# 8 Magnetic order at finite temperature

We start this chapter with a quick summary of the milestones that we have encountered on the way to justifying ferromagnetism in real systems. In the first weeks of the course we have shown how magnetic moments are created at the atomic level according to the Hund’s rules via the **intra-atomic** exchange interaction. Then we have seen that atomic magnetic moments of the free ion, deduced assuming a *spherically symmetric* surrounding (Hund’s rules), are generally reduced when the atom is “put” in a crystal field of ligands. Successively, we have reviewed some mechanisms that give rise to **inter-atomic** exchange interaction in metal oxides of the 3d series. In the the following chapters we have defined the different regimes that characterize the time evolution of magnetic moments in the presence of an external field, including the approach to thermal equilibrium.

Even limiting ourselves to what discussed so far, it is clear that ferromagnetism is not the rule but rather an *exception*, in the sense that many factors that are encountered in ordinary materials
usually prevent the formation of magnetic moments or that of a ferromagnetic inter-atomic coupling.
All the above-mentioned properties have been deduced neglecting thermal fluctuations or, in other words, they are *ground-state* properties. Now we are going to discuss the consequences of introducing temperature. The general trend is that thermal fluctuations destroy the ground-state ferromagnetism (when present). In the same line as before, we will define the conditions under which ferromagnetism can “survive” at finite temperatures and at *thermodynamic equilibrium*.

## 8.1 From paramagnetism to ferromagnetism

The fist thermodynamic system that we are going to consider is a paramagnet, namely an*ensemble*of non-interacting magnetic atoms or molecules. Hund’s rules allow determining the ground-state multiplet of the free ion, characterized by the total angular momentum \(\hat{\mathbf J}\) (orbital plus spin contribution) which results from all the unpaired electrons. Transition-metal (3d) and rare-earth ions (4f) have been the main focus of our attention. When one atom belonging to the first family of elements is embedded in a solid (insulator or conductor) the orbital contribution gets quenched

^{29}. This does not happen in rare-earths, for which the magnetic moment is proportional to \(\hat{\mathbf J}\). Henceforth, we will not make this distinction so that \(\hat{\mathbf S}\) will always be used to indicate the angular-momentum operator that is proportional to the magnetic moment

^{30}.

Typically, the ground-state multiplet is separated from the first excited one by an energy gap of the order of the intra-atomic exchange interaction (\(\sim\) eV).
The Zeeman energy is of the order of 0.1 meV \(\sim 1\) K for one-Tesla applied field and, therefore, comparable to the thermal energy (\(k_{\rm B}T\)).
These two facts imply that only the ground-state multiplet shall be populated at realistic temperatures but the Zeeman splitting of its levels needs to be treated in the framework of statistical physics. Choosing as a quantization axis for the spin the direction along which the \(B\) field is applied, the Zeeman Hamiltonian reads
\[\begin{equation}
\label{Ham_Zeeman_S_eff}
\mathcal{H}_{{\rm Z}} = g\,\mu_{\rm B}\, B \hat{S}^z \,,
\end{equation}\]
where \(g\) is a generic Land'e factor and \(\hat{\mathbf S}\) the effective spin meant as explained above.

