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arXiv:1509.08219 · Physical Review A

Topological superfluids on a square optical lattice with non-Abelian gauge fields: Effects of next-nearest-neighbor hopping in the BCS-BEC evolution

M. Iskin

Published Jan 12, 2016• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.93.013608

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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 study the system of attractively interacting fermions hopping in a square lattice with any linear combinations of Rashba or Dresselhaus spin-orbit coupling in a normal Zeeman field. By imposing self-consistence equations at half filling with zero chemical potential, we find that there are three phases: band insulator (BI), superfluid (SF), and topological superfluid (TSF) with a Chern number C=2C=2C=2. The C=2C=2C=2 TSF happens in small Zeeman fields and weak interactions which is in the experimentally most easily accessible regime. The transition from the BI to the SF is a first-order one due to the multiminima structure of the ground state energy landscape. There is a class of topological phase transitions (TPTs) from the SF to the C=2C=2C=2 TSF at the low critical field hc1{h}_{c1}hc1​, then another one from the C=2C=2C=2 TSF to the BI at the upper critical field hc2{h}_{c2}hc2​. We derive effective actions to describe these two classes of topological phase transitions, then use them to study the Majorana edge modes and the zero modes inside the vortex core of the C=2C=2C=2 TSF near both hc1{h}_{c1}hc1​ and hc2{h}_{c2}hc2​, especially exploring their spatial and spin structures. We find that the edge modes decay into the bulk with oscillating behaviors and determine both the decay and oscillating lengths. We compute the bulk spectra and map out the Berry curvature distribution in momentum space near both hc1{h}_{c1}hc1​ and hc2{h}_{c2}hc2​. We elaborate some intriguing bulk-Berry curvature-edge-vortex correspondences. We also discuss the competitions between SFs and charge density wave states in more general cases. A possible classification scheme of all the TPTs in a square lattice is outlined. Comparisons with previous works on related systems are discussed. Possible experimental implications in cold atoms in an optical lattice are given.
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 analyze the effects of in- and out-of-plane Zeeman fields on the BCS--Bose-Einstein condensation (BEC) evolution of a Fermi gas with equal Rashba-Dresselhaus (ERD) spin-orbit coupling (SOC). We show that the ground state of the system involves gapless superfluid phases that can be distinguished with respect to the topology of the momentum-space regions with zero excitation energy. For the BCS-like uniform superfluid phases with zero center-of-mass momentum, the zeros may correspond to one or two doubly degenerate spheres, two or four spheres, two or four concave spheroids, or one or two doubly degenerate circles, depending on the combination of Zeeman fields and SOC. Such changes in the topology signal a quantum phase transition between distinct superfluid phases and leave their signatures on some thermodynamic quantities. We also analyze the possibility of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)--like nonuniform superfluid phases with finite center-of-mass momentum and obtain an even richer phase diagram.
The two-dimensional spin-imbalanced Fermi gas subject to sss-wave pairing and spin-orbit coupling is considered a promising platform for realizing a topological chiral-ppp-wave superfluid. In the BCS limit of sss-wave pairing, i.e., when Cooper pairs are only weakly bound, the system enters the topological phase via a second-order transition driven by increasing the Zeeman spin-splitting energy. Stronger attractive two-particle interactions cause the system to undergo the BCS-BEC crossover, in the course of which the topological transition becomes first order. As a result, topological and nontopological superfluids coexist in spatially separated domains in an extended region of phase space spanned by the strength of sss-wave interactions and the Zeeman energy. Here we investigate this phase-coexistence region theoretically using a zero-temperature mean-field approach. Exact numerical results are presented to illustrate basic physical characteristics of the coexisting phases and to validate an approximate analytical description derived for weak spin-orbit coupling. Besides extending our current understanding of spin-imbalanced superfluid Fermi systems, the present approach also provides a platform for future studies of unconventional Majorana excitations that, according to topology, should be present at the internal interface between coexisting topological and nontopological superfluid parts of the system.
We investigate the BCS-BEC crossover in three-dimensional degenerate Fermi gases in the presence of spin-orbit coupling (SOC) and Zeeman field. We show that the superfluid order parameter destroyed by a large Zeeman field can be restored by the SOC. With increasing strengths of the Zeeman field, there is a series of topological quantum phase transitions from a nontopological superfluid state with fully gapped fermionic spectrum to a topological superfluid state with four topologically protected Fermi points (i.e., nodes in the quasiparticle excitation gap) and then to a second topological superfluid state with only two Fermi points. The quasiparticle excitations near the Fermi points realize the long-sought low-temperature analog of Weyl fermions of particle physics. We show that the topological phase transitions can be probed using the experimentally realized momentum-resolved photoemission spectroscopy.