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arXiv:1006.0453 · Physical Review Letters

Non-Abelian Anyons from Degenerate Landau Levels of Ultracold Atoms in Artificial Gauge Potentials

Michele Burrello, Andrea Trombettoni

Published Sep 17, 2010• 50 citations
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Abstract

We show that non-Abelian potentials acting on ultracold gases with two hyperfine levels can give rise to ground states with non-Abelian excitations. We consider a realistic gauge potential for which the Landau levels can be exactly determined: The non-Abelian part of the vector potential makes the Landau levels nondegenerate. In the presence of strong repulsive interactions, deformed Laughlin ground states occur in general. However, at the degeneracy points of the Landau levels, non-Abelian quantum Hall states appear: These ground states, including deformed Moore-Read states (characterized by Ising anyons as quasiholes), are studied for both fermionic and bosonic gases.

DOI
10.1103/physrevlett.105.125304

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We show that non-Abelian potentials acting on ultracold gases with two hyperfine levels can give rise to ground states with non-Abelian excitations. We consider a realistic gauge potential for which the Landau levels can be exactly determined: The non-Abelian part of the vector potential makes the Landau levels nondegenerate. In the presence of strong repulsive interactions, deformed Laughlin ground states occur in general. However, at the degeneracy points of the Landau levels, non-Abelian quantum Hall states appear: These ground states, including deformed Moore-Read states (characterized by Ising anyons as quasiholes), are studied for both fermionic and bosonic gases.
We study ultracold atoms subjected to U(2) non-Abelian potentials: we consider gauge potentials having, in the Abelian limit, degenerate Landau levels and we then investigate the effect of general homogeneous non-Abelian terms. The conditions under which the structure of degenerate Landau levels is preserved are classified and discussed. The typical gauge potentials preserving the Landau levels are characterized by a fictitious magnetic field and by an effective spin-orbit interaction (e.g., obtained through the rotation of two-dimensional atomic gases coupled with a tripod scheme). The single-particle energy spectrum can be analytically determined for a class of gauge potentials, whose physical implementation is discussed. The corresponding Landau levels are deformed by the non-Abelian contribution of the potential and their spin degeneracy is split. The related deformed quantum Hall states for fermions and bosons (in the presence of strong intraspecies interaction) are determined far from and at the degeneracy points of the Landau levels, where non-Abelian states appear. We present a discussion of the effect of the angular momentum, as well as results for U(3) gauge potentials.
The present thesis studies a variety of cold atomic systems in artificial gauge fields. In the first part we focus on fractional quantum Hall effects, asking whether interesting topological states can be realized with cold atoms. We start by making a close connection to solid-state systems and first consider fermionic atoms with dipolar interactions. Assuming the system to be in the Laughlin state, we evaluate the energy gap in the thermodynamic limit as a measure for the robustness of the state. We show that it can be increased by additionally applying a non-Abelian gauge field squeezing the Landau levels. We then switch to bosonic systems with repulsive contact interactions. Artificial magnetic fields for cold bosons have extensively been discussed before in the context of rotating Bose gases. We follow a different approach where the gauge field is due to an atom-laser coupling. Thus, transitions between different dressed states have to be included. They are shown to break the cylindrical symmetry of the system. Modifying the Laughlin state and the Moore-Read state accordingly, we determine the parameter regimes where they are good representations for the ground state of the system obtained via exact diagonalization. One of the most interesting feature of fractional quantum Hall states is the anyonic behavior of their excitations. We therefore also study quasiholes in the Laughlin state and the modified Laughlin state. They are shown to posses anyonic properties, which become manifest even in small systems. Moreover, the dynamics of a single quasihole causes visible traces in the density of the system which allow to clearly distinguish the Laughlin regime from less correlated phases. In the latter, a sequence of collapses and revivals of the quasihole can be observed, which is absent in the Laughlin regime. Extending our study to bosonic systems with a pesudospin-1/2 degree of freedom, we discuss the formation of strongly correlated spin singlets. Strikingly, at filling v=4/3, the system is described by a state with non-Abelian excitations, which is constructed as the zero-energy ground state of repulsive three-body contact interactions. Systems with internal degrees of freedom also allow for implementing artificial spin-orbit coupling. It is shown to give rise to a variety of incompressible states. In the second part of the thesis, we concentrate on condensed system. Bose-Einstein condensates with spin-orbit coupling are shown to have a degeneracy on the mean-field level, which is lifted by quantum and thermal fluctuations. The system becomes experimentally feasible in three dimensions, where the condensate depletion remains finite, and thus allow for an experimental observation of this order-by-disorder mechanism. Finally, we study the influence of Abelian and non-Abelian gauge fields on the quantum phase transitions of bosons in a square optical lattice. Re-entrant superfluid phases and superfluids at finite momenta are interesting properties featured by such systems.
