We contrast topological insulating phases of SU(3) and SU(2) fermions in optical lattices under artificial magnetic fields. For this purpose, we compare the phase diagram of chemical potential vs. color-flip fields for SU(3) colored fermions ( D 10 representation) for fixed color-orbit coupling to the phase diagram of chemical potential vs. spin-flip fields for SU(2) spin- 1 / 2 fermions ( D 1 representation) for fixed spin-orbit coupling. To classify the emergent topological phases, we obtain the Chern matrix and identify three distinct topological invariants for SU(3) colored fermions: the charge-charge, the color-charge and the color-color Chern numbers. This is in sharp contrast to the SU(2) spin- 1 / 2 case, where only two topological invariants are necessary: the charge-charge and the spin-charge Chern numbers. Although we focus on SU(3) colored and SU(2) spin- 1 / 2 fermions, our results indicate the existence of three independent Chern numbers associated with the quantum Hall response for higher fermionic irreducible representations D p and D p 0 ( p > 1) of SU(2) and SU(3), respectively. We suggest to investigate this comparison using the internal states of ultracold Fermi atoms ytterbium-173 ( 173 Yb ) and strontium-87 ( 87 Sr ) .
We contrast topological insulating phases of SU(3) and SU(2) fermions in optical lattices under artificial magnetic fields. For this purpose, we compare the phase diagram of chemical potential vs. color-flip fields for SU(3) colored fermions ( D 10 representation) for fixed color-orbit coupling to the phase diagram of chemical potential vs. spin-flip fields for SU(2) spin- 1 / 2 fermions ( D 1 representation) for fixed spin-orbit coupling. To classify the emergent topological phases, we obtain the Chern matrix and identify three distinct topological invariants for SU(3) colored fermions: the charge-charge, the color-charge and the color-color Chern numbers. This is in sharp contrast to the SU(2) spin- 1 / 2 case, where only two topological invariants are necessary: the charge-charge and the spin-charge Chern numbers. Although we focus on SU(3) colored and SU(2) spin- 1 / 2 fermions, our results indicate the existence of three independent Chern numbers associated with the quantum Hall response for higher fermionic irreducible representations D p and D p 0 ( p > 1) of SU(2) and SU(3), respectively. We suggest to investigate this comparison using the internal states of ultracold Fermi atoms ytterbium-173 ( 173 Yb ) and strontium-87 ( 87 Sr ) .
We discuss the emergence of topological color insulators in optical lattices as quantum phases of SU(3) ultra-cold neutral fermions. We construct the Chern matrix and classify all insulating phases in terms of three topological invariants: the charge-charge, the color-charge and the color-color Chern numbers. Our classification transcends that of SU(2) systems which require only the charge-charge (charge-Hall) and spin-charge (spin-Hall) Chern numbers. To illustrate the topological classification of the insulating phases of SU(3) fermions, we construct phase diagrams of chemical potential and color-orbit parameter versus color-flip fields for fixed magnetic flux ratio.
We study ultracold color fermions with three internal states, red, green, and blue, with SU(3) symmetry in optical lattices, when color-orbit coupling and color-flip fields are present. This system corresponds to a generalization of two internal state fermions with SU(2) symmetry in the presence of spin-orbit coupling and spin-flipping Zeeman fields. We investigate the eigenspectrum and Chern numbers to describe different topological phases that emerge in the phase diagrams of color-orbit coupled fermions in optical lattices. We obtain the phases as a function of artificial magnetic, color-orbit, and color-flip fields that can be independently controlled. For fixed artificial magnetic flux ratio, we identify topological quantum phases and phase transitions in the phase diagrams of chemical potential versus color-flip fields or color-orbit coupling, where the chirality and number of midgap edge states changes. The topologically nontrivial phases are classified in three groups: The first group has total nonzero chirality and exhibits only the quantum charge Hall effect; the second group has total nonzero chirality and exhibits both quantum charge and quantum color Hall effects; and the third group has total zero chirality but exhibits the quantum color Hall effect. These phases are generalizations of the quantum Hall and quantum spin Hall phases for charged spin-1/2 fermions. Lastly, we also describe the color density of states and a staircase structure in the total and color filling factors versus chemical potential for fixed color-orbit coupling, color-flip field, and magnetic flux ratio. We show the existence of incompressible states at rational filling factors precisely given by a gap labelling theorem that relates the filling factors to the magnetic flux ratio and topological quantum numbers.
