arXiv:cond-mat/9509010 · Physical review. B, Condensed matter

Impurity energy level in the Haldane gap

Wei Wang, Shaojin Qin, Zhong‐Yi Lu, Lu Yu, Zhao-Bin Su

Published Jan 1, 1996• 23 citations

Abstract

An impurity bond ${\mathit{J}}^{\ensuremath{'}}$ in a periodic one-dimensional antiferromagnetic spin-1 chain with exchange J is considered. Using the numerical density matrix renormalization-group method, we find an impurity energy level in the Haldane gap, corresponding to a bound state near the impurity bond. When ${\mathit{J}}^{\ensuremath{'}}$J the level changes gradually from the edge of the Haldane gap to the ground-state energy as the deviation dev=(J-${\mathit{J}}^{\ensuremath{'}}$)/J changes from 0 to 1. It seems that there is no threshold. Yet, there is a threshold when ${\mathit{J}}^{\ensuremath{'}}$\ensuremath{\gtrsim}J. The impurity level appears only when the deviation dev=(${\mathit{J}}^{\ensuremath{'}}$-J)/${\mathit{J}}^{\ensuremath{'}}$ is greater than ${\mathit{B}}_{\mathit{c}}$, which is close to 0.3 in our calculation. \textcopyright{} 1996 The American Physical Society.

DOI: 10.1103/physrevb.53.40

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Impurity state in the Haldane gap for an S=1 Heisenberg antiferromagnetic chain with bond doping

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arXiv:cond-mat/9509010 · Physical review. B, Condensed matter

Impurity energy level in the Haldane gap

Wei Wang, Shaojin Qin, Zhong‐Yi Lu, Lu Yu, Zhao-Bin Su

Published Jan 1, 1996• 23 citations

arXiv HTML PDF

Abstract

An impurity bond

{\mathit{J}}^{\ensuremath{'}}

in a periodic one-dimensional antiferromagnetic spin-1 chain with exchange J is considered. Using the numerical density matrix renormalization-group method, we find an impurity energy level in the Haldane gap, corresponding to a bound state near the impurity bond. When

{\mathit{J}}^{\ensuremath{'}}

J the level changes gradually from the edge of the Haldane gap to the ground-state energy as the deviation dev=(J-

{\mathit{J}}^{\ensuremath{'}}

)/J changes from 0 to 1. It seems that there is no threshold. Yet, there is a threshold when

{\mathit{J}}^{\ensuremath{'}}

\ensuremath{\gtrsim}J. The impurity level appears only when the deviation dev=(

{\mathit{J}}^{\ensuremath{'}}

-J)/

{\mathit{J}}^{\ensuremath{'}}

is greater than

{\mathit{B}}_{\mathit{c}}

, which is close to 0.3 in our calculation. \textcopyright{} 1996 The American Physical Society.

DOI: 10.1103/physrevb.53.40

Related papers

Impurity state in the Haldane gap for an S=1 Heisenberg antiferromagnetic chain with bond doping

Xiaoqun Wang, Steffen Mallwitz

Using an impurity density-matrix renormalization-group scheme, we establish a reliable picture of how the low-lying energy levels of an $S=1$ 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)].Using an impurity density-matrix renormalization-group scheme, we establish a reliable picture of how the low-lying energy levels of an $S=1$ 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)].

arXiv:cond-mat/9508132Physical review. B, Condensed matterJan 1, 199637 citations

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The s=1 Heisenberg antiferromagnet is studied in the presence of two kinds of local impurities. First, a perturbed antiferromagnetic bond with J'\ensuremath{\ne}J at the center of an even-length open chain is considered. Using the density-matrix renormalization-group method we find that, for sufficiently strong or weak J', a bound state is localized at the impurity site, giving rise to an energy level in the Haldane gap. The energy of the bound state is in agreement with perturbative results, based on s=1/2 chain-end excitations, both in the weak- and strong-coupling limit. In a region around the uniform limit, J'=J, no states are found with energy below the Haldane gap. Second, a s=1/2 impurity at the center of an otherwise even-length open chain is considered. The coupling to the s=1/2 impurity is varied. Bound states in the Haldane gap are found only for sufficiently weak (antiferromagnetic) coupling. For a s=1/2 impurity coupled with a strong (antiferromagnetic) bond, no states are found in the Haldane gap. Our results are in good qualitative agreement with recent experiments on doped NENP and ${\mathrm{Y}}_{2}$ ${\mathrm{BaNiO}}_{5}$ .The s=1 Heisenberg antiferromagnet is studied in the presence of two kinds of local impurities. First, a perturbed antiferromagnetic bond with J'\ensuremath{\ne}J at the center of an even-length open chain is considered. Using the density-matrix renormalization-group method we find that, for sufficiently strong or weak J', a bound state is localized at the impurity site, giving rise to an energy level in the Haldane gap. The energy of the bound state is in agreement with perturbative results, based on s=1/2 chain-end excitations, both in the weak- and strong-coupling limit. In a region around the uniform limit, J'=J, no states are found with energy below the Haldane gap. Second, a s=1/2 impurity at the center of an otherwise even-length open chain is considered. The coupling to the s=1/2 impurity is varied. Bound states in the Haldane gap are found only for sufficiently weak (antiferromagnetic) coupling. For a s=1/2 impurity coupled with a strong (antiferromagnetic) bond, no states are found in the Haldane gap. Our results are in good qualitative agreement with recent experiments on doped NENP and ${\mathrm{Y}}_{2}$${\mathrm{BaNiO}}_{5}$.

