arXiv:cond-mat/0601254

Scale perturbation in valence bond ground states

E. Rico

Published Jan 12, 2006• 0 citations

Abstract

A simple and efficient method for calculating the ground state for a class of antiferromagnet systems is presented. It combines the valence bond structure of the ground state for this class of systems and real space renormalization group. As an example, the entire Haldane phase for a spin-1 chain is described, from the AKLT model through to the Heisenberg model, ending at the critical WZNW $SU_2(2)$ model. The picture that emerges from this new method leads us to consider the relationship between the hidden topological order that characterizes the Haldane phase and potential applications in quantum communication technology.

DOI: 10.48550/arxiv.cond-mat/0601254

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arXiv:cond-mat/0601254

Scale perturbation in valence bond ground states

E. Rico

Published Jan 12, 2006• 0 citations

arXiv HTML PDF

Abstract

SU_2(2)

model. The picture that emerges from this new method leads us to consider the relationship between the hidden topological order that characterizes the Haldane phase and potential applications in quantum communication technology.

DOI: 10.48550/arxiv.cond-mat/0601254

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Ground-State Phase Diagram of the Spin- $\frac{\bf 1}{\bf 2}$ - $\frac{\bf 1}{\bf 2}$ -1-1 ChainGround-State Phase Diagram of the Spin-$\frac{\bf 1}{\bf 2}$-$\frac{\bf 1}{\bf 2}$-1-1 Chain

Toshiya Hikihara, Takashi Tonegawa, Makoto Kaburagi +3 more

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SU_2(2)

We study the ground-state properties of antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ chains with half-integer spins ranging from

S=\frac{3}{2}

S=\frac{11}{2}

using the density-matrix renormalization group method. We map out the ground-state phase diagrams as a function of

\frac{{J}_{2}}{{J}_{1}}

containing topological phases with alternating $\frac{2S\ensuremath{-}1}{2}$ and

\frac{2S+1}{2}

valence bonds. We identify these topological phases and their boundaries by calculating the string order parameter, the dimer order parameter, and the spin gap for those $\mathrm{high}\text{\ensuremath{-}}S$ systems in the thermodynamic limit (finite-size scaling). We find that these topological regions narrow down inversely with

S

and converge to a single point at

\frac{{J}_{2}}{{J}_{1}}=\frac{1}{4}

in the classical limit---a critical threshold between commensurate and incommensurate orders. In addition, we extend the discussion of the Majumder-Ghosh state, previously noted only for

S=\frac{1}{2}

, and speculate about its possible presence as a ground state in half-integer high-spin systems over a substantial range of

\frac{{J}_{2}}{{J}_{1}}

values.

Ground-State Phase Diagram of the Spin-

\frac{\bf 1}{\bf 2}

\frac{\bf 1}{\bf 2}

-1-1 Chain

We investigate the ground-state phase diagram of the quantum mixed spin chain in which two

S\!=\!\frac{1}{2}

and two S =1 spins are arranged alternatively and isotropic exchange interactions between nearest-neighboring pairs of spins are assumed. Guided by our previous results [J. Phys. Soc. Jpn. 67 (1998) 1000] for a few limiting cases of the exchange interactions, we carry out a density-matrix renormalization-group calculation to determine the ground-state phase diagram in detail. From the gapless points of the singlet-triplet energy gap, five boundary lines are determined, which divide the phase diagram into six regions. In three regions the ground states with the Haldane-phase character appear, while in the remaining three regions those with the dimer character appear. We also present a characterization of the phases concerning how the ground state in each phase is represented in the framework of the valence-bond-solid (VBS) picture. Making use of the matrix-product wave function, we classify the VBS pictures and interpret the ground-state phase diagram in terms of symmetries schematically related to the VBS pictures. We investigate the ground-state nearest-neighboring two-spin correlation functions and a kind of string order parameters, by which we confirm the validity of the characterization.