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Physical Review B

Antiferromagnetic Heisenberg spin-1 chain: Magnetic susceptibility of the Haldane chain described using scaling

Jean Souletie, Marc Drillon, Pierre Rabu, Swapan Kumar Pati

Published Aug 18, 2004• 16 citations
Publisher

Abstract

The phenomenological expression $\ensuremath{\chi}T∕(N{g}^{2}{\ensuremath{\mu}}_{B}^{2}∕k)={C}_{1n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}{W}_{1n}∕T)+{C}_{2n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}{W}_{2n}∕T)$ describes very accurately the temperature dependence of the magnetic susceptibility computed for antiferromagnetic rings of Heisenberg spins $S=1$, whose size $n$ is even and ranges from 6 to 20. This expression has been obtained through a strategy justified by scaling considerations together with finite size numerical calculations. For $n$ large, the coefficients of the expression converge towards ${C}_{1}=0.125$, ${W}_{1}=0.451J$, ${C}_{2}=0.564$, ${W}_{2}=1.793J$ ($J$ is the exchange constant), which are appropriate for describing the susceptibility of the spin-1 Haldane chain. The Curie constant, the paramagnetic Curie--Weiss temperature, the correlation length at $T=0$ and the Haldane gap are found to be closely related to these coefficients. With this expression, a very good description of the magnetic behavior of ${\mathrm{Y}}_{2}\mathrm{Ba}\mathrm{Ni}{\mathrm{O}}_{5}$ and of $\mathrm{Ni}{({\mathrm{C}}_{2}{\mathrm{H}}_{8}{\mathrm{N}}_{2})}_{2}\mathrm{N}{\mathrm{O}}_{2}\mathrm{Cl}{\mathrm{O}}_{4}$ (NENP), the archetype of the Haldane gap systems, is achieved over the whole temperature range.

DOI
10.1103/physrevb.70.054410

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Critical behavior of the S=1 antiferromagnetic Heisenberg chain (abstract)

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The spectral properties, and associated critical singularities, of spin-1 Heisenberg antiferromagnetic chains are of high current interest, both theoretical and experimental. Theoretical studies using the single mode approximation and also a Monte Carlo calculation indicate that the spectrum has a gap at k=π, as predicted by Haldane. They also predict the unusual feature that the excited state continuum is asymmetric about the mid-point of the Brillouin zone, k=π/2. However, neither approach is sufficiently accurate to predict reliably the magnitude of the gap at k=0 which was not discussed by Haldane. Our approach is to use static (spectral) calculations of excited states of chains of N=4 through 14 spins for all values of k, a novel feature not associated with previous finite-size calculations, to discuss the dispersion spectra of the S=1 Heisenberg chain. In addition, we have calculated dynamical properties, in particular S(q,ω), exactly for finite chains up to N=14 spins. This allows us to estimate the gap at k=0 to good accuracy, and we find that to within this accuracy the gap at k=0 is twice the Haldane gap at k=π. This result is important in connection with very recent neutron scattering and magnetization measurements. In addition, we have calculated the correlation length and critical singularities associated with dynamical quantities.

