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Register a new userUltracold atomic gases in non-Abelian gauge potentials: The case of constant Wilson loop
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Nowadays it is experimentally feasible to create artificial, and in particular, non-Abelian gauge potentials for ultracold atoms trapped in optical lattices. Motivated by this fact, we investigate the fundamental properties of an ultracold Fermi gas in a non-Abelian U(2) gauge potential characterized by a constant Wilson loop. Under this specific condition, the energy spectrum exhibits a robust band structure with large gaps and reveals a new fractal figure. The transverse conductivity is related to topological invariants and is shown to be quantized when the Fermi energy lies inside a gap of the spectrum. We demonstrate that the analogue of the integer quantum Hall effect for neutral atoms survives the non-Abelian coupling and leads to a striking fractal phase diagram. Moreover, this coupling induces an anomalous Hall effect as observed in graphene.
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A detailed description of the method for analytical evaluation of the three-loop contributions to renormalization group functions is presented. This method is employed to calculate the charge renormalization function and anomalous dimensions for non-Abelian gauge theories with fermions in the three-loop approximation. A three-loop expression for the effective charge of QCD is given. Charge renormalization effects in the SU(4)-supersymmetric gauge model is shown to vanish at this level. A complete list of required formulas is given in Appendix. The above-mentioned results of three-loop calculations have been published by the present authors (with A.Yu., Zharkov and L.V., Avdeev) in 1980 in Physics Letters B. The present text, which treats the subject in more details and contains a lot of calculational techniques, has also been published in 1980 as the JINR Communication E2-80-483.
Quantum states can acquire a phase called the Berry phase after adiabatically traversing a closed loop in parameter space. This phase is geometric -- dependent on the path -- not the rate of motion. Wilczek and Zee extended this concept to include non-Abelian phases that can be characterized the Wilson loop, a gauge independent quantity. Here we quantum-engineer a non-Abelian SU(2) gauge field for an atomic Bose-Einstein Condensate (BEC), in a 5-dimensional synthetic parameter space derived from internal atomic degrees of freedom. By slowly encircling a topological monopole, we observed the adiabatic Wilczek-Zee phase that we characterize in terms of the Wilson loop.
Nowadays it is experimentally feasible to create artificial, and in particular, non-Abelian gauge potentials for ultracold atoms trapped in optical lattices. Motivated by this fact, we investigate the fundamental properties of an ultracold Fermi gas in a non-Abelian U(2) gauge potential characterized by a emph{constant} Wilson loop. Under this specific condition, the energy spectrum exhibits a robust band structure with large gaps and reveals a new fractal figure. The transverse conductivity is related to topological invariants and is shown to be quantized when the Fermi energy lies inside a gap of the spectrum. We demonstrate that the analogue of the integer quantum Hall effect for neutral atoms survives the non-Abelian coupling and leads to a striking fractal phase diagram. Moreover, this coupling induces an anomalous Hall effect as observed in graphene.
We present a general analytical formalism to determine the energy spectrum of a quantum particle in a cubic lattice subject to translationally invariant commensurate magnetic fluxes and in the presence of a general space-independent non-Abelian gauge potential. We first review and analyze the case of purely Abelian potentials, showing also that the so-called Hasegawa gauge yields a decomposition of the Hamiltonian into sub-matrices having minimal dimension. Explicit expressions for such matrices are derived, also for general anisotropic fluxes. Later on, we show that the introduction of a translational invariant non-Abelian coupling for multi-component spinors does not affect the dimension of the minimal Hamiltonian blocks, nor the dimension of the magnetic Brillouin zone. General formulas are presented for the U(2) case and explicit examples are investigated involving $pi$ and $2pi/3$ magnetic fluxes. Finally, we numerically study the effect of random flux perturbations.
Motivated by the possibility of creating non-Abelian fields using cold atoms in optical lattices, we explore the richness and complexity of non-interacting two-dimensional electron gases (2DEGs) in a lattice, subjected to such fields. In the continuum limit, a non-Abelian system characterized by a two-component magnetic flux describes a harmonic oscillator existing in two different charge states (mimicking a particle-hole pair) where the coupling between the states is determined by the non-Abelian parameter, namely the difference between the two components of the magnetic flux. A key feature of the non-Abelian system is a splitting of the Landau energy levels, which broaden into bands, as the spectrum depends explicitly on the transverse momentum. These Landau bands result in a coarse-grained moth, a continuum version of the generalized Hofstadter butterfly. Furthermore, the bands overlap, leading to effective relativistic effects. Importantly, similar features also characterize the corresponding two-dimensional lattice problem when at least one of the components of the magnetic flux is an irrational number. The lattice system with two competing magnetic fluxes penetrating the unit cell provides a rich environment in which to study localization phenomena. Some unique aspects of the transport properties of the non-Abelian system are the possibility of inducing localization by varying the quasimomentum, and the absence of localization of certain zero-energy states exhibiting a linear energy-momentum relation. Furthermore, non-Abelian systems provide an interesting localization scenario where the localization transition is accompanied by a transition from relativistic to non-relativistic theory.
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