arXiv:cond-mat/9807187 · Physical review. B, Condensed matter

Staggered-spin contribution to nuclear spin-lattice relaxation in two-leg antiferromagnetic spin- 12 ladders

Dmitri A. Ivanov, Patrick A. Lee

Published Feb 15, 1999• 11 citations

Abstract

We study the nuclear spin-lattice relaxation rate ${1/T}_{1}$ in the two-leg antiferromagnetic spin-1/2 Heisenberg ladder. More specifically, we consider the contribution to ${1/T}_{1}$ from the processes with momentum transfer $(\ensuremath{\pi},\ensuremath{\pi}).$ In the limit of weak coupling between the two chains, this contribution is of activation type with gap $2\ensuremath{\Delta}$ at low temperatures $(\ensuremath{\Delta}$ is the spin gap), but crosses over to a slowly decaying temperature dependence at the crossover temperature $T\ensuremath{\approx}\ensuremath{\Delta}.$ This crossover possibly explains the recent high-temperature NMR results on ladder-containing cuprates by Imai et al.

DOI: 10.1103/physrevb.59.4803

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Nuclear Spin Relaxation Rates in Two-Leg Spin Ladders

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Spin dynamics and transport in gapped one-dimensional Heisenberg antiferromagnets at nonzero temperatures

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arXiv:cond-mat/9807187 · Physical review. B, Condensed matter

Staggered-spin contribution to nuclear spin-lattice relaxation in two-leg antiferromagnetic spin- 12 ladders

Dmitri A. Ivanov, Patrick A. Lee

Published Feb 15, 1999• 11 citations

arXiv HTML PDF

Abstract

We study the nuclear spin-lattice relaxation rate

{1/T}_{1}

in the two-leg antiferromagnetic spin-1/2 Heisenberg ladder. More specifically, we consider the contribution to

{1/T}_{1}

from the processes with momentum transfer

(\ensuremath{\pi},\ensuremath{\pi}).

In the limit of weak coupling between the two chains, this contribution is of activation type with gap

2\ensuremath{\Delta}

at low temperatures

(\ensuremath{\Delta}

is the spin gap), but crosses over to a slowly decaying temperature dependence at the crossover temperature

T\ensuremath{\approx}\ensuremath{\Delta}.

This crossover possibly explains the recent high-temperature NMR results on ladder-containing cuprates by Imai et al.

DOI: 10.1103/physrevb.59.4803

Related papers

On the Estimate of the Spin-Gap in Quasi-1D Heisenberg Antiferromagnets from Nuclear Spin-Lattice Relaxation

Roberto Melzi, P. Carretta

We present a careful analysis of the temperature dependence of the nuclear spin-lattice relaxation rate l/ T 1 in gapped quasi-1D Heisenberg antiferromagnets. It is found that in order to estimate the value of the gap correctly from l/ T 1 , the peculiar features of the dispersion curve for the triplet excitations must be taken into account. The temperature dependence of l/ T 1 due to two-magnon processes is reported for different values of the ratio r = J ⌊ / J ‖ between the superexchange constants in a 2-leg ladder. As an illustrative example, we compare our results to the experimental findings for 63 Cu l/ T 1 in the dimerized chains and 2-leg ladders contained in Sr 14 Cu 24 O 41We present a careful analysis of the temperature dependence of the nuclear spin-lattice relaxation rate l/ T 1 in gapped quasi-1D Heisenberg antiferromagnets. It is found that in order to estimate the value of the gap correctly from l/ T 1 , the peculiar features of the dispersion curve for the triplet excitations must be taken into account. The temperature dependence of l/ T 1 due to two-magnon processes is reported for different values of the ratio r = J ⌊ / J ‖ between the superexchange constants in a 2-leg ladder. As an illustrative example, we compare our results to the experimental findings for 63 Cu l/ T 1 in the dimerized chains and 2-leg ladders contained in Sr 14 Cu 24 O 41

