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Physical review. B, Condensed matter

Effects of edges inS=1 Heisenberg antiferromagnetic chains

Seiji Miyashita, Shoji Yamamoto

Published Jul 1, 1993• 123 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.48.913

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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 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 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.
Thermodynamic properties of S=1 antiferromagnetic Heisenberg chains with free and periodic boundaries are investigated by a quantum Monte Carlo method. In particular, temperature dependences of the specific heat, the magnetic susceptibility, and the hidden order parameter, which are inherent to the Haldane phase, are investigated. The specific heat turns out to have a peak at a temperature Tpeak{\mathit{T}}_{\mathrm{peak}}Tpeak​\ensuremath{\sim}2\ensuremath{\Delta}, where \ensuremath{\Delta} is the energy gap of the present model, although the temperature dependence of the specific heat is very similar to the Schottky type. In open chains, due to the fourfold degeneracy of the ground state, the magnetic susceptibility shows a Curie-like divergence regardless of the number of spins in the chain. The amplitude of the Curie-like divergence is consistent with the edge moments (S=1/2). On the other hand, it is confirmed that the long-range hidden order exists only in the ground state. How the hidden order grows as the temperature goes to zero (T\ensuremath{\rightarrow}0) is investigated by introducing an intrinsic correlation length ${\ensuremath{\xi}}_{\mathit{D}}$ of the ordering, which diverges as T\ensuremath{\rightarrow}0. A qualitative difference in fluctuations between the ground state and the finite-temperature states is discussed making use of snapshots of the transformed two-dimensional Ising system on which a Monte Carlo simulation is being performed.
Full magnetization curves of S=1 antiferromagnetic Heisenberg chains with periodic and open boundaries are investigated by a quantum Monte Carlo method. The boundary condition has a significant effect on the magnetization process especially in the field range where the Zeeman energy is less than the Haldane gap. The magnetization of the periodic chain in the range is strongly suppressed at low temperatures because the ground state is nonmagnetic. On the other hand, the magnetic ground of the open chain bring about finite magnetization. The difference between both the magnetization curves is also investigated in detail by subtracting the magnetization of the periodic chain with L spins from one of the open chains with L+1 spins. It agrees well with the double of the Brillouin function for S=1/2, rather than the Brillouun function for S=1.
Effects of an uniaxial single-ion anisotropy tsumi{\mathit{tsum}}_{\mathit{i}}tsumi​Dz{\mathit{D}}_{\mathit{z}}Dz​(Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​)2{)}^{2})2 on the magnetic properties of an S=1 antiferromagnetic Heisenberg chain with open boundaries are studied by a quantum Monte Carlo method. In the ground stae, how the anisotropy affects the staggered magnetization 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 and the spin correlation functions 〈S1z{\mathit{S}}_{1}^{\mathit{z}}S1z​Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 and 〈S1x{\mathit{S}}_{1}^{\mathit{x}}S1x​Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​〉 is investigated. Amplitudes of 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 decrease with with increase of Dz{\mathit{D}}_{\mathit{z}}Dz​. The correlation length ${\ensuremath{\xi}}_{\mathit{S}\mathit{S}}^{\mathit{z}}$ of 〈S1z{\mathit{S}}_{1}^{\mathit{z}}S1z​Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 monotonously decreases with increase of Dz{\mathit{D}}_{\mathit{z}}Dz​, while the correlation length ${\ensuremath{\xi}}_{\mathit{S}\mathit{S}}^{\mathit{x}}$ of 〈S1x{\mathit{S}}_{1}^{\mathit{x}}S1x​Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​〉 has the opposite but weak Dz{\mathit{D}}_{\mathit{z}}Dz​ dependence. The fourfold degeneracy of the ground state also holds with nonzero Dz{\mathit{D}}_{\mathit{z}}Dz​'s in the thermodynamic limit although it is lifted up in finite chains. At finite temperatures, the magnetic susceptibilities along the z axis (${\mathrm{\ensuremath{\chi}}}^{\mathit{z}}$) and the x axis (${\mathrm{\ensuremath{\chi}}}^{\mathit{x}}$) are calculated. Temperature dependences of the bulk susceptibilites, ${\mathrm{\ensuremath{\chi}}}_{\mathrm{\ensuremath{\infty}}}^{\mathit{z}}$ and ${\mathrm{\ensuremath{\chi}}}_{\mathrm{\ensuremath{\infty}}}^{\mathit{x}}$, are consistent with the previous experimental result on 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). Finite-size corrections, ${\mathrm{\ensuremath{\chi}}}_{0}^{\mathit{z}}$ and ${\mathrm{\ensuremath{\chi}}}_{0}^{\mathit{x}}$, and the Curie susceptibilities resulting from chain ends, ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}^{\mathit{z}}$ and ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}^{\mathit{x}}$, are also investigated in detail. Both ${\mathrm{\ensuremath{\chi}}}_{0}$ and ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}$ diverge as temperature goes to zero and they are combined through a relation ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}$=${\mathrm{\ensuremath{\chi}}}_{0}$+${\mathrm{\ensuremath{\chi}}}_{\mathrm{\ensuremath{\infty}}}$. Temperature dependences of ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}$ are dominated by the edge effect at low temperatures, while at high temperatures they are attributed to an S=1 free spin.
Ground-state properties of S =1 antiferromagnetic Heisenberg chains with bond alternation, H = J ∑ i {1-(-1) i δ} S i · S i +1 , are studied by a quantum Monte Carlo method. Spin correlation between chain ends on the open chain appears for δ<0.25, while it vanishes for δ≥0.25. At δ≃0.25, the spin correlation length has a peak and the S =1/2 edge states for the open chain disappear. The fourfold degeneracy of the ground state of the open chain holds for δ<0.25, while it is removed for δ≥0.25. All the results support the continuous phase transition at δ≃0.25 demonstrated in the previous works.
Thermodynamic properties of S=2 antiferromagnetic Heisenberg chains are studied not only under the periodic boundary condition but also under the open one employing a quantum Monte Carlo method. Temperature and size dependences of the energy, the specific heat, and the magnetic susceptibility are calculated and edge effects on them are investigated in detail. The specific heat shows a well-pronounced Schottky anomaly but the maximum is located at a temperature much larger than the Haldane gap of the system. As temperature goes to zero, the magnetic susceptibility vanishes for the periodic chains, while it diverges for the open chains. The edge contribution in the open-chain susceptibility is attributed to the two S=1 effective spins localized at the chain ends at low temperatures, while to an S=2 free spin at high temperatures. This is a visualization of a quantum-classical crossover and an evidence that the present model is a Haldane system. High-temperature behaviors of the thermodynamic quantities are also discussed with the help of a series-expansion method. \textcopyright{} 1996 The American Physical Society.
We compute the ground-state spin-spin correlation function for the one-dimensional antiferromagnetic Heisenberg model of spin 1/2, 1, and 3/2. The spin-1 chain has a disordered ground state with a correlation length of 6.2 lattice spacings whereas the spin-1/2 and the spin-3/2 chains both have (ln(r)${)}^{\mathrm{\ensuremath{\sigma}}}$/r decaying correlation functions. The logarithmic corrections are different for the spin-1/2 and the spin-3/2 chains. Finite-size scaling is used to analyze the data for chains of up to 128 spins.