Classical Heisenberg model

For the classical Heisenberg model and using the notations of Eq. 1, the correlation function F is given by ... 

The Ising model is also a classical model and Eqs. 2, 3 and 6 are still valid when Ui is replaced by CTj. Equation 3 gives the parallel susceptibility (when the magnetic field is applied along the z axis) with the Curie constant per spin C = g ii S /ks,. The main difference with the classical Heisenberg model is the expression of the correlation function r [14] ... 

The result 6. was obtained by bounding 5b from above by the X, of the classical Heisenberg model and bounding it from below by a construct related to the periodic case. As the difference was 10%, Theodorou used the classical Heisenberg model for his calculations. The full P(J) is shown in Fig. 1 the fit to di Salvo s data in Fig. 2. 

One way forward from this point is to carry out calculations of Q " ( e ) for a selection of configurations ( spin spirals, two impurities in a ferromagnet, magnetically ordered superceUs, etc.) by making some assumptions about the most dominant fluctuations. One then fits the set of Q " ( e ) s to a simple functional form. Typically a classical Heisenberg model, Q " ( e ) = is set up... 

The quantum-to-classical mapping follows the same procedure as for the clean bilayer quantum Heisenberg model above. The result is an unusual diluted three-dimensional classical Heisenberg model. Because the impurities in the quantum system are quenched (time-independent), the equivalent classical Heisenberg model has line defects parallel to the imaginary time direction. The classical Hamiltonian is given by... 

A particularly simple and yet accurate approach for the evaluation of the Curie temperature on an ab initio level consists in a mapping of the itinerant spin-polarized electron system onto an effective classical Heisenberg model (EHM) "... 

The Yamanouchi-Kotani basis is best suited if we want to solve the Heisenberg problem in the complete spin space. However, the number of spins that can be handled this way, soon reaches an end due to the rapid growth of the spin space dimension f(S,N). Even with the present day computers, the maximum number of spins that can be treated clusters around N = 30. For larger values of N one must resort to approximate treatments, one of which, as described hereafter, is based on the idea of resonating valence bonds (RVB) coming from the classical VB model developed by Pauling and Wheland back in the early 1930 s [37, 51]. In essence,... 

In the approach of micromagnetism the quantum mechanical spin operators are substituted by classical vectors, S. Thus, the exchange energy in the Heisenberg model [34]... 

Starting from the classical Heisenberg-Dirac-Van Vleck (HD W) model based on quantum mechanical interactions, all the other models can also be founded on statistical calculations. 

The same local update algorithm can be applied to systems with longer-range interactions and with coupling constants that vary from bond to bond. For more complex classical models, such as Heisenberg models, local updates will no longer consist of simple spin flips, but of arbitrary rotations of the local spin vectors. 

The electromagnetic spectrum is a quantum effect and the width of a spectral feature is traceable to the Heisenberg uncertainty principle. The mechanical spectrum is a classical resonance effect and the width of a feature indicates a range of closely related r values for the model elements. 

In 1913 Niels Bohr proposed a system of rules that defined a specific set of discrete orbits for the electrons of an atom with a given atomic number. These rules required the electrons to exist only in these orbits, so that they did not radiate energy continuously as in classical electromagnetism. This model was extended first by Sommerfeld and then by Goudsmit and Uhlenbeck. In 1925 Heisenberg, and in 1926 Schrn dinger, proposed a matrix or wave mechanics theory that has developed into quantum mechanics, in which all of these properties are included. In this theory the state of the electron is described by a wave function from which the electron s properties can be deduced. 

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