On Moore-Read states

I always find quite exciting when fundamental (and sometime abstract) results of pure mathematics and quantum field theory can be directly related to condensed matter or statistical mechanics problems with a clear physical interpretation and motivation. For instance, in our recent paper on “Clustering properties, Jack polynomials and unitary conformal field theories” (arXiv:0904.3702), we study the relation between the characterization of a class of symmetric polynomials with particular analytical properties (the clustering properties, specified later) and the correlation function of conserved currents in 2D massless quantum field theories, the so-called conformal field theories. The results we obtained have been directly inspired and motivated by the study of non-Abelian states in Fractional Quantum Hall systems (FQH).

The basics of FQH have been discussed extensively in many posts. In this respect I mention the post of Zlatko Papic which makes a nice introduction to the use of trial many-body wave functions in FQH. Among these functions, he introduces the Moore-Read states which represent the paradigm for non-Abelian states.

In this post I would like to start from discussing more in details some properties of the Moore-Read trial wave-functions. These are described by the so-called Pfaffian state and represent the simplest realization of pairing between spin polarized electrons.

Consider the polynomial of the complex variables :

.

The above function is a Moore-Read state describing  bosons at filling fraction . To see this, it is convenient to consider the system on a sphere with a radial magnetic field generating a flux . The position of -th particle on the sphere can then be represented as a complex variable which is its stereographic projection. Each particle in the lowest Landau level (LLL) has orbital angular momentum and the single-particle basis state take, besides an unimportant geometric factor, the form  , where  is the angular momentum quantum number. The total flux is .

One can easily verify that the function satisfies the following clustering properties: let be the position where, say, the particles and  meet; the  vanishes as  when the third particle  approaches. Because of this simple property, one can show that the Moore-Read states is the zero-energy ground state of a three-body projection Hamiltonian. This Hamiltonian is believed to capture the physics of the LLL where the effective hamiltonian is reduced to the interaction between particles. A natural generalization of the Moore-Read states is obtained by looking for symmetric polynomials which vanishes when at least  particles approach at the same point. This yields the series of the so-called  Read-Rezayi states which describe bosons at filling fraction  .

Now, the crucial observation is that symmetric polynomials with such properties can be generated by correlation functions of certain conformal field theories (CFTs). Roughly speaking, the approach of the conformal field theory is to compute correlation functions (of a 2D massless quantum field theory) by studying the infinite constraints imposed by the conformal symmetry and by, more in general, other infinite additional symmetries. From the Noether theorem at each symmetry it is associated a set conserved current which act as a generator of this symmetry. These currents generate an algebra whose representations form the Hilbert space of the theory.

Among the different families of CFTs, a central role in condensed matter is played by the so-called parafermionic theories which are CFT with an additional symmetry. Because of the particular form of the symmetry algebra, it turns out that the current correlation functions can be used to generate the polynomials with  clustering properties. The simplest realization of these parafermionic theories produces the  Read-Rezayi states. Note that the connection between FQH/CFT Model is a key point to understand the non-Abelian states as much of the theory underlying these is based on the monodromy properties of the CFT conformal blocks.

After the success of the series of the Read-Rezayi states, there has been an intense activity to scan the other natural direction of generalizing these states. We have seen that the symmetric polynomials associated to the  Read-Rezayi states vanishes as  when the  particles approaches. Now, what happen if now the symmetric polynomials with  clustering properties vanishes more generally as  , integer? This has a direct clear physical interpretation as you are imposing that any cluster of  particles has relative angular momentum less than . These states will then describe bosons at filling fraction .

As said above, the problem of generating such polynomials put in communication two different domains of research: from one side, the classification of well defined parafermionic theories, from the other side the theory of symmetric polynomials spanned by Jack polynomials with negative parameter.

In our paper we have investigated the interplay between these two approaches, putting in evidence some unexpected properties and proposing possible candidate for new non-abelian states… I hope that with this post I attired your attention to give a look at it!

To know more:

1. S.H. Simon, E.H. Rezayi, and N.R. Cooper, Phys. Rev. B 75, 075318 (2007).

2. B.A. Bernevig and F.D.M. Haldane, Phys. Rev. Lett. 100, 246802 (2008).

3. Vl.S. Dotsenko, J.L. Jacobsen and R. Santachiara, Nucl.Phys. B 656 259-324 (2003); Nucl.Phys. B 664 477-511 (2003); Nucl.Phys. B 679 464-494 (2004); Phys.Lett. B 584 186-191 (2004).