arXiv:cond-mat/9802260 · Physical Review Letters

Order from Disorder: Nonmagnetic Impurities in the Spin-Gap Phase of the Cuprates

C. Pépin, Patrick A. Lee

Published Sep 28, 1998• 44 citations

Abstract

We solve the problem of $N$ nonmagnetic impurities in the staggered flux phase of the Heisenberg model which we assume to be a good mean-field approximation for the spin-gap phase of the cuprates. The density of states is evaluated exactly in the unitary scattering limit and is proportional to $1/\ensuremath{\omega}{ln}^{2}(|\ensuremath{\omega}|/D)$, in analogy with the 1D case of doped spin-Peierls and two-leg ladder compounds. We argue that the system exhibits a quasi-long-range order at $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ with instantaneous spin-spin correlations decreasing as ${n}_{i}/{ln}^{2}({n}_{i}{R}_{\mathrm{ij}})$ for large distances ${R}_{\mathrm{ij}}$ and we predict enhanced low energy fluctuations in neutron scattering.

DOI: 10.1103/physrevlett.81.2779

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Impurity-induced staggered polarization and antiferromagnetic order in spin- 12 Heisenberg two-leg ladder compound SrCu2O3: Extensive Cu NMR and NQR studies

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Nonmagnetic Impurities in Spin Gap Systems

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The effect of non-magnetic impurities is discussed for both the two-leg Heisenberg ladder system and other spin liquids with excitation gap. It is shown that the random depletion of spins introduces a random Berry phase term to the non-linear sigma model. The classical nature of the antiferromagnetic correlation is enhanced by the topological decoherence, and the staggered susceptibility shows more singular behavior at low temperatures than the uniform antiferromagnetic Heisenberg chain.

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Nonmagnetic impurities in spin-gapped and gapless Heisenberg antiferromagnets

Anders W. Sandvik, Elbio R. Dagotto, Douglas J. Scalapino

Physical review. B, Condensed matterNov 1, 199749 citations

Impurity Induced Antiferromagnetism in Spin-1/2 Heisenberg Two-Leg Ladder Systems

Hidetoshi Fukuyama, Naoto Nagaosa, Masako Saitō +1 more

Based on the phase Hamiltonian representation of a spin-1/2 Heisenberg two-leg ladder system coupled antiferromagnetically, which has a spin gap if clean, it is shown that a non-magnetic impurity replacing a spin induces a staggered spin modulation. This results in the long range order of antiferromagnetism in the presence of three-dimensional coupling between the ladders. This will be in accordance with a recent experimental observation by Azuma et al. on Sr(Cu 1- x Zn x ) 2 O 3 .

Journal of the Physical Society of JapanAug 15, 199673 citations

Disappearance of the Spin Gap in a Zn-Doped 2-Leg Ladder Compound Sr(Cu1-xZnx)2O3

Masaki Azuma, Mikio Takano, Roger S. Eccleston

arXiv:cond-mat/9706170Journal of the Physical Society of JapanMar 15, 199840 citations

arXiv:cond-mat/9802260 · Physical Review Letters

Order from Disorder: Nonmagnetic Impurities in the Spin-Gap Phase of the Cuprates

C. Pépin, Patrick A. Lee

Published Sep 28, 1998• 44 citations

arXiv HTML PDF

Abstract

We solve the problem of

N

nonmagnetic impurities in the staggered flux phase of the Heisenberg model which we assume to be a good mean-field approximation for the spin-gap phase of the cuprates. The density of states is evaluated exactly in the unitary scattering limit and is proportional to

1/\ensuremath{\omega}{ln}^{2}(|\ensuremath{\omega}|/D)

, in analogy with the 1D case of doped spin-Peierls and two-leg ladder compounds. We argue that the system exhibits a quasi-long-range order at

T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0

with instantaneous spin-spin correlations decreasing as

{n}_{i}/{ln}^{2}({n}_{i}{R}_{\mathrm{ij}})

for large distances

{R}_{\mathrm{ij}}

and we predict enhanced low energy fluctuations in neutron scattering.

