One-dimensional Ising Ferromagnet Research Articles

Distribution of kinks in an Ising ferromagnet after annealing and the generalized Kibble-Zurek mechanism

We consider the annealing dynamics of a one-dimensional Ising ferromagnet induced by a temperature quench in finite time. In the limit of slow cooling, the asymptotic two-point correlator is analytically found under Glauber dynamics, and the distribution of the number of kinks in the final state is shown to be consistent with a Poissonian distribution. The mean kink number, the variance, and the third centered moment take the same value and obey a universal power-law scaling with the quench time in which the temperature is varied. The universal power-law scaling of cumulants is corroborated by numerical simulations based on Glauber dynamics for moderate cooling times away from the asymptotic limit, when the kink-number distribution takes a binomial form. We analyze the relation of these results to physics beyond the Kibble-Zurek mechanism for critical dynamics, using the kink number distribution to assess adiabaticity and its breakdown. We consider linear, nonlinear, and exponential cooling schedules, among which the latter provides the most efficient shortcuts to cooling in a given quench time. The non-thermal behavior of the final state is established by considering the trace norm distance to a canonical Gibbs state.

Non-coplanar Model States in Quantum Magnetism Applications of the High-Order Coupled Cluster Method

Coplanar model states for applications of the coupled cluster method (CCM) to problems in quantum magnetism are those in which all spins lie in a plane, whereas three-dimensional (3D) model states are, by contrast, non-coplanar ones in which all the spins do not lie in any single plane. A crucial first step in applying the CCM to any such lattice quantum spin system is to perform a passive rotation of the local spin axes so that all spins in the model state appear mathematically to point in the same (say, downwards z-)direction. Whereas this process leads to terms with only real coefficients in the rotated Hamiltonian for coplanar model states, an additional complication arises for 3D model states where the corresponding coefficients can become complex-valued. We show here for the first time how high-order implementations of the CCM can be performed for such Hamiltonians. We explain in detail why the extension of the computational implementation of the CCM when going from coplanar to 3D model states is a non-trivial task that has not hitherto been undertaken. To illustrate these new developments, we present results for three cases: (a) the spin-half one-dimensional Ising ferromagnet in an applied transverse magnetic field (as an exactly solvable test model to use as a yardstick for the viability and accuracy of our new methodology); (b) the spin-half triangular-lattice Heisenberg antiferromagnet in the presence of an external magnetic field; and (c) the spin-S triangular-lattice XXZ antiferromagnet in the presence of an external magnetic field, for the cases frac{1}{2} le S le 5 . For 3D model states the sets of algebraic CCM equations for the ket- and bra-state correlation coefficients become complex-valued, but ground-state expectation values of all physical observables are manifestly real numbers, as required, and as we explicitly demonstrate in all three applications. Indeed, excellent correspondence is seen with the results of other methods, where they exist, for these systems. In particular, our CCM results demonstrate explicitly that coplanar ordering is favoured over non-coplanar ordering for the triangular-lattice spin-half Heisenberg antiferromagnet at all values of the applied external magnetic field, whereas for the anisotropic XXZ model non-coplanar ordering can be favoured in some regions of the parameter space. Specifically, we present a precise determination of the boundary (i.e., the critical value of the XXZ anisotropy parameter Delta ) between a 3D ground state and a coplanar ground state for the XXZ model for values for the external magnetic field near to saturation, for values of the spin quantum number S le 5. Although the CCM calculations are computationally intensive for this frustrated model, especially for high spin quantum numbers, our accurate new results certainly improve our understanding of it.

Growth probability and formation time of the individual Oosawa-Kasai protein fibril.

Protein fibrils are currently of great academic and practical interest because of their involvement in scores of severe human diseases and their promising use in various high-technology devices. The Oosawa-Kasai (OK) model of protein self-assembly into fibrils has been widely used to gain mechanistic insight into the process of fibril formation and growth. Here this model is employed to obtain exact and mathematically simple expressions for the probability P_{n} of an individual fibril of n protein monomers to grow to a macroscopically large size and for the mean time τ_{n} that such a fibril needs for its formation. These expressions quantify the increase of P_{n} and the decrease of τ_{n} with increasing the concentration of monomeric protein in the solution. When used for analysis of experimental P_{n} and τ_{n} data, they make it possible to determine the parameters characterizing fibril nucleation and growth in the framework of the OK model. Finally, an expression is found for the mean time of the first appearance of an n-sized fibril in the protein solution. The results obtained are applicable to the formation of other aggregates corresponding to the OK fibrils, such as the one-dimensional Kossel-Stranski crystals and Ising ferromagnets.

