Spin-Wave Correlations Research Articles

Emergence of Spatial Spin-Wave Correlations in a Cold Atomic Gas

Rydberg spin-waves are optically excited in a quasi-one-dimensional atomic sample of Rb atoms. Pairwise spin-wave correlations are observed by a spatially selective transfer of the quantum state onto a light field and photoelectric correlation measurements of the light. The correlations are interpreted in terms of the dephasing of multiply excited spin-waves by long-range Rydberg interactions.

Renormalization Theory for Ordered Phase of 2D XY Magnet with Dipolar Interaction

In the ordered phase of 2D XY ferromagnet the dipolar interaction between spins induces a relevant interaction between spin-waves. Although the density of spin-waves is small at low temperature, this long-range interaction correlates two spatially separated spin-waves. Anomalous scaling of spin-wave correlation functions is found in both statics and dynamics cases.

Exact scaling of spin-wave correlations in the 2D XY ferromagnet with dipolar forces.

Exact scaling of spin-wave correlations in the 2D XY ferromagnet with dipolar forces.

Correlation functions of the one-dimensional Hubbard model

We have investigated the critical behavior of the Hubbard model with attraction. To investigate the critical behavior of the model with repulsion, it is sufficient to use a unitary electron-hole transformation [3,8], and then the SCP correlation function will correspond to the spin-wave correlation function (the parameters of the models can be readily recalculated). The quasione-dimensional generalization of the investigated system can be used as a model of a high-temperature superconductor with “chemical” pairing mechanism [16,25].

Static form factors for the classical easy-plane ferromagnetic chain with an in-plane magnetic field: Application to CsNiF3

We present low-temperature expressions for the correlation functions of the classical easy-plane Heisenberg chain in a field. Approaching the critical field for domain-wall transition, the in-plane domain-wall form factors increase in strength and the in-plane spin-wave correlations display damped oscillations. The out-of-plane spin-wave correlations decay with a growing correlation length and the form factor is increased in intensity. We propose to look for this effect in CsNi${\mathrm{F}}_{3}$ by means of neutrons.

Renormalization of Large-Wave-Vector Magnons in Ferromagnetic CrBr3Studied by Inelastic Neutron Scattering: Spin-Wave Correlation Effects

The renormalization of large-wave-vector magnons in Cr${\mathrm{Br}}_{3}$ has been carefully studied using inelastic neutron scattering. The results are in disagreement with the first-order approximation to Dyson's spin-wave theory, which predicts that the spin-wave energy shift $\frac{\ensuremath{\Delta}E}{E}$ is independent of wave vector. Over part of the Brillouin zone our data can be fitted to a constant $\frac{\ensuremath{\Delta}E}{E}$. However, in the region $q=0.4$ to 0.6 ${\mathrm{\AA{}}}^{\ensuremath{-}1}$ in the $\ensuremath{\Delta}$ direction, there is strong evidence for a wave-vector dependence of $\frac{\ensuremath{\Delta}E}{E}$, such as would arise from the higher-order spin-wave interactions, i.e., spin-wave correlation effects. Qualitative comparison is made to the predictions of $t$-matrix calculations, which treat the interactions to all orders (and are thus capable of reproducing Dyson's results). The observed $q$ dependence agrees with these (qualitative) predictions, giving evidence for the direct observation of the effect of correlations between spin waves. Unfortunately, quantitative comparisons between theory and experiment are not possible at present, because of the difficulty of calculating the two-spin-wave $t$ matrix for this system.

Field Dependence of the Magnetization and Spin-Wave Correlations in Ferromagnetic CrBr3

Using the ${\mathrm{Cr}}^{53}$ NMR, the field dependence of the magnetization $M({T}_{0},H)$ at elevated temperatures of single-crystal ferromagnetic Cr${\mathrm{Br}}_{3}$ has been determined. While an accurate fit to the $M({T}_{0},H)\ensuremath{-}\mathrm{v}\mathrm{s}\ensuremath{-}H$ data can be obtained using a two-exchange-parameter first-order renormalized spin-wave theory, the parameters required differ appreciably from those required for an accurate fit to the previously obtained $M(T,0)\ensuremath{-}\mathrm{v}\mathrm{s}\ensuremath{-}T$ data with the same theory. A $t$-matrix two-parameter-model theory, correct to lowest order in the magnon density, but to all orders in the magnon-magnon interaction, was constructed. Although both the first-order and $t$-matrix corrections to the spin-wave energies are sizable, the full $t$-matrix results only change the large first-order corrections to $M(T,H)$ by 15%. However, even if one employs the full $t$-matrix renormalization there are no pairs of values of the two exchange parameters which simultaneously fit the $M(T,0)\ensuremath{-}\mathrm{v}\mathrm{s}\ensuremath{-}T$ and $M({T}_{0},H)\ensuremath{-}\mathrm{v}\mathrm{s}\ensuremath{-}H$ data. We attribute the inability of the more sophisticated theory to provide agreement with measurements of more than one thermodynamic function to be an inadequacy of the two-parameter model rather than an intrinsic failure of the theoretical approach.

