Editing Classical XY model (section)

===One dimension===
As in any 'nearest-neighbor' [[n-vector model|''n''-vector model]] with free (non-periodic) boundary conditions, if the external field is zero, there exists a simple exact solution. In the free boundary conditions case, the Hamiltonian is
<math display="block">H(\mathbf{s}) = - J [\cos(\theta_1-\theta_2)+\cdots+\cos(\theta_{L-1}-\theta_L)]</math>
therefore the partition function factorizes under the change of coordinates
<math display="block">\theta_j=\theta_j'+\theta_{j-1}\qquad j\ge 2</math>
This gives 
<math display="block">\begin{align}
Z & = \int_{-\pi}^\pi d\theta_1 \cdots d\theta_L \; e^{\beta J\cos(\theta_1-\theta_2)} \cdots e^{\beta J\cos(\theta_{L-1}-\theta_L)} \\
& = 2\pi \prod_{j=2}^L\int_{-\pi}^\pi d\theta'_j \;e^{\beta J\cos\theta'_j} = (2\pi) \left[\int_{-\pi}^\pi d\theta'_j \;e^{\beta J\cos\theta'_j}\right]^{L-1} = (2\pi)^L (I_0 (\beta J))^{L-1}
\end{align}</math>
where <math>I_0</math> is the [[modified Bessel function]] of the first kind. The partition function can be used to find several important thermodynamic quantities. For example, in the thermodynamic limit (<math>L\to \infty</math>), the [[Helmholtz free energy|free energy]] per spin is
<math display="block">f(\beta,h=0)=-\lim_{L\to \infty} \frac{1}{\beta L} \ln Z = - \frac{1}{\beta} \ln [2\pi I_0(\beta J)]</math>
Using the properties of the modified Bessel functions, the specific heat (per spin) can be expressed as<ref>{{cite journal |last=Badalian|first=D. |year=1996|title=On the thermodynamics of classical spins with isotrop Heisenberg interaction in one-dimensional quasi-periodic structures |journal=Physica B|volume=226|issue=4 |pages=385–390 |doi=10.1016/0921-4526(96)00283-9 |bibcode=1996PhyB..226..385B}}</ref>
<math display="block"> \frac{c}{k_{\rm B}} = \lim_{L\to \infty} \frac{1}{L(k_{\rm B} T)^2} \frac{\partial^2}{\partial \beta^2} (\ln Z) = K^2 \left(1 - \frac{\mu}{K} - \mu^2\right) </math>
where <math>K = J/k_{\rm B} T</math>, and <math>\mu</math> is the short-range correlation function, 
[[File:1D XY Specific Heat.svg|275px|thumb|Exact specific heat per spin in the one-dimensional XY model|alt=]]<math display="block">\mu(K) = \langle \cos(\theta - \theta') \rangle = \frac{I_1(K)}{I_0(K)}</math>

Even in the thermodynamic limit, there is no divergence in the specific heat. Indeed, like the one-dimensional Ising model, the one-dimensional XY model has no phase transitions at finite temperature.

The same computation for periodic boundary condition (and still {{math|''h'' {{=}} 0}}) requires the [[Transfer-matrix method (statistical mechanics)|transfer matrix formalism]], though the result is the same.<ref>{{cite journal |last=Mattis|first=D.C.|year=1984 | title=Transfer matrix in plane-rotator model | journal=Physics Letters A|volume=104 A| issue=6–7|pages=357–360|bibcode=1984PhLA..104..357M|doi=10.1016/0375-9601(84)90816-8}}</ref>{{hidden
|(Click "show" at right to see the details of the transfer matrix formalism.)
|2=

The partition function can be evaluated as 
<math display="block">Z = \text{tr}\left\{ \prod_{i=1}^N \oint d\theta_i e^{\beta J\cos(\theta_i - \theta_{i+1})} \right\} </math>
which can be treated as the trace of a matrix, namely a product of matrices (scalars, in this case). The trace of a matrix is simply the sum of its eigenvalues, and in the thermodynamic limit <math>L\to \infty</math> only the largest eigenvalue will survive, so the partition function can be written as a repeated product of this maximal eigenvalue. This requires solving the eigenvalue problem
<math display="block">\oint d\theta' \exp\{\beta J\cos(\theta' - \theta)\} \psi(\theta') = z_i \psi(\theta) </math>
Note the expansion 
<math display="block">\exp\{\beta J\cos(\theta-\theta')\} = \sum_{n=-\infty}^\infty I_n(\beta J) e^{in(\theta-\theta')} = \sum_{n=-\infty}^\infty \omega_n \psi^*_n(\theta') \psi_n(\theta) </math>
which represents a diagonal matrix representation in the basis of its plane-wave eigenfunctions <math>\psi = \exp(in\theta)</math>. The eigenvalues of the matrix is simply are modified Bessel functions evaluated at <math>\beta J</math>, namely <math>\omega_n = 2\pi I_n (\beta J)</math>. For any particular value of <math>\beta J</math>, these modified Bessel functions satisfy <math>I_0 > I_1 > I_2 > \cdots </math> and <math>I_{-n}(\beta J) = I_n(\beta J)</math>. Therefore in the thermodynamic limit the eigenvalue <math>I_0</math> will dominate the trace, and so <math>Z = [2\pi I_0(\beta J)]^L</math>.}}

This transfer matrix approach is also required when using free boundary conditions, but with an applied field <math>h \neq 0</math>. If the applied field <math>h</math> is small enough that it can be treated as a perturbation to the system in zero-field, then the [[magnetic susceptibility]] <math>\chi\equiv\partial M/\partial h</math> can be estimated. This is done by using the eigenstates computed by the transfer matrix approach and computing the energy shift with second-order [[perturbation theory]], then comparing with the free-energy expansion <math>F=F_0 - \frac{1}{2} \chi h^2</math>. One finds <ref>{{cite book|first1=D. C.| last1=Mattis|title=The Theory of Magnetism II|publisher=Springer Series in Solid-State Physics|year=1985 |isbn=978-3-642-82405-0}}</ref>
<math display="block">\chi(h\to 0) = \frac{C}{T} \frac{1+\mu}{1-\mu}</math>
where <math>C</math> is the [[Curie constant]] (a value typically associated with the susceptibility in magnetic materials). This expression is also true for the one-dimensional Ising model, with the replacement <math>\mu = \tanh K</math>.