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arXiv:0704.2181 · Physical Review B

Density-matrix renormalization group study of the bond-alternating S=1∕2 Heisenberg ladder with ferro-antiferromagnetic couplings

J. O. de Almeida, Miguel Angel Martin-Delgado, Germán Sierra

Published Nov 21, 2007• 24 citations
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Abstract

We obtain the phase diagram in the parameter space $({J}^{\ensuremath{'}}∕J,\ensuremath{\gamma})$ and an accurate estimate of the critical line separating the different phases. We show several measurements of the magnetization, dimerization, nearest neighbor 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 nonvanishing phase order parameter characterizing Haldane phases. All these results will be compared in the limit ${J}^{\ensuremath{'}}∕J⪢1$ with the behavior of the $\mathbf{S}=1$ 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 $\mathbf{S}=1$ BAHC.

DOI
10.1103/physrevb.76.184428

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We obtain the phase diagram in the parameter space (J\ensuremath′∕J,\ensuremathγ)({J}^{\ensuremath{'}}∕J,\ensuremath{\gamma})(J\ensuremath′∕J,\ensuremathγ) and an accurate estimate of the critical line separating the different phases. We show several measurements of the magnetization, dimerization, nearest neighbor 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 nonvanishing phase order parameter characterizing Haldane phases. All these results will be compared in the limit J\ensuremath′∕J⪢1{J}^{\ensuremath{'}}∕J⪢1J\ensuremath′∕J⪢1 with the behavior of the S=1\mathbf{S}=1S=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 S=1\mathbf{S}=1S=1 BAHC.
Using the density matrix renormalization group method, we determine the phase diagram of a frustrated bond-alternating S = 1/2 Heisenberg ladder with ferro-antiferromagnetic couplings at zero temperature. With the interactions between spins along the rungs set, we identify three spin-gapped phases (the Haldane phase, the singlet phase and the dimer phase) in the whole parameter range. The analysis of our data shows that two-leg spin bond-alternating ladders have a rich phase diagram if both rung and diagonal couplings are taken into account.
Dimerized spin-1/2 ladders exhibit a variety of phase structures, which depend on the intra-chain and inter-chain spin exchange energies as well as on the dimerization pattern of the ladder. Using the density matrix renormalization group (DMRG) algorithm, we study critical properties of the bond-alternating two-leg Heisenberg spin ladder with diagonal interaction J ×. Two types of spin systems, staggered dimerized antiferromagnetic ladder and columnar dimerized ferro-antiferromagnetic couplings ladder, are investigated. To clarify the phase transition behaviors, we simultaneously analyze the string order parameter (SOP), the twisted order parameter (TOP), as well as a measurement of the quantum information analysis. Based on measuring this different observables, we establish the phase diagram accurately and give the fitting functions of the phase boundaries. In addition, the phase transition of cross-coupled spin ladder (in the absence of intrinsic dimerization) is also discussed.
It is conjectured that the Haldane phase of the S=1S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1S=1 antiferromagnetic Heisenberg chain and the S=1/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2S=1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chain is stable against any strength of randomness, because of imposed breakdown of translational symmetry. This conjecture is confirmed by the density matrix renormalization group calculation of the string order parameter and the energy gap distribution.
We determine the existence of critical lines in dimerized quantum spin ladders in their phase diagram of coupling constants using the finite-size density matrix renormalization group algorithm. We consider both staggered and columnar dimerization patterns, and antiferromagnetic and ferromagnetic interleg 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 content.
Using density-matrix renormalization group calculations, we investigate the general phase diagram of the frustrated two-leg ladder with Heisenberg interactions along legs, rungs, and diagonals. We confirm that all antiferromagnetic gapped states belong to the same universality class as the Haldane phase. In a three-dimensional phase diagram, we determine a continuous surface with singularities in the string-order parameter or its first derivative, corresponding to a transition between two Haldane phases with different topological order. Some parts of this transition surface are critical with zero gap and vanishing string-order parameter. In the complementary parts, the transition is first order with finite gap and string order. The boundary of this surface with the ferromagnetic region is a critical end line, when the surface is critical, and a triple line anywhere else. Part of this boundary coincides with the exactly soluble model proposed by Dmitriev, Krivnov, and Ovchinnikov [Phys. Rev. B 56, 5985 (1997)].
We study the one-dimensional Heisenberg spin-1∕21∕21∕2 alternating bond chain in which the nearest neighbor exchange couplings are ferromagnetic (FM) and antiferromagnetic (AF) alternatively. By using exact diagonalization and the density matrix renormalization group method, we discuss how the system approaches the AF uniform spin-1 chain under certain conditions. When the ratio of AF to FM coupling strengths, $\ensuremath{\alpha}$ $(\ensuremath{\alpha}={J}_{AF}∕{J}_{F})$, is very small, physical properties of the alternating bond chain such as spin-spin correlation, the string correlation function, and the spin density coincide with those of the AF uniform spin-1 chain. The edge state problem is discussed in the present model with the small $\ensuremath{\alpha}$ limit. In addition, the Haldane gap of the AF uniform spin-1 chain is four times the gap of the system considered.