DOI: 10.1103/physrevlett.81.2779

Related papers

Impurity-induced staggered polarization and antiferromagnetic order in spin- 12 Heisenberg two-leg ladder compound SrCu2O3: Extensive Cu NMR and NQR studies

Shigeki Ohsugi, Y. Tokunaga, Kenji Ishida +4 more

We report characteristics of impurity-induced staggered polarization (IISP) and antiferromagnetic long-range order (AF-LRO) in the gapped spin-1/2 Heisenberg two-leg ladder compound ${\mathrm{SrCu}}_{2}{\mathrm{O}}_{3}$ (Sr123). We have carried out comprehensive NMR and NQR investigations on three impurity-doped systems, $\mathrm{Sr}({\mathrm{Cu}}_{1\ensuremath{-}x}{M}_{x}{)}_{2}{\mathrm{O}}_{3} (M=\mathrm{Z}\mathrm{n},\mathrm{}\mathrm{Ni})$ with $x<~0.02$ and ${\mathrm{Sr}}_{1\ensuremath{-}x}{\mathrm{La}}_{x}{\mathrm{Cu}}_{2}{\mathrm{O}}_{3}$ with $x<~0.03.$ Either the Zn or Ni impurity that is nonmagnetic depletes a single spin on the ladders, whereas the La impurity is believed to dope electrons onto the ladders. The width of the Lorentzian Cu NMR spectrum increases with the increase in impurity content x and follows the Curie-like temperature (T) dependence as $W/T.$ The W's for the Zn- and Ni-doped samples (M doping) are larger than for the La-doped one (La doping). The NMR spectra were fit by assuming that unpaired spin ${S}_{0}=1/2$ induced next to impurity on the rung for the Zn and Ni doping ${(S}_{0}=1/4$ for the La doping) creates the staggered spin polarization along the leg, which decreases exponentially from ${S}_{0}.$ In Sr123, an instantaneous spin-correlation length ${\ensuremath{\xi}}_{0}$ was theoretically predicted as ${\ensuremath{\xi}}_{0}/a\ensuremath{\sim}3\ensuremath{-}8,$ where a is the lattice spacing between the Cu sites along the leg. However, a correlation length ${\ensuremath{\xi}}_{s}/a$ estimated from the IISP along the leg was found to be much longer than ${\ensuremath{\xi}}_{0}/a$ in $x=0.001$ and 0.005. The notable result is that ${\ensuremath{\xi}}_{s}/a$ that was found to be T independent is scaled to mean distances ${D}_{\mathrm{AV}}=1/(2x)$ between the Zn and Ni impurities and ${D}_{\mathrm{AV}}=1/x$ between the La impurities. When ${D}_{\mathrm{AV}}=500$ for $x=0.001$ (Zn doping), ${\ensuremath{\xi}}_{s}/a\ensuremath{\sim}50$ is estimated. The significantly broadened NQR spectrum has provided unambiguous evidence for the AF-LRO in the Zn and Ni doping (x=0.01 and 0.02). Rather uniform AF moments at the middle Cu sites between the impurities are estimated to be about $0.04{\ensuremath{\mu}}_{B}$ at 1.4 K along the a axis. By assuming that exponential decay constants of AF moments are equivalent to ${\ensuremath{\xi}}_{s}/a$'s for the IISP, the size of an AF moment next to the impurity is deduced as ${S}_{\mathrm{AF}}\ensuremath{\sim}1/4.$ We propose that these exponential distributions of IISP and AF moments along the two-leg suggest that an interladder interaction is in a weakly coupled quasi-one-dimensional (WC-Q1D) regime. The formula of ${T}_{N}{=J}_{0}\mathrm{exp}(\ensuremath{-}{D}_{\mathrm{AV}}/({\ensuremath{\xi}}_{s}/a))$ based on the WC-Q1D model explains ${T}_{N}(\mathrm{exp})=3$ K $(x=0.01)$ and 5.8 K $(x=0.02)$ quantitatively and predicts to be as small as ${T}_{N}=0.09$ K for $x=0.001$ using ${J}_{0}=2000$ K. On the other hand, there is no evidence of AF-LRO for the La doping $(x=0.02$ and 0.03) down to 1.4 K, nevertheless their ${\ensuremath{\xi}}_{s}/a$'s are almost equivalent to those in the Zn and Ni doping $(x=0.01$ and 0.02). We remark that the Q1D-IISP is dramatically enhanced by the interladder interaction even though so weak, once the impurity breaks up the quantum coherence in the short-range resonating valence bond (RVB) state with the gap. On the one hand, we propose that ${T}_{N}$ is determined by a strength of the interladder interaction and a size of ${S}_{0}.