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## Main Question or Discussion Point

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 thats 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!

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 thats 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!