Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review B

Stationary and dynamical aspects of two-magnon states in disordered ferromagnetic chains

Elton Malta Nascimento, F. A. B. F. de Moura, M. L. Lyra

Published Dec 14, 2005• 3 citations
Publisher

Abstract

We study the nature of collective two-spin excitations in disordered $S=1∕2$ ferromagnetic chains. Using a direct diagonalization scheme, we characterize the two-magnon eigenstates by computing their spacial extent, two-point correlation and the average distance between the excited spins within the allowed energy band. We found that, due to the effective excitation interaction imposed by the exclusion rule, the low-energy two-magnon states display strong spin-spin correlations as compared to the more localized high-energy states. We further solve the time-dependent Schroedinger equation to follow the time evolution of an initially localized two-magnon state. We show that the effective one-magnon wave packet develops power-law tails with distinct exponents for the left and right tails, with the distribution function of the spin-spin distance decaying as $P(d)\ensuremath{\propto}1∕d$. We show that the average distance between the two excited spins evolves in time diffusively, while the wave-packet dispersion evolves superdiffusively.

DOI
10.1103/physrevb.72.224420

Related papers

Magnon delocalization in ferromagnetic chains with long-range correlated disorder

R. P. A. Lima, M. L. Lyra, Elton Malta Nascimento +1 more

We study one-magnon excitations in a random ferromagnetic Heisenberg chain with long-range correlations in the coupling constant distribution. By employing an exact diagonalization procedure, we compute the localization length of all one-magnon states within the band of allowed energies E. The random distribution of coupling constants was assumed to have a power spectrum decaying as $S(k)\ensuremath{\propto}{1/k}^{\ensuremath{\alpha}}.$ We found that for $\ensuremath{\alpha}<1,$ one-magnon excitations remain exponentially localized with the localization length $\ensuremath{\xi}$ diverging as $1/E.$ For $\ensuremath{\alpha}=1$ a faster divergence of $\ensuremath{\xi}$ is obtained. For any $\ensuremath{\alpha}>1,$ a phase of delocalized magnons emerges at the bottom of the band. We characterize the scaling behavior of the localization length on all regimes and relate it with the scaling properties of the long-range correlated exchange coupling distribution.

arXiv:cond-mat/0112274•Physical review. B, Condensed matter•Feb 13, 2002•20 citations

Delocalization and spin-wave dynamics in ferromagnetic chains with long-range correlated random exchange

F. A. B. F. de Moura, Mauricio D. Coutinho-Filho, Ernesto P. Raposo +1 more

We study the one-dimensional quantum Heisenberg ferromagnet with exchange couplings exhibiting long-range correlated disorder with power spectrum proportional to ${1/k}^{\ensuremath{\alpha}},$ where k is the wave vector of the modulations on the random coupling landscape. By using the renormalization group, integration of the equations of motion, and exact diagonalization, we compute the spin-wave localization length and the mean-square displacement of the wave packet. We find that, associated with the emergence of extended spin waves in the low-energy region for \ensuremath{\alpha}>1, the wave-packet mean-square displacement changes from a long-time super diffusive behavior for \ensuremath{\alpha}1 to a long-time ballistic behavior for \ensuremath{\alpha}>1. At the vicinity of \ensuremath{\alpha}=1, the mobility edge separating the extended and localized phases is shown to scale with the degree of correlation as ${E}_{c}\ensuremath{\propto}(\ensuremath{\alpha}\ensuremath{-}{1)}^{1/3}.$

arXiv:cond-mat/0206038•Physical review. B, Condensed matter•Jul 8, 2002•60 citations

Few-magnon excitations in a frustrated spin- S ferromagnetic chain with single-ion anisotropy

