- Source: Square lattice Ising model
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. An analytical solution for the general case for
H
≠
0
{\displaystyle H\neq 0}
has yet to be found.
Defining the partition function
Consider a 2D Ising model on a square lattice
Λ
{\displaystyle \Lambda }
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
J
{\displaystyle J}
and the vertical coupling
J
∗
{\displaystyle J^{*}}
are not equal. With
β
=
1
k
T
{\displaystyle \textstyle \beta ={\frac {1}{kT}}}
and absolute temperature
T
{\displaystyle T}
and the Boltzmann constant
k
{\displaystyle k}
, the partition function
Z
N
(
K
≡
β
J
,
L
≡
β
J
∗
)
=
∑
{
σ
}
exp
(
K
∑
⟨
i
j
⟩
H
σ
i
σ
j
+
L
∑
⟨
i
j
⟩
V
σ
i
σ
j
)
.
{\displaystyle Z_{N}(K\equiv \beta J,L\equiv \beta J^{*})=\sum _{\{\sigma \}}\exp \left(K\sum _{\langle ij\rangle _{H}}\sigma _{i}\sigma _{j}+L\sum _{\langle ij\rangle _{V}}\sigma _{i}\sigma _{j}\right).}
Critical temperature
The critical temperature
T
c
{\displaystyle T_{\text{c}}}
can be obtained from the Kramers–Wannier duality relation. Denoting the free energy per site as
F
(
K
,
L
)
{\displaystyle F(K,L)}
, one has:
β
F
(
K
∗
,
L
∗
)
=
β
F
(
K
,
L
)
+
1
2
log
[
sinh
(
2
K
)
sinh
(
2
L
)
]
{\displaystyle \beta F\left(K^{*},L^{*}\right)=\beta F\left(K,L\right)+{\frac {1}{2}}\log {\big [}\sinh \left(2K\right)\sinh \left(2L\right){\big ]}}
where
sinh
(
2
K
∗
)
sinh
(
2
L
)
=
1
{\displaystyle \sinh \left(2K^{*}\right)\sinh \left(2L\right)=1}
sinh
(
2
L
∗
)
sinh
(
2
K
)
=
1
{\displaystyle \sinh \left(2L^{*}\right)\sinh \left(2K\right)=1}
Assuming that there is only one critical line in the (K, L) plane, the duality relation implies that this is given by:
sinh
(
2
K
)
sinh
(
2
L
)
=
1
{\displaystyle \sinh \left(2K\right)\sinh \left(2L\right)=1}
For the isotropic case
J
=
J
∗
{\displaystyle J=J^{*}}
, one finds the famous relation for the critical temperature
T
c
{\displaystyle T_{c}}
k
T
c
J
=
2
ln
(
1
+
2
)
≈
2.26918531421
{\displaystyle {\frac {kT_{\text{c}}}{J}}={\frac {2}{\ln(1+{\sqrt {2}})}}\approx 2.26918531421}
Dual lattice
Consider a configuration of spins
{
σ
}
{\displaystyle \{\sigma \}}
on the square lattice
Λ
{\displaystyle \Lambda }
. Let r and s denote the number of unlike neighbours in the vertical and horizontal directions respectively. Then the summand in
Z
N
{\displaystyle Z_{N}}
corresponding to
{
σ
}
{\displaystyle \{\sigma \}}
is given by
e
K
(
N
−
2
s
)
+
L
(
N
−
2
r
)
{\displaystyle e^{K(N-2s)+L(N-2r)}}
Construct a dual lattice
Λ
D
{\displaystyle \Lambda _{D}}
as depicted in the diagram. For every configuration
{
σ
}
{\displaystyle \{\sigma \}}
, 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
Λ
{\displaystyle \Lambda }
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
Z
N
(
K
,
L
)
=
2
e
N
(
K
+
L
)
∑
P
⊂
Λ
D
e
−
2
L
r
−
2
K
s
{\displaystyle Z_{N}(K,L)=2e^{N(K+L)}\sum _{P\subset \Lambda _{D}}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
T
→
0
,
e
−
K
,
e
−
L
→
0
{\displaystyle T\rightarrow 0,\ \ e^{-K},e^{-L}\rightarrow 0}
, so that
Z
N
(
K
,
L
)
=
2
e
N
(
K
+
L
)
∑
P
⊂
Λ
D
e
−
2
L
r
−
2
K
s
{\displaystyle Z_{N}(K,L)=2e^{N(K+L)}\sum _{P\subset \Lambda _{D}}e^{-2Lr-2Ks}}
defines a low temperature expansion of
Z
N
(
K
,
L
)
{\displaystyle Z_{N}(K,L)}
.
