1. Model systems with interaction
Ising spin chain (with open boundary conditions)
Classical variable
𝑆 𝑖 =±1 J
𝑆 𝑛
𝑆 𝑛+1
𝑆 𝑛−1
𝐻=− 1 2 𝑖,𝑗 𝐽 𝑖𝑗 𝑆 𝑖 𝑆 𝑗
𝐽 𝑖𝑗 = 𝐽 𝑓𝑜𝑟 𝑖,𝑗 𝑛𝑒𝑎𝑟𝑒𝑠𝑡 𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑟𝑠 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
𝐻=−𝐽 𝑆 1 𝑆 2 + 𝑆 2 𝑆 3 + …+ 𝑆 𝑁−1 𝑆 𝑁
𝑍= 𝛼 𝑒 −𝛽 𝐸 𝛼
= 𝑆 1 , 𝑆 2 , …, 𝑆 𝑁 𝑒 𝛽𝐽 𝑆 1 𝑆 2 + 𝑆 2 𝑆 3 + …+ 𝑆 𝑁−1 𝑆 𝑁

2. 𝑍= 𝑆 1 , 𝑆 2 , …, 𝑆 𝑁 𝑒 𝛽𝐽 𝑆 1 𝑆 2 + 𝑆 2 𝑆 3 + …+ 𝑆 𝑁−1 𝑆 𝑁
= 𝑆 2 , …, 𝑆 𝑁 𝑒 𝛽𝐽 𝑆 2 + 𝑆 2 𝑆 3 + …+ 𝑆 𝑁−1 𝑆 𝑁 + 𝑒 𝛽𝐽 − 𝑆 2 + 𝑆 2 𝑆 3 + …+ 𝑆 𝑁−1 𝑆 𝑁
= 𝑆 3 , …, 𝑆 𝑁 𝑒 𝛽𝐽 1+ 𝑆 3 + …+ 𝑆 𝑁−1 𝑆 𝑁 + 𝑒 𝛽𝐽 −1+ 𝑆 3 + …+ 𝑆 𝑁−1 𝑆 𝑁 + 𝑒 𝛽𝐽 −1− 𝑆 3 + …+ 𝑆 𝑁−1 𝑆 𝑁 + 𝑒 𝛽𝐽 1− 𝑆 3 + …+ 𝑆 𝑁−1 𝑆 𝑁
= 𝑆 3 , …, 𝑆 𝑁 (𝑒 𝛽𝐽 + 𝑒 −𝛽𝐽 )( 𝑒 𝛽𝐽 𝑆 3 + …+ 𝑆 𝑁−1 𝑆 𝑁 + 𝑒 𝛽𝐽 −𝑆 3 + …+ 𝑆 𝑁−1 𝑆 𝑁 )
= 2cosh 𝛽𝐽 𝑆 3 , …, 𝑆 𝑁 ( 𝑒 𝛽𝐽 𝑆 3 + 𝑆 3 𝑆 4 …+ 𝑆 𝑁−1 𝑆 𝑁 + 𝑒 𝛽𝐽 −𝑆 3 +𝑆 3 𝑆 4 …+ 𝑆 𝑁−1 𝑆 𝑁 )
= 2cosh 𝛽𝐽 𝑆 4 , …, 𝑆 𝑁 ( 𝑒 𝛽𝐽 1+ 𝑆 4 …+ 𝑆 𝑁−1 𝑆 𝑁 + 𝑒 𝛽𝐽 −1+ 𝑆 4 …+ 𝑆 𝑁−1 𝑆 𝑁 𝑒 𝛽𝐽 −1− 𝑆 4 …+ 𝑆 𝑁−1 𝑆 𝑁 + 𝑒 𝛽𝐽 1− 𝑆 4 …+ 𝑆 𝑁−1 𝑆 𝑁 )
= 2cosh 𝛽𝐽 𝑆 4 , …, 𝑆 𝑁 (𝑒 𝛽𝐽 + 𝑒 −𝛽𝐽 )( 𝑒 𝛽𝐽 𝑆 4 + …+ 𝑆 𝑁−1 𝑆 𝑁 + 𝑒 𝛽𝐽 −𝑆 4 + …+ 𝑆 𝑁−1 𝑆 𝑁 )
= 2cosh 𝛽𝐽 𝑆 4 , …, 𝑆 𝑁 ( 𝑒 𝛽𝐽 𝑆 4 + …+ 𝑆 𝑁−1 𝑆 𝑁 + 𝑒 𝛽𝐽 −𝑆 4 + …+ 𝑆 𝑁−1 𝑆 𝑁 )

