Classical Heisenberg Chain Research Articles

Subdiffusive spin transport in disordered classical Heisenberg chains

Subdiffusive spin transport in disordered classical Heisenberg chains

Long-lived solitons and their signatures in the classical Heisenberg chain.

Motivated by the Kardar-Parisi-Zhang (KPZ) scaling recently observed in the classical ferromagnetic Heisenberg chain, we investigate the role of solitonic excitations in this model. We find that the Heisenberg chain, although well known to be nonintegrable, supports a two-parameter family of long-lived solitons. We connect these to the exact soliton solutions of the integrable Ishimori chain with ln(1+S_{i}·S_{j}) interactions. We explicitly construct infinitely long-lived stationary solitons, and provide an adiabatic construction procedure for moving soliton solutions, which shows that Ishimori solitons have a long-lived Heisenberg counterpart when they are not too narrow and not too fast moving. Finally, we demonstrate their presence in thermal states of the Heisenberg chain, even when the typical soliton width is larger than the spin correlation length, and argue that these excitations likely underlie the KPZ scaling.

Anomalous dynamics and equilibration in the classical Heisenberg chain

The search for departures from standard hydrodynamics in many-body systems has yielded a number of promising leads, especially in low dimension. Here we study one of the simplest classical interacting lattice models, the nearest-neighbour Heisenberg chain, with temperature as tuning parameter. Our numerics expose strikingly different spin dynamics between the antiferromagnet, where it is largely diffusive, and the ferromagnet, where we observe strong evidence either of spin super-diffusion or an extremely slow crossover to diffusion. This difference also governs the equilibration after a quench, and, remarkably, is apparent even at very high temperatures.

Winding Up Quantum Spin Helices: How Avoided Level Crossings Exile Classical Topological Protection.

A magnetic helix can be wound into a classical Heisenberg chain by fixing one end while rotating the other one. We show that in quantum Heisenberg chains of finite length, the magnetization slips back to the trivial state beyond a finite turning angle. Avoided level crossings thus undermine classical topological protection. Yet, for special values of the axial Heisenberg anisotropy, stable spin helices form again, which are nonlocally entangled. Away from these sweet spots, spin helices can be stabilized dynamically or by dissipation. For half-integer spin chains of odd length, a spin slippage state and its Kramers partner define a qubit with a nontrivial Berry connection.

Light-Cone Spreading of Perturbations and the Butterfly Effect in a Classical Spin Chain.

We find that the effects of a localized perturbation in a chaotic classical many-body system-the classical Heisenberg chain at infinite temperature-spread ballistically with a finite speed even when the local spin dynamics is diffusive. We study two complementary aspects of this butterfly effect: the rapid growth of the perturbation, and its simultaneous ballistic (light-cone) spread, as characterized by the Lyapunov exponents and the butterfly speed, respectively. We connect this to recent studies of the out-of-time-ordered commutators (OTOC), which have been proposed as an indicator of chaos in a quantum system. We provide a straightforward identification of the OTOC with a natural correlator in our system and demonstrate that many of its interesting qualitative features are present in the classical system. Finally, by analyzing the scaling forms, we relate the growth, spread, and propagation of the perturbation with the growth of one-dimensional interfaces described by the Kardar-Parisi-Zhang equation.

Low dimensional magnetism in MnNb2−xVxO6

The structural and magnetic properties of MnNb2−xVxO6 samples were studied. These materials belong to the AB2O6 class, which presents low-dimensional magnetism. Curie-Weiss fitting gives a negative Weiss temperature that is a characteristic of dominant antiferromagnetic interactions in paramagnetic region. Rietveld refinements of neutron diffraction data show that magnetic moments are aligned antiferromagnetically within the zig-zag chains running along the c axis. Unlike other ANb2O6 type systems crystallizing also in Pbcn space group and presenting ferromagnetic Ising chains, MnNb2−xVxO6 samples exhibit antiferromagnetic type order along the chains. At 1.5K, the Mn atoms are found to carry an ordered magnetic moment of about 4.3μB. The Mn chains are antiferromagnetic and do not present significant anisotropy. These characteristics and the high spin allowed us to model the system as weakly interacting classical Heisenberg chains, estimating the values of both intra- and interchain exchange constants evolution with x content.