$ We report characteristics of impurity-induced staggered polarization (IISP) and antiferromagnetic long-range order (AF-LRO) in the gapped spin-1/2 Heisenberg two-leg ladder compound ${\mathrm{SrCu}}_{2}{\mathrm{O}}_{3}$ (Sr123). We have carried out comprehensive NMR and NQR investigations on three impurity-doped systems, $\mathrm{Sr}({\mathrm{Cu}}_{1\ensuremath{-}x}{M}_{x}{)}_{2}{\mathrm{O}}_{3} (M=\mathrm{Z}\mathrm{n},\mathrm{}\mathrm{Ni})$ with $x<~0.02$ and ${\mathrm{Sr}}_{1\ensuremath{-}x}{\mathrm{La}}_{x}{\mathrm{Cu}}_{2}{\mathrm{O}}_{3}$ with $x<~0.03.$ Either the Zn or Ni impurity that is nonmagnetic depletes a single spin on the ladders, whereas the La impurity is believed to dope electrons onto the ladders. The width of the Lorentzian Cu NMR spectrum increases with the increase in impurity content x and follows the Curie-like temperature (T) dependence as $W/T.$ The W's for the Zn- and Ni-doped samples (M doping) are larger than for the La-doped one (La doping). The NMR spectra were fit by assuming that unpaired spin ${S}_{0}=1/2$ induced next to impurity on the rung for the Zn and Ni doping ${(S}_{0}=1/4$ for the La doping) creates the staggered spin polarization along the leg, which decreases exponentially from ${S}_{0}.$ In Sr123, an instantaneous spin-correlation length ${\ensuremath{\xi}}_{0}$ was theoretically predicted as ${\ensuremath{\xi}}_{0}/a\ensuremath{\sim}3\ensuremath{-}8,$ where a is the lattice spacing between the Cu sites along the leg. However, a correlation length ${\ensuremath{\xi}}_{s}/a$ estimated from the IISP along the leg was found to be much longer than ${\ensuremath{\xi}}_{0}/a$ in $x=0.001$ and 0.005. The notable result is that ${\ensuremath{\xi}}_{s}/a$ that was found to be T independent is scaled to mean distances ${D}_{\mathrm{AV}}=1/(2x)$ between the Zn and Ni impurities and ${D}_{\mathrm{AV}}=1/x$ between the La impurities. When ${D}_{\mathrm{AV}}=500$ for $x=0.001$ (Zn doping), ${\ensuremath{\xi}}_{s}/a\ensuremath{\sim}50$ is estimated. The significantly broadened NQR spectrum has provided unambiguous evidence for the AF-LRO in the Zn and Ni doping (x=0.01 and 0.02). Rather uniform AF moments at the middle Cu sites between the impurities are estimated to be about $0.04{\ensuremath{\mu}}_{B}$ at 1.4 K along the a axis. By assuming that exponential decay constants of AF moments are equivalent to ${\ensuremath{\xi}}_{s}/a$'s for the IISP, the size of an AF moment next to the impurity is deduced as ${S}_{\mathrm{AF}}\ensuremath{\sim}1/4.$ We propose that these exponential distributions of IISP and AF moments along the two-leg suggest that an interladder interaction is in a weakly coupled quasi-one-dimensional (WC-Q1D) regime. The formula of ${T}_{N}{=J}_{0}\mathrm{exp}(\ensuremath{-}{D}_{\mathrm{AV}}/({\ensuremath{\xi}}_{s}/a))$ based on the WC-Q1D model explains ${T}_{N}(\mathrm{exp})=3$ K $(x=0.01)$ and 5.8 K $(x=0.02)$ quantitatively and predicts to be as small as ${T}_{N}=0.09$ K for $x=0.001$ using ${J}_{0}=2000$ K. On the other hand, there is no evidence of AF-LRO for the La doping $(x=0.02$ and 0.03) down to 1.4 K, nevertheless their ${\ensuremath{\xi}}_{s}/a$'s are almost equivalent to those in the Zn and Ni doping $(x=0.01$ and 0.02). We remark that the Q1D-IISP is dramatically enhanced by the interladder interaction even though so weak, once the impurity breaks up the quantum coherence in the short-range resonating valence bond (RVB) state with the gap. On the one hand, we propose that ${T}_{N}$ is determined by a strength of the interladder interaction and a size of ${S}_{0}.$