arXiv:cond-mat/9904074Kluwer Academic Publishers eBooksMar 29, 20060 citations

Nuclear Spin Relaxation Rates in Two-Leg Spin Ladders

Félix Naef, Xiaoqun Wang

Using the transfer-matrix density-matrix renormalization group method, we study the nuclear spin relaxation rate ${1/T}_{1}$ in the two-leg $s\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}$ ladder as a function of the interchain $({J}_{\ensuremath{\perp}})$ and intrachain $({J}_{\ensuremath{\parallel}})$ couplings. In particular, we separate the ${q}_{y}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ and $\ensuremath{\pi}$ contributions and show that the latter contribute significantly to the copper relaxation rate ${}^{63}({1/T}_{1})$ in the experimentally relevant coupling and temperature range. We compare our results to both theoretical predictions and experimental measures on ladder materials.Using the transfer-matrix density-matrix renormalization group method, we study the nuclear spin relaxation rate ${1/T}_{1}$ in the two-leg $s\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}$ ladder as a function of the interchain $({J}_{\ensuremath{\perp}})$ and intrachain $({J}_{\ensuremath{\parallel}})$ couplings. In particular, we separate the ${q}_{y}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ and $\ensuremath{\pi}$ contributions and show that the latter contribute significantly to the copper relaxation rate ${}^{63}({1/T}_{1})$ in the experimentally relevant coupling and temperature range. We compare our results to both theoretical predictions and experimental measures on ladder materials.

arXiv:cond-mat/9907117Physical Review LettersFeb 7, 200018 citations

Spin dynamics and transport in gapped one-dimensional Heisenberg antiferromagnets at nonzero temperatures

Kedar Damle, Subir Sachdev A

We discuss the theory of the nonzero temperature $(T)$ spin dynamics and transport in one-dimensional Heisenberg antiferromagnets with a gap $\ensuremath{\Delta}$. For $T\ensuremath{\ll}\ensuremath{\Delta}$, we develop a semiclassical picture of thermally excited particles. Multiple inelastic collisions between the particles are crucial, and are described by a two-particle $\mathcal{S}$ matrix which is shown to have a superuniversal form at low momenta. This is established by computations on the $O(3)$ $\ensuremath{\sigma}$ model, and strong- and weak-coupling expansions (the latter using a Majorana fermion representation) for the two-leg $S=1/2$ Heisenberg antiferromagnetic ladder. As an aside, we note that the strong-coupling calculation reveals an $S=1$ , two-particle bound state which leads to the presence of a second peak in the $T=0$ inelastic neutron-scattering (INS) cross section for a range of values of momentum transfer. We obtain exact, or numerically exact, universal expressions for the thermal broadening of the quasiparticle peak in the INS cross section, the spin diffusivity, and for the field dependence of the NMR relaxation rate ${1/T}_{1}$ of the effective semiclassical model; these are expected to be asymptotically exact for the quantum antiferromagnets in the limit $T\ensuremath{\ll}\ensuremath{\Delta}$. The results for ${1/T}_{1}$ are compared with the experimental findings of Takigawa et al. [Phys. Rev. Lett. 76, 2173 (1996)] and the agreement is quite good. In the regime $\ensuremath{\Delta}<T<$(a typical microscopic exchange) and we argue that a complementary description in terms of semiclassical waves applies, and give some exact results for the thermodynamics and dynamics.We discuss the theory of the nonzero temperature $(T)$ spin dynamics and transport in one-dimensional Heisenberg antiferromagnets with a gap $\ensuremath{\Delta}$. For $T\ensuremath{\ll}\ensuremath{\Delta}$, we develop a semiclassical picture of thermally excited particles. Multiple inelastic collisions between the particles are crucial, and are described by a two-particle $\mathcal{S}$ matrix which is shown to have a superuniversal form at low momenta. This is established by computations on the $O(3)$ $\ensuremath{\sigma}$ model, and strong- and weak-coupling expansions (the latter using a Majorana fermion representation) for the two-leg $S=1/2$ Heisenberg antiferromagnetic ladder. As an aside, we note that the strong-coupling calculation reveals an $S=1$, two-particle bound state which leads to the presence of a second peak in the $T=0$ inelastic neutron-scattering (INS) cross section for a range of values of momentum transfer. We obtain exact, or numerically exact, universal expressions for the thermal broadening of the quasiparticle peak in the INS cross section, the spin diffusivity, and for the field dependence of the NMR relaxation rate ${1/T}_{1}$ of the effective semiclassical model; these are expected to be asymptotically exact for the quantum antiferromagnets in the limit $T\ensuremath{\ll}\ensuremath{\Delta}$. The results for ${1/T}_{1}$ are compared with the experimental findings of Takigawa et al. [Phys. Rev. Lett. 76, 2173 (1996)] and the agreement is quite good. In the regime $\ensuremath{\Delta}<T<$(a typical microscopic exchange) and we argue that a complementary description in terms of semiclassical waves applies, and give some exact results for the thermodynamics and dynamics.

arXiv:cond-mat/9711014Physical review. B, Condensed matterApr 1, 1998175 citations