Contour Methods for Long-Range Ising Models: Weakening Nearest-Neighbor Interactions and Adding Decaying Fields

We consider ferromagnetic long-range Ising models which display phase transitions. They are long-range one-dimensional Ising ferromagnets, in which the interaction is given by $J_{x,y} = J(|x-y|)\equiv \frac{1}{|x-y|^{2-\alpha}}$ with $\alpha \in [0, 1)$, in particular, $J(1)=1$. For this class of models one way in which one can prove the phase transition is via a kind of Peierls contour argument, using the adaptation of the Fr\"ohlich-Spencer contours for $\alpha \neq 0$, proposed by Cassandro, Ferrari, Merola and Presutti. As proved by Fr\"ohlich and Spencer for $\alpha=0$ and conjectured by Cassandro et al for the region they could treat, $\alpha \in (0,\alpha_{+})$ for $\alpha_+=\log(3)/\log(2)-1$, although in the literature dealing with contour methods for these models it is generally assumed that $J(1)\gg1$, we can show that this condition can be removed in the contour analysis. In addition, combining our theorem with a recent result of Littin and Picco we prove the persistence of the contour proof of the phase transition for any $\alpha \in [0,1)$. Moreover, we show that when we add a magnetic field decaying to zero, given by $h_x= h_*\cdot(1+|x|)^{-\gamma}$ and $\gamma >\max\{1-\alpha, 1-\alpha^* \}$ where $\alpha^*\approx 0.2714$, the transition still persists.

Dynamical structure factor of magnetic Bloch oscillations at finite temperatures

Domain-walls in one-dimensional Ising ferromagnets can undergo Bloch oscillations when subjected to a skew magnetic field. Such oscillations imply finite temperature non-dispersive low-frequency peaks in the dynamical structure factor which can be probed in neutron scattering. We study in detail the spectral weight of these peaks. Using an analytical approach based on an approximate treatment of a gas of spin-cluster excitations we give an explicit expression for the momentum- and temperature-dependence of the spectral weights. Generally the spectral weights increase with temperature and approaches the same order of magnitude as the spin-wave spectral weights at high temperatures. We compare the analytical expression to numerical exact diagonalizations and find that it can, without any adjustable parameters, account for the temperature and momentum-transfer dependence of the numerically obtained spectral weights in the parameter regime where the ratio of magnetic fields $h_x/h_z \ll 1$ and the temperature is $h_x < T <\sim J_z/2$. We also carry out numerical calculations pertinent to the material CoNb$_2$O$_6$, and find qualitatively similar results.

Finite-size scaling above the upper critical dimension in Ising models with long-range interactions

The correlation length plays a pivotal role in finite-size scaling and hyperscaling at continuous phase transitions. Below the upper critical dimension, where the correlation length is proportional to the system length, both finite-size scaling and hyperscaling take conventional forms. Above the upper critical dimension these forms break down and a new scaling scenario appears. Here we investigate this scaling behaviour in one-dimensional Ising ferromagnets with long-range interactions. We show that the correlation length scales as a non-trivial power of the linear system size and investigate the scaling forms. For interactions of sufficiently long range, the disparity between the correlation length and the system length can be made arbitrarily large, while maintaining the new scaling scenarios. We also investigate the behavior of the correlation function above the upper critical dimension and the modifications imposed by the new scaling scenario onto the associated Fisher relation.

Comment on “Phase transition in a one-dimensional Ising ferromagnet at zero temperature using Glauber dynamics with a synchronous updating mode”

Sznajd-Weron [Phys. Rev. E 82, 031120 (2010)] suggested that the one-dimensional Ising model subject to the zero-temperature synchronous Glauber dynamics exhibits a discontinuous phase transition. We show here that the phase transition instead is of a continuous nature, and we identify critical exponents, β ≈ 0, ν ≈ 1, and z ≈ 2, via a systematic finite-size scaling analysis.

Phase transition in a one-dimensional Ising ferromagnet at zero temperature using Glauber dynamics with a synchronous updating mode

In the past decade low-temperature Glauber dynamics for the one-dimensional Ising system has been several times observed experimentally and occurred to be one of the most important theoretical approaches in a field of molecular nanomagnets. On the other hand, it has been shown recently that Glauber dynamics with the Metropolis flipping probability for the zero-temperature Ising ferromagnet under synchronous updating can lead surprisingly to the antiferromagnetic steady state. In this paper the generalized class of Glauber dynamics at zero temperature will be considered and the relaxation into the ground state, after a quench from high temperature, will be investigated. Using Monte Carlo simulations and a mean field approach, discontinuous phase transition between ferromagnetic and antiferromagnetic phases for a one-dimensional ferromagnet will be shown.

