Exploring Electron Splitting in Materials: Spinons and Orbitons in QFT

[Philipsmett]

What does it mean that an electron in a material can split into a spinon or an orbiton? Does QFT have a spinon or orbiton field?

 

[king vitamin]

It's not correct to think of the microscopic electrons literally splitting when you have fractionalized quantum numbers in a material. The resulting excitations (call them spinons and orbitons if you like) are really many-body composites of the degrees of freedom in the system. The electrons in a system form some sort of strongly-interacting "liquid" in the ground state, and the exact excitations above this very nontrivial state can contain delocalized quasiparticles which separately carry fractions of the electron charge/spin/etc. I think of the total charge/spin being shared among all of the electrons, so that locally fractions can be made, but this is pretty heuristic.

But you must create enough quasiparticles that their total charges add up to the electron charge. For example, in the $\nu = 1/3$ fractional quantum Hall effect, you must create multiples of three quasiparticles at once, since each quasiparticle carries a third of the electron charge.

Does QFT have a spinon or orbiton field?

It can, in the sense that you can consider an effective QFT at low energies, and in the QFT the "fundamental" fields are the spinon/orbiton fields. Note that when you have fractionalization, you necessarily have some dynamic gauge fields at low-energy too (which are usually emergent from the original microscopic Hamiltonian you started with). In certain cases and approximations you can actually derive the low-energy QFT from the microscopic model of electrons, and see these fields emerge.

 

[Philipsmett]

It's not correct to think of the microscopic electrons literally splitting when you have fractionalized quantum numbers in a material. The resulting excitations (call them spinons and orbitons if you like) are really many-body composites of the degrees of freedom in the system. The electrons in a system form some sort of strongly-interacting "liquid" in the ground state, and the exact excitations above this very nontrivial state can contain delocalized quasiparticles which separately carry fractions of the electron charge/spin/etc. I think of the total charge/spin being shared among all of the electrons, so that locally fractions can be made, but this is pretty heuristic.

But you must create enough quasiparticles that their total charges add up to the electron charge. For example, in the $\nu = 1/3$ fractional quantum Hall effect, you must create multiples of three quasiparticles at once, since each quasiparticle carries a third of the electron charge.
It can, in the sense that you can consider an effective QFT at low energies, and in the QFT the "fundamental" fields are the spinon/orbiton fields. Note that when you have fractionalization, you necessarily have some dynamic gauge fields at low-energy too (which are usually emergent from the original microscopic Hamiltonian you started with). In certain cases and approximations you can actually derive the low-energy QFT from the microscopic model of electrons, and see these fields emerge.

But how do electrons repel each other if they are divided into quasi-particles and only the holon repels because it carries a charge?

 

[king vitamin]

Once again, you should not view the orbitons/halons as split up versions of the original electrons. They are extremely complicated superpositions of a large number of the original electrons.

As an example, let's look at the $\nu = 1/3$ fractional quantum Hall effect (FQHE) again. I claimed that the quasiparticles carry a third of an electron charge. What does the wave function of this quasiparticle look like in terms of the original electrons? For realistic materials, there is no exact solution, but the "Laughlin wavefunctions" are known to provide very accurate approximations to the actual ones (and they are exact if one considers simplified models of electron-electron interactions). Up to a normalization constant, the Laughlin wave function describing the ground state of the $\nu = 1/3$ FQHE is
$$
\psi_0(z_1,z_2,...,z_N) = \prod_{j<k}^N (z_j - z_k)^3 \exp\left( - \frac{1}{4 l_B^2} \sum_{m=1}^N |z_m|^2 \right)
$$
Here, $z_m = x_m + i y_m$ is the position of the $m$th electron, where we write the positions as complex numbers for convenience, and we have a total of $N$ electrons. We also have a length scale $l_B$ (related to the applied magnetic field for the FQHE, $l_B = \sqrt{\hbar c/eB}$).

Now, what does the fundamental excitation (the "Laughlin quasihole"), which has charge $+e/3$, look like? A single excitation at position $z_0$ has wave function
$$
\psi_{\mathrm{qh}} = \left[\prod_{j=1}^N (z_j - z_0) \right] \psi_0(z_1,z_2,...,z_N)
$$
As you can see, although this describes a localized excitation above the ground state, it is described by an enormously complicated function of all of the electrons in the system. Even in describing its difference from the ground state wave function $\psi_0$, it involves a product over all of the electron positions. It is certainly not some sort of "fraction" of the original electrons!

 

[Philipsmett]

Once again, you should not view the orbitons/halons as split up versions of the original electrons. They are extremely complicated superpositions of a large number of the original electrons.