3. 𝐺=− 𝑘 𝐵 𝑇 ln 2 − 𝑘 𝐵 𝑇 𝑁−1 ln 2cosh 𝛽𝐽
𝑍= 2cosh 𝛽𝐽 𝑁−2 𝑆 𝑁 𝑒 𝛽𝐽 𝑆 𝑁 + 𝑒 −𝛽𝐽 𝑆 𝑁 = 2cosh 𝛽𝐽 𝑁−2 𝑆 𝑁 2 cosh 𝛽𝐽 𝑆 𝑁 = 2cosh 𝛽𝐽 𝑁−2 𝑆 𝑁 2 cosh 𝛽𝐽
𝑍=2 2cosh 𝛽𝐽 𝑁−1
𝐺=− 𝑘 𝐵 𝑇 ln 2 − 𝑘 𝐵 𝑇 𝑁−1 ln 2cosh 𝛽𝐽
𝑔= 𝐺 𝑁 𝑁→∞ =− 𝑘 𝐵 𝑇 ln 2cosh 𝛽𝐽
Entropy per spin:
s= 𝑆 𝑁 =− 𝜕𝑔 𝜕𝑇 = 𝑘 𝐵 ln 2cosh 𝛽𝐽 − 𝑘 𝐵 𝑇 2 sinh 𝛽𝐽 2cosh 𝛽𝐽 𝐽 𝑘 𝐵 𝑇 2
ln 2
𝑠 𝑘 𝐵 𝑇
𝐽 𝑘 𝐵 =1 𝐾
For
s= 𝑘 𝐵 ln 2cosh 𝛽𝐽 − 𝐽 𝑇 tanh 𝛽𝐽
T->∞ , 𝛽→0
s= 𝑘 𝐵 ln 2 1
s ≈𝑘 𝐵 ln e 𝛽𝐽 − 𝐽 𝑇 =0
T->0 , 𝛽→∞ 2

4. Internal energy per spin:
We recall from thermodynamics 𝑈=𝐺+𝑇𝑆+ 𝜇 0 𝑉𝑀𝐻
which reduces for H=0 to
𝑈=𝐺+𝑇𝑆
𝑢= 𝑈 𝑁 =𝑔+𝑇𝑠
=− 𝑘 𝐵 𝑇 ln 2cosh 𝛽𝐽 + 𝑘 𝐵 𝑇 ln 2cosh 𝛽𝐽 − J tanh 𝛽𝐽 𝑇 𝑢
𝐽 𝑘 𝐵 =1 𝐾
For
𝑢=− J tanh 𝛽𝐽
T->∞ , 𝛽→0
𝑢=0 1
T->0 , 𝛽→∞
𝑢=-J 2 𝑐 𝑇
Heat capacity per spin:
𝑐= 𝜕𝑢 𝜕𝑇 =− J 𝜕 𝜕𝑇 tanh 𝛽𝐽
= k B 𝐽 𝑘 𝐵 𝑇 cosh 𝐽 𝑘 𝐵 𝑇 2

5. Spin-spin correlation function
𝑺 𝒂 𝑺 𝒂+𝒃 measures the statistical correlation between the spins (random
variables) at position a and a+b a
a+b
Intuitive properties
𝑺 𝒂 𝑺 𝒂 =1
No interaction between spins
(spins are uncorrelated)
𝑺 𝒂 𝑺 𝒂+𝒃 = 𝑺 𝒂 𝑺 𝒂+𝒃 =0
for 𝑏≠0
Interaction creates finite T-dependent correlation on characteristic
length scale 𝜉(𝑇)
Let’s calculate 𝑺 𝒂 𝑺 𝒂+𝒃 for chain of nearest neighbor interacting Ising spins
𝑺 𝒂 𝑺 𝒂+𝒃 = 𝑆 1 , 𝑆 2 , …, 𝑆 𝑁 𝑆 𝑎 𝑆 𝑎+𝑏 𝑒 𝛽𝐽 𝑆 1 𝑆 2 + 𝑆 2 𝑆 3 + …+ 𝑆 𝑁−1 𝑆 𝑁 𝑆 1 , 𝑆 2 , …, 𝑆 𝑁 𝑒 𝛽𝐽 𝑆 1 𝑆 2 + 𝑆 2 𝑆 3 + …+ 𝑆 𝑁−1 𝑆 𝑁