High-temperature expansion
Since
σ
σ
′
=
±
1
{\displaystyle \sigma \sigma '=\pm 1}
one has
e
K
σ
σ
′
=
cosh
K
+
sinh
K
(
σ
σ
′
)
=
cosh
K
(
1
+
tanh
K
(
σ
σ
′
)
)
.
{\displaystyle e^{K\sigma \sigma '}=\cosh K+\sinh K(\sigma \sigma ')=\cosh K(1+\tanh K(\sigma \sigma ')).}
Therefore
Z
N
(
K
,
L
)
=
(
cosh
K
cosh
L
)
N
∑
{
σ
}
∏
⟨
i
j
⟩
H
(
1
+
v
σ
i
σ
j
)
∏
⟨
i
j
⟩
V
(
1
+
w
σ
i
σ
j
)
{\displaystyle Z_{N}(K,L)=(\cosh K\cosh L)^{N}\sum _{\{\sigma \}}\prod _{\langle ij\rangle _{H}}(1+v\sigma _{i}\sigma _{j})\prod _{\langle ij\rangle _{V}}(1+w\sigma _{i}\sigma _{j})}
where
v
=
tanh
K
{\displaystyle v=\tanh K}
and
w
=
tanh
L
{\displaystyle w=\tanh L}
. Since there are N horizontal and vertical edges, there are a total of
2
2
N
{\displaystyle 2^{2N}}
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
v
σ
i
σ
j
{\displaystyle v\sigma _{i}\sigma _{j}}
(or
w
σ
i
σ
j
)
{\displaystyle w\sigma _{i}\sigma _{j})}
is chosen in the product. Summing over the configurations, using
∑
σ
i
=
±
1
σ
i
n
=
{
0
for
n
odd
2
for
n
even
{\displaystyle \sum _{\sigma _{i}=\pm 1}\sigma _{i}^{n}={\begin{cases}0&{\mbox{for }}n{\mbox{ odd}}\\2&{\mbox{for }}n{\mbox{ even}}\end{cases}}}
shows that only configurations with an even number of lines at each vertex (polygons) will contribute to the partition function, giving
Z
N
(
K
,
L
)
=
2
N
(
cosh
K
cosh
L
)
N
∑
P
⊂
Λ
v
r
w
s
{\displaystyle Z_{N}(K,L)=2^{N}(\cosh K\cosh L)^{N}\sum _{P\subset \Lambda }v^{r}w^{s}}
where the sum is over all polygons in the lattice. Since tanh K, tanh L
→
0
{\displaystyle \rightarrow 0}
as
T
→
∞
{\displaystyle T\rightarrow \infty }
, this gives the high temperature expansion of
Z
N
(
K
,
L
)
{\displaystyle Z_{N}(K,L)}
.
The two expansions can be related using the Kramers–Wannier duality.
Exact solution
The free energy per site in the limit
N
→
∞
{\displaystyle N\to \infty }
is given as follows. Define the parameter
k
{\displaystyle k}
as
k
=
1
sinh
(
2
K
)
sinh
(
2
L
)
{\displaystyle k={\frac {1}{\sinh \left(2K\right)\sinh \left(2L\right)}}}
The Helmholtz free energy per site
F
{\displaystyle F}
can be expressed as
−
β
F
=
log
(
2
)
2
+
1
2
π
∫
0
π
log
[
cosh
(
2
K
)
cosh
(
2
L
)
+
1
k
1
+
k
2
−
2
k
cos
(
2
θ
)
]
d
θ
{\displaystyle -\beta F={\frac {\log(2)}{2}}+{\frac {1}{2\pi }}\int _{0}^{\pi }\log \left[\cosh \left(2K\right)\cosh \left(2L\right)+{\frac {1}{k}}{\sqrt {1+k^{2}-2k\cos(2\theta )}}\right]d\theta }
For the isotropic case
J
=
J
∗
{\displaystyle J=J^{*}}
, from the above expression one finds for the internal energy per site:
U
=
−
J
coth
(
2
β
J
)
[
1
+
2
π
(
2
tanh
2
(
2
β
J
)
−
1
)
∫
0
π
/
2
1
1
−
4
k
(
1
+
k
)
−
2
sin
2
(
θ
)
d
θ
]
{\displaystyle U=-J\coth(2\beta J)\left[1+{\frac {2}{\pi }}(2\tanh ^{2}(2\beta J)-1)\int _{0}^{\pi /2}{\frac {1}{\sqrt {1-4k(1+k)^{-2}\sin ^{2}(\theta )}}}d\theta \right]}
and the spontaneous magnetization is, for
T
<
T
c
{\displaystyle T
,
M
=
[
1
−
sinh
−
4
(
2
β
J
)
]
1
/
8
{\displaystyle M=\left[1-\sinh ^{-4}(2\beta J)\right]^{1/8}}
and
M
=
0
{\displaystyle M=0}
for
T
≥
T
c
{\displaystyle T\geq T_{\text{c}}}
.