One-dimensional Ising ferromagnet frustrated by long-range interactions at finite temperatures

We consider a one-dimensional lattice of Ising-type variables where the ferromagnetic exchange interaction $J$ between neighboring sites is frustrated by a long-ranged antiferromagnetic interaction of strength $g$ between the sites $i$ and $j$, decaying as ${|i\ensuremath{-}j|}^{\ensuremath{-}\ensuremath{\alpha}}$, with $\ensuremath{\alpha}&gt;1$. For $\ensuremath{\alpha}$ smaller than a certain threshold ${\ensuremath{\alpha}}_{0}$, which is larger than 2 and depends on the ratio $J/g$, the ground state consists of an ordered sequence of segments with equal length and alternating magnetization. The width of the segments depends on both $\ensuremath{\alpha}$ and the ratio $J/g$. Our Monte Carlo study shows that the on-site magnetization vanishes at finite temperatures and finds no indication of any phase transition. Yet, the modulation present in the ground state is recovered at finite temperatures in the two-point correlation function, which oscillates in space with a characteristic spatial period. The latter depends on $\ensuremath{\alpha}$ and $J/g$ and decreases smoothly from the ground-state value as the temperature is increased. Such an oscillation of the correlation function is exponentially damped over a characteristic spatial scale, the correlation length, which asymptotically diverges roughly as the inverse of the temperature as $T=0$ is approached. This suggests that the long-range interaction causes the Ising chain to fall into a universality class consistent with an underlying continuous symmetry. The ${e}^{\ensuremath{\Delta}/T}$ temperature dependence of the correlation length and the uniform ferromagnetic ground state, characteristic of the $g=0$ discrete Ising symmetry, are recovered for $\ensuremath{\alpha}&gt;{\ensuremath{\alpha}}_{0}$.

Extended scaling for ferromagnetic Ising models with zero-temperature transitions

We study the second-moment correlation length and the reduced susceptibility of two ferromagnetic Ising models with zero-temperature ordering. By introducing a scaling variable motivated by high-temperature series expansions, we are able to scale data for the one-dimensional Ising ferromagnet rigorously over the entire temperature range. Analogous scaling expressions are then applied to the two-dimensional fully frustrated Villain model where excellent finite-size scaling over the entire temperature range is achieved. Thus we broaden the applicability of the extended scaling method to Ising systems having a zero-temperature critical point.

Inflow versus outflow zero-temperature dynamics in one dimension

It has been suggested that Glauber (inflow) and Sznajd (outflow) zero-temperature dynamics for the one-dimensional Ising ferromagnet with nearest-neighbor interactions are equivalent. Here we compare the two dynamics from the analytical and computational points of view. We use the method of mapping an Ising spin system onto the dimer RSA model and show that already this simple mapping allows us to see the differences between inflow and outflow zero-temperature dynamics. Then we investigate both dynamics with synchronous, partially synchronous, and random sequential updating using the Monte Carlo technique and compare both dynamics in the number of persistent spins, clusters, mean relaxation time, and relaxation time distribution.

Persistence in systems with conserved order parameter

We consider the low-temperature coarsening dynamics of a one-dimensional Ising ferromagnet with conserved Kawasaki-like dynamics in the domain representation. Domains diffuse with size-dependent diffusion constant, $D(l) \propto l^\gamma$ with $\gamma = -1$. We generalize this model to arbitrary $\gamma$, and derive an expression for the domain density, $N(t) \sim t^{-\phi}$ with $\phi=1/(2-\gamma)$, using a scaling argument. We also investigate numerically the persistence exponent $\theta$ characterizing the power-law decay of the number, $N_p(t)$, of persistent (unflipped) spins at time $t$, and find $N_{p}(t)\sim t^{-\theta}$ where $\theta$ depends on $\gamma$. We show how the results for $\phi$ and $\theta$ are related to similar calculations in diffusion-limited cluster-cluster aggregation (DLCA) where clusters with size-dependent diffusion constant diffuse through an immobile `empty' phase and aggregate irreversibly on impact. Simulations show that, while $\phi$ is the same in both models, $\theta$ is different except for $\gamma=0$. We also investigate models that interpolate between symmetric domain diffusion and DLCA.

Semiclassical dynamics of domain walls in the one-dimensional Ising ferromagnet in a transverse field

We investigate analytically and numerically the dynamics of domain walls in a spin chain with ferromagnetic Ising interaction and subject to an external magnetic field perpendicular to the easy magnetization axis (transverse field Ising model). The analytical results obtained within the continuum approximation and numerical simulations performed for discrete classical model are used to analyze the quantum properties of domain walls using the semiclassical approximation. We show that the domain wall spectrum shows a band structure consisting of 2$S$ non-intersecting zones.

