Infinite Density Matrix Renormalization Group Research Articles

Topological characterization of fractional quantum Hall ground states from microscopic Hamiltonians.

We show how to numerically calculate several quantities that characterize topological order starting from a microscopic fractional quantum Hall Hamiltonian. To find the set of degenerate ground states, we employ the infinite density matrix renormalization group method based on the matrix-product state representation of fractional quantum Hall states on an infinite cylinder. To study localized quasiparticles of a chosen topological charge, we use pairs of degenerate ground states as boundary conditions for the infinite density matrix renormalization group. We then show that the wave function obtained on the infinite cylinder geometry can be adapted to a torus of arbitrary modular parameter, which allows us to explicitly calculate the non-Abelian Berry connection associated with the modular T transformation. As a result, the quantum dimensions, topological spins, quasiparticle charges, chiral central charge, and Hall viscosity of the phase can be obtained using data contained entirely in the entanglement spectrum of an infinite cylinder.

A Classical Background for the Wave Function Prediction in the Infinite System Density Matrix Renormalization Group Method

We report a physical background of the wave function prediction in the infinite system density matrix renormalization group (DMRG) method, from the view point of two-dimensional vertex model, a typical lattice model in statistical mechanics. Singular value decomposition applied to rectangular corner transfer matrices naturally draws matrix product representation for the maximal eigenvector of the row-to-row transfer matrix. The wave function prediction can be expressed as the insertion of an approximate half-column transfer matrix. This insertion process is in accordance with the scheme proposed by McCulloch recently.

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 of a two-dimensional classical lattice model. A state vector created from the upper or the lower half of a finite size cluster approximates the largest-eigenvalue eigenvector. Decomposition of this state vector into the MPS gives a way of extending the MPS recursively. The extension process is a special case of the product wave function renormalization group (PWFRG) method, that accelerates the numerical calculation of the infinite system density matrix renormalization group (DMRG) method. As a result, we successfully give the physical interpretation of the PWFRG method, and obtain its appropriate initial condition.