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Note for projected coherent state approximation of bosonic FQHE in a spherical geometry

T. Mizusaki

Published Mar 5, 2009• 0 citations
Publisher

Abstract

Recently a bosonic counterpart of the Fractional Quantum Hall Effect (FQHE) has been theoretically pointed out in fast rotating Bose-Einstein Condensates (BEC). In this study, exact diagonalization has played an important role but it can not be applied to a large boson system. To overcome this restriction, a variational method by the spin projected boson coherent state is promising. In this Note, this method is presented in detail.

DOI
10.34360/00001403

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•Jan 1, 2005•0 citations
Recently a bosonic counterpart of the Fractional Quantum Hall Effect (FQHE) has been theoretically pointed out in fast rotating Bose-Einstein Condensates (BEC). In this study, exact diagonalization has played an important role but it can not be applied to a large boson system. To overcome this restriction, a variational method by the spin projected boson coherent state is promising. In this Note, this method is presented in detail.
We use a Chern Simons Landau-Ginzburg (CSLG) framework related to hierarchies of composite bosons to describe 2D harmonically trapped fast rotating Bose gases in Fractional Quantum Hall Effect (FQHE) states. The predicted values for ννν (ratio of particle to vortex numbers) are ννν===pq{{p}\over{q}}qp​ (ppp, qqq are any integers) with even product pqpqpq, including numerically favored values previously found and predicting a richer set of values. We show that those values can be understood from a bosonic analog of the law of the corresponding states relevant to the electronic FQHE. A tentative global phase diagram for the bosonic system for ννν$
We study the quantum Hall phases that appear in the dilute limit of rotating Bose-Einstein condensates. By exact diagonalization in a spherical geometry we obtain the ground state and low-lying excited states of a small number of bosons as a function of the filling fraction nu, the ratio of the number of bosons to the number of vortices. We show the occurrence of the Jain principal sequence of incompressible liquids for nu=2/3,3/4,4/3,5/4 in addition to the Laughlin state nu=1/2 as well as the Pfaffian state for nu=1. We give gap estimates by finite-size scaling of both charged and neutral excitations.
We study the quantum Hall phases that appear in the fast-rotation limit for Bose-Einstein condensates of spinless bosonic atoms. We use exact diagonalization in a spherical geometry to obtain low-lying states of a small number of bosons as a function of the angular momentum. This allows us to understand or guess the physics at a given filling fraction $\ensuremath{\nu}$, the ratio of the number of bosons to the number of vortices. This is also the filling factor of the lowest Landau level. In addition to the well-known Bose-Laughlin state at $\ensuremath{\nu}=1∕2$, we give evidence for the Jain principal sequence of incompressible states at $\ensuremath{\nu}=p∕(p\ifmmode\pm\else\textpm\fi{}1)$ for a few values of ppp. There is a collective mode in these states whose phenomenology is in agreement with standard arguments coming, e.g., from the composite-fermion picture. At filling factor 1, the potential Fermi sea of composite fermions is replaced by a paired state, the Moore-Read state. This is most clearly seen from the half-flux nature of elementary excitations. We find that the hierarchy picture does not extend up to the point of transition towards a vortex lattice. While we cannot come to any definite conclusions, we investigate the clustered Read-Rezayi states and show evidence for incompressible states at the expected ratio of flux versus number of Bose particles.
The close theoretical analogy between the physics of rapidly rotating atomic Bose condensates and the quantum Hall effect (i.e. a two-dimensional electron gas in a strong magnetic field) was first pointed out ten years ago. As a consequence of this analogy, a large number of strongly correlated quantum-Hall-type states have been predicted to occur in rotating Bose systems, and suggestions have been made for how to manipulate and observe their fractional quasiparticle excitations. Due to a very rapid development in experimental techniques over the past years, experiments on BEC now appear to be close to reaching the quantum Hall regime. This paper reviews the theoretical and experimental work done to date in exploring quantum Hall physics in cold bosonic gases. Future perspectives are discussed briefly, in particular the idea of exploiting some of these strongly correlated states in the context of topological quantum computing.
We have studied collective spin density excitations of fractional quantum Hall effect (FQHE) in rotating Bose-Einstein condensation for the three filling fractions of first series of Jain's composite fermion sequences. We have considered, short-ranged contact interactions between the Bose atoms as well as long range Coulomb interactions to compare the nature of the spectras with FQHE of electrons. Using Monte Carlo method for finite but large number of particles, the lowest order collective modes of spin-reversed sectors is calculated here, by computing the energy differences of the respective excitons from the fully polarized ground states.
The system of neutral bosons in rapidly rotating atomic trap is equivalent to charged bosons coupled to a magnetic field, which has opened up the possibility of fractional quantum Hall effect (FQHE) for bosons interacting with a short range interaction. We present numerical evidences for incompressible liquids at filling fractions ν = 1/2, 2/3, 3/4. Motivated by the composite fermion (CF) theory of the FQHE for electrons, we show how interacting bosons can be mapped into non-interacting spinless fermions carrying one vortex each. This provides an elegant explanation of the fractional quantum Hall states in this system. We use overlap calculations to test how the CF model fits with exact numerical results. At ν = 1, the fermionization appears to overcompensate for the repulsive interaction between bosons producing an attractive interaction between fermions, as evidence by a pairing of fermions. In that case, the Pfaffian state is a good candidate to describe the ground state of such system.