NMR relaxation in the spin-1 Heisenberg chain

Sylvain Capponi, Maxime Dupont, Anders W. Sandvik +1 more

We consider the isotropic $S=1$ Heisenberg chain with a finite Haldane gap $\mathrm{\ensuremath{\Delta}}$ and use state-of-the-art numerical techniques to investigate its dynamical properties at finite temperature, focusing on the nuclear spin-lattice relaxation rate $1/{T}_{1}$ measured in nuclear magnetic resonance (NMR) experiments, for instance. In particular, we analyze the contributions from modes with momenta close to $q\ensuremath{\approx}0$ and $q\ensuremath{\approx}\ensuremath{\pi}$ as a function of temperature. At high-temperature we observe spin diffusion, while at low-temperature we argue that a simple activated behavior $1/{T}_{1}\ensuremath{\propto}exp(\ensuremath{-}\mathrm{\ensuremath{\Delta}}/T)$ can be observed only at temperatures much smaller than the gap $\mathrm{\ensuremath{\Delta}}$.We consider the isotropic $S=1$ Heisenberg chain with a finite Haldane gap $\mathrm{\ensuremath{\Delta}}$ and use state-of-the-art numerical techniques to investigate its dynamical properties at finite temperature, focusing on the nuclear spin-lattice relaxation rate $1/{T}_{1}$ measured in nuclear magnetic resonance (NMR) experiments, for instance. In particular, we analyze the contributions from modes with momenta close to $q\ensuremath{\approx}0$ and $q\ensuremath{\approx}\ensuremath{\pi}$ as a function of temperature. At high-temperature we observe spin diffusion, while at low-temperature we argue that a simple activated behavior $1/{T}_{1}\ensuremath{\propto}exp(\ensuremath{-}\mathrm{\ensuremath{\Delta}}/T)$ can be observed only at temperatures much smaller than the gap $\mathrm{\ensuremath{\Delta}}$.

arXiv:1905.12697Physical review. B./Physical review. BSep 9, 201918 citations

Observation of a Spin Gap in Sr Cu 2 O 3 Comprising Spin-½ Quasi-1D Two-Leg Ladders

Masaki Azuma, Zenji Hiroi, M. Takano +2 more

Magnetic properties of Sr ${\mathrm{Cu}}_{2}$ ${\mathrm{O}}_{3}$ and ${\mathrm{Sr}}_{2}$ ${\mathrm{Cu}}_{3}$ ${\mathrm{O}}_{5}$ , containing two-leg and three-leg $S=\frac{1}{2}$ ladders made of antiferromagnetic Cu-O-Cu linear bonds, were investigated. The susceptibility of the two-leg ladder material, Sr ${\mathrm{Cu}}_{2}$ ${\mathrm{O}}_{3}$ , is characteristic of thermal excitation from a nonmagnetic ground state with a spin gap of about 420 K, while the susceptibility of ${\mathrm{Sr}}_{2}$ ${\mathrm{Cu}}_{3}$ ${\mathrm{O}}_{5}$ , containing three-leg ladders, reflects a gapless spin excitation spectrum. The temperature dependence of the nuclear-spin lattice relaxation rate, $\frac{1}{{T}_{1}}$ , of $^{63}\mathrm{Cu}$ NMR also indicates the existence of a spin gap only for Sr ${\mathrm{Cu}}_{2}$ ${\mathrm{O}}_{3}$ . The spin gap we observed for Sr ${\mathrm{Cu}}_{2}$ ${\mathrm{O}}_{3}$ confirms a recent theoretical prediction for the magnetic behavior of spin = \textonehalf{} antiferromagnetically coupled Heisenberg chains.Magnetic properties of Sr${\mathrm{Cu}}_{2}$${\mathrm{O}}_{3}$ and ${\mathrm{Sr}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{5}$, containing two-leg and three-leg $S=\frac{1}{2}$ ladders made of antiferromagnetic Cu-O-Cu linear bonds, were investigated. The susceptibility of the two-leg ladder material, Sr${\mathrm{Cu}}_{2}$${\mathrm{O}}_{3}$, is characteristic of thermal excitation from a nonmagnetic ground state with a spin gap of about 420 K, while the susceptibility of ${\mathrm{Sr}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{5}$, containing three-leg ladders, reflects a gapless spin excitation spectrum. The temperature dependence of the nuclear-spin lattice relaxation rate, $\frac{1}{{T}_{1}}$, of $^{63}\mathrm{Cu}$ NMR also indicates the existence of a spin gap only for Sr${\mathrm{Cu}}_{2}$${\mathrm{O}}_{3}$. The spin gap we observed for Sr${\mathrm{Cu}}_{2}$${\mathrm{O}}_{3}$ confirms a recent theoretical prediction for the magnetic behavior of spin = \textonehalf{} antiferromagnetically coupled Heisenberg chains.

