The Matrix Product Approach to Quantum Spin Ladders

We present a manifestly rotational invariant formulation of the matrix product method valid for spin chains and ladders. We apply it to 2 legged spin ladders with spins 1/2, 1 and 3/2 and different magnetic structures labelled by the exchange coupling constants, which can be ferromagnetic or antiferromagnetic along the legs and the rungs of the ladder We compute ground state energy densities, correlation lengths and string order parameters. We present numerical evidence of the duality properties of the 3 different non ferromagnetic spin 1/2 ladders. We show that the long range topological order characteristic of isolated spin 1 chains is broken by the interchain coupling. The string order correlation function decays exponentially with a finite correlation length that we compute. A physical picture of the spin 1 ladder is given in terms of a collection of resonating spin 1 chains. Finally for ladders with spin equal or greater than 3/2 we define a class of AKLT states whose matrix product coefficients are given by 9-j symbols.Comment: REVTEX file, 16 pages, 12 figures, 6 Table

Universality Classes of Diagonal Quantum Spin Ladders

We find the classification of diagonal spin ladders depending on a characteristic integer $N_p$ in terms of ferrimagnetic, gapped and critical phases. We use the finite algorithm DMRG, non-linear sigma model and bosonization techniques to prove our results. We find stoichiometric contents in cuprate $CuO_2$ planes that allow for the existence of weakly interacting diagonal ladders.Comment: REVTEX4 file, 3 color figures, 1 tabl

Critical Lines and Massive Phases in Quantum Spin Ladders with Dimerization

We determine the existence of critical lines in dimerized quantum spin ladders in their phase diagram of coupling constants using the finite-size DMRG algorithm. We consider both staggered and columnar dimerization patterns, and antiferromagnetic and ferromagnetic inter-leg couplings. The existence of critical phases depends on the precise combination of these patterns. The nature of the massive phases separating the critical lines are characterized with generalized string order parameters that determine their valence bond solid (VBS) content.Comment: 9 pages 10 figure

DMRG study of the Bond Alternating \textbf{S}=1/2 Heisenberg ladder with Ferro-Antiferromagnetic couplings

We obtain the phase diagram in the parameter space $(J'/J, \gamma)$ and an accurate estimate of the critical line separating the different phases. We show several measuments of the magnetization, dimerization, nearest neighbours correlation, and density of energy in the different zones of the phase diagram, as well as a measurement of the string order parameter proposed as the non vanishing phase order parameter characterizing Haldane phases. All these results will be compared in the limit $J'/J\gg 1$ with the behaviour of the $\textbf{S}=1$ Bond Alternated Heisenberg Chain (BAHC). The analysis of our data supports the existence of a dimer phase separated by a critical line from a Haldane one, which has exactly the same nature as the Haldane phase in the $\textbf{S}=1$ BAHC.Comment: Version 4. 8 pages, 15 figures (12 figures in document

Quantum Spins and Quasiperiodicity: a real space renormalization group approach

We study the antiferromagnetic spin-1/2 Heisenberg model on a two-dimensional bipartite quasiperiodic structure, the octagonal tiling -- the aperiodic equivalent of the square lattice for periodic systems. An approximate block spin renormalization scheme is described for this problem. The ground state energy and local staggered magnetizations for this system are calculated, and compared with the results of a recent Quantum Monte Carlo calculation for the tiling. It is conjectured that the ground state energy is exactly equal to that of the quantum antiferromagnet on the square lattice.Comment: To appear in Physical Review Letter