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I know what paramagnetism is. But first I want to know about the paramagnetic current and then the above-mentioned correlation?

Actually, I am working on a paper on superconductivity where I have seen the term: Scalapino, White, and Zhang.

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Typically the current operator is a sum of to parts: a so-called "paramagnetic" term and a "diamagnetic" one. This is just notation, and bears no real meaning, since the true, physical current operator is the sum of them both.

For example, consider an electron gas in an external electromagnetic field:

$H=\int d^3 x \frac{1}{2m} \psi^\dagger (\mathbf x) (-i \nabla -e/c \mathbf A(\mathbf x,t))^2 \psi (\mathbf x) + e \phi(\mathbf x,t) \psi^\dagger (\mathbf x) \psi (\mathbf x)$

Then the current operator is

$\mathbf J(\mathbf x)= \frac{e}{2m} \psi^\dagger (\mathbf x) (-i \nabla -e/c \mathbf A(\mathbf x,t)) \psi (\mathbf x) +h.c$ (where h.c is hermitian conjugate)

The term in the current operator which is proportional to A is the diamagnetic term, the other is the paramagnetic one. As you can see, each term on its own is not gauge invariant (there is an explicit dependence on A in the diamagnetic one!). Only the sum is.

The paramagnetic current correlation function is just the correlation function of the paramagnetic parts of the current operator. The correlation function often appears when things are calculated to first order in A.

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