Singlet Halperin State Construction

In summary, the conversation discusses the construction of the singlet Halperin state in David Tong's Fractional Quantum Hall Effect notes. The confusion lies in 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." The last two factors refer to ##\prod_{i<j \ \text{odd}}(z_{i}-z_{j})## and ##\prod_{k<l \ \text{even}}(z_{k}-z_{l})##, which are the slater determinants for spin up and spin down. However, there is confusion about whether these factors are antisymmetric, which could potentially cause a problem since the
  • #1
thatboi
122
18
Hi all,
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!
 

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