# Tight binding hamiltonian matrix

• gizzmo
In summary, when working with fermions, the tight binding Hamiltonian matrix often takes the form of a four by four matrix with elements of +t and -t, representing the basis states |\uparrow,\downarrow>, |\downarrow,\uparrow>, |\uparrow\downarrow,0>, and |0,\uparrow\downarrow>. This is due to the antisymmetric nature of fermionic wave functions. To calculate these elements from the actual Hamiltonian, one must first write down the basis states in second quantized notation and carefully consider the normal ordering of operators and contractions. Therefore, it is important to provide a description of the system being studied and to explain any discrepancies in calculations.
gizzmo
Can somebody explain to me why, when we work with fermions, the tight binding Hamiltonian matrix has a form
0 0 -t -t
0 0 +t +t
-t +t 0 0
-t +t 0 0
the basis are |\uparrow,\downarrow>, |\downarrow,\uparrow>, |\uparrow\downarrow,0>, |0,\uparrow\downarrow>,
Why there is +t and -t? (I think that this has something to do with the fact the the fermionic wave function is antisymmetric. But can somebody give me an example how to calculate this elements from the actual Hamiltonian. I always get -t.)

Tight binding is a very general method, and there are many systems where it doesn't have that form. You really should describe the system you are studying before asking a question like that. Also, when you make a statement of the form "I always get this answer and it's wrong" you should describe what it is that you're doing or you're just asking people to guess as what you have done wrong.

You need to write down your basis states in second quantized notation:
$$|\uparrow,\downarrow\rangle = c_{1\uparrow}^\dagger c_{2\downarrow}^\dagger |0\rangle$$
and you need to pick a normal ordering for your operators and make sure you are consistent when writing out your states. Then write out each inner product carefully and write out the contractions.

## 1. What is a tight binding Hamiltonian matrix?

A tight binding Hamiltonian matrix is a mathematical representation of the energy levels of electrons in a solid material. It takes into account the interactions between neighboring atoms and the energy required for electrons to move between them.

## 2. How is a tight binding Hamiltonian matrix calculated?

The calculation of a tight binding Hamiltonian matrix involves determining the overlap of atomic orbitals and the strength of their interactions. This information is then used to construct the matrix, which can be solved to determine the energy levels of the electrons.

## 3. What are the advantages of using a tight binding Hamiltonian matrix?

A tight binding Hamiltonian matrix is a useful tool for understanding the electronic properties of materials. It can provide insights into the behavior of electrons in complex systems and can be used to predict material properties such as conductivity and band structure.

## 4. What are the limitations of the tight binding Hamiltonian matrix?

While a tight binding Hamiltonian matrix is a valuable tool, it has some limitations. It is based on simplifying assumptions and may not accurately capture the full complexity of a material. Additionally, it may not be suitable for materials with strong electron-electron interactions or for systems with a large number of atoms.

## 5. How is the tight binding Hamiltonian matrix used in practical applications?

The tight binding Hamiltonian matrix is used in a variety of practical applications, such as in the design of new materials for electronic devices and in the simulation of complex systems. It can also be used in theoretical studies to understand the behavior of electrons in different materials and under different conditions.

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