I Singlet Halperin State Construction

  • I
  • Thread starter Thread starter thatboi
  • Start date Start date
  • Tags Tags
    Singlet State
thatboi
Messages
130
Reaction score
20
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!
 
I am not sure if this falls under classical physics or quantum physics or somewhere else (so feel free to put it in the right section), but is there any micro state of the universe one can think of which if evolved under the current laws of nature, inevitably results in outcomes such as a table levitating? That example is just a random one I decided to choose but I'm really asking about any event that would seem like a "miracle" to the ordinary person (i.e. any event that doesn't seem to...
Not an expert in QM. AFAIK, Schrödinger's equation is quite different from the classical wave equation. The former is an equation for the dynamics of the state of a (quantum?) system, the latter is an equation for the dynamics of a (classical) degree of freedom. As a matter of fact, Schrödinger's equation is first order in time derivatives, while the classical wave equation is second order. But, AFAIK, Schrödinger's equation is a wave equation; only its interpretation makes it non-classical...
Back
Top