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Physics of integer-spin antiferromagnetic chains: Haldane gaps and edge states

Th. Jolicœur, O. Golinelli

The antiferromagnetic Heisenberg spin chain with integer spin has a short-range magnetic order and an excitation energy gap above the ground state. This so-called Haldane gap is proportional to the exchange coupling constant J of the Heisenberg chain. In this study, we discuss recent results about the spin dependence of the Haldane gap and conjecture an analytical formula valid asymptotically for large spin values. We then discuss the robustness of the edge states of the spin-1 chain by studying a spin-1 ladder using the density matrix renormalization group (DMRG) algorithm. We show that the peculiar hidden topological order of the spin-1 chain disappears smoothly by increasing the ladder rung coupling without any intervening phase transition. This is evidence for the fragile character of the topological order of the spin-1 chain.The antiferromagnetic Heisenberg spin chain with integer spin has a short-range magnetic order and an excitation energy gap above the ground state. This so-called Haldane gap is proportional to the exchange coupling constant J of the Heisenberg chain. In this study, we discuss recent results about the spin dependence of the Haldane gap and conjecture an analytical formula valid asymptotically for large spin values. We then discuss the robustness of the edge states of the spin-1 chain by studying a spin-1 ladder using the density matrix renormalization group (DMRG) algorithm. We show that the peculiar hidden topological order of the spin-1 chain disappears smoothly by increasing the ladder rung coupling without any intervening phase transition. This is evidence for the fragile character of the topological order of the spin-1 chain.

arXiv:1906.12207Comptes Rendus ChimieJun 1, 201913 citations

Impurity-Bond Effect on the Spin-1 Antiferromagnetic Heisenberg Chain

Makoto Kaburagi, Isao Harada, Takashi Tonegawa

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.

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Makoto Kaburagi, Isao Hrada, Takashi Tonegawa

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.

Journal of the Physical Society of JapanJan 15, 19940 citations

Numerical evidence of a spin- 1∕2 chain approaching a spin-1 chain

Hsiang-Hsuan Hung, Chang-De Gong

We study the one-dimensional Heisenberg spin- $1∕2$ alternating bond chain in which the nearest neighbor exchange couplings are ferromagnetic (FM) and antiferromagnetic (AF) alternatively. By using exact diagonalization and the density matrix renormalization group method, we discuss how the system approaches the AF uniform spin-1 chain under certain conditions. When the ratio of AF to FM coupling strengths, $\ensuremath{\alpha}$ $(\ensuremath{\alpha}={J}_{AF}∕{J}_{F})$, is very small, physical properties of the alternating bond chain such as spin-spin correlation, the string correlation function, and the spin density coincide with those of the AF uniform spin-1 chain. The edge state problem is discussed in the present model with the small $\ensuremath{\alpha}$ limit. In addition, the Haldane gap of the AF uniform spin-1 chain is four times the gap of the system considered.We study the one-dimensional Heisenberg spin-$1∕2$ alternating bond chain in which the nearest neighbor exchange couplings are ferromagnetic (FM) and antiferromagnetic (AF) alternatively. By using exact diagonalization and the density matrix renormalization group method, we discuss how the system approaches the AF uniform spin-1 chain under certain conditions. When the ratio of AF to FM coupling strengths, $\ensuremath{\alpha}$ $(\ensuremath{\alpha}={J}_{AF}∕{J}_{F})$, is very small, physical properties of the alternating bond chain such as spin-spin correlation, the string correlation function, and the spin density coincide with those of the AF uniform spin-1 chain. The edge state problem is discussed in the present model with the small $\ensuremath{\alpha}$ limit. In addition, the Haldane gap of the AF uniform spin-1 chain is four times the gap of the system considered.

arXiv:cond-mat/0408690Physical Review BFeb 22, 200519 citations

An impurity bond

{\mathit{J}}^{\ensuremath{'}}

{\mathit{J}}^{\ensuremath{'}}

J the level changes gradually from the edge of the Haldane gap to the ground-state energy as the deviation dev=(J-

{\mathit{J}}^{\ensuremath{'}}

)/J changes from 0 to 1. It seems that there is no threshold. Yet, there is a threshold when

{\mathit{J}}^{\ensuremath{'}}

\ensuremath{\gtrsim}J. The impurity level appears only when the deviation dev=(

{\mathit{J}}^{\ensuremath{'}}

-J)/

{\mathit{J}}^{\ensuremath{'}}

is greater than

{\mathit{B}}_{\mathit{c}}

, which is close to 0.3 in our calculation. \textcopyright{} 1996 The American Physical Society.

Using an impurity density-matrix renormalization-group scheme, we establish a reliable picture of how the low-lying energy levels of an

S=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)].

{\mathrm{Y}}_{2}

{\mathrm{BaNiO}}_{5}

We study the one-dimensional Heisenberg spin-

1∕2

alternating bond chain in which the nearest neighbor exchange couplings are ferromagnetic (FM) and antiferromagnetic (AF) alternatively. By using exact diagonalization and the density matrix renormalization group method, we discuss how the system approaches the AF uniform spin-1 chain under certain conditions. When the ratio of AF to FM coupling strengths, $\ensuremath{\alpha}$ $(\ensuremath{\alpha}={J}_{AF}∕{J}_{F})$, is very small, physical properties of the alternating bond chain such as spin-spin correlation, the string correlation function, and the spin density coincide with those of the AF uniform spin-1 chain. The edge state problem is discussed in the present model with the small $\ensuremath{\alpha}$ limit. In addition, the Haldane gap of the AF uniform spin-1 chain is four times the gap of the system considered.