The corresponding partition function is given by the relation
\[\begin{equation}
\label{Part_fct_Zeeman}
\mathcal{Z} = \sum_M {\rm e} ^{-\beta E_M}
\end{equation}\]
where \(\beta=1/(k_{\rm B}T)\) and \(E_M= g\, \mu_{\rm B}\, B M\) with \(M=-S\dots S\).
The partition function is related to the free energy
\[\begin{equation}
\label{Free_energy_Zeeman}
F=-k_{\rm B}T \ln\left(\mathcal{Z} \right)
\end{equation}\]
from which the average magnetic moment per atom can be computed to obtain
\[\begin{equation}
\label{Brillouin_III}
m = -\frac{\partial F}{\partial B} =-g \mu_{\rm B} \langle \hat{S}^z \rangle =
g \mu_{\rm B} \, S \, \mathcal{B}_S \left[ \frac{g \mu_{\rm B} \, S\, B}{k_{\rm B} T} \right] \, ,
\end{equation}\]
with the *Brillouin function*
\[\begin{equation}
\tag{8.1}
%\label{Brillouin_III}
\mathcal{B}_S \left( \alpha \right)
= \frac{2S+1}{2S} \text{coth} \left(\frac{2S+1}{2S} \alpha\right)
- \frac{1}{2S} \text{coth} \left(\frac{\alpha}{2S} \right)
\quad\text{and}\quad \alpha=\frac{g \mu_{\rm B} \, S \, B}{k_{\rm B} T} \, .
\end{equation}\]
The derivation of the Brillouin function for a generic \(S\) can be found in the literature. We do not reproduce all the passages here because it is a relatively long calculation.
Note that the Brillouin function equals the hyperbolic tangent that for a spin \(S=1/2\).
Moreover, \(\mathcal{B}_S \left( \alpha \right)\) is only a function of the ratio \(B/T\) and not of these two external parameters independently.

Expanded for small arguments \(\alpha\), Eq.(8.1) reduces to

\[\begin{equation}
m = \frac{ (g\mu_{\rm B})^2 S(S+1)}{3k_{\rm B}\,T} \, B\,.
\end{equation}\]
The pre-factor of \(B\) on the right-hand side is the magnetic susceptibility of a paramagnet, defined with SI units as \(\chi =\mu_0 \partial M/\partial B\) (computed in \(B=0\)), and obeys the famous Curie law
\[\begin{equation}
\label{Curie-law}
\chi = \frac{\mu_0}{a^3} \frac{C}{T}
\end{equation}\]
with Curie constant equal to
\[\begin{equation}
C=\frac{ (g\mu_{\rm B})^2 S(S+1)}{3k_{\rm B}} \,.
\end{equation}\]
Note that, through \(g\) and \(S\), the Curie constant contains information about the ground-state multiplet of individual magnetic ions in the paramagnetic phase.
For this reason the Curie constant has played an important historical role in confirming the quantum-mechanical description of matter.

For the forthcoming discussion it is important to remark that in the derivation of the Brillouin function \(\mathcal{B}_S \left( \alpha \right)\)
we have implicitly used the knowledge of i) the eigenstates of the atom in the presence of an external,
applied field \(E_M\) and ii) the way of performing *thermal averages* for a quantum system.

### Spin Hamiltonian as quantum N-body problem

While the intra-atomic exchange energy is of the order of \(4-10\) eV\(\sim 10^5\) Kelvin, the inter-atomic exchange interaction is about \(10-50\) meV \(\sim 100-500\) Kelvin. Thus, depending on the material and the temperature range of interest, a statistical-mechanic treatment is also required to study the cooperative effects arising from this type of coupling between different magnetic moments in a solid. The competition between this inter-atomic exchange interaction and thermal fluctuations is indeed responsible for the loss of ferromagnetism above a certain temperature, called Curie temperature \(T_{\rm c}\). The underlying
Table reports the values of that critical temperature for few selected magnetic materials.

**Typical transition temperatures of some ferromagnets**

Fe | Co | Ni | Fe\(_3\)O\(_4\) | Nd\(_2\)Fe\(_{14}\) B | Gd | Dy | EuO | EuS | |
---|---|---|---|---|---|---|---|---|---|

\(T_{\rm c}\)[K] | 1043 | 1388 | 627 | 858 | 593 | 292 | 88 | 69 | 16.5 |

(Some typical values of the Curie temperature \(T_{\rm c}\) for few selected ferromagnets more values can be found in Wikipedia).

