Calculating exciton Green's function on a tight binding chain

In summary, the paper discusses the calculation of the exciton Green's function in a tight-binding chain model, which is essential for understanding the behavior of excitons in low-dimensional systems. The authors derive the Green's function using mathematical techniques pertinent to tight-binding models, analyze its properties, and explore implications for exciton dynamics and localization. The study provides insights into the influence of lattice structure and interactions on excitonic states, highlighting the relevance of these findings for future research in condensed matter physics.
  • #1
physicsxanime
13
3
Homework Statement
On a 1-D tight binding chain, when you put an electron from a site in the valence band to another site in the conduction band, you will can create an exciton if there is Hubbard attraction between the electron and hole. I would like to calculate the green's function of such two particle excited state.
The Hamiltonian of interest:
$$H = t\sum c^\dagger_{n+1}c_{n} - t\sum v^\dagger_{n+1}v_{n} + h.c. + U\sum c^\dagger_{n}c_{n}v^\dagger_{n}v_{n}$$
Relevant Equations
Lehmann representation of Green's function $$G(x,x';E) = \sum_n \frac{<\Psi^N_0|\psi(x)|\Psi^{N+1}_n><\Psi^{N+1}_n|\psi^\dagger(x')|\Psi^N_0>}{E + E^N_0-E^{N+1}_n +i\eta}$$
Honestly, I have no real idea. I know for sure the equation connects the initial state ##|\Psi^N_0>##to a final state ##|\Psi^{N+1}_0>##, ##E## is the energy and ##E^N_0## etc are the energy of the initial state and final state. I also know that these energy are related to conduction band and valence band energy like ##2t\cos(ka)##. But I don't really know what ##\psi(x)## represent here, how to get the wave functions ##|\Psi^{N+1}_0>##, and what it is summing over.

As you can see, I have a severe lack of knowledge here, so any help will be greatly appreciated and if possible, point to me some introductory resources on topics of condensed matter and many particle green's function.
 
Physics news on Phys.org
  • #2

FAQ: Calculating exciton Green's function on a tight binding chain

What is an exciton Green's function?

An exciton Green's function is a mathematical tool used to describe the properties and dynamics of excitons, which are bound states of an electron and a hole. In the context of condensed matter physics, the Green's function provides information about the excitonic states, their energies, and how they evolve over time. It is essential for understanding exciton dynamics in various materials, including semiconductors and nanostructures.

How do you calculate the Green's function for a tight binding chain?

To calculate the Green's function for a tight binding chain, one typically starts by defining the Hamiltonian of the system, which describes the hopping of particles between lattice sites. The Green's function can be computed using the equation G(E) = (E - H)^(-1), where E is the energy and H is the Hamiltonian matrix. For a tight binding chain, the Hamiltonian is usually tridiagonal, making the computation more straightforward using methods such as matrix inversion or recursion relations.

What are the key parameters in a tight binding model?

The key parameters in a tight binding model include the hopping amplitude (t), which represents the probability of an electron hopping from one site to its nearest neighbor, and the on-site energy (ε), which denotes the energy level of the electrons at each lattice site. The ratio of these parameters can significantly influence the band structure and excitonic properties of the system.

What physical phenomena can be analyzed using the exciton Green's function?

The exciton Green's function can be used to analyze various physical phenomena, including exciton binding energies, exciton lifetimes, and the effects of external fields on exciton states. It is also useful for studying the response of excitons to perturbations, such as electron-electron interactions and disorder, which can affect their stability and dynamics in a material.

What are the limitations of using the tight binding model for exciton calculations?

While the tight binding model provides a simplified framework for studying excitons, it has limitations, such as the assumption of nearest-neighbor interactions and neglecting long-range interactions and electron correlations. Additionally, it may not accurately capture the behavior of excitons in more complex materials or in the presence of strong external fields, requiring more advanced models or numerical techniques for a complete description.

Similar threads

Assignment on the tight binding model
Replies
1
Views
2K
Differentiation of functional integral (Blundell Quantum field theory)
Replies
1
Views
1K
Tight binding model (covalent bonding)
Replies
6
Views
3K
I Green's function boundary conditions
Replies
1
Views
725
Most Probable energy after infinite square well expands
Replies
2
Views
927
A Green's function calculation of an infinite lattice with periodicity in 1D
Replies
1
Views
1K
Is the given wave function a stationary state?
Replies
3
Views
2K
Understanding Green's second identity and the reciprocity theorem
Replies
3
Views
4K
I Hartree-Fock: Feynman diagrams vs perturbation theory
Replies
9
Views
2K
How to get general solution via Green's function?
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top