arXiv:cond-mat/9909388 · Physical review. B, Condensed matter

Finite-size spectrum, magnon interactions, and magnetization of S=1 Heisenberg spin chains

Jizhong Lou, Shaojin Qin, Tai Kai Ng, Zhao-Bin Su, Ian K. Affleck

Published Aug 1, 2000• 29 citations

Abstract

We report on density-matrix renormalization-group and analytical work on $S=1$ antiferromagnetic Heisenberg spin chains. We study the finite-size behavior within the framework of the nonlinear $\ensuremath{\sigma}$ model. We study the effect of magnon-magnon interactions on the finite-size spectrum and on the magnetization curve close to the critical magnetic field, determine the magnon scattering length, and compare it to the prediction from the nonlinear $\ensuremath{\sigma}$ model.

DOI: 10.1103/physrevb.62.3786

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arXiv:cond-mat/9909388 · Physical review. B, Condensed matter

Finite-size spectrum, magnon interactions, and magnetization of S=1 Heisenberg spin chains

Jizhong Lou, Shaojin Qin, Tai Kai Ng, Zhao-Bin Su, Ian K. Affleck

Published Aug 1, 2000• 29 citations

arXiv HTML PDF

Abstract

We report on density-matrix renormalization-group and analytical work on

S=1

antiferromagnetic Heisenberg spin chains. We study the finite-size behavior within the framework of the nonlinear

\ensuremath{\sigma}

model. We study the effect of magnon-magnon interactions on the finite-size spectrum and on the magnetization curve close to the critical magnetic field, determine the magnon scattering length, and compare it to the prediction from the nonlinear

\ensuremath{\sigma}

model.

DOI: 10.1103/physrevb.62.3786

Related papers

Magnetization of the S=2 antiferromagnetic Heisenberg chain near the critical magnetic field

Jizhong Lou, Shaojin Qin, Zhao-Bin Su

Using the density-matrix renormalization-group (DMRG) technique, we carry out a large scale numerical calculation for the $S=2$ antiferromagnetic Heisenberg (AFH) chain. We obtain the magnetization curve of $S=2$ chain near the critical field. The corrections to the magnetization due to the magnon-magnon interaction are the same order for $S=2$ and $S=1$ chains. By comparing the experimental magnetization with our calculation, we find that the gap obtained experimentally is about $0.08J$ which is close to the accurate DMRG estimation for the $S=2$ AFH chain. Numerical errors in our large scale calculation are also discussed.Using the density-matrix renormalization-group (DMRG) technique, we carry out a large scale numerical calculation for the $S=2$ antiferromagnetic Heisenberg (AFH) chain. We obtain the magnetization curve of $S=2$ chain near the critical field. The corrections to the magnetization due to the magnon-magnon interaction are the same order for $S=2$ and $S=1$ chains. By comparing the experimental magnetization with our calculation, we find that the gap obtained experimentally is about $0.08J$ which is close to the accurate DMRG estimation for the $S=2$ AFH chain. Numerical errors in our large scale calculation are also discussed.

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We report a density-matrix renormalization-group (DMRG) study of S=1/2 antiferromagnetic Heisenberg spin chains. We show the Friedel oscillations in open spin chains within bosonization theory. The bosonization predictions are in excellent agreement with the DMRG calculation. The chain length dependence of the ground state and several low-energy excited state energies are compared with conformal field theory results. Based upon the DMRG calculations for various quantities, the interplay between the system size and the number of optimal states kept in the DMRG is carefully discussed. \textcopyright{} 1996 The American Physical Society.We report a density-matrix renormalization-group (DMRG) study of S=1/2 antiferromagnetic Heisenberg spin chains. We show the Friedel oscillations in open spin chains within bosonization theory. The bosonization predictions are in excellent agreement with the DMRG calculation. The chain length dependence of the ground state and several low-energy excited state energies are compared with conformal field theory results. Based upon the DMRG calculations for various quantities, the interplay between the system size and the number of optimal states kept in the DMRG is carefully discussed. \textcopyright{} 1996 The American Physical Society.

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Numerical renormalization-group study of low-lying eigenstates of the antiferromagnetic S =1 Heisenberg chain