Jiawei Li, Ye Cao, Ning Wu

We study few-magnon excitations in a finite-size spin-$S$ chain with ferromagnetic nearest-neighbor (NN) interaction $J>0$ and antiferromagnetic next-nearest-neighbor (NNN) interaction ${J}^{\ensuremath{'}}<0$, in the presence of the single-ion (SI) anisotropy $D$. We first reveal the condition for the emergence of zero-excitation-energy states. In the isotropic case with $\mathrm{\ensuremath{\Delta}}={\mathrm{\ensuremath{\Delta}}}^{\ensuremath{'}}=1$ ($\mathrm{\ensuremath{\Delta}}$ and ${\mathrm{\ensuremath{\Delta}}}^{\ensuremath{'}}$ are the corresponding anisotropy parameters), a threshold of $J/|{J}^{\ensuremath{'}}|$, above which the ground state is ferromagnetic, is determined by exact diagonalization for short chains up to 12 sites. Using a set of exact two-magnon Bloch states, we then map the two-magnon problem to a single-particle one on an effective open chain with both NN and NNN hoppings. The whole two-magnon excitation spectrum is calculated for large systems, and the commensurate-incommensurate transition in the lowest-lying mode is found to exhibit different behaviors between $S=1/2$ and higher spins due to the interplay of the SI anisotropy and the NNN interaction. For the commensurate momentum $k=\ensuremath{-}\ensuremath{\pi}$, the effective lattice is decoupled into two NN open chains that can be exactly solved via a plane-wave ansatz. Based on this, we analytically identify in the ${\mathrm{\ensuremath{\Delta}}}^{\ensuremath{'}}\ensuremath{-}D/|{J}^{\ensuremath{'}}|$ plane the regions supporting the SI or NNN exchange two-magnon bound states near the edge of the band. In particular, we prove that there always exists a lower-lying NN exchange two-magnon bound state near the band edge for arbitrary $S\ensuremath{\ge}1/2$. Finally, we numerically calculate the $n$-magnon spectra for $S=1/2$ with $n\ensuremath{\le}5$ by using a spin-operator matrix element method. The corresponding $n$-magnon commensurate instability regions are determined for finite chains, and consistent results with prior literature are observed.

arXiv:2401.14101•Physical review. B./Physical review. B•May 2, 2024•3 citations

Bound states and correlation functions in the easy-plane Heisenberg ferromagnetic chain

H J Jensen, Hans C. Fogedby

The two-magnon excitation spectrum of the quantum spin-one easy-plane Heisenberg ferromagnetic chain is discussed. The spin-spin correlation functions are calculated within a systematic perturbation theory in terms of the anisotropy, and contributions from the two-magnon bound states are discussed. The results are compared with the predictions for non-interacting spin waves.

Journal of Physics C Solid State Physics•Feb 28, 1987•0 citations

Vanishing spin stiffness in weakly disordered two-dimensional Heisenberg ferromagnets

Jacopo Niedda, Aldo Coraggio, Giacomo Bracci-Testasecca +1 more

We show that a small fraction of antiferromagnetic bonds qualitatively alters the long-wavelength dynamics of two-dimensional Heisenberg ferromagnets. Although the classical ground state remains magnetized, weak bond frustration generates logarithmically correlated spatial fluctuations of the local spin stiffness, despite the microscopic disorder being short ranged. A replica field theory calculation shows that the effective disorder strength grows under coarse graining, while the spin stiffness decreases, yielding anomalously soft magnons with a scale-dependent dynamical exponent $z > 2$. Numerical diagonalization of the semiclassical spin-wave Hamiltonian confirms the anomalous low-energy scaling. The flow is toward an infinite-disorder, zero stiffness regime.

arXiv:2607.20036•Jul 22, 2026•0 citations

Long-time tails and anomalous slowing down in the relaxation of spatially inhomogeneous excitations in quantum spin chains