Konsistente Temperaturentwicklungen f�r den Ferromagneten

Ferromagnetism is studied inDyson's model of ideal spinwaves and in the Heisenberg model by means of temperature dependentGreen's functions. As is well known the low temperature magnetization of the free spinwave gas has a series expansion inT (3+2n)/2. The “Hartree-Fock” approximation forGreen's function gives an additional term inT 4 forDyson's model, further contributions toT 4 are furnished by higher approximations for the two spinwave correlation and the summation of all ladder graphs leads back toDyson's result, though the contribution of three and higher spin wave correlations remains obscure. For the Heisenberg model a discussion of the temperature dependence of 〈S i z S j z 〉-correlations, total energy and specific heat shows that the well-known decoupling process (-“Hartree”-approximation for the spin operator two point function-) is not consistent unlessT≪T c .

Statistical Mechanics of Ferromagnetism

The fact that reliable statistical mechanical theories of ferromagnetism existed only in the very low temperature region (spin wave theory) and in the very high temperature region (Kramers and Opechowski) has seriously hampered the development of magnetic theory. The Weiss molecular field theory neglects all spin-spin correlations and therefore gives too high a Curie temperature, misrepresents the details of the phase transition, and is unreliable as a basis for analyzing any effect depending on spin interactions. Cluster methods attempt to introduce correlation, but they are both inadequate and sometimes run into severe difficulties, such as the appearance of an anti-Curie temperature (in the Bethe-Peierls-Weiss theory). The source of such difficulties is demonstrated by the formalism of P. W. Kasteleijn and J. Van Kranendonck [Physica 22, 317 (1956)] who showed that the cluster theories correspond to the choice of a self-inconsistent two-particle density matrix. Furthermore the spin wave theory indicates that the interactions which are dominant in establishing spin-spin correlations are iterated interactions through enormously many long and complicated paths connecting the spins. Cluster methods are unable to include these long paths, and stress instead the direct short-path interactions. The need for a many-body theory is therefore indicated. One such many-body theory is the Green function treatment of Bogoliubov and Tyablikov. That treatment is equivalent to the random phase approximation. The simplest formulation starts with the spin wave theory of Dyson. The destruction operator for a Dyson spin wave is ak≡12NS12 ∑ j S(Rj) exp(ik·Rj). Similarly Dyson introduces the Fourier components of the z components of spin Skz≡ ∑ j Sz(Rj) exp(ik·Rj). Unfortunately the spin waves defined in terms of these operators are neither orthogonal nor are they eigenstates of the Hamiltonian. In fact the commutation relations are [Skz,aλ]=ak+λ, [ak,aλ+]=(1/NS)Sk−λzand [H,ak]=−μHak−(2SN)−1 Σλ (ελ·k−ελ)(Sλzak−λ−aλSk−λz),where εk is the energy of a simple spin wave, expressed as a function of k; for small k it is proportional to k2. The random phase approximation ``decouples'' the awkward higher-order terms in the right-hand members of these commutation relations. In lowest order this decoupling consists of replacing Sλz by N〈Sz〉δλ0, where 〈Sz〉 is the average value of the z component of spin, to be evaluated self-consistently. Then the commutation relations become [ak,aλ+]=(1/S)〈Sz〉δλkand [H,ak]=−[μH+εk〈Sz〉/S]ak. The first of these states that the quasi-particles (magnons) produced by the creation operators ak+ are bosons. The second commutation-relation states that the energy of such a quasi-particle is μH+εk〈Sz〉/S. Thus the average number of such magnons, of wave vector k, is 〈nk〉=〈Sz〉/S[expβ(μH+εk〈Sz〉S−1)−1]−1,which is the result given by Tyablikov1 and by Englert. It differs from the simple spin wave theory by reducing the energy of each spin wave proportionally to the average magnetization. The Green function method (or RPA) gives a result which forms a useful interpolation throughout the entire temperature range. It agrees to second order in 1/T with the Kramers-Opechowski series in the high temperature (paramagnetic) region. Near the Curie temperature the theory reduces to the Weiss theory, and in the very low temperature region it agrees with the simple spin wave theory. The corrections to the simple spin wave theory at slightly higher temperatures are incorrectly given, however, as the RPA approximation predicts a correction to the magnetization varying as T3, whereas Dyson has shown that the lowest order correction varies as T4. This error in the theory arises from the decoupling procedure, which ignores spin-wave correlation effects in replacing Sλz by zero (if λ≠0). A direct approach to a statistical theory consists in a series expansion of the partition sum, a diagrammatic representation of the terms, and a partial summation of those terms corresponding to the long, circuitous paths connecting and correlating two spins. Such a theory has been carried out by Brout and co-workers and by Horwitz and Callen. The results have been published, or are in publication in detailed form elsewhere for the Ising model, and the extension to the Heisenberg model will be presented in a separate more extensive publication. Again the theory provides a useful interpolation between the low and high temperature region. Evaluation of the theory at low temperatures, by R. Stinchcomb, indicates that it properly gives the Dyson dynamical (T4) correction.