6. To calculate 𝑺 𝒂 𝑺 𝒂+𝒃 we use a few tricks
Let’s switch from 𝑯=−𝑱 𝒊=𝟏 𝑵−𝟏 𝑺 𝒊 𝑺 𝒊+𝟏 to the more general Hamiltonian
𝑯=− 𝒊=𝟏 𝑵−𝟏 𝑱 𝒊 𝑺 𝒊 𝑺 𝒊+𝟏
The partition function for the generalized Hamiltonian reads
𝑍=2 𝑖=1 𝑁−1 2cosh 𝛽 𝐽 𝑖
Let’s first calculate the correlation function 𝑺 𝒂 𝑺 𝒂+𝟏
S a S a+1 = 1 𝑍 𝑆 1 , 𝑆 2 , …, 𝑆 𝑁 𝑆 𝑎 𝑆 𝑎+1 𝑒 𝛽 𝑖=1 𝑁−1 𝐽 𝑖 𝑆 𝑖 𝑆 𝑖+1
= 1 𝑍 1 𝛽 𝜕 𝜕 𝐽 𝑎 𝑆 1 , 𝑆 2 , …, 𝑆 𝑁 𝑒 𝛽 𝑖=1 𝑁−1 𝐽 𝑖 𝑆 𝑖 𝑆 𝑖+1
= 1 𝑍 1 𝛽 𝜕𝑍 𝜕 𝐽 𝑎
= 𝛽 sinh 𝛽 𝐽 𝑎 cosh 𝛽 𝐽 𝑎 𝛽 =tanh𝛽 𝐽 𝑎
For the special case 𝐽 𝑎 =𝐽 we obtain
S a S a+1 =tanh𝛽𝐽

7. Let’s now calculate the correlation function 𝑺 𝒂 𝑺 𝒂+𝟐
Per definition
S a S a+2 = 1 𝑍 𝑆 1 , 𝑆 2 , …, 𝑆 𝑁 𝑆 𝑎 𝑆 𝑎+2 𝑒 𝛽 𝑖=1 𝑁−1 𝐽 𝑖 𝑆 𝑖 𝑆 𝑖+1
We use the trick
𝑆 𝑎+1 𝑆 𝑎+1 = 𝑆 𝑎+1 2 =1
𝑆 𝑎 𝑆 𝑎+2 = 𝑆 𝑎 ∗𝟏∗𝑆 𝑎+2 = 𝑆 𝑎 𝑆 𝑎+1 𝑆 𝑎+1 𝑆 𝑎+2
S a S a+2 = 1 𝑍 𝑆 1 , 𝑆 2 , …, 𝑆 𝑁 𝑆 𝑎 𝑆 𝑎+1 𝑆 𝑎+1 𝑆 𝑎+2 𝑒 𝛽 𝑖=1 𝑁−1 𝐽 𝑖 𝑆 𝑖 𝑆 𝑖+1
= 1 𝑍 1 𝛽 2 𝜕 2 𝑍 𝜕 𝐽 𝑎 𝜕 𝐽 𝑎+1
=tanh𝛽 𝐽 𝑎 tanh𝛽 𝐽 𝑎+1
For the special case 𝐽 𝑎 = 𝐽 𝑎+1 =𝐽
T->0
S a S a+𝑏 𝑏
we obtain
S a S a+2 = tanh 𝛽𝐽 2
S a S a+𝑏 = tanh 𝛽𝐽 𝑏