Restricting ourselves to ferromagnetic inter-atomic exchange interactions, a system of coupled magnetic moments arranged in a lattice can be described by the Hamiltonian
\[\begin{equation}
\label{spin_Ham_general}
\tag{8.2}
\mathcal{H}=-\frac{1}{2}J\sum_{|{\underline n}-{\underline n}'|=1} \hat{\mathbf S}({\underline n})\cdot \hat{\mathbf S}({\underline n}')
+g \mu_{\rm B} B\sum_{{\underline n}} \hat{S}^z({\underline n})\,.
\end{equation}\]
The dimension of the Hilbert space associated with this quantum many-body problem
scales as \((2S+1)^N\), \(N\) being the number of magnetic moments (spins) in the lattice.
Due to such an **exponential** dependence on \(N\), the exact treatment of a system of many coupled spins
becomes intractable – even numerically – as far as the number of spins approaches that of
realistic *extended* systems^{31}.
In practice, one can try to circumvent this problem in several ways:

1. Reduce the many-body problem to a single-particle problem. This corresponds to the mean-field approximation (MFA).

2. Simplify the problem replacing the quantum-spin operators by classical vectors.

3. Take advantage of specific symmetries in the problem under investigation and use
a Hamiltonian which can easily be diagonalized.

4. Consider only a selected family of excitations of the ground state,
which can have either *local* (domain walls) or *non-local* (spin waves) character.

### Non-analyticity in the magnetization curve

The goal of this chapter is to define the conditions for the persistence of ferromagnetism at finite temperatures in the framework of equilibrium thermodynamics.
In other words, we will assume that the considered *ensembles* of interacting magnetic moments have had enough time to equilibrate, i.e., to populate different configurations according to the Boltzmann distribution. Before entering the details of specific models, it is useful to provide an operative definition of *ferromagnetism* itself. To this aim, we start from the paramagnetic response of a spin 1/2 given by the corresponding Brillouin function
\[\begin{equation}
\label{Brillouin_one-half}
\tag{8.3}
m = g \mu_{\rm B} \, \frac{1}{2} \, \mathcal{B}_{1/2} \left[ \frac{g \mu_{\rm B} \, B}{2k_{\rm B} T} \right] = g \mu_{\rm B} \, \frac{1}{2} \, \tanh \left[ \frac{g \mu_{\rm B} \, B}{2k_{\rm B} T} \right]\,.
\end{equation}\]
As the temperature is decreased, it is evident that the Brillouin function approaches a step-like function taking the values
\[\begin{equation}
m =
\begin{cases}
& -\frac{1}{2} \,g \mu_{\rm B} \qquad\text{for}\quad B<0 \quad \text{and} \quad T=0 \\
& +\frac{1}{2} \,g \mu_{\rm B} \qquad\text{for}\quad B>0 \quad \text{and} \quad T=0 \,.
\end{cases}
\end{equation}\]
This behavior can be easily understood thinking that, when the thermal energy \(k_{\rm B} T\) becomes significantly smaller than the Zeeman splitting between the two levels
\(|1/2,+1/2\rangle\) and \(|1/2,-1/2\rangle\) produced by the external field \(B\), practically only the lowest-lying state is populated.
In this condition, the magnetization is just the slope of the lowest-lying energy level as a function of \(B\). As already noticed when discussing the Landau-Zener formula, the two energy levels associated with the Zeeman energy of a spin 1/2 consist of two straight lines that cross each other in \(B=0\); therefore, they are consistent with a step-like magnetization curve at \(T=0\). In particular, the ground-state energy as a function of the applied field is **not analytic** in \(B=0\) and, thus, the magnetization in this point needs to be defined taking the left and right derivative of the energy
\[\begin{equation}
\tag{8.4}
\begin{split}
& m = -\lim_{B\rightarrow 0^-} \frac{\partial E_{\rm gs}}{\partial B} =-\frac{1}{2} \,g \mu_{\rm B} \qquad\text{(left)}\\
& m = -\lim_{B\rightarrow 0^+} \frac{\partial E_{\rm gs}}{\partial B} =+\frac{1}{2} \,g \mu_{\rm B} \qquad\text{(right)} \,.
\end{split}
\end{equation}\]
When \(N\) spins are coupled ferromagnetically as in the Hamiltonian (8.2) the ground-state multiplet will contain the two states
\[\begin{equation}
|\underbrace{+S,+S,\dots,+S}_\text{$N$ spins} \rangle \qquad \text{and} \qquad
|\underbrace{-S,-S,\dots,-S}_\text{$N$ spins} \rangle \, .
\end{equation}\]
in which the eigenvalue of \(\hat{S}^z({\underline n})\) takes the extreme values, either \(M=+S\) or \(M=-S\), for every spin. The state with all the spin projections \(M=+S\) is the ground state for \(B<0\), while the state with all the spin projections \(M=-S\) is the ground state for \(B>0\). In other words, the behavior of the lowest-lying energy level is qualitative analogous to the one of a spin 1/2 in an external field, but with the slope of the two lines being \(\pm S g\mu_{\rm B} N\). As a result, the ground-state magnetization curve per spin (per atom) behaves as a step-like function also for a system of \(N\) spin coupled ferromagnetically.

