arXiv:cond-mat/9610100 · Physical review. B, Condensed matter

Finite-size scaling for low-energy excitations in integer Heisenberg spin chains

Shaojin Qin, Yu-Liang Liu, Lu Yu

Published Feb 1, 1997• 30 citations

Abstract

In this paper we study the finite-size scaling for low-energy excitations of S=1 and S=2 Heisenberg chains, using the density matrix renormalization-group technique. A crossover from 1/L behavior (with L as the chain length) for medium chain length to 1/${\mathrm{L}}^{2}$ scaling for long chain length is found for excitations in the continuum band as the length of the open chain increases. Topological spin S=1/2 excitations are shown to give rise to the two lowest energy states for both open and periodic S=1 chains. In periodic chains these two excitations are ``confined'' next to each other, while for open chains they are two free-edge 1/2 spins. The finite-size scaling of the two lowest energy excitations of open S=2 chains is determined by coupling the two free-edge S=1 spins. The gap and correlation length for S=2 open Heisenberg chains are shown to be 0.082 (in units of the exchange J) and 47, respectively.

DOI: 10.1103/physrevb.55.2721

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arXiv:cond-mat/9610100 · Physical review. B, Condensed matter

Finite-size scaling for low-energy excitations in integer Heisenberg spin chains

Shaojin Qin, Yu-Liang Liu, Lu Yu

Published Feb 1, 1997• 30 citations

arXiv HTML PDF

Abstract

{\mathrm{L}}^{2}

scaling for long chain length is found for excitations in the continuum band as the length of the open chain increases. Topological spin S=1/2 excitations are shown to give rise to the two lowest energy states for both open and periodic S=1 chains. In periodic chains these two excitations are ``confined'' next to each other, while for open chains they are two free-edge 1/2 spins. The finite-size scaling of the two lowest energy excitations of open S=2 chains is determined by coupling the two free-edge S=1 spins. The gap and correlation length for S=2 open Heisenberg chains are shown to be 0.082 (in units of the exchange J) and 47, respectively.

DOI: 10.1103/physrevb.55.2721

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Seiji Miyashita, Shoji Yamamoto

Properties of the ground state of open chains of the S=1 Heisenberg antiferromagnet are studied by a quantum Monte Carlo method. Size (length) dependences of the ground-state energy, the energy gap, the staggered magnetization 〈 ${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$ 〉 and the spin-correlation function 〈 ${\mathit{S}}_{1}^{\mathit{z}}$ ${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$ 〉 are investigated under free, fixed, and periodic boundary conditions. The size dependences are significant in chains with less than 25 spins, where the magnitude of the staggered magnetization is rather large. However, the fourfold degeneracy of the ground state and the exponential decay of correlations that are inherent to the Haldane system are recovered when the chain length becomes long enough. Both the staggered magnetization and the spin-correlation function decay exponentially. The correlation length is about six in lattice spacing, which agrees with the value obtained in the bulk.Properties of the ground state of open chains of the S=1 Heisenberg antiferromagnet are studied by a quantum Monte Carlo method. Size (length) dependences of the ground-state energy, the energy gap, the staggered magnetization 〈${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$〉 and the spin-correlation function 〈${\mathit{S}}_{1}^{\mathit{z}}$${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$〉 are investigated under free, fixed, and periodic boundary conditions. The size dependences are significant in chains with less than 25 spins, where the magnitude of the staggered magnetization is rather large. However, the fourfold degeneracy of the ground state and the exponential decay of correlations that are inherent to the Haldane system are recovered when the chain length becomes long enough. Both the staggered magnetization and the spin-correlation function decay exponentially. The correlation length is about six in lattice spacing, which agrees with the value obtained in the bulk.

Physical review. B, Condensed matterJul 1, 1993123 citations

{\mathrm{L}}^{2}

5/2.

For open chains with spin magnitudes

S=5/2

S=5,

we verify that end states with fractional spin quantum numbers ${S}^{\ensuremath{'}}$ exist and are visible in the structure of the low-energy excitation spectrum even when the chain length is much smaller than the correlation length \ensuremath{\xi}. The robustness of end states points to the possibility of the existence of general hidden topological order in quantum spin chains with spin magnitudes

S>~1.

We study the finite-size behavior of the low-lying excitations of spin-

\frac{1}{2}

Heisenberg chains with dimerization and next-to-nearest neighbors interaction

{J}_{2}

. The numerical analysis, performed using density-matrix renormalization group, confirms previous exact diagonalization results and shows that, for different values of the dimerization parameter $\ensuremath{\delta}$, the elementary triplet and singlet excitations present a clear scaling behavior in a wide range of $l=L∕\ensuremath{\xi}$ (where

L

is the length of the chain and $\ensuremath{\xi}$ is the correlation length). At

{J}_{2}={J}_{2c}

, where no logarithmic corrections are present, we compare the numerical results with finite-size predictions for the sine-Gordon model obtained using L\"uscher's theory. For small $\ensuremath{\delta}$ we find a very good agreement for $l\ensuremath{\gtrsim}4$ or 7 depending on the excitation considered.

{\mathit{S}}_{\mathit{i}}^{\mathit{z}}

〉 and the spin-correlation function 〈

{\mathit{S}}_{1}^{\mathit{z}}

{\mathit{S}}_{\mathit{i}}^{\mathit{z}}

〉 are investigated under free, fixed, and periodic boundary conditions. The size dependences are significant in chains with less than 25 spins, where the magnitude of the staggered magnetization is rather large. However, the fourfold degeneracy of the ground state and the exponential decay of correlations that are inherent to the Haldane system are recovered when the chain length becomes long enough. Both the staggered magnetization and the spin-correlation function decay exponentially. The correlation length is about six in lattice spacing, which agrees with the value obtained in the bulk.