Physical review. B, Condensed matterAug 1, 199937 citations

Nonmagnetic Impurities in Spin Gap Systems

Naoto Nagaosa, Akira Furusaki, Manfred Sigrist +1 more

arXiv:cond-mat/9609016Journal of the Physical Society of JapanDec 15, 199657 citations

Impurities in the weakly coupled quantum spin chains Sr2CuO3 and SrCuO2

Koushik Karmakar, Rabindranath Bag, M. Skoulatos +2 more

We study the effect of nonmagnetic ${\mathrm{Zn}}^{2+}$ (spin-0) and magnetic ${\mathrm{Ni}}^{2+}$ (spin-1) impurities on the ground state and low-lying excitations of the quasi-one-dimensional spin- $1/2$ Heisenberg antiferromagnet ${\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}$ using inelastic neutron scattering, specific heat, and bulk magnetization measurements. We show that 1% ${\mathrm{Ni}}^{2+}$ doping in ${\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}$ results in a sizable spin gap in the spinon excitations, analogous to the case of Ni doped ${\mathrm{SrCuO}}_{2}$ previously reported [Simutis et al., Phys. Rev. Lett. 111, 067204 (2013)]. However, a similar level of ${\mathrm{Zn}}^{2+}$ doping in ${\mathrm{SrCuO}}_{2}$ , investigated here for comparison, did not reveal any signs of a spin gap. Magnetic ordering temperature was found to be suppressed in the presence of both ${\mathrm{Zn}}^{2+}$ and ${\mathrm{Ni}}^{2+}$ impurities; however, the rate of suppression due to ${\mathrm{Ni}}^{2+}$ was found to be much more pronounced than for ${\mathrm{Zn}}^{2+}$ . Effect of magnetic field on the ordering temperature is investigated. We found that with increasing magnetic field, not only the magnetic ordering temperature gradually increases but the size of specific heat anomaly associated with the magnetic ordering also progressively enhances, which can be qualitatively understood as due to the field induced suppression of quantum fluctuations.We study the effect of nonmagnetic ${\mathrm{Zn}}^{2+}$ (spin-0) and magnetic ${\mathrm{Ni}}^{2+}$ (spin-1) impurities on the ground state and low-lying excitations of the quasi-one-dimensional spin-$1/2$ Heisenberg antiferromagnet ${\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}$ using inelastic neutron scattering, specific heat, and bulk magnetization measurements. We show that 1% ${\mathrm{Ni}}^{2+}$ doping in ${\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}$ results in a sizable spin gap in the spinon excitations, analogous to the case of Ni doped ${\mathrm{SrCuO}}_{2}$ previously reported [Simutis et al., Phys. Rev. Lett. 111, 067204 (2013)]. However, a similar level of ${\mathrm{Zn}}^{2+}$ doping in ${\mathrm{SrCuO}}_{2}$, investigated here for comparison, did not reveal any signs of a spin gap. Magnetic ordering temperature was found to be suppressed in the presence of both ${\mathrm{Zn}}^{2+}$ and ${\mathrm{Ni}}^{2+}$ impurities; however, the rate of suppression due to ${\mathrm{Ni}}^{2+}$ was found to be much more pronounced than for ${\mathrm{Zn}}^{2+}$. Effect of magnetic field on the ordering temperature is investigated. We found that with increasing magnetic field, not only the magnetic ordering temperature gradually increases but the size of specific heat anomaly associated with the magnetic ordering also progressively enhances, which can be qualitatively understood as due to the field induced suppression of quantum fluctuations.

