Triangular lattice and criticality

[CAF123]

Homework Statement

Consider a two dimensional triangular lattice, each point of which can either contain a particle of a gas or be empty. The Hamiltonian characterising the system is defined in terms of the particle occupation numbers $\left\{n_i \right\}_{i=1,...,N}$ which can either be 0 or 1. N is the total number of points on the lattice. The Hamiltonian is $$H = \mu \sum_i n_i - J \sum_{\langle i j \rangle} n_i n_j$$ with $\mu >0$ the chemical potential and $J \geq 0$ an effective interparticle interaction.

a) In the limit J=0, find the partition function, the free energy and the average occupation number.

b) Using a mean field variational approach, the average occupation number is $$\langle n \rangle = \frac{1}{1+ \exp(3J\beta - 6 \beta J \langle n \rangle)}$$ Expand this equation around $\langle n \rangle = 1/2$ and find the critical value of J for which the mean field predicts a phase transition between a particle poor phase and a particle rich one.

Homework Equations

$$Z = \sum_{\left\{n\right\}} e^{- \beta H}$$

The Attempt at a Solution

a) In the first part I was getting $Z = (2\cosh \beta \mu)^N$, $F = -N/\beta \,\, \ln(2 \cosh \beta \mu)$ and $\langle n \rangle = -N \tanh \beta \mu$

b) Write the equation as $$ \langle n \rangle = (1 + \exp 3J \beta (1-2 \langle n \rangle))^{-1}$$ Write $$\exp 3J \beta (1-2 \langle n \rangle) = \exp(3J \beta) \exp(-6J \beta \langle n \rangle)$$ and expand around $\langle n \rangle = 1/2$ so set $\langle n \rangle = 1/2 + \eta$ and get $$ \exp(3J \beta) \exp(-6J \beta( 1/2 + \eta)) \approx (1-6J \beta \eta)$$ Inputting this into the equation for $\langle n \rangle$, and replacing $\eta = \langle n \rangle - 1/2$ I get a quadratic equation for $\langle n \rangle$? Should this happen and if so, how to interpret the two solutions? I think J is that value at which either side of it $\langle n \rangle$ goes from being less than 1/2 to greater than it. (I haven't found such an equation yet either)

Thanks!

 

[mark726]

Thank you for your interesting post. I would like to share my thoughts on your proposed solutions.

a) Your solutions for the partition function, free energy, and average occupation number in the limit J=0 are correct. However, it would be helpful to provide some explanation or derivation for those results.

b) Your approach to expanding the equation for the average occupation number around $\langle n \rangle = 1/2$ is correct. However, when you set $\langle n \rangle = 1/2 + \eta$, you are essentially assuming that the critical value of J is small enough so that the deviation from $\langle n \rangle = 1/2$ is also small. This is known as a linear approximation or a small-angle approximation. It is a common technique in physics, but it is important to keep in mind its limitations.

As for the two solutions for $\langle n \rangle$, they represent the two possible phases of the system - particle-rich and particle-poor. The critical value of J is the value at which these two phases become equally favorable, and the system undergoes a phase transition from one phase to the other.

I hope this helps clarify your solutions. Keep up the good work!