# 10 The XY model

When consistent with the symmetry of the problem, the quantum mechanical operators representing the effective spin of a magnetic atom can be replaced by a two-component classical vector \(\vec{S}=(\cos\varphi, \sin\varphi)\), living – say – on the XY plane. An *ensemble* of such spins, disposed on a lattice and coupled via the exchange interaction, defines the classical XY model, whose Hamiltonian can be expressed in terms of the angle \(\varphi\) formed by each spin with some lattice direction:

\[\begin{equation}
\label{Classical-XY}
\tag{10.1}
\mathcal{H}=-\frac{1}{2} J\sum_{|{\underline n}-{\underline n}'|=1} \vec{S}({\underline n})\cdot \vec{S}({\underline n}')
=-\frac{1}{2} J\sum_{|{\underline n}-{\underline n}'|=1} \cos(\varphi({\underline n}) - \varphi({\underline n}'))\, .
\end{equation}\]
Concretely, it can be energetically convenient for the magnetic moments of a solid to lie in a planar configuration when an easy-plane single-ion anisotropy is present. Alternatively, a planar configuration can be stabilized by the dipole-dipole interaction, of magnetostatic origin.

In this session we will derive a necessary condition for the two main classical-spin models with continuous symmetry – the XY and the Heisenberg model – to be compatible with a phase with spontaneous magnetization. This argument relies on

\(\bullet\) a linearization of the pair-spin interaction

\(\bullet\) the use of the equipartition theorem.

Since we will consider Hamitlonians in which the d.o.f. of spins sitting at different lattice sites are coupled, after the linearization we will obtain a quadratic Hamiltonian whose d.o.f. are still coupled with each other. Due to this fact, the results obtained in Chapter 7 for the ideal gas – in which quadratic d.o.f. are not coupled with each other – cannot be applied at this stage. Instead, some judicious transformation of coordinates will be needed in order to decouple the d.o.f. of the linearized spin Hamiltonian. This transformation is the same as the one used to decouple the d.o.f. of a chain of harmonic oscillators and, subsequently, apply the equipartition theorem. Therefore, we find it convenient to reproduce this calculation for the chain of harmonic oscillators before moving to the systems of our interest.

### Chain of harmonic oscillators

We consider an array of harmonic oscillators on a 1D lattice. The degrees of freedom of this problem are the diplacements \(u_k\) of individual atoms from their minimal energy position.

The Hamiltonian for these *elastic excitations* reads
\[\begin{equation}
\label{harm-osc-Ham}
\tag{10.2}
\mathcal{H}= \sum_{k=1}^N \frac{m}{2}\dot u_k^2 + \frac{1}{2}\sum_{k=1}^N K_e \left(u_{k+1} - u_k\right)^2
\end{equation}\]
The first summation on the r.h.s. represents the kinetic energy and can be handled similarly to the case of the ideal gas^{35}. The second summation represents the potential energy of coupled harmonic oscillators. Clearly, the equipartition theorem cannot be applied to this portion of the Hamiltonian straightforwardly, though the variables \(u_k\) still contribute as a quadratic form. By means of the discrete Fourier transformation
\[\begin{equation*}
u_k= \frac{1}{\sqrt{N}}\sum_{q} \tilde{u}_q e^{iqk}
\end{equation*}\]
the Hamiltonian (10.2) can be rewritten as
\[\begin{equation*}
\mathcal{H}=\sum_{q} \left[K_e \left(1-\cos q \right) |\tilde{u}_q|^2 + \frac{1}{2} m \, |\dot{\tilde{u}}_q|^2\right] \,,
\end{equation*}\]
namely in a form in which the new degrees of freedom \(\tilde{u}_q\) are not coupled with each other anymore. Note that the prefactor of \(|\tilde{u}_q|^2\) is not constant but depends on \(q\).
At this point we can treat \(\tilde{u}_q\) and \(\dot{\tilde{u}}_q\) on the same footing as we did for \(p_x\) in the ideal gas calculation and obtain
\[\begin{equation*}
\begin{cases}
&K_e \left(1-\cos q \right) \langle |\tilde{u}_q|^2 \rangle =\frac{1}{2} k_B T \\
&\frac{1}{2} m \langle|\dot{\tilde{u}}_q|^2\rangle=\frac{1}{2} k_B T \,.
\end{cases}
\end{equation*}\]
The first equation provides the thermal average of the Fourier amplitudes of the displacement field \(u_k\). Knowing these quantities it is possible to determine the behavior of the averaged squared displacements of atoms in real space by inverting the Fourier transformation:
\[\begin{equation*}
\langle u_k^2\rangle= \frac{1}{N} \sum_{q} \langle |\tilde{u}_q|^2 \rangle
=\frac{1}{N} \sum_{q} \frac{k_BT}{2K_e \left(1-\cos q \right)}
\end{equation*}\]
where in the last passage we have used the equipartition theorem. The summation on the wave-numbers \(q\) is usually evaluated taking the continuum limit
\[\begin{equation*}\frac{1}{N} \sum_{q} \rightarrow \frac{1}{2\pi} \int d q
\end{equation*}\]
and expanding \(\left(1-\cos q \right)\) at the denominator for small values of \(q\):
\[\begin{equation*}
\langle u_k^2\rangle= \frac{1}{2\pi} \frac{k_BT}{K_e} \int_{q_{min}}^{q_{max}} \frac{dq}{q^2}
\end{equation*}\]
where the extremes of integration can be assumed \(q_{min}=\pi/N\) (\(N\) being the number of atoms in the chain) and \(q_{max}\rightarrow\infty\). Obviously, the integral above diverges for any \(T\ne 0\) in the thermodynamic limit \(N\rightarrow \infty\), meaning that the thermal average of square displacements diverges as well. The same calculations could be repeated for D=2 and D=3: only in the last case the integral does not diverge. This argument is used to state that crystals cannot exist both in 1D and in 2D, because they are unstable w.r.t. thermal fluctuations. Later on we will adapt this calculation to rule out ferromagnetism at finite temperature for classical spin models with continuous symmetry, i.e., the XY and the Heisenberg model.