Gersh O. Berim, S. I. Berim, G. G. Cabrera

Exact analytic calculations in spin-1/2 $\mathrm{XY}$ chains show the presence of long-time tails in the asymptotic dynamics of spatially inhomogeneous excitations. The decay of inhomogeneities for $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty},$ is given in the form of a power law $(t/{\ensuremath{\tau}}_{Q}{)}^{\ensuremath{-}{\ensuremath{\nu}}_{Q}},$ where the relaxation time ${\ensuremath{\tau}}_{Q}$ and the exponent ${\ensuremath{\nu}}_{Q}$ depend on the wave vector Q, characterizing the spatial modulation of the initial excitation. We consider several variants of the $\mathrm{XY}$ model (dimerized, with staggered magnetic field, with bond alternation, and with isotropic and uniform interactions), that are grouped into two families, whether the energy spectrum has a gap or not. Once the initial condition is given, the nonequilibrium problem for the magnetization is solved in closed form, without any other assumption. The long-time behavior for $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ can be obtained systematically in a form of an asymptotic series through the stationary phase method. We found that gapped models show critical behavior with respect to Q, in the sense that there exist critical values ${Q}_{c}$ where the relaxation time ${\ensuremath{\tau}}_{Q}$ diverges and the exponent ${\ensuremath{\nu}}_{Q}$ changes discontinuously. At those points, a slowing down of the relaxation process is induced, similarly to phenomena occurring near phase transitions. Long-lived excitations are identified as incommensurate spin density waves that emerge in gapped systems, as a consequence of both approximate nesting of the spectrum and the degeneracy of some stationary points. In contrast, gapless models do not present the above anomalies as a function of the wave vector Q.