arXiv:1706.01262Physical review. B./Physical review. BJun 28, 201711 citations

Nonmagnetic impurities in spin-gapped and gapless Heisenberg antiferromagnets

Anders W. Sandvik, Elbio R. Dagotto, Douglas J. Scalapino

We discuss the effects of nonmagnetic impurities (static holes) in the $S=1/2$ Heisenberg model on a two-dimensional (2D) square lattice, and on two- and three-leg ladders. Results of quantum Monte Carlo simulations show that a free localized moment is induced around an isolated impurity only in the spin-gapped two-leg ladder, in agreement with previous theoretical expectations. The localization length of the $S=1/2$ impurity moment in a two-leg ladder with isotropic couplings is $\ensuremath{\approx}1.5$ lattice spacings. In gapless ladders (odd number of legs) and in the 2D lattice, no free moments are induced. However, in the 2D system, which has antiferromagnetic long-range order at $T=0$ and a gap for longitudinal fluctuations, the impurity leads to a localized distortion of the magnetization in the direction of the broken symmetry, with total ${S}^{z}=1/2.$ The shape of the distortion is in close agreement with previous spin-wave calculations. For all the systems, a staggered moment is induced by the impurity. For the gapped two-leg ladder this is exponentially localized within a length equal to the spin correlation length of the ladder ($\ensuremath{\approx}3.2$ lattice spacings). For the three-leg ladder, the integrated staggered moment diverges as the system size goes to infinity, but the staggered magnetization at a given site appears to vanish. In 2D, we discuss the behavior seen in finite systems both with and without additional symmetry-breaking mechanisms. We also discuss the effect of impurities on the NMR Knight shift and the bulk magnetic susceptibility.We discuss the effects of nonmagnetic impurities (static holes) in the $S=1/2$ Heisenberg model on a two-dimensional (2D) square lattice, and on two- and three-leg ladders. Results of quantum Monte Carlo simulations show that a free localized moment is induced around an isolated impurity only in the spin-gapped two-leg ladder, in agreement with previous theoretical expectations. The localization length of the $S=1/2$ impurity moment in a two-leg ladder with isotropic couplings is $\ensuremath{\approx}1.5$ lattice spacings. In gapless ladders (odd number of legs) and in the 2D lattice, no free moments are induced. However, in the 2D system, which has antiferromagnetic long-range order at $T=0$ and a gap for longitudinal fluctuations, the impurity leads to a localized distortion of the magnetization in the direction of the broken symmetry, with total ${S}^{z}=1/2.$ The shape of the distortion is in close agreement with previous spin-wave calculations. For all the systems, a staggered moment is induced by the impurity. For the gapped two-leg ladder this is exponentially localized within a length equal to the spin correlation length of the ladder ($\ensuremath{\approx}3.2$ lattice spacings). For the three-leg ladder, the integrated staggered moment diverges as the system size goes to infinity, but the staggered magnetization at a given site appears to vanish. In 2D, we discuss the behavior seen in finite systems both with and without additional symmetry-breaking mechanisms. We also discuss the effect of impurities on the NMR Knight shift and the bulk magnetic susceptibility.

Physical review. B, Condensed matterNov 1, 199749 citations

Impurity Induced Antiferromagnetism in Spin-1/2 Heisenberg Two-Leg Ladder Systems

Hidetoshi Fukuyama, Naoto Nagaosa, Masako Saitō +1 more

Journal of the Physical Society of JapanAug 15, 199673 citations

Disappearance of the Spin Gap in a Zn-Doped 2-Leg Ladder Compound Sr(Cu1-xZnx)2O3

Masaki Azuma, Mikio Takano, Roger S. Eccleston

An inelastic neutron scattering study was performed on a Zn-substituted spin-1/2 Heisenberg 2-leg ladder compound Sr(Cu1-xZnx)2O3 (x < 0.04) to investigate nonmagnetic impurity effects on the quantum spin system with a large spin gap of about 400 K. The magnitude of the spin gap was found to be independent of Zn concentration, 33 meV, while the integrated magnetic scattering which corresponds to the singlet-triplet excitation decreased monotonically with increasing x. On account of the total-sum rule, this result supports the existence of a finite in-gap density of states at E = 0 which was suggested from the T-linear magnetic specific heat.An inelastic neutron scattering study was performed on a Zn-substituted spin-1/2 Heisenberg 2-leg ladder compound Sr(Cu1-xZnx)2O3 (x < 0.04) to investigate nonmagnetic impurity effects on the quantum spin system with a large spin gap of about 400 K. The magnitude of the spin gap was found to be independent of Zn concentration, 33 meV, while the integrated magnetic scattering which corresponds to the singlet-triplet excitation decreased monotonically with increasing x. On account of the total-sum rule, this result supports the existence of a finite in-gap density of states at E = 0 which was suggested from the T-linear magnetic specific heat.

