Quantum Hall-like effect for cold atoms in non-Abelian gauge potentials

We study the transport of cold fermionic atoms trapped in optical lattices in the presence of artificial Abelian or non-Abelian gauge potentials. Such external potentials can be created in optical lattices in which atom tunneling is laser assisted and described by commutative or non-commutative tunneling operators. We show that the Hall-like transverse conductivity of such systems is quantized by relating the transverse conductivity to topological invariants known as Chern numbers. We show that this quantization is robust in non-Abelian potentials. The different integer values of this conductivity are explicitly computed for a specific non-Abelian system which leads to a fractal phase diagram. During the last decades, remarkable quantum phenomena have been discovered such as the Aharonov-Bohm effect [1], the geometric phases [2, 3], and the integer quantum Hall effect (IQHE) [4, 5]. These important phenomena manifest themselves when gauge potentials are present in non-trivial topological spaces, in which cases the wave functions describing the system may acquire topological or geometrical phases. They are elegantly described in terms of differential geometry and topology in which the topological phases are related to the concept of holonomy [6]. In the IQHE, Hall’s transverse conductivity of two-dimensional electronic systems submitted to high magnetic fields turns out to be quantized according to σH = −C (e/h) where C is a topologically invariant integer called the first Chern number [6–8]. Since the proportionality factor is a combination of the electric charge e and Planck’s constant h, such quantization phenomena play a key role in metrology [4]. Very recently, great experimental advances have been performed on the control of cold atomic systems [9–12]. In such systems, artificial gauge potentials can be created in optical lattices in order to reproduce the dynamics of periodically constrained fermionic atoms submitted to the analogue of a magnetic field. Such systems would thus offer the possibility to observe the corresponding fractal energy spectrum known as the Hofstadter butterfly [13, 14]. More recently the realization of non-Abelian gauge potentials has also been envisaged, allowing the observation of a non-Abelian Aharonov-Bohm effect [15], magnetic monopoles [16], and particular metal-insulator transitions [17]. The purpose of this Letter is to report the possibility of an integer quantum Hall-like effect in the transport of cold fermionic atoms trapped in optical lattices submitted to artificial gauge potentials. We consider this effect in a family of non-Abelian gauge potentials, which contains Abelian potentials as particular cases. In spite of the new topological context provided by the non-Abelian gauge structure, we show that an integer quantum Halllike effect is possible and that the transverse conductivity is indeed quantized in terms of Chern numbers in systems of fermionic cold atoms in optical lattices with artificial Abelian or non-Abelian gauge potentials. Since the current induced in our system is electrically neutral, this quantized conductivity has the distinctive units of the inverse of Planck’s constant, which could be important for metrological purposes. In this Letter, we first define the system Hamiltonian characterized by its non-Abelian gauge structure and evaluate the transverse conductivity with Kubo’s formula which gives the linear response of the system to an external static perturbation. We then express the Hall-like conductivity in terms of Chern numbers in order to prove the quantization of this physical quantity. We compute these topological invariants for a p-1 ar X iv :c on dm at /0 60 94 72 v3 [ co nd -m at .m es -h al l] 2 3 M ay 2 00 7 N. Goldman P. Gaspard specific system which leads to a fractal phase diagram. We then discuss possible experimental setups for the validation of our theoretical considerations. We are interested in systems described by the following Hamiltonian