Notes
References
Baxter, Rodney J. (1982), Exactly solved models in statistical mechanics (PDF), London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-083180-7, MR 0690578
Baxter, Rodney J. (2016). "The bulk, surface and corner free energies of the square lattice Ising model". Journal of Physics A: Mathematical and Theoretical. 50 (1). IOP Publishing: 014001. arXiv:1606.02029. doi:10.1088/1751-8113/50/1/014001. ISSN 1751-8113. S2CID 2467419.
Kurt Binder (2001) [1994], "Ising model", Encyclopedia of Mathematics, EMS Press
BRUSH, STEPHEN G. (1967-10-01). "History of the Lenz-Ising Model". Reviews of Modern Physics. 39 (4). American Physical Society (APS): 883–893. Bibcode:1967RvMP...39..883B. doi:10.1103/revmodphys.39.883. ISSN 0034-6861.
Huang, Kerson (1987), Statistical mechanics (2nd edition), Wiley, ISBN 978-0471815181
Hucht, Alfred (2021). "The square lattice Ising model on the rectangle III: Hankel and Toeplitz determinants". Journal of Physics A: Mathematical and Theoretical. 54 (37). IOP Publishing: 375201. arXiv:2103.10776. Bibcode:2021JPhA...54K5201H. doi:10.1088/1751-8121/ac0983. ISSN 1751-8113. S2CID 232290629.
Ising, Ernst (1925), "Beitrag zur Theorie des Ferromagnetismus", Z. Phys., 31 (1): 253–258, Bibcode:1925ZPhy...31..253I, doi:10.1007/BF02980577, S2CID 122157319
Itzykson, Claude; Drouffe, Jean-Michel (1989), Théorie statistique des champs, Volume 1, Savoirs actuels (CNRS), EDP Sciences Editions, ISBN 978-2868833600
Itzykson, Claude; Drouffe, Jean-Michel (1989), Statistical field theory, Volume 1: From Brownian motion to renormalization and lattice gauge theory, Cambridge University Press, ISBN 978-0521408059
Barry M. McCoy and Tai Tsun Wu (1973), The Two-Dimensional Ising Model. Harvard University Press, Cambridge Massachusetts, ISBN 0-674-91440-6
Montroll, Elliott W.; Potts, Renfrey B.; Ward, John C. (1963), "Correlations and spontaneous magnetization of the two-dimensional Ising model", Journal of Mathematical Physics, 4 (2): 308–322, Bibcode:1963JMP.....4..308M, doi:10.1063/1.1703955, ISSN 0022-2488, MR 0148406, archived from the original on 2013-01-12
Onsager, Lars (1944), "Crystal statistics. I. A two-dimensional model with an order-disorder transition", Phys. Rev., Series II, 65 (3–4): 117–149, Bibcode:1944PhRv...65..117O, doi:10.1103/PhysRev.65.117, MR 0010315
Onsager, Lars (1949), "Discussion", Supplemento al Nuovo Cimento, 6: 261
John Palmer (2007), Planar Ising Correlations. Birkhäuser, Boston, ISBN 978-0-8176-4248-8.
Yang, C. N. (1952), "The spontaneous magnetization of a two-dimensional Ising model", Physical Review, Series II, 85 (5): 808–816, Bibcode:1952PhRv...85..808Y, doi:10.1103/PhysRev.85.808, MR 0051740
Kata Kunci Pencarian:
- Model Ising
- Ising model
- Square lattice Ising model
- Kramers–Wannier duality
- Potts model
- Bethe lattice
- Canonical ensemble
- Eight-vertex model
- ZN model
- Quantum Heisenberg model
- Statistical mechanics