arXiv:cond-mat/0105594•Physical review. B, Condensed matter•Sep 4, 2002•13 citations
We study the nature of collective two-spin excitations in disordered S=1∕2S=1∕2S=1∕2 ferromagnetic chains. Using a direct diagonalization scheme, we characterize the two-magnon eigenstates by computing their spacial extent, two-point correlation and the average distance between the excited spins within the allowed energy band. We found that, due to the effective excitation interaction imposed by the exclusion rule, the low-energy two-magnon states display strong spin-spin correlations as compared to the more localized high-energy states. We further solve the time-dependent Schroedinger equation to follow the time evolution of an initially localized two-magnon state. We show that the effective one-magnon wave packet develops power-law tails with distinct exponents for the left and right tails, with the distribution function of the spin-spin distance decaying as P(d)\ensuremath∝1∕dP(d)\ensuremath{\propto}1∕dP(d)\ensuremath∝1∕d. We show that the average distance between the two excited spins evolves in time diffusively, while the wave-packet dispersion evolves superdiffusively.
We study one-magnon excitations in a random ferromagnetic Heisenberg chain with long-range correlations in the coupling constant distribution. By employing an exact diagonalization procedure, we compute the localization length of all one-magnon states within the band of allowed energies E. The random distribution of coupling constants was assumed to have a power spectrum decaying as $S(k)\ensuremath{\propto}{1/k}^{\ensuremath{\alpha}}.$ We found that for $\ensuremath{\alpha}<1,$ one-magnon excitations remain exponentially localized with the localization length $\ensuremath{\xi}$ diverging as 1/E.1/E.1/E. For $\ensuremath{\alpha}=1$ a faster divergence of $\ensuremath{\xi}$ is obtained. For any $\ensuremath{\alpha}>1,$ a phase of delocalized magnons emerges at the bottom of the band. We characterize the scaling behavior of the localization length on all regimes and relate it with the scaling properties of the long-range correlated exchange coupling distribution.
We study the one-dimensional quantum Heisenberg ferromagnet with exchange couplings exhibiting long-range correlated disorder with power spectrum proportional to ${1/k}^{\ensuremath{\alpha}},$ where k is the wave vector of the modulations on the random coupling landscape. By using the renormalization group, integration of the equations of motion, and exact diagonalization, we compute the spin-wave localization length and the mean-square displacement of the wave packet. We find that, associated with the emergence of extended spin waves in the low-energy region for \ensuremath{\alpha}>1, the wave-packet mean-square displacement changes from a long-time super diffusive behavior for \ensuremath{\alpha}1 to a long-time ballistic behavior for \ensuremath{\alpha}>1. At the vicinity of \ensuremath{\alpha}=1, the mobility edge separating the extended and localized phases is shown to scale with the degree of correlation as ${E}_{c}\ensuremath{\propto}(\ensuremath{\alpha}\ensuremath{-}{1)}^{1/3}.$
We study few-magnon excitations in a finite-size spin-SSS chain with ferromagnetic nearest-neighbor (NN) interaction J>0J>0J>0 and antiferromagnetic next-nearest-neighbor (NNN) interaction ${J}^{\ensuremath{'}}<0$, in the presence of the single-ion (SI) anisotropy DDD. We first reveal the condition for the emergence of zero-excitation-energy states. In the isotropic case with $\mathrm{\ensuremath{\Delta}}={\mathrm{\ensuremath{\Delta}}}^{\ensuremath{'}}=1$ ($\mathrm{\ensuremath{\Delta}}$ and ${\mathrm{\ensuremath{\Delta}}}^{\ensuremath{'}}$ are the corresponding anisotropy parameters), a threshold of $J/|{J}^{\ensuremath{'}}|$, above which the ground state is ferromagnetic, is determined by exact diagonalization for short chains up to 12 sites. Using a set of exact two-magnon Bloch states, we then map the two-magnon problem to a single-particle one on an effective open chain with both NN and NNN hoppings. The whole two-magnon excitation spectrum is calculated for large systems, and the commensurate-incommensurate transition in the lowest-lying mode is found to exhibit different behaviors between S=1/2S=1/2S=1/2 and higher spins due to the interplay of the SI anisotropy and the NNN interaction. For the commensurate momentum $k=\ensuremath{-}\ensuremath{\pi}$, the effective lattice is decoupled into two NN open chains that can be exactly solved via a plane-wave ansatz. Based on this, we analytically identify in the ${\mathrm{\ensuremath{\Delta}}}^{\ensuremath{'}}\ensuremath{-}D/|{J}^{\ensuremath{'}}|$ plane the regions supporting the SI or NNN exchange two-magnon bound states near the edge of the band. In particular, we prove that there always exists a lower-lying NN exchange two-magnon bound state near the band edge for arbitrary $S\ensuremath{\ge}1/2$. Finally, we numerically calculate the nnn-magnon spectra for S=1/2S=1/2S=1/2 with $n\ensuremath{\le}5$ by using a spin-operator matrix element method. The corresponding nnn-magnon commensurate instability regions are determined for finite chains, and consistent results with prior literature are observed.
We show that a small fraction of antiferromagnetic bonds qualitatively alters the long-wavelength dynamics of two-dimensional Heisenberg ferromagnets. Although the classical ground state remains magnetized, weak bond frustration generates logarithmically correlated spatial fluctuations of the local spin stiffness, despite the microscopic disorder being short ranged. A replica field theory calculation shows that the effective disorder strength grows under coarse graining, while the spin stiffness decreases, yielding anomalously soft magnons with a scale-dependent dynamical exponent z>2z > 2z>2. Numerical diagonalization of the semiclassical spin-wave Hamiltonian confirms the anomalous low-energy scaling. The flow is toward an infinite-disorder, zero stiffness regime.
Exact analytic calculations in spin-1/2 XY\mathrm{XY}XY chains show the presence of long-time tails in the asymptotic dynamics of spatially inhomogeneous excitations. The decay of inhomogeneities for $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty},$ is given in the form of a power law $(t/{\ensuremath{\tau}}_{Q}{)}^{\ensuremath{-}{\ensuremath{\nu}}_{Q}},$ where the relaxation time ${\ensuremath{\tau}}_{Q}$ and the exponent ${\ensuremath{\nu}}_{Q}$ depend on the wave vector Q, characterizing the spatial modulation of the initial excitation. We consider several variants of the XY\mathrm{XY}XY model (dimerized, with staggered magnetic field, with bond alternation, and with isotropic and uniform interactions), that are grouped into two families, whether the energy spectrum has a gap or not. Once the initial condition is given, the nonequilibrium problem for the magnetization is solved in closed form, without any other assumption. The long-time behavior for $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ can be obtained systematically in a form of an asymptotic series through the stationary phase method. We found that gapped models show critical behavior with respect to Q, in the sense that there exist critical values Qc{Q}_{c}Qc​ where the relaxation time ${\ensuremath{\tau}}_{Q}$ diverges and the exponent ${\ensuremath{\nu}}_{Q}$ changes discontinuously. At those points, a slowing down of the relaxation process is induced, similarly to phenomena occurring near phase transitions. Long-lived excitations are identified as incommensurate spin density waves that emerge in gapped systems, as a consequence of both approximate nesting of the spectrum and the degeneracy of some stationary points. In contrast, gapless models do not present the above anomalies as a function of the wave vector Q.