arXiv:cond-mat/9706170Journal of the Physical Society of JapanMar 15, 199840 citations

We solve the problem of

N

1/\ensuremath{\omega}{ln}^{2}(|\ensuremath{\omega}|/D)

, in analogy with the 1D case of doped spin-Peierls and two-leg ladder compounds. We argue that the system exhibits a quasi-long-range order at

T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0

with instantaneous spin-spin correlations decreasing as

{n}_{i}/{ln}^{2}({n}_{i}{R}_{\mathrm{ij}})

for large distances

{R}_{\mathrm{ij}}

and we predict enhanced low energy fluctuations in neutron scattering.

We report characteristics of impurity-induced staggered polarization (IISP) and antiferromagnetic long-range order (AF-LRO) in the gapped spin-1/2 Heisenberg two-leg ladder compound

{\mathrm{SrCu}}_{2}{\mathrm{O}}_{3}

(Sr123). We have carried out comprehensive NMR and NQR investigations on three impurity-doped systems, $\mathrm{Sr}({\mathrm{Cu}}_{1\ensuremath{-}x}{M}_{x}{)}_{2}{\mathrm{O}}_{3} (M=\mathrm{Z}\mathrm{n},\mathrm{}\mathrm{Ni})$ with

x<~0.02

and ${\mathrm{Sr}}_{1\ensuremath{-}x}{\mathrm{La}}_{x}{\mathrm{Cu}}_{2}{\mathrm{O}}_{3}$ with

x<~0.03.

Either the Zn or Ni impurity that is nonmagnetic depletes a single spin on the ladders, whereas the La impurity is believed to dope electrons onto the ladders. The width of the Lorentzian Cu NMR spectrum increases with the increase in impurity content x and follows the Curie-like temperature (T) dependence as

W/T.

The W's for the Zn- and Ni-doped samples (M doping) are larger than for the La-doped one (La doping). The NMR spectra were fit by assuming that unpaired spin

{S}_{0}=1/2

induced next to impurity on the rung for the Zn and Ni doping

{(S}_{0}=1/4

for the La doping) creates the staggered spin polarization along the leg, which decreases exponentially from

{S}_{0}.

In Sr123, an instantaneous spin-correlation length ${\ensuremath{\xi}}_{0}$ was theoretically predicted as ${\ensuremath{\xi}}_{0}/a\ensuremath{\sim}3\ensuremath{-}8,$ where a is the lattice spacing between the Cu sites along the leg. However, a correlation length ${\ensuremath{\xi}}_{s}/a$ estimated from the IISP along the leg was found to be much longer than ${\ensuremath{\xi}}_{0}/a$ in

x=0.001

and 0.005. The notable result is that ${\ensuremath{\xi}}_{s}/a$ that was found to be T independent is scaled to mean distances

{D}_{\mathrm{AV}}=1/(2x)

between the Zn and Ni impurities and

{D}_{\mathrm{AV}}=1/x

between the La impurities. When

{D}_{\mathrm{AV}}=500

for

x=0.001

(Zn doping), ${\ensuremath{\xi}}_{s}/a\ensuremath{\sim}50$ is estimated. The significantly broadened NQR spectrum has provided unambiguous evidence for the AF-LRO in the Zn and Ni doping (x=0.01 and 0.02). Rather uniform AF moments at the middle Cu sites between the impurities are estimated to be about $0.04{\ensuremath{\mu}}_{B}$ at 1.4 K along the a axis. By assuming that exponential decay constants of AF moments are equivalent to ${\ensuremath{\xi}}_{s}/a$'s for the IISP, the size of an AF moment next to the impurity is deduced as ${S}_{\mathrm{AF}}\ensuremath{\sim}1/4.$ We propose that these exponential distributions of IISP and AF moments along the two-leg suggest that an interladder interaction is in a weakly coupled quasi-one-dimensional (WC-Q1D) regime. The formula of ${T}_{N}{=J}_{0}\mathrm{exp}(\ensuremath{-}{D}_{\mathrm{AV}}/({\ensuremath{\xi}}_{s}/a))$ based on the WC-Q1D model explains

