arXiv:cond-mat/0611467 · AIP conference proceedings

How to Calculate Correlation Functions of Heisenberg Chains

Rob Hagemans

Published Jan 1, 2006• 9 citations

Abstract

We describe a method for calculating dynamical spin‐spin correlation functions in the isotropic and anisotropic antiferromagnetic Heisenberg models. Our method is able to produce results with high accuracy over the full parameter space.

DOI: 10.1063/1.2222273

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arXiv:cond-mat/0611467 · AIP conference proceedings

How to Calculate Correlation Functions of Heisenberg Chains

Rob Hagemans

Published Jan 1, 2006• 9 citations

arXiv HTML PDF

Abstract

DOI: 10.1063/1.2222273

Related papers

Dynamics of the Two-Dimensional Finite Heisenberg Model

Chikao Kawabata, Masaki Takeuchi, Masuo Suzuki

Journal of the Physical Society of JapanOct 1, 19760 citations

Correlation functions in Heisenberg magnetic chains: Quantum effects at low temperatures

H.-J. Mikeska

A theoretical approach to the one-dimensional isotropic Heisenberg model at low temperatures is presented which uses rotationally invariant combinations of spin operators only and is thus adapted to treat correctly effects resulting from the absence of long-range order in these systems. Analytic expressions for space- and time-dependent spin correlation functions are given for small values of $\frac{T}{J{S}^{2}}$ and $\frac{1}{S}$ . The results include an expression for the spin correlation function in the antiferromagnetic ground state and a discussion of changes in line shape, due to quantum effects, for short-wavelength magnons as observed in neutron scattering experiments.A theoretical approach to the one-dimensional isotropic Heisenberg model at low temperatures is presented which uses rotationally invariant combinations of spin operators only and is thus adapted to treat correctly effects resulting from the absence of long-range order in these systems. Analytic expressions for space- and time-dependent spin correlation functions are given for small values of $\frac{T}{J{S}^{2}}$ and $\frac{1}{S}$. The results include an expression for the spin correlation function in the antiferromagnetic ground state and a discussion of changes in line shape, due to quantum effects, for short-wavelength magnons as observed in neutron scattering experiments.

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We use the density-matrix renormalization-group technique developed by White to calculate the spin correlation functions 〈 ${\mathit{S}}_{\mathit{n}+\mathit{l}}^{\mathit{z}}$ ${\mathit{S}}_{\mathit{n}}^{\mathit{z}}$ 〉=(-1 ${)}^{\mathit{l}}$ \ensuremath{\omega}(l,N) for isotropic Heisenberg rings up to N=70 sites. The correlation functions for large l and N are found to obey the scaling relation \ensuremath{\omega}(l,N)=\ensuremath{\omega}(l,\ensuremath{\infty})${\mathit{f}}_{\mathit{XY}}^{\mathrm{\ensuremath{\alpha}}}$(l/N) proposed by Kaplan et al., which is used to determine \ensuremath{\omega}(l,\ensuremath{\infty}). The asymptotic correlation function \ensuremath{\omega}(l,\ensuremath{\infty}) and the magnetic structure factor S(q=\ensuremath{\pi}) show logarithmic corrections consistent with \ensuremath{\omega}(l,\ensuremath{\infty})\ensuremath{\sim}a\ensuremath{\surd}lncl /l, where c is related to the cut-off dependent coupling constant ${\mathit{g}}_{\mathrm{eff}}$ ( ${\mathit{l}}_{0}$ )=1/ln( ${\mathit{cl}}_{0}$ ), as predicted by field theoretical treatments.We use the density-matrix renormalization-group technique developed by White to calculate the spin correlation functions 〈${\mathit{S}}_{\mathit{n}+\mathit{l}}^{\mathit{z}}$ ${\mathit{S}}_{\mathit{n}}^{\mathit{z}}$〉=(-1${)}^{\mathit{l}}$\ensuremath{\omega}(l,N) for isotropic Heisenberg rings up to N=70 sites. The correlation functions for large l and N are found to obey the scaling relation \ensuremath{\omega}(l,N)=\ensuremath{\omega}(l,\ensuremath{\infty})${\mathit{f}}_{\mathit{XY}}^{\mathrm{\ensuremath{\alpha}}}$(l/N) proposed by Kaplan et al., which is used to determine \ensuremath{\omega}(l,\ensuremath{\infty}). The asymptotic correlation function \ensuremath{\omega}(l,\ensuremath{\infty}) and the magnetic structure factor S(q=\ensuremath{\pi}) show logarithmic corrections consistent with \ensuremath{\omega}(l,\ensuremath{\infty})\ensuremath{\sim}a\ensuremath{\surd}lncl /l, where c is related to the cut-off dependent coupling constant ${\mathit{g}}_{\mathrm{eff}}$(${\mathit{l}}_{0}$)=1/ln(${\mathit{cl}}_{0}$), as predicted by field theoretical treatments.

