Classical XY model

The classical XY model (sometimes also called classical rotor (rotator) model or O(2) model) is a lattice model of statistical mechanics. It is the special case of the n-vector model for n = 2.

Definition

Given a D-dimensional lattice Λ, per each lattice site j ∈ Λ there is a two-dimensional, unit-length vector sj = (cos θj, sin θj)

The spin configuration, s = (sj)j ∈ Λ is an assignment of the angle π < θjπ for each j ∈ Λ.

Given a translation-invariant interaction Jij = J(ij) and a point dependent external field ${\displaystyle \mathbf {h} _{j}=(h_{j},0)}$, the configuration energy is

${\displaystyle H(\mathbf {s} )=-\sum _{i\neq j}J_{ij}\;\mathbf {s} _{i}\cdot \mathbf {s} _{j}-\sum _{j}\mathbf {h} _{j}\cdot \mathbf {s} _{j}=-\sum _{i\neq j}J_{ij}\;\cos(\theta _{i}-\theta _{j})-\sum _{j}h_{j}\cos \theta _{j}}$

The case in which Jij = 0 except for ij nearest neighbor is called nearest neighbor case.

The configuration probability is given by the Boltzmann distribution with inverse temperature β ≥ 0:

${\displaystyle P(\mathbf {s} )={\frac {e^{-\beta H(\mathbf {s} )}}{Z}}\qquad Z=\int _{[-\pi ,\pi ]^{\Lambda }}\prod _{j\in \Lambda }d\theta _{j}\;e^{-\beta H(\mathbf {s} )}.}$

where Z is the normalization, or partition function.[1] The notation ${\displaystyle \langle A(\mathbf {s} )\rangle }$ indicates the expectation of the random variable A(s) in the infinite volume limit, after periodic boundary conditions have been imposed.

General properties

${\displaystyle \langle \mathbf {s} _{i}\cdot \mathbf {s} _{j}\rangle _{J,2\beta }\leq \langle \sigma _{i}\sigma _{j}\rangle _{J,\beta }}$
Hence the critical β of the XY model cannot be smaller than the double of the critical temperature of the Ising model
${\displaystyle \beta _{c}^{XY}\geq 2\beta _{c}^{\rm {Is}}}$

One dimension

• In case of long range interaction, ${\displaystyle J_{x,y}\sim |x-y|^{-\alpha },}$ the thermodynamic limit is well defined if α > 1; the magnetization remains zero if α ≥ 2; but the magnetization is positive, at low enough temperature, if 1 < α < 2 (infrared bounds).
• As in any 'nearest-neighbor' n-vector model with free boundary conditions, if the external field is zero, there exists a simple exact solution. In the free boundary conditions case, the Hamiltonian is
${\displaystyle H(\mathbf {s} )=-J[\cos(\theta _{1}-\theta _{2})+\cdots +\cos(\theta _{L-1}-\theta _{L})]}$
therefore the partition function factorizes under the change of coordinates
${\displaystyle \theta _{j}=\theta _{j}'+\theta _{j-1}\qquad j\geq 2}$
That gives
{\displaystyle {\begin{aligned}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}\end{aligned}}}
Finally
${\displaystyle f(\beta ,0)=-{\frac {1}{\beta }}\ln \int _{-\pi }^{\pi }d\theta '_{j}\;e^{\beta J\cos \theta '_{j}}=-{\frac {1}{\beta }}\ln[2\pi I_{0}(\beta J)]}$
where ${\displaystyle I_{0}}$ is the modified Bessel function of the first kind.
The same computation for periodic boundary condition (and still h = 0) requires the transfer matrix formalism.[4]