Yang-Lee edge singularity of a one-dimensional Ising ferromagnet with arbitrary spin

It is shown that for a one-dimensional lattice system in a purely imaginary magnetic field, if the interaction is finite range, the nature of the Yang-Lee edge singularity is universal, independent of the spin and interaction strengths. The edge singularity corresponds to the twofold degeneracy of the largest eigenvalues of the transfer matrix. For the Blume-Emery-Griffiths ferromagnet, the tricritical point and the edge pseudosingularity may exist. The tricritical point corresponds to the triple degeneracy of the eigenvalues. The edge pseudosingularity corresponds to the twofold degeneracy of the nonlargest eigenvalues.

The effect of chain length on the temperature dependence of a spin-cluster resonance signal in a one-dimensional Ising ferromagnet

The temperature dependence of the high-field spin cluster resonance observed in the one-dimensional ferromagnet FeTAC at low temperatures is shown to depend on the numbers of ferrous ions in the chains. The longer the chain, the lower is the temperature at which deviations from the previously obtained linear relationship between In(intensity) and 1/T occur. Calculations for the distribution of chain lengths obtained by the random substitution of a nonmagnetic impurity are made for several impurity concentrations and compared with the experimental data points for a pure FeTAC sample. The results indicate that, in this case, other types of crystal defect must limit the chain lengths.

Spin-cluster-resonance spectra in an S=2 one-dimensional Ising ferromagnet.

The electron-paramagnetic-resonance technique has been used to obtain an unambiguous identification of resonances between spin-cluster states from the distinctive satellite spectra observed in the ranges 10--36 GHz and 2.6--4.2 K from a large single crystal of the one-dimensional Ising ferromagnet [(${\mathrm{CH}}_{3}$${)}_{3}$NH]${\mathrm{FeCl}}_{3}$\ensuremath{\cdot}${2\mathrm{H}}_{2}$O, known as FeTAC. The frequency and temperature dependencies of the spectra are explained by the addition of the term \ensuremath{\Delta}${\mathit{S}}_{\mathit{x}}$ to the Zeeman term in the effective S=1/2 spin Hamiltonian describing the first set of excited cluster states. The spin-cluster levels are the eigenvalues of the resulting tridiagonal matrix, and lead to a satellite structure in FeTAC which is independent of the chain length for chains of more than about 30 ${\mathrm{Fe}}^{2+}$ ions.

The effect of Co doping on spin cluster resonance in the one-dimensional Ising ferromagnet FeTAC

Electron paramagnetic resonance measurements at 36 GHz and in the range 12–18 GHz were made on a single crystal of the 1D Ising ferromagnet [(CH3)3 NH] FeCl3&amp;sdot;2H2O, known as FeTAC, containing a nominal 10% of the isomorphous compound CoTAC. While the 4.2 K spectra obtained with the external field parallel to the chain axis were similar to the spin cluster resonance spectra previously identified in pure FeTAC, differences in the low-field satellite structure were observed, which are tentatively attributed to differences in the dipolar interaction and to shorter ferrous chains. An appreciable decrease in the strength of the demagnetizing field is associated with the tendency of the Co2+ moments to align perpendicularly to the chain axis.

Spin-cluster resonance at 9 GHz in the one-dimensional Ising ferromagnet

Low-temperature magnetic-resonance spectra at 9.2--9.3 GHz in single crystals of the one-dimensional Ising ferromagnet FeTAC, [(${\mathrm{CH}}_{3}$${)}_{3}$NH] ${\mathrm{FeCl}}_{3}$\ensuremath{\cdot}${2\mathrm{H}}_{2}$O, were strongly anisotropic with minima in the field occurring for H parallel to the easy axis. A ``strong'' signal in this orientation with a g value of 8.8 at 5.5 K (after correction for internal demagnetization and exchange fields) is identified with transitions between neighboring spin-cluster resonances. Possible origins of broad and weak lines, with maximum corrected g values of 13 and 3.7 are discussed. Spectra from samples of different shape are consistent with sample-dependent demagnetization fields.

Single crystal susceptibility measurements of a one-dimensional Ising ferromagnet, FeTAB

FeTAB, [(( CH 3) 3 NH) FeBr 3·2 H 2 O], is the ferrous bromide member of a family of one-dimensional isostructural compounds which consist of chains of well isolated bihalide-bridged metal ions. The previously studied chloride analog, FeTAC, has been shown to be an excellent example of a one-dimensional Ising ferromagnet. In this paper we present the single crystal susceptibilities of FeTAB. The intrachain exchange strength has been determined to be 5.4 K, a factor of three lower than that of FeTAC. The interchain exchange strength is found to be z' J' / k = -0.04 K. The critical ratio ( kT c / J) for FeTAB is 0.24 and is therefore intermediate between those of the corresponding iron and cobalt chlorides, FeTAC ( kT c/ J = 0.18) and CoTAC ( kT c/ J = 0.30).

High-m spin excitations in the one-dimensional Ising ferromagnet

High-m spin excitations in the one-dimensional Ising ferromagnet