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arXiv:1308.0750 · Physical Review A

Topological phase transitions driven by non-Abelian gauge potentials in optical square lattices

Michele Burrello, Ion Cosma Fulga, Emilio Alba, Luca Lepori, Andrea Trombettoni

Published Nov 15, 2013• 21 citations
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Abstract

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.

DOI
10.1103/physreva.88.053619

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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.
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.
We theoretically investigate a tight-binding model of fermions hopping on the square-octagon lattice which consists of a square lattice with plaquette corners themselves decorated by squares. Upon the inclusion of second-neighbor spin-orbit coupling or non-Abelian gauge fields, time-reversal symmetric topological Z2{Z}_{2}Z2​ band insulators are realized. Additional insulating and gapless phases are also realized via the non-Abelian gauge fields. Some of the phase transitions involve topological changes to the Fermi surface. The stability of the topological phases to various symmetry-breaking terms is investigated via the entanglement spectrum. Our results enlarge the number of known exactly solvable models of Z2{Z}_{2}Z2​ band insulators and are potentially relevant to the realization and identification of topological phases in both the solid-state and cold atomic gases.
Motivated by the recent progress in engineering artificial non-Abelian gauge fields for ultracold fermions in optical lattices, we investigate the time-reversal-invariant Hofstadter–Hubbard model. We include an additional staggered lattice potential and an artificial Rashba-type spin–orbit coupling term available in experiment. Without interactions, the system can be either a (semi)-metal, a normal or a topological insulator, and we present the non-Abelian generalization of the Hofstadter butterfly. Using a combination of real-space dynamical mean-field theory (RDMFT), analytical arguments, and Monte-Carlo simulations we study the effect of strong on-site interactions. We determine the interacting phase diagram, and discuss a scenario of an interaction-induced transition from a normal to a topological insulator. At half-filling and large interactions, the system is described by a quantum spin Hamiltonian, which exhibits exotic magnetic order due to the interplay of Rashba-type spin–orbit coupling and the artificial time-reversal-invariant magnetic field term. We determine the magnetic phase diagram: both for the itinerant model using RDMFT and for the corresponding spin model in the classical limit using Monte-Carlo simulations.
Motivated by the recent progress in engineering artificial non-Abelian gauge fields for ultracold fermions in optical lattices, we investigate the time-reversal-invariant Hofstadter–Hubbard model. We include an additional staggered lattice potential and an artificial Rashba-type spin–orbit coupling term available in experiment. Without interactions, the system can be either a (semi)-metal, a normal or a topological insulator, and we present the non-Abelian generalization of the Hofstadter butterfly. Using a combination of real-space dynamical mean-field theory (RDMFT), analytical arguments, and Monte-Carlo simulations we study the effect of strong on-site interactions. We determine the interacting phase diagram, and discuss a scenario of an interaction-induced transition from a normal to a topological insulator. At half-filling and large interactions, the system is described by a quantum spin Hamiltonian, which exhibits exotic magnetic order due to the interplay of Rashba-type spin–orbit coupling and the artificial time-reversal-invariant magnetic field term. We determine the magnetic phase diagram: both for the itinerant model using RDMFT and for the corresponding spin model in the classical limit using Monte-Carlo simulations.
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 method of synthetic gauge potentials opens up a new avenue for our understanding and discovering novel quantum states of matter. We investigate the topological quantum phase transition of Fermi gases trapped in a honeycomb lattice in the presence of a synthetic non-Abelian gauge potential. We develop a systematic fermionic effective field theory to describe a topological quantum phase transition tuned by the non-Abelian gauge potential and explore its various important experimental consequences. Numerical calculations on lattice scales are performed to compare with the results achieved by the fermionic effective field theory. Several possible experimental detection methods of topological quantum phase transition are proposed. In contrast to condensed matter experiments where only gauge invariant quantities can be measured, both gauge invariant and non-gauge invariant quantities can be measured by experimentally generating various non-Abelian gauges corresponding to the same set of Wilson loops.