# 9 The Ising model

When consistent with the symmetry of the problem, two-value classical spins, \(S^z=\pm 1\), can be assumed:
\[\begin{equation}
\label{Ising-Ham}
\tag{9.1}
\mathcal{H}%\left[\left\{ S^z({\underline n})\right\}\right]
=-\frac{1}{2}J\sum_{|{\underline n}-{\underline n}'|=1} {S}^z({\underline n})\, {S}^z({\underline n}')
+g \mu_{\rm B} B\sum_{{\underline n}} {S}^z({\underline n})\,.
\end{equation}\]
This approximation is obviously justified in the limit in which the anisotropy \(D\) is significantly larger than other energies at play (\(J\), \(k_{\rm B} T\), etc.).
Another instance is realized when the full degeneracy of the total angular momentum of unpaired electrons of a magnetic atom in the gas phase (spherically symmetric environment, Hund’s rules) is reduced to the minimal two-fold degeneracy for \(B=0\) in the solid phase (Kramers doublet). In this case, magnetism can be described with an *effective* spin one-half.
Beside its application to magnetism, the Ising Hamiltonian (9.1) is used in many different contexts, ranging from biophysics to social sciences.

### Mean-field approximation

Assuming that the reader has encountered the mean-field approximation (MFA) in different courses, we refresh here only the aspects that are relevant to our discussion on ferromagnetism at finite \(T\). The MFA is a simplified treatment of a many-body problem, which consists in replacing the original problem with its *best* single-particle counterpart. For magnetic systems, the reference single-particle problem is the paramagnet, which can be regarded as the equivalent of the “ideal gas” in the study of statistical thermodynamics. In formula, the MFA of Hamiltonian (9.1) reads
\[\begin{equation}
\label{Ising-Ham-MFA}
\tag{9.2}
\mathcal{H}
=-\frac{1}{2}J\sum_{|{\underline n}-{\underline n}'|=1} {S}^z({\underline n})\, {S}^z({\underline n}')
+g \mu_{\rm B} B\sum_{{\underline n}} {S}^z({\underline n})\simeq
g \mu_{\rm B} B^{\rm eff}\sum_{{\underline n}} {S}^z({\underline n})
\end{equation}\]

where the effective (Weiss) field \(B^{\rm eff}\) depends parametrically on the single-particle averages of the \(z\) spin projection
\[\begin{equation}
\label{MFA-averages}
\tag{9.3}
s_{\rm av} = \langle {S}^z({\underline n}) \rangle \,.
\end{equation}\]
Setting equal to zero terms like
\[\begin{equation}
\label{MFA-fluct-zero}
\tag{9.4}
\left[s_{\rm av} - {S}^z({\underline n}) \right] \left[s_{\rm av} - {S}^z({\underline n'}) \right] =0 \,,
\end{equation}\]
technically called *fluctuations*, makes it possible to rewrite the Hamiltonian (9.1) as the Hamiltonian of a paramagnet (with two energy levels, for the Ising model).
In fact, in this way, the product of pairspins can be expressed in terms of the average \(s_{\rm av}\) and single-particle contributions
\[\begin{equation}
\label{MFA-pairspin}
\tag{9.5}
{S}^z({\underline n}) {S}^z({\underline n'}) = - s_{\rm av}^2 + \left[{S}^z({\underline n}) + {S}^z({\underline n'}) \right] s_{\rm av}\,.
\end{equation}\]
Within the MFA the spontaneous magnetization behaves as follows
\[\begin{equation}
\label{spontaneous-mag-MFA}
\tag{9.6}
\begin{split}
&\lim_{B\rightarrow 0^+} m(B,T) \ne 0 \qquad \text{for} \quad T < T_{\rm c} \\
&\lim_{B\rightarrow 0^+} m(B,T) = 0 \qquad \text{for} \quad T > T_{\rm c}
\end{split}
\end{equation}\]
where the critical temperature is defined as \(k_{\rm B}T_{\rm c} = z_n J\), with \(z_n\) number of nearest neighbors of each spin.
Below \(T > T_{\rm c}\), the spontaneous magnetization is predicted to behave critically
\[\begin{equation}
m(T,0)\sim \left(T_{\rm c}-T\right)^{\frac{1}{2}}\,.
\end{equation}\]
In summary, the MFA predicts the occurrence of a phase with spontaneous magnetization at finite temperature, which is realized below a material-dependent \(T_{\rm c}\).
In the following we will discuss some limitations of this approach that are mainly rooted in the crudeness of the approximation in Eq.(9.4).

