Composite fermions on a torus

Abstract

We achieve an explicit construction of the lowest Landau level (LLL) projected wave functions for composite fermions in the periodic (torus) geometry. To this end, we first demonstrate how the vortex attachment of the composite fermion (CF) theory can be accomplished in the torus geometry to produce the "unprojected" wave functions satisfying the correct (quasi-)periodic boundary conditions. We then consider two methods for projecting these wave functions into the LLL. The direct projection produces valid wave functions but can be implemented only for very small systems. The more powerful and more useful projection method of Jain and Kamilla fails in the torus geometry because it does not preserve the periodic boundary conditions and thus takes us out of the original Hilbert space. We have succeeded in constructing a modified projection method that is consistent with both the periodic boundary conditions and the general structure of the CF theory. This method is valid for a large class of states of composite fermions, called "proper states," which includes the incompressible ground states at electron filling factors ν = n 2pn+1, their charged and neutral excitations, and also the quasidegenerate ground states at arbitrary filling factors of the form ν = ν * 2pν * +1 , where n and p are integers and ν * is the CF filling factor. Comparison with exact results known for small systems for the ground and excited states at filling factors ν = 1/3, 2/5 and 3/7 demonstrates our LLL-projected wave functions to be extremely accurate representations of the actual Coulomb eigenstates. Our construction enables the study of large systems of composite fermions on the torus, thereby opening the possibility of investigating numerous interesting questions and phenomena. CONTENTS

Hall viscosity of composite fermions

“…For this purpose we use LLL wave functions constructed in Ref. [22], but appropriately re-expressed in τ -gauge. The numerical results agree with Eq.…”

“…The PWJ projection cannot be represented by an operator, but it is possible to show that the PWJ projected wave functions are also modular covariant; we refer to the work by Fremling [34] for a detailed proof. Interestingly the condition for modular covariance of the wave function is the same as that for the validity of the PWJ projection, namely that the states are proper states, where proper states correspond to configurations for which there are no unoccupied CF-orbitals directly beneath an occupied CF-orbital [22]. (The PWJ projection does not preserve the quasi-periodic boundary conditions for non-proper states.…”

“…The wave functions for general FQH states at ν = n/(2pn ± 1) in Ref. [22] and for the CF Fermi sea in Refs. [38][39][40][41] were first constructed in the symmetric gauge.…”

“…It is now straightforward to apply the modified PWJ projection [35,36] as shown in Ref. [22]. For the ν = n 2pn+1 states, the full LLL wave function is written as:…”

“…These were originally constructed in Ref. 22, but here we use a slightly different (though equivalent) form, using the so-called τ -gauge, which is crucial for our analytical proof. (Both the symmetric and the τ gauges are equally good for Monte Carlo evaluations.)…”