Two dimensions

• In the case of long range interaction, ${\displaystyle J_{x,y}\sim |x-y|^{-\alpha }}$, the thermodynamic limit is well defined if α > 2; the magnetization remains zero if α ≥ 4; but the magnetization is positive at low enough temperature if 2 < α < 4 (Kunz and Pfister; alternatively, infrared bounds).
• At high temperature and nearest neighbour interaction, the spontaneous magnetization vanishes:
${\displaystyle M(\beta ):=|\langle \mathbf {s} _{i}\rangle |=0}$
Besides, cluster expansion shows that the spin correlations cluster exponentially fast: for instance
${\displaystyle |\langle \mathbf {s} _{i}\cdot \mathbf {s} _{j}\rangle |\leq C(\beta )e^{-c(\beta )|i-j|}}$
• At low temperature and nearest neighbour interaction, i.e. β ≫ 1, the spontaneous magnetization remains zero (see the Mermin–Wagner theorem),
${\displaystyle M(\beta ):=|\langle \mathbf {s} _{i}\rangle |=0}$
but the decay of the correlations is only power law: Fröhlich and Spencer[5] found the lower bound
${\displaystyle |\langle \mathbf {s} _{i}\cdot \mathbf {s} _{j}\rangle |\geq {\frac {C(\beta )}{1+|i-j|^{\eta (\beta )}}}}$
while McBryan and Spencer found the upper bound, for any ${\displaystyle \epsilon >0}$
${\displaystyle |\langle \mathbf {s} _{i}\cdot \mathbf {s} _{j}\rangle |\leq {\frac {C(\beta ,\epsilon )}{1+|i-j|^{\eta (\beta ,\epsilon )}}}}$

The fact that at high temperature correlations decay exponentially fast, while at low temperatures decay with power law, even though in both regimes M(β) = 0, is called Kosterlitz–Thouless transition.

The continuous version of the XY model is often used to model systems that possess order parameters with the same kinds of symmetry, e.g. superfluid helium, hexatic liquid crystals. This is what makes them peculiar from other phase transitions which are always accompanied with a symmetry breaking. Topological defects in the XY model lead to a vortex-unbinding transition from the low-temperature phase to the high-temperature disordered phase.

Three and higher dimensions

Independently of the range of the interaction, at low enough temperature the magnetization is positive.

• At high temperature, the spontaneous magnetization vanishes:
${\displaystyle M(\beta ):=|\langle \mathbf {s} _{i}\rangle |=0}$
Besides, cluster expansion shows that the spin correlations cluster exponentially fast: for instance
${\displaystyle |\langle \mathbf {s} _{i}\cdot \mathbf {s} _{j}\rangle |\leq C(\beta )e^{-c(\beta )|i-j|}}$
• At low temperature, infrared bound shows that the spontaneous magnetization is strictly positive:
${\displaystyle M(\beta ):=|\langle \mathbf {s} _{i}\rangle |>0}$
Besides, there exists a 1-parameter family of extremal states, ${\displaystyle \langle \;\cdot \;\rangle ^{\theta }}$, such that
${\displaystyle \langle \mathbf {s} _{i}\rangle ^{\theta }=M(\beta )(\cos \theta ,\sin \theta )}$
but, conjecturally, in each of these extremal states the truncated correlations decay algebraically.

In general, the XY model can be seen as a specialization of Stanley's n-vector model [6]

Notes

1. ^ Chaikin, P.M.; Lubensky, T.C. (2000). Principles of Condensed Matter Physics. Cambridge University Press. ISBN 978-0521794503.
2. ^ Ginibre, J. (1970). "General formulation of Griffiths' inequalities". Comm. Math. Phys. 16 (4): 310–328. Bibcode:1970CMaPh..16..310G. doi:10.1007/BF01646537.
3. ^ Aizenman, M.; Simon, B. (1980). "A comparison of plane rotor and Ising models". Phys. Lett. A. 76. Bibcode:1980PhLA...76..281A. doi:10.1016/0375-9601(80)90493-4.
4. ^ Mattis, D.C. (1984). "Transfer matrix in plane-rotator model". Phys. Lett. 104 A. Bibcode:1984PhLA..104..357M. doi:10.1016/0375-9601(84)90816-8.
5. ^ Fröhlich, J.; Spencer, T. (1981). "The Kosterlitz–Thouless transition in two-dimensional abelian spin systems and the Coulomb gas". Comm. Math. Phys. 81 (4): 527–602. Bibcode:1981CMaPh..81..527F. doi:10.1007/bf01208273.
6. ^ Stanley, H.E. "Dependence of Critical Properties on Dimensionality of Spins". Phys. Rev. Lett. 20: 589. Bibcode:1968PhRvL..20..589S. doi:10.1103/PhysRevLett.20.589.