Magnetism – CSINSS

The sources of the magnetic field
are electric currents and the changing electric fields according to classical electromagnetic theory.
The source of the magnetic phenomena in solids can be considered as the microscopic currents due to the motion of the electrons in the atoms.
However, Niels Bohr in 1911 and 1919 Johanna H. van Leeuwen showed that the diamagnetism cannot exist in classical physics.
Therefore, magnetism can only be understood by using quantum mechanics.
Magnetism is closely related to the angular momentum of electrons in atoms.
Thus, microscopic theory of magnetism is based on the quantum mechanics of the angular momentum which has two sources:
orbital motion and spin.
Ferromagnetic materials can have spontaneous magnetization even in the absence of magnetic field.
This magnetization is created by the strong exchange interaction between the magnetic moments of atoms [1].

Spontaneous magnetic order at low temperatures is a fundamental issue in solid state physics.
Ferromagnets, antiferromagnets, liquid crystals and superconductors are in ordered phase as solid state itself.
All these regular structures have some common basic features. For example,
the magnitude of the order parameter exhibits a striking difference below the critical temperature $T_C$ .
The order parameter defined for each magnetic order is zero above the critical temperature ( $T>T_C$ ) and
it takes a value other than zero under below critical temperature ( $T<T_C$ ).
In other words, critical temperature gives information about whether the system is in regular phase or not.
Magnetization is the order parameter for ferromagnetic materials. At low temperatures, magnetization created
by spontaneous alignment of magnetic moments in the same direction is reduced to zero at the Curie temperature [2].

Spin Models

Spin models not only help us to understand magnetism but also they are used in other branches of science such as
quantum mechanics, statistical mechanics and biology. In order to model the magnetic systems,
it is convenient to write the Hamiltonian in terms of spin variables. Spins are located on the sites
of a regular lattice in the models that we will discuss in the following subsections.
We can be interested in lattices in three dimensions (such as simple cubic, face centered cubic)
and also square, triangular and hexagonal lattices in two dimensions [3].

Spin-1/2 Ising Model

Ising model is one of the most successful magnetic model for describing the systems which consists of
interacting magnetic moments. In the Ising model, a classical spin variable $S_i$ with allowed values of $\pm 1$
is placed on each site of the lattice. The Hamiltonian of the Ising model is
Spin-1/2 Ising model

(1) $\begin{equation*} \mathcal{H}=\displaystyle-J\sum_{<ij>} S_i S_j -H\sum_i S_i \end{equation*}$

where the first term represents the cooperative behavior of the system and $J$ corresponds
to exchange interaction term. While positive $J$ values favor the parallel alignment of the spins,
negative $J$ favors antiparallel alignment. $<ij>$ represents the summation over nearest-neighbour spins.
If necessary, further-neighbour interactions can be included to the Hamitonian.
The second term of equation 1 is the Zeeman term which represents the interaction between
the magnetic field, $H$ , and the spins. Ising Hamiltonian reduces to a Hamiltonian of a paramagnet for
$J=0$ . In this case, spins do not interact and thus no phase transition occurs.

Ising model was firstly introduced by Wilhelm Lenz (1888-1957) in 1920 and it is exactly solved in one dimension by
Ernest Ising [4]. One dimensional Ising model corresponds to a special case since phase transion occurs only at zero temperature.
Also, one dimensional Ising model can be solved exactly by transfer matrix, series expansion and renormalisation group.
Lars Onsager performed the solution of two dimensional Ising model at zero magnetic field excatly 1944 and he has demonstrated that
there occurs a phase transition in two dimensions [5]. Two dimensional Ising model in the presence of a magnetic field and
even three dimensional Ising model at zero magnetic field remain unsolved analytically.

Spin-1 Ising Model

Higher-spin Ising models are convenient for the system with more than two states.
For instance, spin-1 Ising Hamiltonian is:

(2) $\begin{equation*} \mathcal{H}=\displaystyle-J\sum_{<ij>} S_i S_j-K\sum_{<ij>} S_i^2 S_j^2-D\sum_i S_i^2-H\sum_i S_i \end{equation*}$

where $D$ is crystal field parameter, $K$ is biquadratic spin-spin interaction term. Spin variable
can take the values of $S_i=0,\pm1$ . Because of the large parameter space, spin-1 Ising model displays
rich critical behavior.

X-Y and Heisenberg Models

Spin vector in Ising model are allowed to take $\pm 1$ along a given direction.
Thus, Ising model is appropriate for magnets which are anisotropic in spin space.
Some physical systems can be well described by Ising model. However, for some magnetic
systems spin variable can exhibit fluctuations from the quantization axis.
For such magnets a more realistic Hamiltonian is

(3) $\begin{equation*} \mathcal{H}=\displaystyle-J_z\sum_{<ij>} S_i^z S_j^z -J_{xy}\sum_{<ij>} (S_i^xS_j^x+S_i^yS_j^y) -H\sum_i S_i^z \end{equation*}$

here $x$ , $y$ and $z$ are the Cartesian axes in spin space. Equation 3 reduces
to Ising Hamiltonian for $J_{xy}=0$ . For the case of
$J_{xy}=J_{z}=J$ Equation 3 takes the form of

(4) $\begin{equation*} \mathcal{H}=\displaystyle-J\sum_{<ij>} \vec{S}_i.\vec{S}_j -H\sum_i S_i. \end{equation*}$

Equation 4 is the Heisenberg model [6]. Heisenberg
model was introduced in 1928. The Hamiltonian of Heisenberg model is
a microscopic Hamiltonian
and this Hamiltonian is described by the exchange which leads to ferromagnetism.
The Heisenberg model exhibits phase transitions for $d>2$ where $d$ is space size.
The X-Y model is obtained by taking $J_z = 0$ in the equation [6].
In this model, the spins are two-dimensional vectors.

References

[1] R. Skomski, Simple Models of Magnetism (Oxford University Press, New York, 2008).
[2] J. M. D. Coey, Magnetism and Magnetic Materials (Cambridge University Press, New York, 2009).
[3] J. M. Yeomans, Statistical Mechanics of Phase Transitions (Oxford University Press, New York, 1992).
[4] E. Ising, Z. Phys. 31 (1925) 253-258.
[5] L. Onsager, Physical Review, Series II, 65 (1944) 117-149.
[6] W. Heisenberg, Z. Phys. 49 (1928) 619-636.