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Entanglement Perturbation Theory for Antiferromagnetic Heisenberg Spin Chains

Lihua Wang, S. G. Chung

A recently developed numerical method, entanglement perturbation theory (EPT), is used to study the antiferromagnetic Heisenberg spin chains with z-axis anisotropy $λ$ and magnetic field B. To demonstrate the accuracy, we first apply EPT to the isotropic spin-1/2 antiferromagnetic Heisenberg model, and find that EPT successfully reproduces the exact Bethe Ansatz results for the ground state energy, the local magnetization, and the spin correlation functions (Bethe ansatz result is available for the first 7 lattice separations). In particular, EPT confirms for the first time the asymptotic behavior of the spin correlation functions predicted by the conformal field theory, which realizes only for lattice separations larger than 1000. Next, turning on the z-axis anisotropy and the magnetic field, the 2-spin and 4-spin correlation functions are calculated, and the results are compared with those obtained by Bosonization and density matrix renormalization group methods. Finally, for the spin-1 antiferromagnetic Heisenberg model, the ground state phase diagram in $λ$ space is determined with help of the Roomany-Wyld RG finite-size-scaling. The results are in good agreement with those obtained by the level-spectroscopy method.A recently developed numerical method, entanglement perturbation theory (EPT), is used to study the antiferromagnetic Heisenberg spin chains with z-axis anisotropy $λ$ and magnetic field B. To demonstrate the accuracy, we first apply EPT to the isotropic spin-1/2 antiferromagnetic Heisenberg model, and find that EPT successfully reproduces the exact Bethe Ansatz results for the ground state energy, the local magnetization, and the spin correlation functions (Bethe ansatz result is available for the first 7 lattice separations). In particular, EPT confirms for the first time the asymptotic behavior of the spin correlation functions predicted by the conformal field theory, which realizes only for lattice separations larger than 1000. Next, turning on the z-axis anisotropy and the magnetic field, the 2-spin and 4-spin correlation functions are calculated, and the results are compared with those obtained by Bosonization and density matrix renormalization group methods. Finally, for the spin-1 antiferromagnetic Heisenberg model, the ground state phase diagram in $λ$ space is determined with help of the Roomany-Wyld RG finite-size-scaling. The results are in good agreement with those obtained by the level-spectroscopy method.

arXiv:1109.6758Journal of the Physical Society of JapanOct 23, 20123 citations

\frac{T}{J{S}^{2}}

and

\frac{1}{S}

. The results include an expression for the spin correlation function in the antiferromagnetic ground state and a discussion of changes in line shape, due to quantum effects, for short-wavelength magnons as observed in neutron scattering experiments.

We use the density-matrix renormalization-group technique developed by White to calculate the spin correlation functions 〈

{\mathit{S}}_{\mathit{n}+\mathit{l}}^{\mathit{z}}

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

〉=(-1

{)}^{\mathit{l}}

\ensuremath{\omega}(l,N) for isotropic Heisenberg rings up to N=70 sites. The correlation functions for large l and N are found to obey the scaling relation \ensuremath{\omega}(l,N)=\ensuremath{\omega}(l,\ensuremath{\infty})${\mathit{f}}_{\mathit{XY}}^{\mathrm{\ensuremath{\alpha}}}$(l/N) proposed by Kaplan et al., which is used to determine \ensuremath{\omega}(l,\ensuremath{\infty}). The asymptotic correlation function \ensuremath{\omega}(l,\ensuremath{\infty}) and the magnetic structure factor S(q=\ensuremath{\pi}) show logarithmic corrections consistent with \ensuremath{\omega}(l,\ensuremath{\infty})\ensuremath{\sim}a\ensuremath{\surd}lncl /l, where c is related to the cut-off dependent coupling constant

{\mathit{g}}_{\mathrm{eff}}

(

{\mathit{l}}_{0}

)=1/ln(

{\mathit{cl}}_{0}

), as predicted by field theoretical treatments.

A recently developed numerical method, entanglement perturbation theory (EPT), is used to study the antiferromagnetic Heisenberg spin chains with z-axis anisotropy

λ

and magnetic field B. To demonstrate the accuracy, we first apply EPT to the isotropic spin-1/2 antiferromagnetic Heisenberg model, and find that EPT successfully reproduces the exact Bethe Ansatz results for the ground state energy, the local magnetization, and the spin correlation functions (Bethe ansatz result is available for the first 7 lattice separations). In particular, EPT confirms for the first time the asymptotic behavior of the spin correlation functions predicted by the conformal field theory, which realizes only for lattice separations larger than 1000. Next, turning on the z-axis anisotropy and the magnetic field, the 2-spin and 4-spin correlation functions are calculated, and the results are compared with those obtained by Bosonization and density matrix renormalization group methods. Finally, for the spin-1 antiferromagnetic Heisenberg model, the ground state phase diagram in

λ

space is determined with help of the Roomany-Wyld RG finite-size-scaling. The results are in good agreement with those obtained by the level-spectroscopy method.