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arXiv:0807.1034 · Physical Review A

Pairing in ultracold Fermi gases in the lowest Landau level

Gunnar Möller, Th. Jolicœur, Nicolas Regnault

Published Mar 11, 2009• 7 citations
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Abstract

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 $s$-wave and $p$-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 $s$- and $p$-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 $\ensuremath{\nu}=2/5$.

DOI
10.1103/physreva.79.033609

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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 \ensuremathν=2/3\ensuremath{\nu}=2/3\ensuremathν=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 \ensuremathν=1/2\ensuremath{\nu}=1/2\ensuremathν=1/2 and the permanent state of Moore and Read for \ensuremathν=1\ensuremath{\nu}=1\ensuremathν=1. For purely repulsive interactions, we also find evidence for a gapped Halperin state at \ensuremathν=2/5\ensuremath{\nu}=2/5\ensuremathν=2/5.
In the limit of very fast rotation atomic Bose-Einstein condensates may entirely reside in the lowest two-dimensional Landau level (LLL). For small enough filling factor of the LLL, one may have formation of fractional quantum Hall states. We investigate the case of bosons with dipolar interactions as may be realized with chromium-52 atoms. We show that at a filling factor equal to unity the ground state is a Moore-Read (or Pfaffian) paired state as is the case of bosons with purely sss-wave scattering interactions. This Pfaffian state is destabilized when the interaction in the sss-wave channel is small enough and the ground state is a stripe phase with unidimensional density modulation. For filling factor 1/3, we show that there is formation of a Fermi sea of ``composite fermions.'' These composites are made of one boson bound with three vortices. This phase has a wide range of stability and the effective mass of the fermions essentially depends only on the scattering amplitude in momentum channels larger or equal to 2. The formation of such a Fermi sea opens up a new possible route to detection of the quantum Hall correlations.
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 $\ensuremath{\nu}=1∕3$. 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.
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 study the physics of a rapidly rotating gas of ultracold atomic bosons, with an internal degree of freedom. We show that in the limit of rapid rotation of the trap the problem exactly maps onto that of noninteracting fermions with spin in the lowest Landau level. The spectrum of the real bosonic system is identical to the one of the effective fermions, with the same eigenvalues and the same density of states. When the ratio of the number of atoms to the spin degeneracy is an integer number, the ground state for the effective fermions is an integer quantum Hall state. The corresponding bosonic state is a fractional quantum Hall liquid whose filling factor ranges in the sequence $\ensuremath{\nu}=1/2,2/3,3/4,\dots{},$ as the spin degeneracy increases. Anyons with 1/2,1/3,1/4,…1/2,1/3,1/4,\dots{}1/2,1/3,1/4,… statistics can be created by inserting lasers with the appropriate polarizations. A special situation appears when the spin degeneracy equals the number of atoms in the gas. The ground state is then the product of a completely antisymmetric spin state and a $\ensuremath{\nu}=1$ Laughlin state. In this case the system exhibits fermionic excitations with fermionic statistics although the real components are bosonic atoms.
We study 'second generation' of fractional quantum Hall states which contain a partially filled composite fermion (CF) Landau level (LL). We explain the role of CF-CF interaction in the incompressibility of the underlying quantum electron liquid. In particular, using exact diagonalization on a Haldane sphere, we determine two- and three-body CF correlation functions for these liquids and show that the CFs form a paired state (instead of a Laughlin liquid) when filling v = 1/3 of their second LL. At v = 1/2 we show that the CFs group into even larger clusters rather than form a Moore—Read paired state. We also address the problem of spin polarization of interacting CFs and predict transition to the partially unpolarized ground state in sufficiently narrow quantum wells and/or weak magnetic fields.
We investigate strongly correlated phases of two-component (or pseudospin-1/21/21/2) Bose gases under rapid rotation through exact diagonalization on a torus geometry. In the case of pseudospin-independent contact interactions, we find the formation of gapped spin-singlet states at the filling factors $\ensuremath{\nu}=k/3+k/3$ (k/3k/3k/3 filling for each component) with integer kkk. We present numerical evidences that the gapped state with k=2k=2k=2 is well described as a non-Abelian spin-singlet (NASS) state, in which excitations feature non-Abelian statistics. Furthermore, we find the phase transition from the product of composite fermion states to the NASS state by changing the ratio of the intercomponent-to-intracomponent interactions.