Synthesis, Structure, and Magnetic Characterization of a Series of Iron(II) Coordination Frameworks with 2, 6‐Bis(1, 2,4‐triazole‐4‐yl)pyridine

AbstractBased on the bis‐triazole ligand 2, 6‐bis(1, 2,4‐triazole‐4‐yl)pyridine (L), the triazole‐iron(II) complexes [Fe(L)2(dca)2(H2O)2]·2H2O (1) (Nadca = sodium dicyanamide), {[Fe(μ2‐L)2(H2O)2]Cl2}n (2), and {[Fe(μ2‐L)2(H2O)2](ClO4)2·L·H2O}n (3) were isolated by solvent diffusion methods. When iron(II) salts and Nadca were used, compound 1 was isolated, which contains mononuclear Fe(L)2(dca)2(H2O)2 units. When FeCl2 or FeClO4 were used, one‐dimensional (1D) cation iron(II) chains (2) and two‐dimensional (2D) cation iron(II) networks (3) were isolated indicating anion directing structural diversity. Moreover, variable‐temperature magnetic susceptibility data of 1–3 were recorded in the temperature range 2–300 K. The magnetic curve of complex 2 was fitted by using the classical spin Heisenberg chain model indicating anti‐ferromagnetic interactions (J = –5.31 cm–1). Obviously complexes 1–3 show no detectable thermal spin crossover behaviors, the lack of spin‐crossover behavior may be correlated with FeN4O2 coordination spheres in 1–3.

Thermally driven classical Heisenberg chain with a spatially varying magnetic field: thermal rectification and negative differential thermal resistance

Thermal rectification and negative differential thermal resistance are two important features that have direct technological relevance. Here, we study the classical one-dimensional Heisenberg model, driven thermally by heat baths attached at the two ends of the system and in the presence of an external magnetic field that varies monotonically in space. Heat conduction in this system is studied using a local energy conserving dynamics. It is found that by suitably tuning the spatially varying magnetic field, the structurally homogeneous symmetric system exhibits both thermal rectification and negative differential thermal resistance. Thermal rectification, in some parameter ranges, shows interesting dependencies on the average temperature T and the system size N—rectification improves as T and N are increased. Using the microscopic dynamics of the spins we present a physical picture to understand thermal rectification as exhibited by this system and provide supporting numerical evidence. Emergence of the negative response in this system can be controlled by tuning the external magnetic field alone, which can have possible applications in the fabrication of novel thermal devices.

Thermal rectification and negative differential thermal resistance in a driven two segment classical Heisenberg chain

Using computer simulation we investigate thermal transport in a two segment classical Heisenberg spin chain with nearest neighbor interaction and in the presence of an external magnetic field. The system is thermally driven by heat baths attached at the two ends and transport properties are studied using energy conserving dynamics. We demonstrate that by properly tuning the parameters thermal rectification can be achieved—the system behaves as a good conductor of heat along one direction but becomes a bad conductor when the thermal gradient is reversed, and crucially depends on nonlinearity and spatial asymmetry. Moreover, suitable tuning of the system parameters gives rise to the counterintuitive and technologically important feature known as negative differential thermal resistance (NDTR). We find that the crucial factor responsible for the emergence of NDTR is a suitable mechanism for impeding the current in the bulk of the system.

Anomalous Tunneling of Spin Wave in Heisenberg Ferromagnet

The ferromagnetic spin wave (FSW) in classical Heisenberg chain exhibits the perfect transmission in the long-wavelength limit in the transmission-reflection problem with an inhomogeneity of exchange integral. In the presence of local magnetic field, on the other hand, FSW undergoes the perfect reflection in the long-wavelength limit. This difference in the long-wavelength limit is attributed to the symmetry property of the scatterers; it is crucial whether the potential preserves or breaks the spin rotation symmetry. Our result implies that the anomalous tunneling (i.e., perfect transmission in the low-energy limit) found both in scalar and spinor BECs is not specific to gapless modes in superfluids but is a common property shared with generic Nambu-Goldstone modes in the presence of a symmetry-preserving potential scatterer.

