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arXiv:1506.08461 · Physical Review B

Chiral currents in one-dimensional fractional quantum Hall states

Eyal Cornfeld, Eran Sela

Published Sep 29, 2015• 65 citations
arXivHTMLPDF

Abstract

We study bosonic and fermionic quantum two-leg ladders with orbital magnetic flux. In such systems, the ratio $\ensuremath{\nu}$ of particle density to magnetic flux shapes the phase space, as in quantum Hall effects. In fermionic (bosonic) ladders, when $\ensuremath{\nu}$ equals one over an odd (even) integer, Laughlin fractional quantum Hall (FQH) states are stabilized for sufficiently long-ranged repulsive interactions. As a signature of these fractional states, we find a unique dependence of the chiral currents on particle density and on magnetic flux. This dependence is characterized by the fractional filling factor $\ensuremath{\nu}$, and forms a stringent test for the realization of FQH states in ladders, using either numerical simulations or future ultracold-atom experiments. The two-leg model is equivalent to a single spinful chain with spin-orbit interactions and a Zeeman magnetic field, and results can thus be directly borrowed from one model to the other.

DOI
10.1103/physrevb.92.115446

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We study bosonic and fermionic quantum two-leg ladders with orbital magnetic flux. In such systems, the ratio \ensuremathν\ensuremath{\nu}\ensuremathν of particle density to magnetic flux shapes the phase space, as in quantum Hall effects. In fermionic (bosonic) ladders, when \ensuremathν\ensuremath{\nu}\ensuremathν equals one over an odd (even) integer, Laughlin fractional quantum Hall (FQH) states are stabilized for sufficiently long-ranged repulsive interactions. As a signature of these fractional states, we find a unique dependence of the chiral currents on particle density and on magnetic flux. This dependence is characterized by the fractional filling factor \ensuremathν\ensuremath{\nu}\ensuremathν, and forms a stringent test for the realization of FQH states in ladders, using either numerical simulations or future ultracold-atom experiments. The two-leg model is equivalent to a single spinful chain with spin-orbit interactions and a Zeeman magnetic field, and results can thus be directly borrowed from one model to the other.
Helical liquids have been experimentally realized in both nanowires and ultracold atomic chains as the result of strong spin–orbit interactions. In both cases the inner degrees of freedom can be considered as an additional space dimension, providing an interpretation of these systems as chiral synthetic ladders, with artificial magnetic fluxes determined by the spin–orbit terms. In this work, we characterize the helical state which appears at filling ν = 1/2: this state is generated by a gap arising in the spin sector of the corresponding Luttinger liquid and it can be interpreted as the one-dimensional (1D) limit of a fractional quantum Hall state of bosonic pairs of fermions. We study its main features, focusing on entanglement properties and correlation functions. The techniques developed here provide a key example for the study of similar quasi-1D systems beyond the semiclassical approximation commonly adopted in the description of the Laughlin-like states.
Ladder models of ultracold atoms offer a versatile platform for the experimental and theoretical study of different phenomena and phases of matter linked to the interplay between artificial gauge fields and interactions.Strongly correlated helical states are known to appear for specific ratios of the particle and magnetic flux densities, and they can often be interpreted as a one-dimensional limit of fractional quantum Hall states, thus being called pretopological.Their signatures, however, are typically hard to observe due to the small gaps characterizing these states.Here we investigate bosonic ladder models at filling factor ν = 1.Based on bosonization, renormalization group, and matrix product state simulations we pinpoint two strongly correlated helical phases appearing at this resonance.We show that one of them can be accessed in systems with two-species hardcore bosons and on-site repulsions only, thus amenable for optical lattice experiments.Its signatures are sizable and stable over a broad range of parameters for realistic system sizes.
We introduce generic bosonic models exemplifying that chiral Meissner currents can persist in insulating phases of matter. We first consider interacting bosons on a two-leg ladder. The total density sector can be gapped in a bosonic Mott insulator at odd-integer filling, while the relative density sector remains superfluid due to interchain hopping. Coupling the relative density to gauge fields yields a pseudospin Meissner effect. We show that the same phase arises if the bosons are replaced by spinful fermions confined in Cooper pairs, and find a dual fermionic Mott insulator with spinon currents. We prove that, by tuning the mean density, the Mott insulator with Meissner currents turns into a low-dimensional bosonic $\ensuremath{\nu}=\frac{1}{2}$ Laughlin state for strong enough repulsive interactions across the ladder rungs. We finally discuss extensions to multileg ladders and bilayers in which spinon superfluids with Meissner currents become possible. We propose two experimental realizations, one with ultracold atoms in the setup of Atala et al. [Nat. Phys. 10, 588 (2014)] and another with Josephson junction arrays. We also address a Bose-Fermi mixture subject to a magnetic field in connection with the pseudogap phase of high-Tc{T}_{c}Tc​ cuprates.
Synthetic ladders pierced by a magnetic field realized with one-dimensional alkaline-earth(-like) fermionic gases represent a promising environment for the investigation of quantum-Hall physics with ultracold atoms. We unveil the existence of a hierarchy of insulating and conducting states which appear at fractional filling factors by means of both analytical and numerical methods based on the density-matrix renormalization group algorithm. Analogously to the original quantum Hall effect, we show that such states can be exploited for constructing a topological Thouless pump where the charge transported after one cycle is quantized to fractional values and demonstrate this behavior with a full many-body time-dependent calculation. We conclude with some results on Laughlin-like states in both fermionic and bosonic systems.
Strongly interacting topological matter exhibits fundamentally new phenomena with potential applications in quantum information technology. Emblematic instances are fractional quantum Hall states, where the interplay of magnetic fields and strong interactions gives rise to fractionally charged quasi-particles, long-ranged entanglement, and anyonic exchange statistics. Progress in engineering synthetic magnetic fields has raised the hope to create these exotic states in controlled quantum systems. However, except for a recent Laughlin state of light, preparing fractional quantum Hall states in engineered systems remains elusive. Here, we realize a fractional quantum Hall (FQH) state with ultracold atoms in an optical lattice. The state is a lattice version of a bosonic ν=1/2ν=1/2ν=1/2 Laughlin state with two particles on sixteen sites. This minimal system already captures many hallmark features of Laughlin-type FQH states: we observe a suppression of two-body interactions, we find a distinctive vortex structure in the density correlations, and we measure a fractional Hall conductivity of σH/σ0=0.6(2)σ_\text{H}/σ_0= 0.6(2)σH​/σ0​=0.6(2) via the bulk response to a magnetic perturbation. Furthermore, by tuning the magnetic field we map out the transition point between the normal and the FQH regime through a spectroscopic probe of the many-body gap. Our work provides a starting point for exploring highly entangled topological matter with ultracold atoms.
In this research, we study the bosonic fractional quantum Hall (FQH) states in a system of ultracold bosons in a two-dimensional optical lattice in the presence of a synthetic magnetic field, described by the bosonic Harper–Hofstadter Hamiltonian. We use the cluster Gutzwiller mean-field and exact diagonalization techniques in our work. We obtain incompressible states as ground states at various filling factors similar to those of the FQH states. We focus in particular on the ν = 1/2 FQH state, and it is characterized by the two-point correlation function and the many-body Chern number. We further investigate the effect of dipolar interaction on the ν = 1/2 FQH state. We find that the dipolar interaction stabilizes the FQH state against the competing superfluid state.