{T}_{N}(\mathrm{exp})=3

(x=0.01)

and 5.8 K

(x=0.02)

quantitatively and predicts to be as small as

{T}_{N}=0.09

K for

x=0.001

using

{J}_{0}=2000

K. On the other hand, there is no evidence of AF-LRO for the La doping

(x=0.02

and 0.03) down to 1.4 K, nevertheless their ${\ensuremath{\xi}}_{s}/a$'s are almost equivalent to those in the Zn and Ni doping

(x=0.01

and 0.02). We remark that the Q1D-IISP is dramatically enhanced by the interladder interaction even though so weak, once the impurity breaks up the quantum coherence in the short-range resonating valence bond (RVB) state with the gap. On the one hand, we propose that

{T}_{N}

is determined by a strength of the interladder interaction and a size of

{S}_{0}.

We study the effect of nonmagnetic

{\mathrm{Zn}}^{2+}

(spin-0) and magnetic

{\mathrm{Ni}}^{2+}

(spin-1) impurities on the ground state and low-lying excitations of the quasi-one-dimensional spin-

1/2

Heisenberg antiferromagnet

{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}

using inelastic neutron scattering, specific heat, and bulk magnetization measurements. We show that 1%

{\mathrm{Ni}}^{2+}

doping in

{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}

results in a sizable spin gap in the spinon excitations, analogous to the case of Ni doped

{\mathrm{SrCuO}}_{2}

previously reported [Simutis et al., Phys. Rev. Lett. 111, 067204 (2013)]. However, a similar level of

{\mathrm{Zn}}^{2+}

doping in

{\mathrm{SrCuO}}_{2}

, investigated here for comparison, did not reveal any signs of a spin gap. Magnetic ordering temperature was found to be suppressed in the presence of both

{\mathrm{Zn}}^{2+}

and

{\mathrm{Ni}}^{2+}

impurities; however, the rate of suppression due to

{\mathrm{Ni}}^{2+}

was found to be much more pronounced than for

{\mathrm{Zn}}^{2+}

. Effect of magnetic field on the ordering temperature is investigated. We found that with increasing magnetic field, not only the magnetic ordering temperature gradually increases but the size of specific heat anomaly associated with the magnetic ordering also progressively enhances, which can be qualitatively understood as due to the field induced suppression of quantum fluctuations.

We discuss the effects of nonmagnetic impurities (static holes) in the

S=1/2

Heisenberg model on a two-dimensional (2D) square lattice, and on two- and three-leg ladders. Results of quantum Monte Carlo simulations show that a free localized moment is induced around an isolated impurity only in the spin-gapped two-leg ladder, in agreement with previous theoretical expectations. The localization length of the

S=1/2

impurity moment in a two-leg ladder with isotropic couplings is $\ensuremath{\approx}1.5$ lattice spacings. In gapless ladders (odd number of legs) and in the 2D lattice, no free moments are induced. However, in the 2D system, which has antiferromagnetic long-range order at

T=0

and a gap for longitudinal fluctuations, the impurity leads to a localized distortion of the magnetization in the direction of the broken symmetry, with total

{S}^{z}=1/2.

The shape of the distortion is in close agreement with previous spin-wave calculations. For all the systems, a staggered moment is induced by the impurity. For the gapped two-leg ladder this is exponentially localized within a length equal to the spin correlation length of the ladder ($\ensuremath{\approx}3.2$ lattice spacings). For the three-leg ladder, the integrated staggered moment diverges as the system size goes to infinity, but the staggered magnetization at a given site appears to vanish. In 2D, we discuss the behavior seen in finite systems both with and without additional symmetry-breaking mechanisms. We also discuss the effect of impurities on the NMR Knight shift and the bulk magnetic susceptibility.