Investigation of frustrated and dimerized classical Heisenberg chains

We have considered the 1D dimerized frustrated antiferromagnetic (ferromagnetic) Heisenberg model with arbitrary spin S. The exact classical magnetic phase diagram at zero temperature is determined using the LK cluster method. The cluster method results show that the classical ground-state phase diagram of the model is very rich, including first-order and second-order phase transitions. In the absence of dimerization, a second-order phase transition occurs between antiferromagnetic (ferromagnetic) and spiral phases at the critical frustration αc=±0.25, a well-known result. In the vicinity of the critical points αc, the exact classical critical exponent of the spiral order parameter is found to be 1/2. In the case of a dimerized chain (δ≠0), the spiral order shows stability and exists in some part of the ground-state phase diagram. We have found two first-order phase boundaries separating antiferromagnetic (uud and duu) phases from the spiral phase.

Frustrated classical Heisenberg model in one dimension with nearest-neighbor biquadratic exchange: Exact solution for the ground-state phase diagram

The ground-state phase diagram is determined exactly for the frustrated classical Heisenberg chain with added nearest-neighbor biquadratic exchange interactions. There appear ferromagnetic, incommensurate-spiral, ``up-up-down-down'' (uudd) phases, and disordered states. The model contains an isotropic version of the antiferromagnetic next-nearest-neighbor Ising model, i.e., an isotropic Ising model. It is closely related to a model proposed for some manganites, suggesting a possible mechanism for the observed uudd state.

Low-field susceptibility of classical Heisenberg chains with arbitrary and differentnearest-neighbour exchange

Interest in molecular magnets continues to grow, offering a link between the atomicand nanoscale properties. The classical Heisenberg model has been effective inmodelling exchange interactions in such systems. In this, the magnetizationand susceptibility are calculated through the partition function, where theHamiltonian contains both Zeeman and exchange energy. For an ensemble ofN spins, this requiresintegrals in 2N dimensions. For two, three and four spin nearest-neighbour chains these integralsreduce to sums of known functions. For the case of the three and four spin chains,the sums are equivalent to results of Joyce. Expanding these sums, the effect ofthe exchange on the linear susceptibility appears as Langevin functions withexchange term arguments. These expressions are generalized here to describe anN spin nearest-neighbour chain, where the exchange between each pair of nearest neighboursis different and arbitrary. For a common exchange constant, this reduces to the result ofFisher. The high-temperature expansion of the Langevin functions for the differentexchange constants leads to agreement with the appropriate high-temperature quantumformula of Schmidt et al, when the spin number is large. Simulations are presented foropen linear chains of three, four and five spins with up to four different exchangeconstants, illustrating how the exchange constants can be retrieved successfully.

Comment on ‘Analytical results for a Bessel function times Legendre polynomials class integrals’

A result is obtained, stemming from Gegenbauer, where the products of certain Bessel functions and exponentials are expressed in terms of an infinite series of spherical Bessel functions and products of associated Legendre functions. Closed form solutions for integrals involving Bessel functions times associated Legendre functions times exponentials, recently elucidated by Neves et al (J. Phys. A: Math. Gen. 39 L293), are then shown to result directly from the orthogonality properties of the associated Legendre functions. This result offers greater flexibility in the treatment of classical Heisenberg chains and may do so in other problems such as occur in electromagnetic diffraction theory.

Continuous and Discrete (Classical) Heisenberg Spin Chain Revised

Most of the work done in the past on the integrability structure of the Classical Heisenberg Chain (CHSC) has been devoted to studying the su(2) case, both at the continuous and at the discrete level. In this paper we address the problem of constructing integrable generalized Spin Chains models, where the relevant field variable is represented by aN◊N matrix whose eigenvalues are theN th roots of unity. To the best of our knowledge, such an extension has never been systematically pursued. In this paper, at first we obtain the continuous N ◊ N generalization of the CHSC through the reduction technique for Poisson-Nijenhuis manifolds, and exhibit some explicit, and hopefully interesting, examples for 3 ◊ 3 and 4 ◊ 4 matrices; then, we discuss the much more difficult discrete case, where a few partial new results are derived and a conjecture is made for the general case.

