Square lattice Ising model explained
In statistical mechanics, the two-dimensional square lattice Ising model is a simple lattice model of interacting magnetic spins. The model is notable for having nontrivial interactions, yet having an analytical solution. The model was solved by Lars Onsager for the special case that the external magnetic field H = 0.[1] An analytical solution for the general case for
has yet to be found.
Defining the partition function
with
N sites and periodic
boundary conditions in both the horizontal and vertical directions, which effectively reduces the
topology of the model to a
torus. Generally, the horizontal
coupling
and the vertical coupling
are not equal. With
and
absolute temperature
and the
Boltzmann constant
, the
partition functionZN(K\equiv\betaJ,L\equiv\betaJ*)=\sum\{\sigma\
} \exp \left(K \sum_ \sigma_i \sigma_j + L \sum_ \sigma_i \sigma_j \right).
Critical temperature
The critical temperature
can be obtained from the
Kramers–Wannier duality relation. Denoting the free energy per site as
, one has:
\betaF\left(K*,L*\right)=\betaF\left(K,L\right)+
log[\sinh\left(2K\right)\sinh\left(2L\right)]
where
\sinh\left(2K*\right)\sinh\left(2L\right)=1
\sinh\left(2L*\right)\sinh\left(2K\right)=1
Assuming that there is only one critical line in the plane, the duality relation implies that this is given by:
\sinh\left(2K\right)\sinh\left(2L\right)=1
For the isotropic case
, one finds the famous relation for the critical temperature
Dual lattice
Consider a configuration of spins
on the square lattice
. Let
r and
s denote the number of unlike neighbours in the vertical and horizontal directions respectively. Then the summand in
corresponding to
is given by
Construct a dual lattice
as depicted in the diagram. For every configuration
, a polygon is associated to the lattice by drawing a line on the edge of the dual lattice if the spins separated by the edge are unlike. Since by traversing a vertex of
the spins need to change an even number of times so that one arrives at the starting point with the same charge, every vertex of the dual lattice is connected to an even number of lines in the configuration, defining a polygon. This reduces the
partition function to
ZN(K,L)=2eN(K+L)
e-2Lr-2Ks
summing over all polygons in the dual lattice, where
r and
s are the number of horizontal and vertical lines in the polygon, with the factor of 2 arising from the inversion of spin configuration.
Low-temperature expansion
At low temperatures, K, L approach infinity, so that as
, so that
ZN(K,L)=2eN(K+L)
e-2Lr-2Ks
defines a low temperature expansion of
.
High-temperature expansion
Since
one has
eK=\coshK+\sinhK(\sigma\sigma')=\coshK(1+\tanhK(\sigma\sigma')).
Therefore
ZN(K,L)=(\coshK\coshL)N\sum\{
} \prod_ (1+v \sigma_i \sigma_j) \prod_(1+w\sigma_i \sigma_j)where
and
. Since there are
N horizontal and vertical edges, there are a total of
terms in the expansion. Every term corresponds to a configuration of lines of the lattice, by associating a line connecting
i and
j if the term
(or
is chosen in the product. Summing over the configurations, using
=\begin{cases}0&fornodd\\
2&forneven\end{cases}
shows that only configurations with an even number of lines at each vertex (polygons) will contribute to the partition function, giving
ZN(K,L)=2N(\coshK\coshL)N\sumPvrws
where the sum is over all polygons in the lattice. Since tanh
K, tanh
L
as
, this gives the high temperature expansion of
.
The two expansions can be related using the Kramers–Wannier duality.
Exact solution
The free energy per site in the limit
is given as follows. Define the parameter
as
k=
| 1 |
\sinh\left(2K\right)\sinh\left(2L\right) |
The Helmholtz free energy per site
can be expressed as
-\betaF=
+
log\left[\cosh\left(2K\right)\cosh\left(2L\right)+
\sqrt{1+k2-2k\cos(2\theta)}\right]d\theta
For the isotropic case
, from the above expression one finds for the internal energy per site:
U=-J\coth(2\betaJ)\left[1+
(2\tanh2(2\betaJ)-1)
| 1 |
\sqrt{1-4k(1+k)-2\sin2(\theta) |
} d\theta \right]and the spontaneous magnetization is, for
,
M=\left[1-\sinh-4(2\betaJ)\right]1/8
and
for
.
References
- Baxter . Rodney J. . The bulk, surface and corner free energies of the square lattice Ising model . Journal of Physics A: Mathematical and Theoretical . IOP Publishing . 50 . 1 . 2016 . 1751-8113 . 10.1088/1751-8113/50/1/014001 . 014001. 1606.02029 . 2467419 .
- BRUSH . STEPHEN G. . History of the Lenz-Ising Model . Reviews of Modern Physics . American Physical Society (APS) . 39 . 4 . 1967-10-01 . 0034-6861 . 10.1103/revmodphys.39.883 . 883–893. 1967RvMP...39..883B .
- Hucht . Alfred . The square lattice Ising model on the rectangle III: Hankel and Toeplitz determinants . Journal of Physics A: Mathematical and Theoretical . IOP Publishing . 54 . 37 . 2021 . 1751-8113 . 10.1088/1751-8121/ac0983 . 375201. 2103.10776 . 2021JPhA...54K5201H . 232290629 .
- Barry M. McCoy and Tai Tsun Wu (1973), The Two-Dimensional Ising Model. Harvard University Press, Cambridge Massachusetts,
- John Palmer (2007), Planar Ising Correlations. Birkhäuser, Boston, .
Notes and References
- Onsager . Lars . 1944-02-01 . Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition . Physical Review . 65 . 3-4 . 117–149 . 10.1103/PhysRev.65.117.