On Moore-Read states
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Raoul Santachiara is a stuff member at Laboratoire de Physique Theorique et Modeles Statistiques, Universite de Paris-Sud. His interests include statistical and mathematical physics, critical phenomena, disordered systems and entanglement properties of many-body systems. Dmitry.
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).
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Dear Raoul,
thanks for the great understandable post! Let me ask you a couple of naive questions:
1) if we focus on filling factor
, which state is more energetically favorable – Laughlin or Moore-Read? (I thought that Laughlin’s is proved to be the lowest energy one by variational methods)
2) why should we ultimately focus only on three-body projection Hamiltonian? Is the logic behind 3-body that
where 
is even?
it seems to be possible to describe non-abelian statistics within 3-body Hamiltonian states,therefore, it might be relevant for describing FQHE with
3) related to (1) and (2): if Laughlin w.f. is energietically favorable for odd
, while Moore-Read – for even 
, why 3-body interactions are important for even 
?
Cheers,
Dmitry.
Thanks for your questions which are far from being naives! (hope I will not have you in some of my seminars
)
Let me answer you by remembering the general idea behind this line of research..The point in this research is to characterize distinct phases of matter. I recall that, at a given filling factor, you can find FQH states described in the asymptotic low-energy limit, and thus independent of the microscopic details details of the effective Hamiltonian and of the wave-function, by different universality classes,i.e. ground-state quantum number, properties of excitations etc etc.
For instance it turns out that the filling 2+1/3 (so the first LL at filling 1/3)the Laughlin states have not a good overlap with the exact ground state. It has been then necessary to investigate other trial wave-functions, as the paired ones which form Lauglins States by pairing electrons, which are representative of other universality classes.
All this just to say that, as in the case of the study of the critical point of statistical models, I cannot in general say, on the basis of a the microscopic hamiltonian, which is the universality class describing my system. So one tries to classify the possible classes on the basis of general “symmetries”, as the properties to be zero-energy eigenstate of specific pseudi-potential hamiltonian..
Dear Raoul,
thanks for the explanations! I guess the ultimate question is why FQHE with odd and even q belong to different universality classes (why they are so amazingly different in other words), but probably nobody has an answer to this question nowadays…
Cheers,
Dmitry.