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Results in Physics

Topological phase transitions driven by real next-nearest-neighbor hopping in cold fermi gases

R. Wang, Y.F. Zhang, C.X. Zhang

Published Jan 1, 2014• 2 citations
Publisher

Abstract

The strength t′ of the next-nearest-neighbor (NNN) hopping in most realized square optical lattices is much smaller than that of the nearest-neighbor hopping, and is usually seen as zero. Recently, both experimental and theoretical works have shown that the magnitude of the NNN hopping can be tuned in a wide range by shaking the optical lattice. In this paper, we study the effect of the real NNN term on topological phase transitions of the cold fermi gases in a two-dimensional square anisotropic optical lattice. We investigate the gapless condition of the system and the topological phase symbolized by the TKNN number. For the real NNN hopping, there exists a critical point tc′ as a function of μ, when 0<t′<tc′, there are only non-Abelian topological phases which are indicted by an odd TKNN number, and when t′>tc′ , the Abelian topological phase appears and the zone of the phase will be widened with the increase of t′. By numerically diagonalizing the Hamiltonian in the real space, the corresponding edge states for different topological phase and Majorana zero modes are discussed.

DOI
10.1016/j.rinp.2014.03.006

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The strength t′ of the next-nearest-neighbor (NNN) hopping in most realized square optical lattices is much smaller than that of the nearest-neighbor hopping, and is usually seen as zero. Recently, both experimental and theoretical works have shown that the magnitude of the NNN hopping can be tuned in a wide range by shaking the optical lattice. In this paper, we study the effect of the real NNN term on topological phase transitions of the cold fermi gases in a two-dimensional square anisotropic optical lattice. We investigate the gapless condition of the system and the topological phase symbolized by the TKNN number. For the real NNN hopping, there exists a critical point tc′ as a function of μ, when 0<t′<tc′, there are only non-Abelian topological phases which are indicted by an odd TKNN number, and when t′>tc′ , the Abelian topological phase appears and the zone of the phase will be widened with the increase of t′. By numerically diagonalizing the Hamiltonian in the real space, the corresponding edge states for different topological phase and Majorana zero modes are discussed.
We investigate the topological phase marked by the Thouless–Kohmoto–Nightingale–Nijs (TKNN) number and the phase transitions driven by the next nearest neighbor (NNN) hopping in noncentrosymmetric cold Fermi gases, both spin-singlet pairing and spin-triplet pairing are considered. There exists a critical for the NNN hopping, at which the quantum phase transition occurs, and the system changes from an Abelian (non-Abelian) phase to a non-Abelian (Abelian) one. By numerically diagonalizing the Hamiltonian in the real space, the energy spectra with edge states for different topological phases and the Majorana zero modes are discussed. Although the spin-triplet pairing does not contribute to the gap closing and the phase diagram, it induces gapless states in the presence of a magnetic field, and the TKNN number in this region is still zero.
We investigate the topological phase marked by the Thouless–Kohmoto–Nightingale–Nijs(TKNN) number and the phase transitions driven by the next nearest neighbor(NNN) hopping in noncentrosymmetric cold Fermi gases, both spinsinglet pairing and spin-triplet pairing are considered. There exists a critical t’c for the NNN hopping, at which the quantum phase transition occurs, and the system changes from an Abelian(non-Abelian) phase to a non-Abelian(Abelian) one. By numerically diagonalizing the Hamiltonian in the real space, the energy spectra with edge states for different topological phases and the Majorana zero modes are discussed. Although the spin-triplet pairing does not contribute to the gap closing and the phase diagram, it induces gapless states in the presence of a magnetic field, and the TKNN number in this region is still zero.
We investigate emergent topological gapless phases in the square-lattice Kitaev model with additional hopping terms. In the presence of nearest-neighbor hopping only, the model is known to exhibit gapless phases with two topological gapless points. When the strength of the newly added next-nearest-neighbor hopping is smaller than a certain value, qualitatively the same phase diagram persists. We find that further increase of the extra hopping results in a new topological phase with four gapless points. We construct a phase diagram to clarify the regions of emergent topological gapless phases as well as topologically trivial ones in the space of the chemical potential and the next-nearest-neighbor hopping strength. We examine the evolution of the gapless phases in the energy dispersions of the bulk as the chemical potential varies. The topological properties of the gapless phases are characterized by the winding numbers of the present gapless points. We also consider the ribbon geometry to examine the corresponding topological edge states. It is revealed that Majorana-fermion edge modes exist as flat bands in topological gapless phases. We also perform the analytical calculation as to Majorana-fermion zero-energy modes and discuss its implications on the numerical results.
We consider a two-component Fermi gas with attractive interactions on a square optical lattice, and study the interplay of Zeeman field, spin-orbit coupling, and next-nearest-neighbor hopping on the ground-state phase diagrams in the entire BCS-BEC evolution. In particular, we first classify and distinguish all possible superfluid phases by the momentum-space topology of their zero-energy quasiparticle-quasihole excitations, and then numerically establish a plethora of quantum phase transitions in between. These transitions are further signaled and evidenced by the changes in the corresponding topological invariant of the system, i.e., its Chern number. Lastly, we find that the superfluid phase exhibits a reentrant structure, separated by a fingering normal phase, the origin of which is traced back to the changes in the single-particle density of states.
We investigate the effect of next-nearest-neighbor hopping on topological quantum phase transitions (QPTs) that are characterized by the number of Majorana zero modes in one-dimensional (1D) p-wave superconducting systems. We also numerically analyze the scaling behavior and the universality of the Berry phase (BP) of the ground state close to the critical point. For critical line (I), the derivative of the ground-state BP is nonanalytic at the phase boundaries. For the phase boundary (II), a noncontractible BP of the ground state itself is also a witness of quantum phase transition.
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.