Journal of Applied Physics•Apr 15, 1991•1 citation
The phenomenological expression \ensuremathχT∕(Ng2\ensuremathμB2∕k)=C1nexp(\ensuremath−W1n∕T)+C2nexp(\ensuremath−W2n∕T)\ensuremath{\chi}T∕(N{g}^{2}{\ensuremath{\mu}}_{B}^{2}∕k)={C}_{1n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}{W}_{1n}∕T)+{C}_{2n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}{W}_{2n}∕T)\ensuremathχT∕(Ng2\ensuremathμB2​∕k)=C1n​exp(\ensuremath−W1n​∕T)+C2n​exp(\ensuremath−W2n​∕T) describes very accurately the temperature dependence of the magnetic susceptibility computed for antiferromagnetic rings of Heisenberg spins S=1S=1S=1, whose size nnn is even and ranges from 6 to 20. This expression has been obtained through a strategy justified by scaling considerations together with finite size numerical calculations. For nnn large, the coefficients of the expression converge towards C1=0.125{C}_{1}=0.125C1​=0.125, W1=0.451J{W}_{1}=0.451JW1​=0.451J, C2=0.564{C}_{2}=0.564C2​=0.564, W2=1.793J{W}_{2}=1.793JW2​=1.793J (JJJ is the exchange constant), which are appropriate for describing the susceptibility of the spin-1 Haldane chain. The Curie constant, the paramagnetic Curie--Weiss temperature, the correlation length at T=0T=0T=0 and the Haldane gap are found to be closely related to these coefficients. With this expression, a very good description of the magnetic behavior of Y2BaNiO5{\mathrm{Y}}_{2}\mathrm{Ba}\mathrm{Ni}{\mathrm{O}}_{5}Y2​BaNiO5​ and of Ni(C2H8N2)2NO2ClO4\mathrm{Ni}{({\mathrm{C}}_{2}{\mathrm{H}}_{8}{\mathrm{N}}_{2})}_{2}\mathrm{N}{\mathrm{O}}_{2}\mathrm{Cl}{\mathrm{O}}_{4}Ni(C2​H8​N2​)2​NO2​ClO4​ (NENP), the archetype of the Haldane gap systems, is achieved over the whole temperature range.
A one-dimensional S=1 Heisenberg antiferromagnet in a magnetic field H (\ensuremath{\parallel}z axis) at T=0 is studied by numerical diagonalizations up to N=16. We give the magnetization curve at the thermodynamic limit, and derive an anomaly at Hc1{\mathit{H}}_{\mathit{c}1}Hc1​ (=\ensuremath{\Delta}), where \ensuremath{\Delta} is the Haldane gap. It is also found that the transverse spin correlation has the asymptotic form 〈S0x{\mathit{S}}_{0}^{\mathit{x}}S0x​Srx{\mathit{S}}_{\mathit{r}}^{\mathit{x}}Srx​〉\ensuremath{\sim}(-1)r{)}^{\mathit{r}})r${\mathit{r}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$, and the transverse staggered susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{st}}^{\mathit{x}\mathit{x}}$ diverges between Hc1{\mathit{H}}_{\mathit{c}1}Hc1​ and Hc2{\mathit{H}}_{\mathit{c}2}Hc2​ (=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 Hc1{\mathit{H}}_{\mathit{c}1}Hc1​ and Hc2{\mathit{H}}_{\mathit{c}2}Hc2​. 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(C2{\mathrm{C}}_{2}C2​H8{\mathrm{H}}_{8}H8​N2{\mathrm{N}}_{2}N2​)2{)}_{2})2​NO2{\mathrm{NO}}_{2}NO2​(ClO4{\mathrm{ClO}}_{4}ClO4​) (NENP) at low temperature.
Linear chains (and rings) of S=12S=\frac{1}{2}S=21​ spins with the anisotropic (Ising-Heisenberg) Hamiltonian $\mathcal{H}=\ensuremath{-}2J\ensuremath{\Sigma}\stackrel{N}{i=1}{{{S}_{i}}^{z}{{S}_{i+1}}^{z}+\ensuremath{\gamma}({{S}_{i}}^{x}{{S}_{i+1}}^{x}+{{S}_{i}}^{y}{{S}_{i+1}}^{y})}\ensuremath{-}g\ensuremath{\beta}\ensuremath{\Sigma}\stackrel{N}{i=1}\mathrm{H}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{i}$ have been studied by exact machine calculations for N=2to11N=2 \mathrm{to} 11N=2to11, $\ensuremath{\gamma}=0 \mathrm{to} 1$ and for ferro- and antiferro-magnetic coupling. The results reveal the dependence on finite size and anisotropy of the spectrum and dispersion laws, of the energy, entropy, and specific heat, of the magnetization and susceptibilities, and of the pair correlations. The limiting $N\ensuremath{\rightarrow}\ensuremath{\infty}$ behavior is accurately indicated, for all $\ensuremath{\gamma}$, in