# Triangular lattice and criticality

1. Apr 14, 2016

### CAF123

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
$$Z = \sum_{\left\{n\right\}} e^{- \beta H}$$

3. 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!

2. Apr 19, 2016