As an example, let's look at the $\nu = 1/3$ fractional quantum Hall effect (FQHE) again. I claimed that the quasiparticles carry a third of an electron charge. What does the wave function of this quasiparticle look like in terms of the original electrons? For realistic materials, there is no exact solution, but the "Laughlin wavefunctions" are known to provide very accurate approximations to the actual ones (and they are exact if one considers simplified models of electron-electron interactions). Up to a normalization constant, the Laughlin wave function describing the ground state of the $\nu = 1/3$ FQHE is
$$
\psi_0(z_1,z_2,...,z_N) = \prod_{j<k}^N (z_j - z_k)^3 \exp\left( - \frac{1}{4 l_B^2} \sum_{m=1}^N |z_m|^2 \right)
$$
Here, $z_m = x_m + i y_m$ is the position of the $m$th electron, where we write the positions as complex numbers for convenience, and we have a total of $N$ electrons. We also have a length scale $l_B$ (related to the applied magnetic field for the FQHE, $l_B = \sqrt{\hbar c/eB}$).

Now, what does the fundamental excitation (the "Laughlin quasihole"), which has charge $+e/3$, look like? A single excitation at position $z_0$ has wave function
$$
\psi_{\mathrm{qh}} = \left[\prod_{j=1}^N (z_j - z_0) \right] \psi_0(z_1,z_2,...,z_N)
$$
As you can see, although this describes a localized excitation above the ground state, it is described by an enormously complicated function of all of the electrons in the system. Even in describing its difference from the ground state wave function $\psi_0$, it involves a product over all of the electron positions. It is certainly not some sort of "fraction" of the original electrons!

Now I realized. thanks

 

[Philipsmett]

It's not correct to think of the microscopic electrons literally splitting when you have fractionalized quantum numbers in a material. The resulting excitations (call them spinons and orbitons if you like) are really many-body composites of the degrees of freedom in the system. The electrons in a system form some sort of strongly-interacting "liquid" in the ground state, and the exact excitations above this very nontrivial state can contain delocalized quasiparticles which separately carry fractions of the electron charge/spin/etc. I think of the total charge/spin being shared among all of the electrons, so that locally fractions can be made, but this is pretty heuristic.

But you must create enough quasiparticles that their total charges add up to the electron charge. For example, in the $\nu = 1/3$ fractional quantum Hall effect, you must create multiples of three quasiparticles at once, since each quasiparticle carries a third of the electron charge.
It can, in the sense that you can consider an effective QFT at low energies, and in the QFT the "fundamental" fields are the spinon/orbiton fields. Note that when you have fractionalization, you necessarily have some dynamic gauge fields at low-energy too (which are usually emergent from the original microscopic Hamiltonian you started with). In certain cases and approximations you can actually derive the low-energy QFT from the microscopic model of electrons, and see these fields emerge.

I apologize.
charge fractionation means that the electron must be divided for example into three parts, which must carry part of the charge? The charge is the ability of matter, it can not exist by itself, it turns out that the holons must have masa and create an electrostatic field around themselves?

 

[Philipsmett]

Is the electron in the material always split into quasiparticles or only in some cases? When two materials contact in which charge fractionation occurred, does electrostatic repulsion occur between quasiparticles (holons)?

 

[king vitamin]

charge fractionation means that the electron must be divided for example into three parts

No, it does not. Please read my posts more carefully. I wrote down a wave function for a charge-$e/3$ object which was made out of full electrons! I also explicitly said "It's not correct to think of the microscopic electrons literally splitting when you have fractionalized quantum numbers in a material." These directly contradict your statement.

 

[king vitamin]

Electrons do not split. This has already been explained to you, and it is getting frustrating for you to repeat this incorrect statement repeatedly in spite of our earnestly trying to help you.

The quasiparticles in a material are collective excitations of a large number of (non-"split") electrons. Whenever you have many particles interacting, like electrons in a material, your low-energy states are better described by quasiparticles rather than electrons. These quasiparticles may have fractions of the quantum numbers held by an individual electron, or more commonly they have the same charge and spin but have some different effective mass.

 

[Philipsmett]

Electrons do not split. This has already been explained to you, and it is getting frustrating for you to repeat this incorrect statement repeatedly in spite of our earnestly trying to help you.

The quasiparticles in a material are collective excitations of a large number of (non-"split") electrons. Whenever you have many particles interacting, like electrons in a material, your low-energy states are better described by quasiparticles rather than electrons. These quasiparticles may have fractions of the quantum numbers held by an individual electron, or more commonly they have the same charge and spin but have some different effective mass.

This is the best explanation! Thanks