We discuss the Chern number spectrum of ultracold fermions in square optical lattices as a function of artificial magnetic, Zeeman, and spin-orbit fields that can be tuned independently. We show the existence of topological quantum phase transitions induced by Zeeman and spin-orbit fields, where the total number and chirality of edge states change for fixed magnetic flux ratio, thus leading to topological-insulator phases which are different from those found at zero Zeeman and spin-orbit fields. We construct phase diagrams of chemical potential versus Zeeman field or spin-orbit coupling and characterize all insulating phases by their topological invariants. Lastly, we obtain a staircase structure in the filling factor versus chemical potential for various Zeeman and spin-orbit fields, showing the existence of incompressible states at rational filling factors derived from a generalized Diophantine equation.
The interplay between disorder, gauge fields, and internal degrees of freedom fundamentally affects localization and topological properties of quantum many-body systems. Motivated by recent experimental realizations of synthetic non-Abelian gauge fields for SU(3) colored fermions, we investigate their localization and topological properties in 1D bichromatic optical lattices consisting of strong and weak laser beams. Describing the non-Abelian gauge field via color-orbit coupling and color-flip (Rabi) fields, we obtain a tight-binding description of trapped SU(3) colored fermions corresponding to a generalized three-color Aubry-André model. We show that these fields explicitly break the conventional self-duality of a simple three-color Aubry-André system. This duality breaking generates mobility regions across the energy spectrum, demonstrating that non-Abelian fields can either enhance or hinder color localization. Using exact diagonalization, density-of-states evaluations, and finite-size scaling of the inverse participation ratio, we obtain phase diagrams that identify regions of extended or localized bulk states. Furthermore, the color-orbit and Rabi fields induce edge states with topological properties. We develop an exact mapping from our 1D disorder model into a 2D color Harper model with a fictitious magnetic flux ratio and dimension controlled by the weak laser beam's phase. Using this mapping, we evaluate topological invariants, such as the charge-charge Chern number, for edge states emerging in energy gaps, revealing the topological insulating nature of several gapped phases. Lastly, we identify that these topological color-insulator phases can energetically neighbor three configurations: two extended, two localized, or one of each. This sharply contrasts with conventional topological insulators, which always neighbor two extended phases.
Motivated by the recent experimental success in realizing synthetic spin-orbit coupling in ultracold atomic systems, we consider N-component atoms coupled to a non-Abelian SU(N) gauge field. More specifically, we focus on the case, referred to here as "SU(3) spin-orbit-coupling," where the internal states of three-component atoms are coupled to their momenta via a matrix structure that involves the Gell-Mann matrices (in contrast to the Pauli matrices in conventional SU(2) spin-orbit-coupled systems). It is shown that the SU(3) spin-orbit-coupling gives rise to qualitatively different phenomena and in particular we find that even a homogeneous SU(3) field on a simple square lattice enables a topologically nontrivial state to exist, while such SU(2) systems always have trivial topology. In deriving this result, we first establish an equivalence between the Hofstadter model with a 1/N Abelian flux per plaquette and a homogeneous SU(N) non-Abelian model. The former is known to have a topological spectrum for N>2, which is thus inherited by the latter. It is explicitly verified by an exact calculation for N=3, where we develop and use a new algebraic method to calculate topological indices in the SU(3) case. Finally, we consider a strip geometry and establish the existence of three gapless edge states-the hallmark feature of such an SU(3) topological insulator.
Motivated by the recent experimental success in realizing synthetic spin-orbit coupling in ultracold atomic systems, we consider N -component atoms coupled to a non-Abelian SU(N) gauge field. More specifically, we focus on the case, referred to here as “SU(3) spin-orbit-coupling,” where the internal states of three-component atoms are coupled to their momenta via a matrix structure that involves the Gell-Mann matrices (in contrast to the Pauli matrices in conventional SU(2) spin-orbitcoupled systems). It is shown that the SU(3) spin-orbit-coupling gives rise to qualitatively different phenomena and in particular we find that even a homogeneous SU(3) field on a simple square lattice enables a topologically non-trivial state to exist, while such SU(2) systems always have trivial topology. In deriving this result, we first establish an equivalence between the Hofstadter model with a 1/N Abelian flux per plaquette and a homogeneous SU(N) non-Abelian model. The former is known to have a topological spectrum for N > 2, which is thus inherited by the latter. It is explicitly verified by an exact calculation for N = 3, where we develop and use a new algebraic method to calculate topological indices in the SU(3) case. Finally, we consider a strip geometry and establish the existence of three gapless edge states – the hallmark feature of such an SU(3) topological insulator.