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Journal of the Physical Society of Japan

Phase Diagram of the Spin-\(\frac{1}{2}\) Bond-Alternating Two-Leg Ladder

Masaaki Nakamura, Takahiro Yamamoto, Kôichirô Ide

Published May 15, 2003• 10 citations
Publisher

Abstract

We study phase transition between two topologically different gapped states in the spin-1/2 bond-alternating two-leg Heisenberg ladder by investigating string order parameters and the level-crossing method based on the exact diagonalization. We determine the phase diagram on the plane of the bond-alternation δ and the rung-leg coupling ratio J ⊥ / J , and conclude that the boundary is given as J ⊥ / J ∝δ 2/3 in the weak coupling region J ⊥ / J , δ≪1.

DOI
10.1143/jpsj.72.1022

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Phase Diagram of the Spin-12\frac{1}{2}21​ Bond-Alternating Two-Leg Ladder
We study phase transition between two topologically different gapped states in the spin-1/2 bond-alternating two-leg Heisenberg ladder by investigating string order parameters and the level-crossing method based on the exact diagonalization. We determine the phase diagram on the plane of the bond-alternation δ and the rung-leg coupling ratio J ⊥ / J , and conclude that the boundary is given as J ⊥ / J ∝δ 2/3 in the weak coupling region J ⊥ / J , δ≪1.
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.
The ground state of the S=1/2S=1/2S=1/2 two-leg ladder with the staggered bond alternations is investigated. We have determined the precise phase diagram on the $\ensuremath{\delta}\ensuremath{-}{J}^{\ensuremath{'}}$ plane $(\ensuremath{\delta}$ and ${J}^{\ensuremath{'}}$ are the magnitudes of the bond alternation and the rung interaction, respectively), employing the level spectroscopic analysis of the numerical diagonalization data obtained by the Lanczos algorithm with the twisted boundary condition. The phase boundary between the leg-dimer state and the rung-dimer state near $(\ensuremath{\delta}{,J}^{\ensuremath{'}})=(0,0)$ is of the form ${J}^{\ensuremath{'}}\ensuremath{\propto}{\ensuremath{\delta}}^{0.69},$ which is consistent with the prediction ${J}^{\ensuremath{'}}\ensuremath{\propto}{\ensuremath{\delta}}^{2/3}$ by Mart\'{\i}n-Delgado, Duleksky, and Sierra, and also by Wang and Nersesyan, but not with ${J}^{\ensuremath{'}}\ensuremath{\propto}{\ensuremath{\delta}}^{2}$ by Cabra and Grynberg.
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)].
Using the density matrix renormalization group method, we have obtained the ground-state phase diagram of a spin-1/2 two-leg antiferromagnetic Heisenberg ladder with a ferromagnetic diagonal interaction ${J}_{\ifmmode\times\else\texttimes\fi{}}.$ The boundary between the two phases found in the system, the Haldane phase and the singlet phase, lies in the frustrated area of the parameter space, i.e., both the rung coupling ${J}_{\ensuremath{\perp}}$ and the diagonal coupling ${J}_{\ifmmode\times\else\texttimes\fi{}}$ are ferromagnetic. The spin gap is found to be insensitive to the change of the ferromagnetic diagonal coupling strength when the ratio of the rung coupling constant ${J}_{\ensuremath{\perp}}$ to the leg coupling constant ${J}_{\ensuremath{\Vert}}$ is 0.5, in agreement with the analysis of recent experimental spin susceptibility data.
The ground-state phase diagram of a Heisenberg $\text{spin}\ensuremath{-}\frac{1}{2}$ system on a two-leg ladder with rung alternation is studied by combining analytical approaches with numerical simulations. For the case of ferromagnetic leg exchanges a unique ferrimagnetic ground state emerges, whereas for the case of antiferromagnetic leg exchanges several different ground states are stabilized depending on the ratio between exchanges along legs and rungs. For the more general case of a honeycomb-ladder model for the case of ferromagnetic leg exchanges besides the usual rung-singlet and saturated ferromagnetic states we obtain a ferrimagnetic Luttinger liquid phase with both linear and quadratic low-energy dispersions and ground-state magnetization continuously changing with system parameters. For the case of antiferromagnetic exchanges along legs, different dimerized states including states with additional topological order are suggested to be realized.