Physical Review LettersDec 19, 1994732 citations

NMR relaxation rates for the spin-1/2 Heisenberg chain

Anders W. Sandvik

The spin-lattice relaxation rate 1/ ${\mathit{T}}_{1}$ and the spin-echo decay rate 1/ ${\mathit{T}}_{2\mathit{G}}$ for the spin-1/2 antiferromagnetic Heisenberg chain are calculated using quantum Monte Carlo and maximum-entropy analytic continuation. The results are compared with recent analytical calculations by Sachdev. If the nuclear hyperfine form factor ${\mathit{A}}_{\mathit{q}}$ is strongly peaked around q=\ensuremath{\pi}, the predicted low-temperature behavior [1/ ${\mathit{T}}_{1}$ \ensuremath{\sim} ${\mathrm{ln}}^{1/2}$ (1/T), 1/ ${\mathit{T}}_{2\mathit{G}}$ \ensuremath{\sim} ${\mathrm{ln}}^{1/2}$ (1/T)/\ensuremath{\surd}T] extends up to temperatures as high as T/J\ensuremath{\approxeq}0.5. If ${\mathit{A}}_{\mathit{q}}$ has significant weight for q\ensuremath{\approxeq}0 there are large contributions from diffusive long-wavelength processes not taken into account in the theory, and very low temperatures are needed in order to observe the asymptotic T\ensuremath{\rightarrow}0 forms.The spin-lattice relaxation rate 1/${\mathit{T}}_{1}$ and the spin-echo decay rate 1/${\mathit{T}}_{2\mathit{G}}$ for the spin-1/2 antiferromagnetic Heisenberg chain are calculated using quantum Monte Carlo and maximum-entropy analytic continuation. The results are compared with recent analytical calculations by Sachdev. If the nuclear hyperfine form factor ${\mathit{A}}_{\mathit{q}}$ is strongly peaked around q=\ensuremath{\pi}, the predicted low-temperature behavior [1/${\mathit{T}}_{1}$\ensuremath{\sim}${\mathrm{ln}}^{1/2}$(1/T), 1/${\mathit{T}}_{2\mathit{G}}$\ensuremath{\sim}${\mathrm{ln}}^{1/2}$(1/T)/\ensuremath{\surd}T] extends up to temperatures as high as T/J\ensuremath{\approxeq}0.5. If ${\mathit{A}}_{\mathit{q}}$ has significant weight for q\ensuremath{\approxeq}0 there are large contributions from diffusive long-wavelength processes not taken into account in the theory, and very low temperatures are needed in order to observe the asymptotic T\ensuremath{\rightarrow}0 forms.

arXiv:cond-mat/9509118Physical review. B, Condensed matterOct 1, 199546 citations

We study the nuclear spin-lattice relaxation rate

{1/T}_{1}

in the two-leg antiferromagnetic spin-1/2 Heisenberg ladder. More specifically, we consider the contribution to

{1/T}_{1}

from the processes with momentum transfer

(\ensuremath{\pi},\ensuremath{\pi}).

In the limit of weak coupling between the two chains, this contribution is of activation type with gap

2\ensuremath{\Delta}

at low temperatures

(\ensuremath{\Delta}

is the spin gap), but crosses over to a slowly decaying temperature dependence at the crossover temperature

T\ensuremath{\approx}\ensuremath{\Delta}.

This crossover possibly explains the recent high-temperature NMR results on ladder-containing cuprates by Imai et al.

Using the transfer-matrix density-matrix renormalization group method, we study the nuclear spin relaxation rate

{1/T}_{1}

in the two-leg

s\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}

ladder as a function of the interchain $({J}_{\ensuremath{\perp}})$ and intrachain $({J}_{\ensuremath{\parallel}})$ couplings. In particular, we separate the

{q}_{y}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0

and $\ensuremath{\pi}$ contributions and show that the latter contribute significantly to the copper relaxation rate

{}^{63}({1/T}_{1})

in the experimentally relevant coupling and temperature range. We compare our results to both theoretical predictions and experimental measures on ladder materials.

