Triangular lattice and criticality

In summary, the conversation discusses a two-dimensional triangular lattice with particles and empty points, and a Hamiltonian that describes the system. The first part of the conversation discusses finding the partition function, free energy, and average occupation number for the system when J=0. The second part discusses using a mean field variational approach to predict a phase transition between particle-rich and particle-poor phases, and the critical value of J at which this transition occurs.
  • #1

CAF123

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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!
 
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  • #2

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!
 

1. What is a triangular lattice?

A triangular lattice is a regular arrangement of points or vertices in a two-dimensional space, forming a repeating pattern of equilateral triangles. It is often used in the study of condensed matter physics, materials science, and statistical mechanics.

2. How is a triangular lattice different from a square lattice?

A triangular lattice has a higher packing density compared to a square lattice, meaning that there are more points per unit area. Additionally, the coordination number (number of nearest neighbors) in a triangular lattice is 6, while in a square lattice it is 4.

3. What is criticality in the context of a triangular lattice?

Criticality refers to the point at which a system undergoes a phase transition, resulting in a dramatic change in its physical properties. In the case of a triangular lattice, this can occur when the lattice is at a certain temperature and the interactions between the particles reach a critical value, causing the system to transition from one phase to another.

4. How is criticality determined in a triangular lattice?

Criticality in a triangular lattice can be determined through the use of various mathematical and computational techniques, such as Monte Carlo simulations and renormalization group theory. These methods allow scientists to analyze the behavior of the system at different temperatures and interactions, and identify the critical point where a phase transition occurs.

5. What are some real-world applications of studying triangular lattices and criticality?

Triangular lattices and criticality have applications in various fields, including materials science, condensed matter physics, and statistical mechanics. For example, understanding criticality in triangular lattices can help researchers design new materials with specific properties, such as superconductors or magnetic materials. It also has implications in understanding the behavior of complex systems, such as social networks and biological systems.

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