1st order vs 2nd order phjase transition#

Critical Phenomena: Correlation length diverges#

Critical Phenomena: Universaility at critical points#

Universality#

Thermodynamics of magnetic systems#

For magnetic systems placed under external field $B$ the change in internal energy consists of an extra term of work done by the external field to displace mangetization $M$

A free energy as a function of intensive variables can be written down following legendre transform:

• Magnetization and Entropy can now be linked to derivatives of free energy

Fluctuations, correlations and susceptibilities#

• Heat capacity $C_v$

Heat capacity is again the familiar expression established in our treatment of canonical ensemble.

• Susceptibility $\chi$

Magnetic susceptibility quantifies response of the ssytem to the variation of magnetic field.

• Correlation function $c(i,j)$ and correlation length

At high temperatures, spins on an ising lattice point up and down randomly. While at low temperatures, all spins tend to align. To quantify the degree of alignment, we can define a quantity named correlation length $\xi$, which can be defined mathematically through correlation function $c(i,j)$

Some notable applications of Ising models.#

1. Ferromagnetism The Ising model is the scalar version of the three-dimensional ‘Heisenberg model’ :
   

2. Binary alloys and lattice gases Each lattice site is occupied either by an atom A or B. Nearest neighbour interac- tions are ${ϵ}_{AA}$, ${ϵ}_{BB}$ and ${ϵ}_{AB}$. We identify $A$ with $s_i=1$ and B with $S_i=-1$. The Hamiltonian then is: $H=-\sum J_{ij}{s}_i{s}_j⟨i,j⟩$ with $J_{ij}={ϵ}_{AB}-0.5({ϵ}_{AA}+{ϵ}_{BB})$. Thus the Ising model describes order-disorder transitions in regard to composition.
   

3. Spin glasses now each bond is assigned an individual coupling constant J and they are drawn from a random distribution. E.g. one can mix ferromagnetic and anti-ferromagnetic couplings. This is an example for a structurally disordered system, on top of which we can have a thermal order-disorder transition.
   

4. Conformations in biomolecules a famous example is the helix-coil transition from biophysics. $s_i$ a hydrogen bond in a DNA-molecules is closed; $s_i$ = −1 the bond is open. The phase tran- sition is between a straight DNA-molecule (helix) and a coiled DNA-molecule. Other examples are the oxygen-binding sites in hemoglobin, chemotactic recep- tors in the receptor fields of bacteria, or the molecules building the bacterial flag- ellum, which undergos a conformational switch if the flagellum is rotated in the other direction (switch from run to tumble phases).
   

5. Neural networks representing the brain $s_i=1$ a synapse is firing, $s_i=-1$ it is resting. The Hopfield model for neu- ral networks is a dynamic version of the Ising model and Boltzmann machines recognise handwriting by using the Ising model.
   

6. Spread of opinions or diseases Spread of opinions, rumours or diseases in a society; these kinds of models are used in socioeconomic physics. If nearest neighbour coupling is sufficiently strong, the system gets ‘infected’.

No phase transition in 1D at T>0#

1D case Start with 1D chain of all spin up configuration and create an isalnd of oposite spins.

• Energy cost is:

• Entropy cost is:

• Free Energy differnece:

Thus free energy for $N\gg 1$ is always negative for any finite temperature. It is always favorable to create grain boundaries due to entropic reasons.
Phase transition in 1D cannot occur at finite temperature!

Phase transition in 2D at $T>0$ realizable#

In 2D and 3D introducing isalnd of spins creates a surface spins sproportional to linear size of the island L which is now a macroscopic quantitiy comparable to N

• Energy cost is:

• Entropy cost using a crude estimate of random walk on cubic lattice is:

• Free Energy differnece:

Analytic Solutions to 1D Ising#

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 $T=\mathrm{\infty }$

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