG with these IRF weights, the Haldane gaps $\ensuremath{\Delta}$ and the ground state energy densities e0 for both S=1 and S=2 isotropic antiferromagnetic Heisenberg quantum spin chains are calculated by keeping up to only m=90 states.
A density-matrix renormalization-group (DMRG) method for highly anisotropic two-dimensional (2D) systems is presented. The method consists of applying the usual DMRG in two steps. In the first step, a pure one-dimensional calculation along the longitudinal direction is made in order to generate a low-energy Hamiltonian. In the second step, the anisotropic 2D lattice is obtained by coupling in the transverse direction the 1D Hamiltonians. The method is applied to the anisotropic quantum spin-half Heisenberg model on a square lattice.
An interaction-round-a-face density-matrix renormalization-group (IRF-DMRG) method is developed for higher integer spin chain models which are rotational invariant. The expressions of the IRF weights associated with the nearest-neighbor spin-S interaction Si⋅Si+1 are derived explicitly. Using the IRF-DMRG with these IRF weights, the Haldane gaps Δ and the ground state energy densities e0 for both S=1 and S=2 isotropic antiferromagnetic Heisenberg quantum spin chains are calculated by keeping up to only m=90 states.
The density matrix renormalization group (DMRG) method and its applications to finite temperatures and two-dimensional systems are reviewed. The basic idea of the original DMRG method, which allows precise study of the ground-state properties and low-energy excitations, is presented for models which include long-range interactions. The DMRG scheme is then applied to the diagonalization of the quantum transfer matrix for one-dimensional systems, and a reliable algorithm at finite temperatures is formulated. Dynamic correlation functions at finite temperatures are calculated from the eigenvectors of the quantum transfer matrix with analytical continuation to the real frequency axis. An application of the DMRG method to two-dimensional quantum systems in a magnetic field is demonstrated and reliable results for quantum Hall systems are presented.