We discuss the theory of the nonzero temperature

(T)

spin dynamics and transport in one-dimensional Heisenberg antiferromagnets with a gap $\ensuremath{\Delta}$. For $T\ensuremath{\ll}\ensuremath{\Delta}$, we develop a semiclassical picture of thermally excited particles. Multiple inelastic collisions between the particles are crucial, and are described by a two-particle

\mathcal{S}

matrix which is shown to have a superuniversal form at low momenta. This is established by computations on the

O(3)

$\ensuremath{\sigma}$ model, and strong- and weak-coupling expansions (the latter using a Majorana fermion representation) for the two-leg

S=1/2

Heisenberg antiferromagnetic ladder. As an aside, we note that the strong-coupling calculation reveals an

S=1

, two-particle bound state which leads to the presence of a second peak in the

T=0

inelastic neutron-scattering (INS) cross section for a range of values of momentum transfer. We obtain exact, or numerically exact, universal expressions for the thermal broadening of the quasiparticle peak in the INS cross section, the spin diffusivity, and for the field dependence of the NMR relaxation rate

{1/T}_{1}

of the effective semiclassical model; these are expected to be asymptotically exact for the quantum antiferromagnets in the limit $T\ensuremath{\ll}\ensuremath{\Delta}$. The results for

{1/T}_{1}

are compared with the experimental findings of Takigawa et al. [Phys. Rev. Lett. 76, 2173 (1996)] and the agreement is quite good. In the regime $\ensuremath{\Delta}<T<$(a typical microscopic exchange) and we argue that a complementary description in terms of semiclassical waves applies, and give some exact results for the thermodynamics and dynamics.

We consider the isotropic

S=1

Heisenberg chain with a finite Haldane gap $\mathrm{\ensuremath{\Delta}}$ and use state-of-the-art numerical techniques to investigate its dynamical properties at finite temperature, focusing on the nuclear spin-lattice relaxation rate

1/{T}_{1}

measured in nuclear magnetic resonance (NMR) experiments, for instance. In particular, we analyze the contributions from modes with momenta close to $q\ensuremath{\approx}0$ and $q\ensuremath{\approx}\ensuremath{\pi}$ as a function of temperature. At high-temperature we observe spin diffusion, while at low-temperature we argue that a simple activated behavior $1/{T}_{1}\ensuremath{\propto}exp(\ensuremath{-}\mathrm{\ensuremath{\Delta}}/T)$ can be observed only at temperatures much smaller than the gap $\mathrm{\ensuremath{\Delta}}$.

Magnetic properties of Sr

{\mathrm{Cu}}_{2}

{\mathrm{O}}_{3}

and

{\mathrm{Sr}}_{2}

{\mathrm{Cu}}_{3}

{\mathrm{O}}_{5}

, containing two-leg and three-leg

S=\frac{1}{2}

ladders made of antiferromagnetic Cu-O-Cu linear bonds, were investigated. The susceptibility of the two-leg ladder material, Sr

{\mathrm{Cu}}_{2}

{\mathrm{O}}_{3}

, is characteristic of thermal excitation from a nonmagnetic ground state with a spin gap of about 420 K, while the susceptibility of

{\mathrm{Sr}}_{2}

{\mathrm{Cu}}_{3}

{\mathrm{O}}_{5}

, containing three-leg ladders, reflects a gapless spin excitation spectrum. The temperature dependence of the nuclear-spin lattice relaxation rate,

\frac{1}{{T}_{1}}

, of

^{63}\mathrm{Cu}

NMR also indicates the existence of a spin gap only for Sr

{\mathrm{Cu}}_{2}

{\mathrm{O}}_{3}

. The spin gap we observed for Sr

{\mathrm{Cu}}_{2}

{\mathrm{O}}_{3}

confirms a recent theoretical prediction for the magnetic behavior of spin = \textonehalf{} antiferromagnetically coupled Heisenberg chains.

The spin-lattice relaxation rate 1/

{\mathit{T}}_{1}

and the spin-echo decay rate 1/

{\mathit{T}}_{2\mathit{G}}

for the spin-1/2 antiferromagnetic Heisenberg chain are calculated using quantum Monte Carlo and maximum-entropy analytic continuation. The results are compared with recent analytical calculations by Sachdev. If the nuclear hyperfine form factor

{\mathit{A}}_{\mathit{q}}

is strongly peaked around q=\ensuremath{\pi}, the predicted low-temperature behavior [1/

{\mathit{T}}_{1}

\ensuremath{\sim}

{\mathrm{ln}}^{1/2}

(1/T), 1/

{\mathit{T}}_{2\mathit{G}}

\ensuremath{\sim}

{\mathrm{ln}}^{1/2}

(1/T)/\ensuremath{\surd}T] extends up to temperatures as high as T/J\ensuremath{\approxeq}0.5. If

{\mathit{A}}_{\mathit{q}}

has significant weight for q\ensuremath{\approxeq}0 there are large contributions from diffusive long-wavelength processes not taken into account in the theory, and very low temperatures are needed in order to observe the asymptotic T\ensuremath{\rightarrow}0 forms.