## 1D model

Probably one of the most striking failure of MFA is the prediction of a magnetically ordered phase below \(T_{\rm c}\) for one-dimensional (1D) systems. In fact, the result in Eq.(9.6) is independent of the magnetic-lattice dimensionality D. The latter corresponds to the number of directions along which the exchange coupling propagates indefinitely. In practice, this dimension may also be different from the actual dimensionality of the considered solid, like in molecular spin chains.

A well-known result of Statistical Physics is that systems whose magnetic lattice has a dimensionality is smaller than 2D cannot sustain spontaneous magnetization at thermodynamic equilibrium.

### Landau argument

Here we provide a heuristic argument presented in the Landau–Lifshitz series that applies to the 1D Ising model and more generally to spin chains with uniaxial anisotropy. We evaluate the variation of the free energy associated with the creation of a domain wall (DW) in a configuration with all the spins parallel to each other.Creating a DW in a spin chain where all the spins point along the same direction increases the energy by a factor \(E_2-E_1=2J\). This DW may occupy \(N\) different positions in the spin chain, so that this set of configurations has an entropy of the order of \(S_2\simeq k_{\rm B} \ln(N)\). The entropy of the ground state vanishes if we assume that the two spins at the boundaries have been forced to point upward (otherwise one has \(S_1=k_{\rm B} \ln(2)\)).
Therefore, the free-energy difference between the two configurations sketched in Fig.9.1 is roughly given by
\[\begin{equation}
\label{Landau_arg}
\tag{9.7}
\Delta F \simeq 2J - k_{\rm B} T\ln (N)\,.
\end{equation}\]
The qualitative behavior of \(\Delta F\) is sketched in Fig.9.2. A characteristic temperature-dependent threshold \(\bar N\) can be defined such that for \(N>\bar N\) the free energy difference \(\Delta F\) is negative and therefore DWs start forming *spontaneously* in the chain. The threshold \(\bar N\) is obtained by requiring \(\Delta F=0\), which gives

\[\begin{equation}
\label{N_bar}
\tag{9.8}
\bar N \simeq \exp\left(2\beta J\right).
\end{equation}\]
Practically, when \(\bar N\) is larger than the number of spins^{34} in the chain \(N\) (low temperature), the ground-state configurations with all spins aligned are also minima of the free energy, since \(\Delta F>0\). In this case, as far as equilibrium properties are concerned, the behavior of the spin chain is reminiscent to that of a two-level paramagnet with magnetic moment \(\mu=N g S \mu_B\). When \(\bar N < N\) (high temperature), instead, the ground-state configurations with all spins aligned do not minimize the free energy and DWs are always present in the system at equilibrium. In the limit of an infinite chain, the same argument can be repeated to justify the presence of an indefinite number of DWs. We refer to this condition – realized in spin chains at higher temperatures – as the *thermodynamic limit* in which the inverse of \(\bar N\) is proportional to the average density of DWs.

### Correlation length

In 1D magnetic systems the averaged pair-spin correlation decays exponentially with the separation between spins. Focusing on the Ising model, in which only the spin component along \(z\) is defined, one has
\[\begin{equation}
\langle S^z_{i} S^z_{i+r}\rangle = {\rm e}^{-r/\xi} \,.
\end{equation}\]
The characteristic scale of this decay defines the correlation length \(\xi\). It can be shown that \(\xi\) is related to the susceptibility measured along the easy axis in zero field by the general equation
\[\begin{equation}
\label{chi_xi}
\tag{9.9}
\chi= 2\, \frac{C}{k_{\rm B}T}\,\xi\, ,
\end{equation}\]
where \(C\) is the Curie constant characterizing the magnetic centers coupled to form the chain. Apart from proportionality factors, \(\bar N\) defined above and indicated with a dot in Fig.9.2 can be identified with the correlation length \(\xi\). Thus, similarly to \(\bar N\), one expects a leading dependence of the Arrhenius type for the correlation length as well:
\[\begin{equation}
\label{exp_xi}
\tag{9.10}
\xi \sim \exp\left(2\beta J\right)\,.
\end{equation}\]

For a finite chain with \(N< \bar N\) (see Fig.9.2) the role of the correlation length in the susceptibility is – roughly speaking – replaced by the chain size \(N\).

