What's the difference between a Luttinger surface and a Fermi surface?

In summary, Luttinger's sum rule states that the concept of a Fermi surface can be applied to interacting electrons, even though solving the quantum problem with Coulomb interactions is difficult. This allows for a picture of states in k-space that form the Fermi surface, which can be adapted for interacting electrons with the same filling law. Luttinger showed that properly defined quantities still apply in this case.
  • #1
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I'm trying to learn about Luttinger's sum rule, but I haven't taken QM yet, and the papers on the topic are very difficult to understand without it. However it seems qualitatively like the Luttinger surface is equivalent to the Fermi surface. If this is not the case, what's the difference? Does anyone know of a source of information on the topic that can be understood without QM?
 
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  • #2
What I gather from Wikipedia: Luttinger's Theorem says the concept of a Fermi Surface can be applied to interacting electrons. Electrons, as you know interact via a 1/(r-r') Coulomb interaction. That makes the quantum problem extremely difficult to solve. Nevertheless, due to screening and other reasons it is often assumed the electrons are not interacting with each other. This leads to the picture of states in k-space that get filled up to form the Fermi surface. It's not a priori clear you can adapt such a picture for interacting electrons. Luttinger showed, that with properly defined quantities, you can still talk about k-space and a Fermi surface and that the same filling law applies.
 

1. What is a Luttinger surface and how is it different from a Fermi surface?

A Luttinger surface is a type of surface that describes the allowed states of electrons in a crystal lattice. It is characterized by its shape and the distribution of energy states. In contrast, a Fermi surface refers to the collection of all the points in momentum space that have the same energy level. This means that while a Luttinger surface describes the allowed states of electrons, a Fermi surface describes the distribution of these states in momentum space.

2. How do Luttinger surfaces and Fermi surfaces affect the properties of materials?

Both Luttinger surfaces and Fermi surfaces play a crucial role in determining the electronic and transport properties of materials. The shape and size of these surfaces can affect the electrical conductivity, magnetic properties, and thermal conductivity of a material. Additionally, the presence of gaps in these surfaces can give rise to unique phenomena such as superconductivity.

3. Can a material have both a Luttinger surface and a Fermi surface?

Yes, a material can have both a Luttinger surface and a Fermi surface. In fact, most materials have both types of surfaces, but their relative sizes and shapes can vary greatly, depending on factors such as the crystal structure and electron interactions within the material.

4. What are some real-world applications of understanding Luttinger surfaces and Fermi surfaces?

Understanding Luttinger surfaces and Fermi surfaces is crucial for developing new materials with specific electronic properties. For example, researchers can use this knowledge to design materials with high electrical conductivity for use in electronic devices, or to create materials with unique magnetic properties for applications in data storage.

5. Can Luttinger surfaces and Fermi surfaces change under different conditions?

Yes, Luttinger surfaces and Fermi surfaces can change under different conditions, such as changes in temperature, pressure, or magnetic field. These changes can alter the shape and size of the surfaces, leading to changes in the material's properties. This is why understanding the behavior of these surfaces under different conditions is crucial for predicting and controlling the properties of materials.

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