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

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 2 𝑆 4 , …, 𝑆 𝑁 ( 𝑒 𝛽𝐽 𝑆 4 + …+ 𝑆 𝑁−1 𝑆 𝑁 + 𝑒 𝛽𝐽 −𝑆 4 + …+ 𝑆 𝑁−1 𝑆 𝑁 )

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

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

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

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 1 𝛽 =tanh𝛽 𝐽 𝑎 For the special case 𝐽 𝑎 =𝐽 we obtain S a S a+1 =tanh𝛽𝐽

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

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

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

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 + 1 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

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 → =1+2 1 1− 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

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