As trivial as they may seem, the limits in Eq.(8.4) suggest a concrete definition of ferromagnetism at finite temperature, provided that the ground-state energy is replaced by the free energy: A system is ferromagnetic if the *spontaneous magnetization*

\[\begin{equation}
\label{spontaneous_mag}
\tag{8.5}
\lim_{B\rightarrow 0^+} m(B,T) = -\frac{1}{N}\lim_{B\rightarrow 0^+}\frac{\partial F}{\partial B}
\end{equation}\]
is different from zero (with the free energy \(F=-\beta^{-1} \ln(\mathcal{Z})\)). We remark that one speaks of ferromagnetism when a sample remains magnetic in the absence of field, that is the reason why the limit for \(B\rightarrow 0^{\pm}\) matters. The existence of atomic magnetic moments warrant only paramagnetism but – as we will see – not necessarily ferromagnetism at finite temperature.

Based on the time-reversal symmetry of spin Hamiltonians in the absence of field (\(B=0\)) it can be argued that the limit (8.5) can only be finite if the magnetization is discontinuous in \(B=0\), i.e., if the derivative \(\partial m/\partial B\) is not analytic. Yet, in *equilibrium thermodynamics* the magnetization is expected to be an analytic function for any finite system. In fact, through the free energy, the magnetization is related to the first derivative of the partition function
\[\begin{equation}
\label{trace_Z_O_eigenstates}
\tag{8.6}
\mathcal{Z}=\sum_{n} e^{-\beta E_n}\,,
\end{equation}\]
\(n\) labeling the states of the system and \(E_n\) being the corresponding energies. Since for a finite system \(n\) is finite, \(\mathcal{Z}\) is a *finite* sum of analytic functions, which has to be analytic. This argument rules out the possibility of having *spontaneous* magnetization at finite temperature for *any* finite system^{32}.
However, formally speaking, an infinite sum of analytic functions can be non-analytic. Think, for instance, of the Fourier series of a square or a saw-tooth wave. Therefore, the thermodynamic limit \(N\rightarrow\infty\) practically underlines any theoretical description of ferromagnetism at finite temperature as an equilibrium phenomenon, namely as resulting from a thermodynamic phase transition. The concept of bistability is indeed broader than this definition of ferromagnetism because it includes the possibility of a sample being magnetic as a result of metastability, that is an out-of-equilibrium situation^{33}.

Partial quenching of orbital momentum can be accounted for in an anisotropic Land'e \(g\) factor.↩︎

This choice is made i) in order to avoid confusion with the exchange interaction (\(J\)) and ii) because – in this context – people often speak about “spin” or “effective spin” to indicate the total single-atom angular momentum.↩︎

Some effective zero-dimensional structures (magnetic clusters or nanoparticles) are also studied in the context of nanomagnetism. For some of these systems, exact diagonalization of the associated quantum problem is still feasible numerically and makes it possible to describe their magnetic behavior at any temperature.↩︎

Note that this statement does not contradict what discussed previously for the ground-state magnetization because in the limit of \(T\rightarrow 0\) the exponential functions appear ng in Eq.(8.6) are not analytic either.↩︎

Roughly speaking, when we use Boltzmann statistic we assume thermodynamic equilibrium, namely we assume that the system under investigation had enough time to visit all its statistically relevant states. Yet, for \(T<T_{\rm c}\) this is true only for subsets of the phase space because of th occurrence of

*ergodicity breaking*, which accompanies spontaneous symmetry breaking.↩︎