The Darboux–Bäcklund transformation for the static 2-dimensional continuum Heisenberg chain

We construct the Darboux-Backlund transformation for the sigma model describing static configurations of the 2-dimensional classical continuum Heisenberg chain. The transformation is characterized by a non-trivial normalization matrix depending on the background solution. In order to obtain the transformation we use a new, more general, spectral problem.

Dynamic properties of a classical anisotropic Heisenberg chain under external magnetic field

We investigate the dynamic properties of a classical anisotropic Heisenberg chain interacting with an external magnetic field at different temperatures. Properties such as time-dependent energy autocorrelation and space–time spin–spin correlation are obtained by solving the dynamic equation S ˙ i = S i × ∇ S i H . While the static spin–spin correlation length decreases as the system is heated up, it increases when an external magnetic field is present. The time-dependent spin–spin correlation decreases when the system is heated up, resulting in a decrease of the spin-diffusion speed. The presence of the magnetic field contributes to the order, and therefore produces an increase of the spin-diffusion speed. In contrast, the single-ion type anisotropy behaves dynamically as a local field, inducing disorder in the chain.

Stochastic motion of solitary excitations on the classical Heisenberg chain

We study stochastic motion of solitary excitations on a classical, discrete, isotropic, ferromagnetic Heisenberg spin chain with nearest-neighbour exchange interactions. Gaussian white noise is coupled to the spins in a way that allows for the noise to be interpreted as a stochastic magnetic field. The noise translates into a collective stochastic force affecting a solitary excitation as a whole. The position of a solitary excitation has to be calculated from the noisy spin configuration, i.e. the position is defined as a function of the spin components. Two examples of such definitions are given, because we want to investigate the dependence of the results on the choice of definition. Using these definitions, we calculate the variance of the position as a function of time and determine the variance from simulations as well. The calculations require knowledge of the shape of the solitary wave. We approximate the shape with that of soliton solutions of the continuum Heisenberg chain, restricting our considerations to solitary waves of large width, in which case this approximation is good. The calculations yield a linear dependence of the variance on time, the slope being determined by parameters describing the shape of the soliton. The two definitions of the position we use provide different results for this slope. The origin of this difference is discussed. With both definitions very good agreement is found between the results of the simulations and the corresponding theoretical results, for not too large time scales.

Low-energy behavior of a random classical Heisenberg chain

Low-energy magnons in a classical Heisenberg chain with random nearest-neighbor interactions are investigated using the negative eigenvalue counting technique. A detailed study is made for a range of concentrations of ``wrong sign'' bonds (i.e., $c<~0.1$ and $c>~0.9)$. It is found that low-energy magnons have an anomalous power-law (2/3) behavior in the spectrum for random sign-fixed magnitude exchange interactions. Combining these and previous results, a phenomenological fit is established describing the concentration dependence of the spectrum for a wide range of values of $c$. A discussion based on scaling arguments is presented to account for the anomalous power law for arbitrary concentrations. The general case where the magnitudes and the signs of the exchange interactions and the magnitudes of the spins are all random quantities is further investigated to check whether the anomalous power-law behavior holds. For uniform (flat), non-singular distributions of the random quantities, the 2/3 power in the spectrum is obtained but for singular distributions and mixed distributions (e.g., the magnitude of the spin has a singular distribution while the magnitude of the exchange interaction has a regular distribution) significant deviations from the 2/3 power law occur.

Instability and Self-Modulation of the Classical Heisenberg Ferromagnet

The helical solution of the one-dimensional classical Heisenberg chain is shown to be unstable against long-wavelength perturbations. Using the fact that the classical Heisenberg chain is exactly integrable, inverse scattering techniques are used to display the growth of the instabilities all the way to nonlinear saturation.