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Two Basic Conjectures in Fractional Quantum Hall Effect: A Mathematical Inquiry

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I will apologize for the length of the question beforehand. Let me first start by stating the physical motivation quickly and then straight to mathematics:

In theory of "(fractional) quantum Hall effect'', the circumstances force the wavefunction to be of the following form $$ \Psi(z_1, \cdots, z_N)=P(z_1, \cdots, z_N)e^{\frac{\sum_{i=1}^N |z_i|^2}{4}} $$ where $z_i\in \mathbb{C}$, $N$ is the number of particles (variables) and most importantly $P(z_1, \cdots, z_N)$ is a polynomial in $N$ variables such that

  • First of all $N\gg 1$ is supposed to be very large. In fact physicists are interested in thermodynamic limit $N\to \infty$ but here this limit (mathematically) is not well-defined at all, so we stick with $N\gg 1$.
  • The coefficients are themselves another point of controversy. All the examples I have seen have coefficients in $\mathbb{Z}$, i.e. $P\in \mathbb{Z}[z_1, \cdots, z_N]$. In fact there are speculations about possible relations between these polynpmials and "Jack polynomials". In any case in the questions which follow feel free to assume the coefficients are in either in $\mathbb{Z}$ or $\mathbb{Q}$ (whichever fits best).
  • And finally $P$ is a homogeneous + symmetric + translation invariant polynomial, where translation invariance means for any $c\in \mathbb{C}$ we have $f(z_1+c, \cdots, z_N+c)=f(z_1, \cdots, z_N)$.

As usual $|\Psi|^2$ should be interpreted as joint probability distribution of $N$-particles after normalization. Hence one defines a "single-particle density" to be $$ \rho(z)=\frac{\int \left(\prod_{i=2}^N dz_i\right) |\Psi(z, z_2, \cdots, z_N)|^2}{\int \left(\prod_{i=1}^N dz_i\right) |\Psi(z_1, z_2, \cdots, z_N)|^2} $$ Physically it is expected for $\rho(z)$ to have the following shape:

enter image description here

where the plane is the complex plane, the center of the disk is the origin and the decay starts far away on a radius $\propto N$ (or maybe a power of $N$?). So here comes the first of the pair of (kind of forgotten) conjectures:

Conjecture I: The density profile of a (potential) FQH state has rotational symmetry in complex plane, and there exists (a maximal) $\zeta\in \mathbb{C}$, proportional to $N\gg 1$, such that $\rho(z)$ is uniform inside $|z|<|\zeta|$ and decays rapidly outside.

Another important physical quantity in theory of FQH effect is filling fraction which is defined as $\nu= 2\pi \rho(0)$. It is known as a trade mark of FQH effect that $\nu$ is rational number less than one, i.e. $\nu\in \mathbb{Q}$. In fact not just any rational number, but only a very special subset of rational numbers (not important to us). So the second conjecture is

Conjecture II: The filling fraction $\nu=2\pi \rho(0)$ is a rational number (of course only when $N\gg 1$).

My purpose for giving you a headache by this long question is

My Question from You: I suspect that there might be some tools in Complex Analysis + Analytic/Algebraic Complex Geometry with which one can (at least partially) address these problems. Does anyone know of the right framework to study whether (under what circumstances) these assertions are (approximately) true.

Two known issues:

  • I say "approximately" because these conjectures were designed for thermodynamic limit $N\to \infty$ but $N$ here is the number of variables of a polynomial! You can't take such limit (it is meaningless done naively). What physicists mean by thermodynamic limit in this problem is: There is a sequence of polynomials $P_{N_1}, P_{N_2}, \cdots$ with $N_1<N_2< \cdots$ and the limit $N\to \infty$ is done in this sense. It is clear in specific examples of FQH states how to define this sequence but it is not at all (at least to me) how to define this sequence in the abstract, for all (potential) FQH polynomials. I can explain how this sequencing is done in examples if it is needed (Ask!)

  • Secondly I say "under which circumstances" because these conjectures were designed for "physically meaningful" FQH states. We introduced potential FQH states to be homogeneous+symmetric+translation invariant. It is known that any physical FQH state (solution to a many-body Hamiltonian) is of this form, but not all potential FQH states are physical (not all are solution to a Hamiltonian). So the above conjectures may work on a subclass of potential FQH states.

Finally my motivation for finding the answer to this question (which most people left alone) is I'm in the process of constructing a (meaningful) moduli space (like a appropriate variation of a Hilbert scheme) for the family of potential FQH states. I have reason to believe that if this family is described as $\phi: E\to M$, then geometrically $\nu$ is strongly correlated to the connected components of $E$. The answer of this question might be what I need to solidify this connection.

Thanks for your patience! P.S. my "tags" of choice may not be the most appropriate, feel free to change them.


This post imported from StackExchange MathOverflow at 2016-09-04 15:21 (UTC), posted by SE-user Hamed

asked Aug 30, 2016 in Mathematics by Hamed (20 points) [ revision history ]
recategorized Sep 4, 2016 by Dilaton
Is homogeneity of $P$ of some fixed degree $d$ or $d$ may vary with $N$? If it may vary, then how exactly does it depend on $N$?

This post imported from StackExchange MathOverflow at 2016-09-04 15:21 (UTC), posted by SE-user fedja
@fedja the degree $d$ of $P$ is dependent on $N$ in a major way. In the example of Laughlin states for example ($\prod_{i<j} (x_i-x_j)^{2m}$) we have $d\propto {N\choose 2}$ (order $N^2$), there is another example Pfaffian $\nu=1$ state with $d\propto N$. In general there is no limitation on how $d$ should depend on $N$; maybe one should assume $d\sim N^p$ for some integer $p$ ($p$ can also be zero in an uninteresting yet possible case).

This post imported from StackExchange MathOverflow at 2016-09-04 15:21 (UTC), posted by SE-user Hamed
Then it looks like the homogeneous+translation invariant+symmetric assumptions are not enough and we will need to use some special features of $P$. Are you interested in the particular polynomials you mentioned for Laughlin states or that case is somehow trivial?

This post imported from StackExchange MathOverflow at 2016-09-04 15:21 (UTC), posted by SE-user fedja
Oh there is no doubt these tri-conditions are not enough (probably not even close)! It's all we've got (till now) though :) Laughlin states have been studied completely. If you think together with some extra conditions one can say more, feel free to insert those extra conditions. In fact this may shed some light on what extra conditions are necessary for a full description.

This post imported from StackExchange MathOverflow at 2016-09-04 15:21 (UTC), posted by SE-user Hamed
In fact I think you are on to something here... One definitely needs some bounds on how $d$ (which is physically the total angular momentum) may depend on $N$. I don't know of any, but I can ask. Intuitively I'm tempted to say $d$ is always at most $O(N^2)$ but I'm not sure (I'll ask).

This post imported from StackExchange MathOverflow at 2016-09-04 15:21 (UTC), posted by SE-user Hamed

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