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thatboi

- 122

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I'm reading through David Tong's Fractional Quantum Hall Effect notes right now and am stumped by how he constructs the singlet Halperin state (the last equation in this document: https://www.damtp.cam.ac.uk/user/tong/qhe/three.pdf, on page 116 as per the document page number at the bottom of each page). Specifically, I do not understand the sentence "It can be seen to be a spin singlet because the last two factors are just Slater determinants for spin up and spin down respectively, which is guaranteed to form a spin singlet." I assume that the "last two factors" are referring to ##\prod_{i<j \ \text{odd}}(z_{i}-z_{j})## and ##\prod_{k<l \ \text{even}}(z_{k}-z_{l})##. My 2 questions are:

i.) How do we see that these are the slater determinants of spin up and spin down? To me, they just look like the vandermonde determinant we see associated with the Laughlin states.

ii.) Aren't these factors antisymmetric? Wouldn't that the be a problem considering the spin states are already antisymmetric?

Thanks!