Lowest-Landau-Level Projection

I. V. Tokatly

Mikael Fremling, Jørgen Fulsebakke, Niall Moran +1 more

We develop a method to efficiently calculate trial wave functions for quantum Hall systems which involve projection onto the lowest Landau level. The method essentially replaces the lowest Landau level projection by projection onto the $M$ lowest eigenstates of a suitably chosen Hamiltonian acting within the lowest Landau level. The resulting ``energy projection'' is a controlled approximation to the exact lowest Landau level projection which improves with increasing $M$. It allows us to study the projected trial wave functions for system sizes close to the maximal sizes that can be reached by exact diagonalization and can be straightforwardly applied in any geometry. As a first application and test case, we study a class of trial wave functions first proposed by Girvin and Jach [Girvin and Jach, Phys. Rev. B 29, 5617 (1984)], which are modifications of the Laughlin states involving a single real parameter. While these modified Laughlin states probably represent the same universality class exemplified by the Laughlin wave functions, we show by extensive numerical work for systems on the sphere and torus that they provide a significant improvement of the variational energy, overlap with the exact wave function and properties of the entanglement spectrum.

Jurgen Michael Honig, Józef Spałek

Jiangke Yang

Recently we proposed a state described by the second Landau level (SLL) projection of the antiholomorphic Pfaffian wavefunction as a candidate for the ground state of the 5/2 fractional quantum Hall effect. In this paper we provide a mathematical proof that, when mapped to the lowest Landau level (LLL), the aforementioned state, which we call the SLL PH-Pfaffian state, becomes exactly a LLL projected orbital angular momentum l = -3 pairing Pfaffian state, which we call the LLL antiholomorphic f-wave pairing Pfaffian state, or the LLL af-Pfaffian state for short. We further prove that there is also an exact mapping between the upstream neutral Majorana fermion edge mode of the SLL PH-Pfaffian state and that of the LLL af-Pfaffian state. However, we find that while there exists an upstream neutral boson edge mode in the LLL af-Pfaffian state, it has no mapped counterpart in the SLL PH-Pfaffian state.

E. Onofri

Abhishek Anand, G. J. Sreejith

Real-space entanglement spectrum (RSES) of a quantum Hall (QH) wave function gives a natural route to infer the nature of its edge excitations. Computation of RSES becomes expensive with increase in number of particles and number of included Landau levels (LL). RSES can be efficiently computed using Monte Carlo (MC) methods for trial states that can be written as products of determinants such as the composite fermion (CF) and parton states. This computational efficiency applies also to the RSES of lowest Landau level (LLL) projected CF and parton states, however, LLL projection to be used here requires approximations that generalize the Jain Kamilla (JK) projection. This work is a careful study of how this approximation should be done. We identify the approximation that is closest in spirit to JK projection, and perform tests of the approximations involved in the projection by comparing the MC results with the RSES obtained from computationally expensive but exact methods. We present the techniques and use them to calculate exact RSES of the exact LLL projected bosonic Jain $2/3$ state in bipartition of systems of sizes up to $N=24$ on the sphere. For the lowest few angular momentum sectors of the RSES, we present evidence to show that MC results closely match the exact spectra. We also compare other plausible projection schemes. We also calculate the exact RSES of the unprojected fermionic Jain $2/5$ state obtained from exact diagonalization of the Trugman-Kivelson Hamiltonian in the two lowest LLs on the sphere. By comparing with the RSES of the unprojected $2/5$ state from Monte Carlo methods, we show that the latter is practically exact.