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Journal of the Physical Society of Japan

Impurity Spin Effect on the Spin-1 Antiferromagnetic Heisenberg Chain

Makoto Kaburagi, Takashi Tonegawa, Minkyu Kang

Published Jan 2, 2003• 1 citation
Publisher

Abstract

Low-lying excited states of the spin-1 antiferromagnetic Heisenberg chain with a spin- S ( S >1) impurity are investigated by means of variational method. It is shown that (1) the phenomenological Hamiltonian which may be useful in the analysis of the experimental results can be derived from the usual Hamiltonian, (2) the origin of the anisotropy in the phenomenological Hamiltonian can be explained from the results obtained by the analytical method, (3) the impurity spin brings about a massive mode in the Haldane gap, (4) the energy levels of the massive mode can be classified into four multiples corresponding to the states with the total spin quantum S = S +1, S and S -1 coupled to triplet of host-system and the state with S = S coupled to the singlet of host-system.

DOI
10.1143/jpsjs.72sb.135

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Low-lying excited states of the spin-1 antiferromagnetic Heisenberg chain with a spin- S ( S >1) impurity are investigated by means of variational method. It is shown that (1) the phenomenological Hamiltonian which may be useful in the analysis of the experimental results can be derived from the usual Hamiltonian, (2) the origin of the anisotropy in the phenomenological Hamiltonian can be explained from the results obtained by the analytical method, (3) the impurity spin brings about a massive mode in the Haldane gap, (4) the energy levels of the massive mode can be classified into four multiples corresponding to the states with the total spin quantum S = S +1, S and S -1 coupled to triplet of host-system and the state with S = S coupled to the singlet of host-system.
Low-lying excited states as well as the ground state of the spin-1 antiferro- magnetic Heisenberg chain with a spin-1/2 impurity are investigated by means of a variational method and a method of numerical diagonalization. It is shown that 1) the impurity spin brings about massive modes in the Haldane gap, 2) when the the impurity-host coupling is sufficiently weak, the phenomenological Hamiltonian used by Hagiwara {\it et al.} in the analysis of ESR experimental results for NENP containing a small amount of spin-1/2 Cu impurities is equivalent to a more realistic Hamiltonian, as far as the energies of the low-lying states are concerned, 3) the results obtained by the variational method are in semi-quantitatively good agreement with those obtained by the numerical diagonalization.
Low-lying excited states of the spin-1 antiferromagnetic Heisenberg chain with an impurity bond are investigated in terms of domain-wall excitations by means of analytical methods as well as a method of numerical diagonalization.It is shown that 1) the impurity bond brings about a massive triplet mode in the Haldane gap, 2) the triplet state comprises three of the four ground states of an open chain, 3) the excited states with the domain wall trapped at the impurity bond, including the massive triplet mode, exhibit a Néel-type configuration of the magnetic moment around the impurity bond.On the basis of the present results, we propose a new energy versus magnetic-field diagram.
Low-lying excited states of the spin-1 antiferromagnetic Heisenberg chain with an impurity bond are investigated in terms of domain-wall excitations by means of analytical methods as well as a method of numerical diagonalization. It is shown that 1) the impurity bond brings about a massive triplet mode in the Haldane gap, 2) the triplet state comprises three of the four ground states of an open chain, 3) the excited states with the domain wall trapped at the impurity bond, including the massive triplet mode, exhibit a Néel-type configuration of the magnetic moment around the impurity bond. On the basis of the present results, we propose a new energy versus magnetic-field diagram.
Low-lying excited states as well as the ground state of the spin-1 antiferromagnetic Heisenberg chain with a spin-1/2 impurity are studied by means of a method of numerical diagonalization. The isotropic nearest-neighbor exchange coupling plus the uniaxial single-ion-type anisotropy energy is assumed for the host-system Hamiltonian. The energy differences between the ground state and the low-lying excited states, which appear in the Haldane gap, are estimated in the thermodynamic limit. The results are used to analyze the electron-spin-resonance experimental data on Ni(C2_22​H8_88​N2_22​)2_22​NO2_22​(ClO4_44​), abbreviated NENP, containing a small amount of spin-1/2 Cu2+^{2+}2+ impurities. It is found that in this system, the impurity-host (\bigl((Cu2+^{2+}2+-Ni2+^{2+}2+)\bigr)) coupling is ferromagnetic and its magnitude is about 5 \,\% of the magnitude of the host-host (\bigl((Ni2+^{2+}2+-Ni2+^{2+}2+)\bigr)) coupling.
We explore low-lying excited states as well as the ground state of the antiferromagnetic S=1 bond-alternating chain with an S=1/2 impurity spin. For the case where the ground-state phase of the host system is the Haldane phase, we review a numerical analysis of the electron-spin-resonance experimental results on the NENP:Cu2+ system. For the case where the ground-state phase of the host system is the dimer phase, on the other hand, we calculate, using the exact-diagonalization method, the dependences of the energy differences between the ground and low-lying excited states upon both the impurity-host exchange constant and the single-ion-type anisotropy constant, and also calculate, using the density-matrix renormalization-group method, the external-magnetic-field dependence of the impurity-spin magnetizaion in the ground state. In these calculations, we keep the NTENP:Cu2+ system in mind to choose the value of the bond-alternation parameter. We find that a few low-lying excited states which are expected from the valence-bond-solid picture appear as the impurity states in the energy gap between the singlet ground and triplet first-excited states (the dimer gap). Furthermore, for certain values of the above constants, we find that the impurity-spin magnetizaion shows a clear jump at a magnetic field which is in the dimer-gap region or in the magnetization-plateau region of the host system, and also that the impurity-spin magnetizaion has a magnetic-field region where it decreases as a function of the magnetic field.
Using an impurity density-matrix renormalization-group scheme, we establish a reliable picture of how the low-lying energy levels of an S=1S=1S=1 Heisenberg antiferromagnetic chain change quantitatively upon bond doping. An impurity state gradually occurs in the Haldane gap as ${J}^{\ensuremath{'}}<J$, while it appears only if $\frac{{J}^{\ensuremath{'}}}{J}>{\ensuremath{\gamma}}_{c}$ with $\frac{1}{{\ensuremath{\gamma}}_{c}}=0.708$ as ${J}^{\ensuremath{'}}>J$. The spin polarization in the impurity state shows a localization-delocalization transition at ${\ensuremath{\gamma}}_{c}$. The system is nonperturbative as $1<~\frac{{J}^{\ensuremath{'}}}{J}<~{\ensuremath{\gamma}}_{c}$. It can explain the appearance of a new state in the Haldane gap recently observed experimentally on ${\mathrm{Y}}_{2\ensuremath{-}x}{\mathrm{Ca}}_{x}\mathrm{BaNi}{\mathrm{O}}_{5}$ [J.F. DiTusa et al., Phys. Rev. Lett. 73, 1857 (1994)].