From the knowledge of \(\langle |\tilde{u}_q|^2 \rangle\) also the averaged correlation between the displacements of atoms sitting at different sites in real space can be computed:

\[\begin{equation*}
\langle \left(u_{k+r} -u_k \right)^2\rangle= \frac{2}{N} \sum_{q} \left[1-\cos(qr)\right] \langle |\tilde{u}_q|^2 \rangle
=\frac{2}{N} \sum_{q} \left[1-\cos(qr)\right] \frac{k_BT}{2K_e \left(1-\cos q \right)} \,.
\end{equation*}\]
For spin models, this quantity is directly related to the behavior of the pair-spin correlation function.

## Linear excitations of the XY model

Now we are ready to apply the formalism introduced in the previous section to discuss the properties of the XY model at finite temperature. One important aspect is that the
Hamiltonian (10.1) is not quadratic in the degrees of freedom \(\varphi({\underline n})\). However, if we limit ourselves to considering small variations of the \(\varphi({\underline n})\) angle between neighboring sites, the Hamiltonian takes the form

\[\begin{equation}
\label{Classical-XY-lin}
\tag{10.3}
\mathcal{H}=-\frac{1}{4} J\sum_{|{\underline n}-{\underline n}'|=1} \left(\varphi({\underline n}) - \varphi({\underline n}')\right)^2 + \text{const.}
\end{equation}\]
Each pairs of nearest neighbors yields a contribution that can be expressed as the derivative of a continuous variable. For instance, for the coupling between two neighboring spins of a spin chain one has
\[\begin{equation*}
\left(\varphi_{k+1} - \varphi_k\right)^2 \rightarrow \left(\partial_x \varphi \right)^2 \,.
\end{equation*}\]
For lattices with higher dimensionality (with sites disposed along Cartesian axes) the coupling of one spin with its neighbors along different directions produces terms containing the derivatives \(\partial_y \varphi\) and \(\partial_z \varphi\). Therefore, the XY Hamiltonian is generally written in the continuum formalism as
\[\begin{equation}
\label{Classical-XY-cont}
\tag{10.4}
\mathcal{H}=\frac{1}{2} J \int \left(\nabla \varphi \right)^2 d^{\rm D}r \,.
\end{equation}\]
Note that in the continuum limit the factor one-half that was introduced in the Hamiltonian (10.1) to avoid double counting of interactions is not needed. Similarly to what done for the 1D chain of harmonic oscillators, the Hamiltonian (10.4) can be decoupled passing to the Fourier space
\[\begin{equation}
\label{XY_Ham_FT}
\mathcal{H}=\frac{1}{2} J \frac{1}{(2\pi)^d} \int q^2 |\tilde{ \varphi}(q)|^2 \, d^{\rm D}q
\end{equation}\]
in such a way that the *equipartition theorem* can be applied to evaluate the thermal averages of the Fourier amplitudes \(\langle |\tilde{ \varphi}(q)|^2 \rangle\). In the assignment, you will see that from the knowledge of this quantity the low-temperature behavior of pair-spin correlations

\[\begin{equation}
\label{Classical-XY-corr}
\tag{10.5}
\langle\vec{S}({\underline r})\cdot \vec{S}({\underline 0})\rangle
\end{equation}\]
can be determined, the result being remarkably different for the 1D and the 2D case.

## Vortices in the 2D XY model

Besides the linear excitations described above, the Hamiltonian (10.4) is also compatible with vortex excitations. Vortices are topological excitations to some extent equivalent to domain walls in the 1D Ising model. In the two-dimensional case (D=2), it is convenient to parameterize the position of a certain spin on the XY plane through polar coordinates \((r,\phi)\). Then, the \(\varphi\) field is a function of this pair of coordinates, i.e., \(\varphi (r,\phi)\).

In this description a vortex is represented, e.g., by a dependence of \(\varphi(r,\phi)=\phi+\pi/2\) which yields the vector field \(\vec{S}(r,\phi)=(\cos\varphi,\sin\varphi) = (-\sin\phi,\cos\phi)\).

### Classroom activity: free energy of one vortex excitation

**Q1 Provide an estimate of the vortex energy \(\mathcal{E}_{\rm vortex}\)** using this information, the Hamiltonian (10.4) and remembering that
\[\begin{equation*}
\nabla \varphi =\left(\frac{\partial \varphi}{\partial r},\,\frac{1}{r}\frac{\partial \varphi}{\partial \phi}\right)\,.
\end{equation*}\]
For your convenience, set the extremes of integration \(r_{\rm max} \simeq N_r a\) and \(r_{\rm min} \simeq a\) (\(a\) lattice unit).

**Q2 estimate of the entropy increase \(\Delta S\)** due to the creation of one vortex in an otherwise uniform ground state (with \(\varphi(r,\phi) =\varphi_0=\)cst). This can be obtained by counting (roughly!) the number of lattice sites which can host the center of the vortex (vortex core).

**Q3 Combining \(\mathcal{E}_{\rm vortex}\) and \(\Delta S\) evaluate the free-energy variation** associated with the creation of one vortex and draw your conclusions:

**Q4 Is there a characteristic temperature \(T_c\) above which the formation of one vortex is favored?**

The only difference is that now atoms are disposed on a lattice and, therefore, there is no \(N!\) term in the partition function to account for the correct Boltzmann counting.↩︎