8. Correlation length The last figure suggests:
Correlation between spins decays with increasing separation of the spin on a characteristic
T-dependent length scale we will call 𝝃(𝑻)
=𝑒 − 𝑏 𝜉 𝑇
S a S a+𝑏 = tanh 𝛽𝐽 𝑏
=𝑒 ln tanh 𝛽𝐽 𝑏
=𝑒 𝑏 ln tanh 𝛽𝐽
tanh(x)
𝜉 𝑇 = 1 ln tanh 𝛽𝐽
Note that tanh 𝛽𝐽 <1 and thus ln tanh 𝛽𝐽<0
Discussion
T->∞ , 𝛽→0
tanh 𝛽𝐽 →0
ln tanh 𝛽𝐽 →∞
𝜉 𝑇→∞ →0 1
tanh 𝛽𝐽= 𝑒 𝛽𝐽 − 𝑒 −𝛽𝐽 𝑒 𝛽𝐽 + 𝑒 −𝛽𝐽 = 𝑒 𝛽𝐽 − 𝑒 −𝛽𝐽 𝑒 𝛽𝐽 1+ 𝑒 −2𝛽𝐽 = 1− 𝑒 −2𝛽𝐽 1+ 𝑒 −2𝛽𝐽
T->0 , 𝛽→∞ 2
≈ 1− 𝑒 −2𝛽𝐽 1+ 𝑒 −2𝛽𝐽 ≈ 1− 𝑒 −2𝛽𝐽 2 ≈1− 2𝑒 −2𝛽𝐽
𝜉 𝑇→0 → 1 2 𝑒 2𝛽𝐽
ln tanh 𝛽𝐽≈ ln (1− 2𝑒 −2𝛽𝐽 )≈ 2𝑒 −2𝛽𝐽

9. At Tc fluctuations on all length scales including
Significance of 𝝃(𝑻) is more transparent for higher spatial dimensions (2d, 3d) where 𝝃(𝑻) diverges at finite temperature Tc
At Tc fluctuations on all length scales including
length scale of the entire sample
𝝃 𝑻= 𝑻 𝑪 →∞
W𝐢𝐭𝐡 𝝃 𝑻 also the susceptibility 𝝌 𝑻 diverges because there is a general relation
between 𝝌 𝑻 and the sum of spin-spin correlation functions S 𝐢 S 𝐣
𝜒 𝑇 = lim 𝐻→0 𝜕𝑀 𝜕𝐻 ∝ lim 𝐻→0 𝜕 𝜕𝐻 𝑎 𝑆 𝑎
∝ lim ℎ→0 𝜕 𝜕ℎ 𝑎 𝑆 1 , 𝑆 2 …, 𝑆 𝑁 𝑆 𝑎 𝑒 𝛽𝐽 𝑖 𝑁 𝑆 𝑖 𝑆 𝑖+1 + 𝛽ℎ 𝑖 𝑁 𝑆 𝑖 𝑆 1 , 𝑆 2 …, 𝑆 𝑁 𝑒 𝛽𝐽 𝑖 𝑁 𝑆 𝑖 𝑆 𝑖+1 +𝛽ℎ 𝑖 𝑁 𝑆 𝑖
= lim ℎ→0 𝑎 𝑆 1 , 𝑆 2 …, 𝑆 𝑁 𝑆 𝑎 𝛽 𝑖 𝑁 𝑆 𝑖 𝑒 𝛽𝐽 𝑖 𝑁 𝑆 𝑖 𝑆 𝑖+1 +𝛽ℎ 𝑖 𝑁 𝑆 𝑖 𝑍− 𝑆 1 , 𝑆 2 …, 𝑆 𝑁 𝛽 𝑖 𝑁 𝑆 𝑖 𝑒 𝛽𝐽 𝑖 𝑁 𝑆 𝑖 𝑆 𝑖+1 +𝛽ℎ 𝑖 𝑁 𝑆 𝑖 𝑆 1 , 𝑆 2 …, 𝑆 𝑁 𝑆 𝑎 𝑒 𝛽𝐽 𝑖 𝑁 𝑆 𝑖 𝑆 𝑖+1 + 𝛽ℎ 𝑖 𝑁 𝑆 𝑖 𝑍 2
= lim ℎ→0 𝑎 𝑆 1 , 𝑆 2 …, 𝑆 𝑁 𝑆 𝑎 𝛽 𝑖 𝑁 𝑆 𝑖 𝑒 𝛽𝐽 𝑖 𝑁 𝑆 𝑖 𝑆 𝑖+1 +𝛽ℎ 𝑖 𝑁 𝑆 𝑖 𝑍 − 𝑆 1 , 𝑆 2 …, 𝑆 𝑁 𝛽 𝑖 𝑁 𝑆 𝑖 𝑒 𝛽𝐽 𝑖 𝑁 𝑆 𝑖 𝑆 𝑖+1 +𝛽ℎ 𝑖 𝑁 𝑆 𝑖 𝑆 1 , 𝑆 2 …, 𝑆 𝑁 𝑆 𝑎 𝑒 𝛽𝐽 𝑖 𝑁 𝑆 𝑖 𝑆 𝑖+1 + 𝛽ℎ 𝑖 𝑁 𝑆 𝑖 𝑍 2

