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arXiv:1811.00136 · Physical Review A

Chern-number spectrum of ultracold fermions in optical lattices tuned independently via artificial magnetic, Zeeman, and spin-orbit fields

Man Hon Yau, C. A. R. Sá de Melo

Published Apr 25, 2019• 5 citations
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Abstract

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.

DOI
10.1103/physreva.99.043625

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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.
We study ultracold color fermions with three internal states, red, green, and blue, with SU(3)\mathrm{SU}(3)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)\mathrm{SU}(2)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-1/21/21/2 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.
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We propose an approach to study the topological properties of matter. In this approach, fermionic atoms are placed in an external magnetic field and in a two-dimensional spin-dependent optical lattice (SDOL) created by intersecting laser beams with a superposition of polarizations. To demonstrate the utility of the SDOL-based technique we compute the topological invariants (Chern numbers) for the SDOL bands as a function of an external magnetic field and show the existence of a rich topology of the energy bands for this system which does not have parity-time-reversal symmetry. We explicitly consider 6LiF=1/2^{6}\mathrm{Li}\phantom{\rule{4pt}{0ex}}F=1/26LiF=1/2 atoms. Using a projection matrix method we observe topological phase transitions among an ordinary insulator, an Abelian topological insulator, and a non-Abelian topological insulator as the external magnetic field strength is varied. On introducing edges for the SDOL we find topological edge states (that are correlated with the band Chern numbers) that simultaneously exhibit nontrivial density and spin currents with both a rotational flow contribution and flow along the edge of the SDOL.
Recent advanced experimental implementations of optical lattices with highly tunable geometry open up new regimes for exploring quantum many-body states of matter that had not been accessible previously. Here we report that a topological fermionic superfluid with higher Chern number emerges spontaneously from sss-wave spin-singlet pairing in an orbital optical lattice when its geometry is tuned to explicitly break reflection symmetry. Qualitatively distinct from the conventional scheme that relies on higher partial-wave pairing, the crucial ingredient of our model is topology originating from mixing higher Wannier orbitals. It leads to unexpected changes in the topological band structure at the single-particle level, i.e., the bands are transformed from possessing two flux-$\ensuremath{\pi}$ Dirac points into a single quadratic touching point with flux $2\ensuremath{\pi}$. Based on such engineered single-particle bands, spin-singlet pairing of ultracold fermions arising from standard sss-wave attractive interaction is found to induce higher Chern number (Chern number of 2) and topologically protected chiral edge modes, all occurring at a higher critical temperatures in relative scales, potentially circumventing one of the major obstacle for its realization in ultracold gases.
We analyze a tight-binding model of ultracold fermions loaded in an optical square lattice and subjected to a synthetic non-Abelian gauge potential featuring both a magnetic field and a translationally invariant SU(2) term. We consider in particular the effect of broken time-reversal symmetry and its role in driving nontrivial topological phase transitions. By varying the spin-orbit coupling parameters, we find both a semimetal-insulator phase transition and a topological phase transition between insulating phases with different numbers of edge states. The spin is not a conserved quantity of the system, and the topological phase transitions can be detected by analyzing its polarization in time-of-flight images, providing a clear diagnostic for the characterization of the topological phases through the partial entanglement between spin and lattice degrees of freedom.