Case-1: $h\ne 0$ and $J=0$#

• Free energy

• Energy. Obtained by ensemble average of Ising Hamiltonian. Which comes down as simply the average of single spin multipled by number of spins

• Entropy

• Heat capacity

We see that heat capacity goes to zero at both $T=0$ and $T=\mathrm{\infty }$

import ipywidgets as widgets
import matplotlib.pyplot as plt
import numpy as np
import scipy as sp
@widgets.interact(kbT=(0.1,10))
def mag1(kbT=1):
    
    x = np.linspace(-10,10,1000)
    plt.plot(x, np.tanh(x/kbT), lw=3, label=f'$k_B T={kbT}$')
    plt.xlabel('h, magnetic field',fontsize=16)
    plt.ylabel('m, magnetization',fontsize=16)
    plt.legend(fontsize=16)

kbT

1.00

Case-2: $h=0$ and $J\ne 0$#

• We use free boundary conditions

• Let ${\tau }_1=s_1$ and ${\tau }_j=s_{j-1}{s}_j$ for $j\ge 2$

• There will be an overall factor of 2 in front of partition function becasue of summation over ${\tau }_1$ $$Z=\sum _{\tau }\prod _{j=1}^N{e}^{K{\tau }_j}=2\prod _{j=2}^N\sum _{{\tau }_j}{e}^{K{\tau }_j}=2\prod _{j=2}^N{Z}_j$$

Case-3: $h\ne$ and $J\ne 0$ and Transfer Matrix technique#

• We write hamiltonain in this symmetric form consisting of sums of $(s_j,s_{j+1})$ terms for presenting partiion function as product of terms. $$Z=\sum _{[s]}\prod _j{e}^{K{s}_j{s}_{j+1}+\frac{1}{2}h(s_j+s_{J+1})}=\sum _{[s]}\prod _{j}T(s_j,s_{j+1})$$

• Transfer matrix has been introduced:

While compared to previous examples partition function did not factor out into single particle contributions we nevertheless have factored the partition function as product of 2 by 2 matrices!

Note the close connection of matrix technique applied to partion functions with mathematical formalism of quantum mechanics!#

In quantum mecchanical notation $T_{j,j+1}=⟨s_j|T|s_{j+1}⟩$ can be seen as an operator that propagates or transfers state from spin $j+1$ to spin state $j$. $$Z=\sum _{[s]}{e}^{-\beta H}=\sum _{s_1,s_2,...s_N}⟨s_1|T|s_2⟩⟨s_2|T|s_3⟩...⟨s_{N-1}|T|s_N⟩$$

Trace of matrix is invariant to unitary trasnformation $U^{-1}TU=D$ which we can use to diagonalize the matrix $T$ which then allows to us to write the N product in terms of two diagonal elements:

Problem is reduced to diagonalizing the transfer matrix

• Thus we have arrived at an exact solution for the one dimension Ising model with external field:

No phase transition at finite $T>0$ is posisble for 1D ising model as free energy remains analytic for T>0