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We present results of a numerical renormalization-group study of the isotropic S=1 Heisenberg chain. The density-matrix renormalization-group techniques used allow us to calculate a variety of properties of the chain with unprecedented accuracy. The ground-state energy per site of the infinite chain is found to be ${\mathit{e}}_{0}$ \ensuremath{\simeq}-1.401 484 038 971(4). Open-ended S=1 chains have effective S=1/2 spins on each end, with exponential decay of the local spin moment away from the ends, with decay length \ensuremath{\xi}\ensuremath{\simeq}6.03(1). The spin-spin correlation function also decays exponentially, and although the correlation length cannot be measured as accurately as the open-end decay length, it appears that the two lengths are identical. The string correlation function shows long-range order, with g(\ensuremath{\infty})\ensuremath{\simeq}-0.374 325 096(2). The excitation energy of the first excited state, a state with one magnon with momentum q=\ensuremath{\pi}, is the Haldane gap, which we find to be \ensuremath{\Delta}\ensuremath{\simeq}0.410 50(2). We find many low-lying excited states, including one- and two-magnon states, for several different chain lengths. The magnons have spin S=1, so the two-magnon states are singlets (S=0), triplets (S=1), and quintuplets (S=2). For magnons with momenta near \ensuremath{\pi}, the magnon-magnon interaction in the triplet channel is shown to be attractive, while in the singlet and quintuplet channels it is repulsive.We present results of a numerical renormalization-group study of the isotropic S=1 Heisenberg chain. The density-matrix renormalization-group techniques used allow us to calculate a variety of properties of the chain with unprecedented accuracy. The ground-state energy per site of the infinite chain is found to be ${\mathit{e}}_{0}$\ensuremath{\simeq}-1.401 484 038 971(4). Open-ended S=1 chains have effective S=1/2 spins on each end, with exponential decay of the local spin moment away from the ends, with decay length \ensuremath{\xi}\ensuremath{\simeq}6.03(1). The spin-spin correlation function also decays exponentially, and although the correlation length cannot be measured as accurately as the open-end decay length, it appears that the two lengths are identical. The string correlation function shows long-range order, with g(\ensuremath{\infty})\ensuremath{\simeq}-0.374 325 096(2). The excitation energy of the first excited state, a state with one magnon with momentum q=\ensuremath{\pi}, is the Haldane gap, which we find to be \ensuremath{\Delta}\ensuremath{\simeq}0.410 50(2). We find many low-lying excited states, including one- and two-magnon states, for several different chain lengths. The magnons have spin S=1, so the two-magnon states are singlets (S=0), triplets (S=1), and quintuplets (S=2). For magnons with momenta near \ensuremath{\pi}, the magnon-magnon interaction in the triplet channel is shown to be attractive, while in the singlet and quintuplet channels it is repulsive.

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We investigate the behavior of antiferromagnetic integer-spin chains in a staggered magnetic field, by means of the density-matrix renormalization group, carefully addressing the role of finite-size effects within the Haldane phase at small fields. In the case of spin $S=2,$ we determine the dependence of the ground-state energy and magnetization on the external field, in the thermodynamic limit, and show how the peculiar finite-size behavior can be connected with the crossover in the ground state from a spin liquid to a polarized N\'eel state.We investigate the behavior of antiferromagnetic integer-spin chains in a staggered magnetic field, by means of the density-matrix renormalization group, carefully addressing the role of finite-size effects within the Haldane phase at small fields. In the case of spin $S=2,$ we determine the dependence of the ground-state energy and magnetization on the external field, in the thermodynamic limit, and show how the peculiar finite-size behavior can be connected with the crossover in the ground state from a spin liquid to a polarized N\'eel state.

arXiv:cond-mat/0107282Physical review. B, Condensed matterOct 19, 200110 citations

We report on density-matrix renormalization-group and analytical work on

S=1

antiferromagnetic Heisenberg spin chains. We study the finite-size behavior within the framework of the nonlinear

\ensuremath{\sigma}

\ensuremath{\sigma}

model.

Using the density-matrix renormalization-group (DMRG) technique, we carry out a large scale numerical calculation for the

S=2

antiferromagnetic Heisenberg (AFH) chain. We obtain the magnetization curve of

S=2

chain near the critical field. The corrections to the magnetization due to the magnon-magnon interaction are the same order for

S=2

and

S=1

chains. By comparing the experimental magnetization with our calculation, we find that the gap obtained experimentally is about

0.08J

which is close to the accurate DMRG estimation for the

S=2

AFH chain. Numerical errors in our large scale calculation are also discussed.

We consider the properties of finite isotropic antiferromagnetic Heisenberg chains with

S=1/2,1,3/2

spins when a weak magnetic field is applied on a few sites, using White's density-matrix renormalization-group method. For the

S=1

chain there exists only one length scale in the system which determines the behavior of the one- and two-point correlation functions both around the local perturbation and near the free boundary. For the critical half-odd-integer spin cases the exponent of the spin-spin correlation function was found to be $\ensuremath{\eta}=1$ and the exponent of the decay of the site magnetization around the perturbed site is ${x}_{m}=\ensuremath{\eta}/2.$ Close to a free boundary, however, the behavior is completely different for

S=1/2

and

S>1/2.

{\mathit{e}}_{0}

\ensuremath{\simeq}-1.401 484 038 971(4). Open-ended S=1 chains have effective S=1/2 spins on each end, with exponential decay of the local spin moment away from the ends, with decay length \ensuremath{\xi}\ensuremath{\simeq}6.03(1). The spin-spin correlation function also decays exponentially, and although the correlation length cannot be measured as accurately as the open-end decay length, it appears that the two lengths are identical. The string correlation function shows long-range order, with g(\ensuremath{\infty})\ensuremath{\simeq}-0.374 325 096(2). The excitation energy of the first excited state, a state with one magnon with momentum q=\ensuremath{\pi}, is the Haldane gap, which we find to be \ensuremath{\Delta}\ensuremath{\simeq}0.410 50(2). We find many low-lying excited states, including one- and two-magnon states, for several different chain lengths. The magnons have spin S=1, so the two-magnon states are singlets (S=0), triplets (S=1), and quintuplets (S=2). For magnons with momenta near \ensuremath{\pi}, the magnon-magnon interaction in the triplet channel is shown to be attractive, while in the singlet and quintuplet channels it is repulsive.

1/2

Heisenberg antiferromagnet; an analytic calculation shows that logarithmic corrections to scaling can sometimes be avoided. We also examine the spin-1 chain at the critical point separating the Haldane gap and dimerized phases. Exponents for the dimer-dimer and the spin-spin correlation functions are consistent with results obtained from bosonization.

S=2,

we determine the dependence of the ground-state energy and magnetization on the external field, in the thermodynamic limit, and show how the peculiar finite-size behavior can be connected with the crossover in the ground state from a spin liquid to a polarized N\'eel state.