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I came across with the following questions in reading Altland and Simons page 186

https://books.google.cl/books?id=0KMkfAMe3JkC&pg=PA188&lpg=PA188&dq=Pauli+paramagnetism+altland&source=bl&ots=8Z4XEHF2MB&sig=ACfU3U1GnQUI-hZqDQLRe808zjVzTAVbnQ&hl=es&sa=X&ved=2ahUKEwijwPaD4qXpAhWWEbkGHdhuCWQQ6AEwAHoECAoQAQ#v=onepage&q=Pauli%20paramagnetism%20altland&f=false

Is $\alpha=l$ the same orbital quantum number related to angular momentum $\hat{L}\to l(l+1)$?

They talk about that fermions couple to a magnetic field by their orbital momentum. Why did not they include that orbital momentum coupling? Are they talking about $$\sim \vec{L}\cdot\vec{B}$$ ?

They write down the hamiltonian $$H_0=\sum_{\alpha, \sigma}a^\dagger_{\alpha\sigma}\epsilon_\alpha a_{\alpha\sigma}$$

What is the expression for $\epsilon_\alpha$ ?

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Answer

$H_0$ is the "non magnetic" Hamiltonian operator, which means the part which is not coupled to the magnetic field. In the purpose of the exercise, you can think the quantum number $\alpha$ as the set of momentum-spin $k,\sigma$. For instance if you have an electron system in a cubic lattice with hopping between nearest neighbours sites, you have $\varepsilon_{k\sigma} = -2t\sum_{\mu} \cos{(\vec{k} \cdot \hat{e}_{\mu})}$ (where $\vec{k}$ is the momentum and $\hat{e}_{\mu}$, $\mu=x,y,z$ is the versor of an axis). Anyway the explicit form is not necessary to solve the exercise.

Solution to your doubt

Your doubt is justified: if you want to give a full quantum treatment of the electron gas coupled to a magnetic field, you not only have to take into account the coupling with the spin $-\mu_B\vec{S}\cdot \vec{B}$, but also the coupling to the orbital angular momentum $-\mu_B\vec{L}\cdot \vec{B}$, which indeed affects $H_0$.

However one can prove that the magnetic susceptibility can be written as $\chi = \chi_L + \chi_S$, where $\chi_S$ is what you get if you consider the spin coupling only, while $\chi_L$ is what you get if you consider the orbital coupling only. A reference is Landau Vol. 5 (which is really precise, but a little tough to understand). In conclusion, the spirit of the exercise is to compute only one of these two contributions, in particular $\chi_S$.

Curiosity

It turns out that $\chi_S$ is positive (paramagnetic term), while $\chi_L$ is negative (diamagnetic term), and that $\chi_L = -\chi_S/3$ if you assume no underlying lattice structure. In conclusion the free electron gas is paramagnetic, and the effect of the term you are neglecting is to reduce the paramagnetic response by a factor $2/3$.

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