the region kT∣J∣> 0.5\frac{\mathrm{kT}}{|J|}>~0.5∣J∣kT​> 0.5 which includes the maxima in the specific heat and susceptibility. The behavior of thermal and magnetic properties of infinite chains at lower temperatures is estimated by extrapolation. For infinite antiferromagnetic chains the ground-state degeneracy, the anisotropy gap, and the magnetization, perpendicular susceptibility, and pair correlations at T=0T=0T=0 are similarly studied. Estimates of the long-range order suggest that it vanishes only at the Heisenberg limit $\ensuremath{\gamma}=1$ and confirm the accuracy of Walker's perturbation series in $\ensuremath{\gamma}$.
The specific heat and magnetic susceptibility of the antiferromagnetic Heisenberg chain have been calculated using a numerical finite-size technique. An agreement with experiment for the quasi-one-dimensional compound CsNiCl3{\mathrm{CsNiCl}}_{3}CsNiCl3​, which displays a Haldane gap, is revealed.
The structure function S(k) for the s=1, Haldane-gap antiferromagnetic chain is measured accurately using the recent density-matrix renormalization-group method, with a chain length of 100. Excellent agreement with the nonlinear-\ensuremath{\sigma}-model prediction is obtained, both at k\ensuremath{\approxeq}\ensuremath{\pi}, where a single-magnon process dominates, and at k\ensuremath{\approxeq}0, where a two-magnon process dominates. We repeat our calculation with crystal-field anisotropy chosen to model Ni(C2{\mathrm{C}}_{2}C2​H8{\mathrm{H}}_{8}H8​N2{\mathrm{N}}_{2}N2​)2{)}_{2})2​NO2{\mathrm{NO}}_{2}NO2​(ClO4{\mathrm{ClO}}_{4}ClO4​), obtaining good agreement with both field-theory predictions and recent experiments. Correlation lengths, gaps, and velocities are determined for both polarizations.
Using the density matrix renormalization group (DMRG) technique, we carry out a large scale numerical calculation for the S=2S=2S=2 antiferromagnetic Heisenberg chain. Performing systematic scaling analysis for both the chain length L and the number of optimal states kept in the iterations m, the Haldane gap $\ensuremath{\Delta}(2)$ is estimated accurately as $(0.0876\ifmmode\pm\else\textpm\fi{}0.0013)J.$ Our systematic analysis for the S=2S=2S=2 chains not only clarifies the controversies arising from various DMRG calculations and Monte Carlo simulations, but also sheds light on how to obtain reliable results from the DMRG calculations for other similar complicated systems.
The spectral properties, and associated critical singularities, of spin-1 Heisenberg antiferromagnetic chains are of high current interest, both theoretical and experimental. Theoretical studies using the single mode approximation and also a Monte Carlo calculation indicate that the spectrum has a gap at k=π, as predicted by Haldane. They also predict the unusual feature that the excited state continuum is asymmetric about the mid-point of the Brillouin zone, k=π/2. However, neither approach is sufficiently accurate to predict reliably the magnitude of the gap at k=0 which was not discussed by Haldane. Our approach is to use static (spectral) calculations of excited states of chains of N=4 through 14 spins for all values of k, a novel feature not associated with previous finite-size calculations, to discuss the dispersion spectra of the S=1 Heisenberg chain. In addition, we have calculated dynamical properties, in particular S(q,ω), exactly for finite chains up to N=14 spins. This allows us to estimate the gap at k=0 to good accuracy, and we find that to within this accuracy the gap at k=0 is twice the Haldane gap at k=π. This result is important in connection with very recent neutron scattering and magnetization measurements. In addition, we have calculated the correlation length and critical singularities associated with dynamical quantities.