10. with const= 𝑁 𝑉 𝑔 𝜇 𝐵 2 𝜇 0 𝐽(𝐽+1) 3
= lim ℎ→0 𝑎 𝑆 1 , 𝑆 2 …, 𝑆 𝑁 𝑆 𝑎 𝛽 𝑖 𝑁 𝑆 𝑖 𝑒 𝛽𝐽 𝑖 𝑁 𝑆 𝑖 𝑆 𝑖+1 +𝛽ℎ 𝑖 𝑁 𝑆 𝑖 𝑍 − 𝑆 1 , 𝑆 2 …, 𝑆 𝑁 𝛽 𝑖 𝑁 𝑆 𝑖 𝑒 𝛽𝐽 𝑖 𝑁 𝑆 𝑖 𝑆 𝑖+1 +𝛽ℎ 𝑖 𝑁 𝑆 𝑖 𝑆 1 , 𝑆 2 …, 𝑆 𝑁 𝑆 𝑎 𝑒 𝛽𝐽 𝑖 𝑁 𝑆 𝑖 𝑆 𝑖+1 + 𝛽ℎ 𝑖 𝑁 𝑆 𝑖 𝑍 ∗ 𝑍
∝ 𝑖 𝑁 𝑆 𝑖 ∝ 𝑀
for h->0
𝑆 𝑎 ->0
for h->0
No long range order for the Ising chain
𝜒 𝑇 =const 𝛽 𝑁 𝑎,𝑖 𝑆 𝑎 𝑆 𝑖
with const= 𝑁 𝑉 𝑔 𝜇 𝐵 2 𝜇 0 𝐽(𝐽+1) 3
Let’s see how this expression reproduces paramagnetic susceptibility in the limit J=0
1 𝑁 𝑎,𝑖 𝑆 𝑎 𝑆 𝑖 = 1 𝑁 𝑎=𝑖 𝑆 𝑎 𝑆 𝑖 𝑁 𝑎 𝑖≠𝑎 𝑆 𝑎 𝑆 𝑖
=1+ 1 N 𝑎 2 𝑖 =1 𝑁 𝑆 𝑎 𝑆 𝑎+𝑗
The 2 originates from the fact that
terms such as
appear twice in the a,i-double sum
𝑆 1 𝑆 2 = 𝑆 2 𝑆 1
=1+2 𝑗=1 𝑁 tanh 𝛽𝐽 𝑗

11. Let’s have a look to the limit of a long (relative to the correlation length) Ising chain
𝑁→∞
1+2 𝑗=1 ∞ tanh 𝛽𝐽 𝑗 =1−2+2 𝑗=0 ∞ tanh 𝛽𝐽 𝑗
1 𝑁 𝑎,𝑖 𝑆 𝑎 𝑆 𝑖 →
= − tanh 𝛽𝐽 −1
=1+2 tanh 𝛽𝐽 1− tanh 𝛽𝐽
=1+2 𝑒 𝛽𝐽 − 𝑒 −𝛽𝐽 𝑒 𝛽𝐽 + 𝑒 −𝛽𝐽 1− 𝑒 𝛽𝐽 − 𝑒 −𝛽𝐽 𝑒 𝛽𝐽 + 𝑒 −𝛽𝐽
=1+2 𝑒 𝛽𝐽 − 𝑒 −𝛽𝐽 𝑒 𝛽𝐽 + 𝑒 −𝛽𝐽 − 𝑒 𝛽𝐽 + 𝑒 −𝛽𝐽
=1+ 𝑒 2𝛽𝐽 −1
𝜒 𝑇 =const 𝛽 𝑁 𝑎,𝑖 𝑆 𝑎 𝑆 𝑖
=const 𝛽 𝑁 𝑒 2𝛽𝐽
𝜒 𝑇 ∞ 1 𝑇
For J=0 this indeed reduced to the Curie-law of interaction free paramagnets

12. For J0 it is interesting to discuss the cases of ferro and antiferromagnetic coupling
𝜒 𝑇 T J
𝜒∝ 𝑒 2𝐽 𝑘 𝐵 𝑇 𝑇
J<0
antiferromagnetic coupling
𝜒 𝑇 T
J=-1/2
J=-1
𝜒∝ 𝑒 − 2 𝐽 𝑘 𝐵 𝑇 𝑇