arXiv:cond-mat/0508354 · Physical Review B

Luttinger liquid parameter for the spin-1 Heisenberg chain in a magnetic field

Ian K. Affleck

Published Oct 28, 2005• 12 citations

Abstract

A recently derived general formula and older numerical results are combined to deduce the behavior of the transverse correlation exponent for the spin-1 Heisenberg antiferromagnetic chain in an applied magnetic field $\ensuremath{\eta}\ensuremath{\approx}1∕2\ensuremath{-}(2.0)m+O({m}^{2})$, where $m$ is the magnetization per site. A comparison with the O(3) nonlinear $\ensuremath{\sigma}$ model is also made.

DOI: 10.1103/physrevb.72.132414

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arXiv:cond-mat/0508354 · Physical Review B

Luttinger liquid parameter for the spin-1 Heisenberg chain in a magnetic field

Ian K. Affleck

Published Oct 28, 2005• 12 citations

arXiv HTML PDF

Abstract

\ensuremath{\eta}\ensuremath{\approx}1∕2\ensuremath{-}(2.0)m+O({m}^{2})

, where

m

is the magnetization per site. A comparison with the O(3) nonlinear

\ensuremath{\sigma}

model is also made.

DOI: 10.1103/physrevb.72.132414

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Gábor Fáth

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Dynamics of a Heisenberg spin chain in the quantum critical regime: NMR experiment versus effective field theory

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Mohamed Moussadek Azzouz

Within a mean-field theory, the disordered spin-1/2 single-chain state is shown to be unstable towards an antiferromagnetically, long-range ordered one, when any finite transverse coupling, ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$, is introduced between chains. A mean-field Hamiltonian is derived from the Heisenberg model, using an extension of the Wigner-Jordan transformation in two dimensions. The staggered magnetization and the ground-state energy are calculated for values of transverse couplings ranging from 0 to 1. The main consequence of these calculations is that the value of ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$, required to have antiferromagnetic long-range order, must only be nonzero. This result is consistent with that of Sakai and Takahashi indicating that the critical transverse coupling should be of order of the inverse of the one-dimensional staggered magnetic susceptibility. Following a qualitative argument, it can also suggest that the Luttinger-liquid state is unstable at least at nearly half-filling for the Hubbard model in the limit of large Coulomb repulsion.Within a mean-field theory, the disordered spin-1/2 single-chain state is shown to be unstable towards an antiferromagnetically, long-range ordered one, when any finite transverse coupling, ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$, is introduced between chains. A mean-field Hamiltonian is derived from the Heisenberg model, using an extension of the Wigner-Jordan transformation in two dimensions. The staggered magnetization and the ground-state energy are calculated for values of transverse couplings ranging from 0 to 1. The main consequence of these calculations is that the value of ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$, required to have antiferromagnetic long-range order, must only be nonzero. This result is consistent with that of Sakai and Takahashi indicating that the critical transverse coupling should be of order of the inverse of the one-dimensional staggered magnetic susceptibility. Following a qualitative argument, it can also suggest that the Luttinger-liquid state is unstable at least at nearly half-filling for the Hubbard model in the limit of large Coulomb repulsion.

Physical review. B, Condensed matterSep 1, 199366 citations

\ensuremath{\eta}\ensuremath{\approx}1∕2\ensuremath{-}(2.0)m+O({m}^{2})

, where

m

is the magnetization per site. A comparison with the O(3) nonlinear

\ensuremath{\sigma}

model is also made.

S=1/2

and

S=1

chains using the density matrix renormalization group method. Critical amplitudes and the scattering length at the chain ends are also determined. Although both systems are Luttinger liquids the characteristic parameters differ considerably.

We have studied the zero- and low-temperature behaviors of anisotropic alternating antiferromagnetic-ferromagnetic Heisenberg spin-

\frac{1}{2}

chains in a transverse magnetic field. Using the analytical spinless fermion approach, the thermodynamic behavior of the model has been studied. We have introduced a different order parameter to distinguish the gapless Luttinger liquid phase from the other gapped phases. The exact diagonalization Lanczos results are also used to compare with the spinless fermion ones. We have found a double peak structure in the specific-heat curves for the region between the two quantum critical points. Using the numerical full diagonalization results, the existence of the double peak structure is confirmed.

{\mathit{H}}_{\mathit{c}1}

(=\ensuremath{\Delta}), where \ensuremath{\Delta} is the Haldane gap. It is also found that the transverse spin correlation has the asymptotic form 〈

{\mathit{S}}_{0}^{\mathit{x}}

{\mathit{S}}_{\mathit{r}}^{\mathit{x}}

〉\ensuremath{\sim}(-1

{)}^{\mathit{r}}

${\mathit{r}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$, and the transverse staggered susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{st}}^{\mathit{x}\mathit{x}}$ diverges between

{\mathit{H}}_{\mathit{c}1}

and

{\mathit{H}}_{\mathit{c}2}

(=4). The exponent \ensuremath{\eta} has a minimum (\ensuremath{\eta}\ensuremath{\simeq}0.3) at magnetization m\ensuremath{\simeq}1/3 and \ensuremath{\eta}\ensuremath{\simeq}0.5 at

{\mathit{H}}_{\mathit{c}1}

and

{\mathit{H}}_{\mathit{c}2}

. If the system is quasi-one-dimensional, even small interchain couplings can create a canted N\'eel order, within a mean-field approximation for interchain interactions. This is consistent with a recent NMR measurement for Ni(

{\mathrm{C}}_{2}

{\mathrm{H}}_{8}

{\mathrm{N}}_{2}

{)}_{2}

{\mathrm{NO}}_{2}

(

{\mathrm{ClO}}_{4}

) (NENP) at low temperature.

^{13}\mathrm{C}

nuclear magnetic resonance relaxation rate ${T}_{1}^{\ensuremath{-}1}$, and the field theoretical approach in the Luttinger liquid regime was performed. We found very good agreement between experiment and theory in the investigated temperature and field range. Our results demonstrate how spin-spin interactions strongly affect the spin dynamics of Heisenberg spin chain compounds in the vicinity of the quantum critical point.