## 2D lattice

An argument similar to the Landau’s one, holding for the 1D Ising model, can be developed for the 2D system as well. In this case we should refer to the possibility of reversing a cluster of spins enclosed in a perimeter of \(l\) lattice sites and embedded in a region of spins all pointing in the same direction, as sketched in Fig.9.3.For simplicity we consider a square lattice and sharp domain walls, meaning that all the spins are assumed to point either along \(S^z=+1\) (outward) or along \(S^z=-1\) (inward). The total cost in terms of exchange energy is of the order of \(2J\, l\). To estimate the entropy variation due to the creation of a reversed cluster in an otherwise uniform spin configuration, we can think of a self-avoiding random walk. Suppose that a walker can move with one step from the center of a square in Fig.9.3 to the center of a neighboring square. At each step the walker has at most three choices of which way to go, since it has to avoid itself (the walker cannot take a step back in the direction where it came from). A possible random walk is highlighted with a thick line in the figure. Based on these simple considerations, we expect the number of closed loops corresponding to the perimeter \(l\) to be of the order \(p^l\), with \(p<3\). As a result, the free-energy variation associated with the flip of a cluster delimited by a perimeter \(l\) is roughly \(\Delta F=2J\, l - k_{\rm B} T \,l \ln p\). Therefore, for \(T< 2J/(k_{\rm B} \ln p)\) the ordered phase – with all the spin aligned along \(S^z=+1\) – should be stable against the formation of large domains with reversed spins. This argument for the existence of an ordered low-temperature phase in this 2D Ising model and, thus, of a finite Curie temperature \(T_{\rm c}\) was first put forward by Peierls – in more precise terms.

### Rigorous results

The Ising model represents a particularly lucky case in which the heuristic arguments given
above can be checked by solving the problem analytically. Even if we will not derive these results, it is useful to recall which steps should be followed to prove rigorously whether a model is consistent with a phase with *spontaneous* magnetization (finite magnetization in zero external field) for \(T\ne 0\) or not. To this end, one has to compute:

1. the partition function
\[\begin{equation}
\label{Part_fct_Ising}
\tag{9.11}
\mathcal{Z}= \mathcal{T}r\left\{e^{-\beta \mathcal{H}\left[\left\{{S}^z({\underline n})\right\}\right]}\right\}
\end{equation}\]
where \(\beta =1/(k_{\rm B} T)\) and the trace is obtained by letting each discrete variable take the two possible values
\({S}^z({\underline n})=\pm 1\) (\(\mathcal{Z}\) is a sum with \(2^N\) terms!)

2. the thermal average of the magnetic moment
\[\begin{equation}
m(T,B)=-\frac{1}{N}\frac{\partial F}{ \partial B}= \frac{1}{N}\frac{1}{\beta} \frac{\partial \ln\mathcal{Z} }{ \partial B}
\end{equation}\]
3. the limit
\[\begin{equation}
\label{spontaneous_mag_Ising}
\tag{9.12}
m(T,0)=\lim_{B\rightarrow 0^+} m(T,B)
\end{equation}\]
and evaluate if there exists a temperature \(T_{\rm c}\) below which the limit (9.12) takes a value different from zero.

This procedure can be carried out analytically for the Ising model in 1D and 2D producing different results:

\(\bullet\) For 1D, no spontaneous magnetization is possible at any finite temperature.

\(\bullet\) For 2D, a spontaneous magnetization appears for \(T<T_{\rm c}\). The transition temperature is given by
\[\begin{equation}
\label{T_c_Onsager}
\tag{9.13}
\sinh\left(\frac{2J}{k_{\rm B}T_{\rm c}}\right)=1 \quad\Rightarrow\quad T_{\rm c} = \frac{2}{\ln(1+\sqrt{2})} \frac{J}{k_{\rm B}} \simeq 2.27 \frac{J}{k_{\rm B}}\,.
\end{equation}\]
The comparison with the MF theory shows that the latter typically overestimates the transition temperature:
The critical temperature of the 2D Ising model reported in Eq.(9.13) has to be compared with \(T_{\rm c}^{\text{MF}}=4 J/k_{\rm B}\) (\(z_n=4\) for a square lattice). Expanding the spontaneous magnetization close to \(T_{\rm c}\) yields
\[\begin{equation}
m(T,0)\sim \left(T_{\rm c}-T\right)^{\frac{1}{8}}\,.
\end{equation}\]
Thus, for the 2D Ising model the exact value of the critical exponent is \(\beta=1/8\), at odds with the MF value \(\beta^{\text{MF}}=1/2\).

Indeed both these exact results obtained for the 1D and 2D Ising model show that the MFA overlooks some important features of the transition from the paramagnetic to the ferromagnetic phase, possibly occurring upon lowering the temperature.

In molecular spin chains \(\bar N\) should be compared with the average number of sites separating two successive defects.↩︎