Edge Transport in 2D Cold Atom Optical Lattices

Abstract

We theoretically study the observable response of edge currents in two dimensional cold atom optical lattices. As an example we use Gutzwiller mean-field theory to relate persistent edge currents surrounding a Mott insulator in a slowly rotating trapped Bose-Hubbard system to time of flight measurements. We briefly discuss an application, the detection of Chern number using edge currents of a topologically ordered optical lattice insulator. [4,5,6,7,8].By combining TOF with externally applied potentials recent work has demonstrated transport in one dimensional optical lattices [9,10]. In a closed two dimensional system the notion of "transport" is less direct. A recent experiment has applied rotation to weak lattices [11] confining bosons. While far from the single band BH limit, this experiment reveals vortex pinning arising from the weak lattice. Recent work [12,13] also suggests that uniform effective magnetic fields (equivalent to rotation) may be applied to optical lattices already in the BH limit. Either implementation, rotation or an effective magnetic field, can be used as an applied potential valuable in establishing persistent currents, and therefore transport, in two dimensional lattices.Concurrent with experimental progress, a variety of cold atom phases have been proposed in two dimensional optical lattices [2]. Some of the proposed lattice models have rich phase diagrams with particularly intriguing or even unknown ground states, including: extended BH models [2,14,15], higher band spin models [16], fractional quantum Hall models [13], and the Kitaev spin model [17,18,19]. We ask how insulating phases arising in two dimensional lattice models can be studied using a combination of externally applied potentials and TOF.Below we argue that trapping leads to edge states which serve as a probe of bulk insulating states. As a concrete and relevant example we study the slowly rotating BH model in detail. Other studies have considered vortex configurations in the superfluid phase of the rotating uniform BH model [20,21]. Here we study edge effects in the Mott insulating phase of the slowly rotating trapped BH model. We propose that diamagnetic response of edge states can indeed be observed thereby offering a quantitative response probe of a variety of bulk two dimensional insulators. We briefly discuss implications for another insulator where edge states can be used to detect the Chern number [17,22,23] of a topologically ordered insulator, the non-Abelian ground state of the Kitaev model.We first note that response to externally applied fields can be obtained at a quantitative level by analyzing TOF measurements. TOF can be related to the momentum density, ρ k , of particles with lattice momentum k originally trapped in an optical lattice. Observation of ρ k (with sufficient accuracy) can be combined with input parameters to restore quantities of the form: J ≡ k W k ρ k , where W k is any function of k which can be accurately determined from input experimental parameters. By defining W k = M k (∂E k /∂k ...

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