# Susceptibilites for the Anderson Impurity Model

• A
Quantenpunk
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!