Susceptibilites for the Anderson Impurity Model

Your name]In summary, the conversation discusses susceptibilities in the context of the Anderson Impurity Model. The paper referenced provides a general definition for a susceptibility "tensor" in terms of spin operators, but the specific definitions for longitudinal, perpendicular, and transverse susceptibilities are not clearly defined. The relationship between these susceptibilities and the response to different magnetic field components is also discussed. The paper also states a specific relationship between two of the susceptibilities, but this is dependent on the assumptions and approximations made in the model. Recommendations for further literature on the topic are also provided.
  • #1
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My question is a rather general one about susceptibilities but specific answers for the Anderson Impurity Model are also appreciated.
In this paper https://arxiv.org/abs/1101.2840 (was also published in the Journal of Physics: Condensed Matter http://iopscience.iop.org/article/10.1088/0953-8984/23/4/045601/meta but it has no open access) the authors specify different kinds of susceptibilities starting right after eq. (17) (not counting the charge susceptibility):
longitudinal ##\chi_l##
perpendicular ##\chi_\perp##
transverse ##\chi^{+-}## or ##\chi_t##.
How are these defined in general? I couldn't find any satisfactory answer in any textbook or article. The only general definition for a susceptibility "tensor" in terms of spin operators I found was
$$\chi_{ij}^{\beta\alpha}(t,t')=-\frac{\mu_0}{V\hbar}\frac{g^2\mu_B^2}{\hbar^2}
\ll S_i^\beta(t);S_j^\alpha(t')\gg$$
when the time dependant magnetization is given by
$$M^\beta(t)-M^\beta(0)=-\frac{1}{V\hbar}\sum_\alpha\int_{-\infty}^\infty
dt'\ll m^\beta(t);m^\alpha(t')\gg B^\alpha(t')$$
where ##B## is an external magnetic field, ##m## are the magnetic moment operators and the indices ##\alpha## and ##\beta## stand for the three room coordinates x,y and z (I think ##i## and ##j## specify the sites but that's not clear from the text, it's from Nolting's many particle theory book). The double angular brackets are some kind of retarded Green's function defined by
$$\ll A(t);B(t')\gg=-i\Theta(t-t')\langle[A(t),B(t')]\rangle.$$
From theses definitions I would assume that ##\chi_l## is the response to the part of the field vector, that is parallel to the spin quantization axis and ##\chi_\perp## is the response to the pependicular part. But what would then be ##\chi_t## and what is about all other mixed entries of ##\ll S_i^\beta(t);S_j^\alpha(t')\gg## (response of parallel magnetization to perpendicular field and vice versa)?
If I had all of these information, I would probably also be able to answer my next question, why
$$\chi_t(0,h)=2\chi_\perp(h)$$
holds (between eqs. (24) and (25)).
It would also be nice, if someone could recommend any kind of literature about these spin susceptibilities to me. Thanks in advance!
 
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  • #2

Thank you for your question about susceptibilities and specifically about the Anderson Impurity Model. The paper you referenced does indeed provide a general definition for a susceptibility "tensor" in terms of spin operators. However, I understand that this may not be clear to you and that you are looking for more information and clarification.

To answer your first question, the longitudinal susceptibility ##\chi_l## is the response of the spin system to a magnetic field component that is parallel to the spin quantization axis. The perpendicular susceptibility ##\chi_\perp## is the response to a magnetic field component that is perpendicular to the spin quantization axis. The transverse susceptibility ##\chi^{+-}## or ##\chi_t## is the response to a magnetic field component that is perpendicular to the spin quantization axis, but in the plane defined by the spin quantization axis and the applied field. In other words, it is the response to a "tilted" magnetic field.

As for the other mixed entries of ##\ll S_i^\beta(t);S_j^\alpha(t')\gg##, they represent the response of the spin system to a combination of magnetic field components, such as the response of parallel magnetization to a perpendicular field and vice versa. These mixed entries are important in understanding the overall response of the spin system to a given magnetic field.

Regarding the relationship between ##\chi_t## and ##\chi_\perp##, the paper you referenced states that ##\chi_t(0,h)=2\chi_\perp(h)## holds for the Anderson Impurity Model at zero temperature and zero frequency. This relationship is derived from the equations in the paper and is not a general result for all spin systems. It is specific to the Anderson Impurity Model and is dependent on the assumptions and approximations made in the model.

In terms of literature, I would recommend looking into textbooks and articles on solid state physics, as well as specific literature on the Anderson Impurity Model. Some examples include "Introduction to Solid State Physics" by Charles Kittel and "The Anderson Model of Localized Magnetism" by Patrick Fulde. Additionally, you can search for articles and papers on spin susceptibilities and the Anderson Impurity Model in databases such as Google Scholar or ScienceDirect.

I hope this helps clarify some of your questions about susceptibilities and the Anderson Impurity Model. If you have any further questions, please do not hesitate to ask.

Best regards
 

1. What is the Anderson Impurity Model?

The Anderson Impurity Model is a theoretical model used in condensed matter physics to describe the electronic properties of a single magnetic impurity in a non-magnetic host material. It was first proposed by physicist Philip W. Anderson in 1961.

2. What are susceptibilities in the context of the Anderson Impurity Model?

Susceptibilities in the Anderson Impurity Model refer to the response of the system to an external perturbation, such as a magnetic field or temperature change. These susceptibilities can provide information about the electronic properties and interactions of the impurity and host material.

3. How are susceptibilities calculated in the Anderson Impurity Model?

There are several methods for calculating susceptibilities in the Anderson Impurity Model, including the numerical renormalization group (NRG) method and the Bethe ansatz approach. These calculations involve solving the equations of motion for the impurity and host electrons and considering the effects of interactions and external perturbations.

4. What are the applications of the Anderson Impurity Model?

The Anderson Impurity Model has been used to study a variety of physical systems, including magnetic materials, heavy fermion systems, and quantum dots. It has also been applied to understand phenomena such as the Kondo effect and the metal-insulator transition. This model provides valuable insights into the behavior of many-body systems and can be used to make predictions about material properties.

5. What are the limitations of the Anderson Impurity Model?

The Anderson Impurity Model is a simplified theoretical model and does not capture all of the complexities of real materials. For example, it does not take into account the effects of lattice vibrations or the full band structure of the host material. Additionally, the model may not accurately describe systems with strong correlations or at very low temperatures. Therefore, the predictions of the Anderson Impurity Model should be interpreted with caution and compared to experimental data.

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