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arXiv:cond-mat/0404093 · Physical Review B

Quantum Hall fractions in ultracold fermionic vapors

Nicolas Regnault, Th. Jolicœur

Published Dec 1, 2004• 24 citations
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Abstract

We study the quantum Hall states that appear in the dilute limit of rotating ultracold fermionic gases when a single hyperfine species is present. We show that the $p$-wave scattering translates into a pure hard-core interaction in the lowest Landau level. The Laughlin wave function is then the exact ground state at filling fraction $\ensuremath{\nu}=1∕3$. We give estimates of some of the gaps of the incompressible liquids for $\ensuremath{\nu}=p∕(2p\ifmmode\pm\else\textpm\fi{}1)$. We estimate the mass of the composite fermions at $\ensuremath{\nu}=1∕2$. The width of the quantum Hall plateaus is discussed by considering the equation of state of the system.

DOI
10.1103/physrevb.70.241307

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We study the quantum Hall states that appear in the dilute limit of rotating ultracold fermionic gases when a single hyperfine species is present. We show that the ppp-wave scattering translates into a pure hard-core interaction in the lowest Landau level. The Laughlin wave function is then the exact ground state at filling fraction \ensuremathν=1∕3\ensuremath{\nu}=1∕3\ensuremathν=1∕3. We give estimates of some of the gaps of the incompressible liquids for \ensuremathν=p∕(2p\ifmmode±\else\textpm\fi1)\ensuremath{\nu}=p∕(2p\ifmmode\pm\else\textpm\fi{}1)\ensuremathν=p∕(2p\ifmmode±\else\textpm\fi1). We estimate the mass of the composite fermions at \ensuremathν=1∕2\ensuremath{\nu}=1∕2\ensuremathν=1∕2. The width of the quantum Hall plateaus is discussed by considering the equation of state of the system.
Atomic vapors can be prepared and manipulated at very low densities and temperatures. When they are rotating, they can reach a quantum Hall regime in which there should be manifestations of the fractional quantum Hall effect. We discuss the appearance of the principal sequence of fractions nu =p/(p+- 1) for bosonic atoms. The termination point of this series is the paired Moore-Read Pfaffian state. Exotic states fill the gap between the paired state and the vortex lattice expected at high filling of the lowest Landau level. In fermionic vapors, the p-wave scattering typical of ultralow energy collisions leads to the hard-core model when restricted to the lowest Landau level.
Atomic vapors can be prepared and manipulated at very low densities and temperatures. When they are rotating, they can reach a quantum Hall regime in which there should be manifestations of the fractional quantum Hall effect. We discuss the appearance of the principal sequence of fractions ν=p/(p±1) for bosonic atoms. The termination point of this series is the paired Moore–Read Pfaffian state. Exotic states fill the gap between the paired state and the vortex lattice expected at high filling of the lowest Landau level. In fermionic vapors, the p-wave scattering typical of ultralow energy collisions leads to the hard-core model when restricted to the lowest Landau level.
We study a rapidly rotating gas of unpolarized spin-1/2 ultracold fermions in the two-dimensional regime when all atoms reside in the lowest Landau level. Due to the presence of the spin degree of freedom both sss-wave and ppp-wave interactions are allowed at ultralow temperatures. We investigate the phase diagram of this system as a function of the filling factor in the lowest Landau level and in terms of the ratio between sss- and ppp-wave interaction strengths. We show that the presence of attractive interactions induces a wide regime of phase separation with formation of maximally compact droplets that are either fully polarized or composed of spin-singlets. In the regime with no phase separation, we give evidence for fractional quantum Hall states. Most notably, we find two distinct singlet states at the filling $\ensuremath{\nu}=2/3$ for different interactions. One of these states is accounted for by the composite fermion theory, while the other one is a paired state for which we identify two competing descriptions with different topological structures. This paired state may be an Abelian liquid of composite spin-singlet Bose molecules with Laughlin correlations. Alternatively, it may be a known non-Abelian paired state, indicated by good overlaps with the corresponding trial wave function. By fine tuning of the scattering lengths it is possible to create the non-Abelian critical Haldane-Rezayi state for $\ensuremath{\nu}=1/2$ and the permanent state of Moore and Read for $\ensuremath{\nu}=1$. For purely repulsive interactions, we also find evidence for a gapped Halperin state at $\ensuremath{\nu}=2/5$.
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 ultracold Rydberg-dressed Bose gases subject to artificial gauge fields in the fractional quantum Hall (FQH) regime. The characteristics of the Rydberg interaction gives rise to interesting many-body ground states different from standard FQH physics in the lowest Landau level (LLL). The non-local but rapidly decreasing interaction potential favors crystalline ground states for very dilute systems. While a simple Wigner crystal becomes energetically favorable compared to the Laughlin liquid for filling fractions ν<1/12\nu<1/12ν<1/12, a correlated crystal of composite particles emerges already for ν≤1/6\nu \leq 1/6ν≤1/6 with a large energy gap to the simple Wigner crystal. The presence of a new length scale, the Rydberg blockade radius aBa_BaB​, gives rise to a bubble crystal phase for ν≲1/4\nu\lesssim 1/4ν≲1/4 when the average particle distance becomes less than aBa_BaB​, which describes the region of saturated, almost constant interaction potential. For larger fillings indications for strongly correlated cluster liquids are found.
We demonstrate the experimental feasibility of incompressible fractional quantum Hall-like states in ultracold two-dimensional rapidly rotating dipolar Fermi gases. In particular, we argue that the state of the system at filling fraction nu = 1/3 is well described by the Laughlin wave function and find a substantial energy gap in the quasiparticle excitation spectrum. Dipolar gases, therefore, appear as natural candidates of systems that allow us to realize these very interesting highly correlated states in future experiments.