We analyze the strongly correlated regime of a two-component trapped ultracold fermionic gas in a synthetic non-Abelian U(2)U(2)U(2) gauge potential, that consists of both a magnetic field and a homogeneous spin-orbit coupling. This gauge potential deforms the Landau levels (LLs) with respect to the Abelian case and exchanges their ordering as a function of the spin-orbit coupling. In view of experimental realizations, we show that a harmonic potential combined with a Zeeman term gives rise to an angular momentum term, which can be used to test the stability of the correlated states obtained through interactions. We derive the Haldane pseudopotentials (HPs) describing the interspecies contact interaction within a lowest LL approximation. Unlike ordinary fractional quantum Hall systems and ultracold bosons with short-range interactions in the same gauge potential, the HPs for sufficiently strong non-Abelian fields show an unconventional nonmonotonic behavior in the relative angular momentum. Exploiting this property, we study the occurrence of new incompressible ground states as a function of the total angular momentum. In the first deformed Landau level (DLL) we obtain Laughlin and Jain states. Instead, in the second DLL three classes of stabilized states appear: Laughlin states, a series of intermediate strongly correlated states, and finally vortices of the integer quantum Hall state. Remarkably, in the intermediate regime, the nonmonotonic HPs of the second DLL induce two-particle correlations which are reminiscent of paired states such as the Haffnian state. Via exact diagonalization in the disk geometry, we compute experimentally relevant observables such as density profiles and correlations, and we study the entanglement spectra as a further tool to characterize the obtained strongly correlated states.
Non-Abelian anyons, strange particles forming the basis of fault-tolerant topological quantum computing, are predicted to emerge in fractional quantum Hall states at even-denominator filling factors. Identifying which topological order is realized has proven to be a challenging problem that requires an accurate, non-perturbative treatment of realistic Landau level mixing, which has so far been missing from theoretical studies. Here we present the first complete solution of non-Abelian quantum Hall problem that includes all electrons and all Landau levels, for spin-polarized electrons at filling factor ν=3/2ν= 3/2ν=3/2. We develop a momentum-resolved neural-network variational wave function, optimized solely on the microscopic Hamiltonian, which gives direct access to the topologically degenerate ground states. Without any physics input about the expected outcome, it discovers a degenerate ground-state manifold whose entanglement reveals anyons with quantum dimension 2\sqrt{2}2​, the hallmark of Ising-type non-Abelian order. By directly computing the overlap with trial wave functions, we further show that Landau level mixing favors the anti-Pfaffian over other candidate states. Our results demonstrate the power of neural-network wave functions in finding and characterizing non-Abelian anyons from first principles, in regimes of strong Landau level mixing, relevant to many recent experiments.
A set of localized, non-Abelian anyons—such as vortices in a p x +i p y superconductor or quasiholes in certain quantum Hall states—gives rise to a macroscopic degeneracy. Such a degeneracy is split in the presence of interactions between the anyons. Here, we show that in two spatial dimensions this splitting selects a unique collective state as ground state of the interacting many-body system. This collective state can be a novel gapped quantum liquid nucleated inside the original parent liquid (of which the anyons are excitations). This physics is of relevance for any quantum Hall plateau realizing a non-Abelian quantum Hall state when moving off the center of the plateau.
We study the fractional quantum Hall phases of a pseudospin-1/2 Bose gas in an artificial gauge field. In addition to an external magnetic field, the gauge field mimics an intrinsic spin–orbit coupling of the Rashba type. While the spin degeneracy of the Landau levels is lifted by the spin–orbit coupling, the crossing of two Landau levels at certain coupling strengths gives rise to a new degeneracy. We therefore take into account two Landau levels and perform exact diagonalization of the many-body Hamiltonian. We study and characterize the quantum Hall phases which occur in the vicinity of the degeneracy point. Notably, we describe the different states appearing at the Laughlin fillings, ν = 1/2 and ν = 1/4. While for these filling factors incompressible phases disappear at the degeneracy point, we find gaps in the spectra of denser systems at ν = 3/2 and ν = 2. For filling factors ν = 2/3 and ν = 4/3, we discuss the connection of the exact ground states to the non-Abelian spin singlet states, obtained as the ground states of ( k + 1)-body contact interactions.