arXiv:1310.7557v2 [cond-mat.quant-gas] 20 Jan 2014 Majorana fermions in quasi-1D and higher dimensional ultracold optical lattices Chunlei Qu 1 , Ming Gong 2 , Yong Xu 1 , Sumanta Tewari 3 , and Chuanwei Zhang 1* 1 Department of Physics, the University of Texas at Dallas, Richardson, TX 75080, USA 2 Department of Physics and Centre for Quantum Coherence, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China 3 Department of Physics and Astronomy, Clemson University, Clemson, South Carolina 29634, USA (Dated: June 20, 2021) We show that Majorana fermions (MFs) exist in two- and three-dimensional (2D,3D) fermionic optical lattices with strictly 1D spin-orbit coupling (SOC) which has already been realized in ex- periments. For a quasi-1D topological BCS superfluid, there are multiple MFs at each end which are topologically protected by a chiral symmetry. In the generalization to higher dimensions, the multiple MFs form a zero energy flat band. An additional experimentally tunable in-plane Zeeman field drives the system to a topological Fulde-Ferrell (FF) superfluid phase. We find that even though the multiple MFs are robust against the in-plane Zeeman field if the order parameters at the different chains are enforced to be identical, they are destroyed in the self-consistently obtained FF phase where the order parameters are inhomogeneous on the boundaries. Our results are useful to guide the experimentalists on searching for MFs in the context of ultracold fermionic atoms. PACS numbers: 03.75.Ss, 67.85.-d, 74.20.Fg Introduction.— MFs, quantum particles which are their own anti-particles, have attracted a lot of atten- tion because of their topological properties and the po- tential applications in fault-tolerant topological quantum computation [1–3]. Many solid state materials have been predicted to be candidates for the realization of MFs [4– 16]. Even though experimental progress in the solid state systems has been made in the past few years and possible signatures of MFs have been observed [17–25], a “smok- ing gun” signature of MFs is still lacking due to many factors influencing the measurement results in solid state materials [26–30]. On the other hand, ultracold atoms provide an ideal playground for the quantum simula- tions of many condensed matter systems because they are clean and highly controllable in the system parameters. The recent realization of SOC in BEC [31–34] and Fermi gases [35, 36] paves a way for the observation of MFS in cold atoms [37–41]. In this context, many schemes for the creation and observation of MFs in a 1D cold atom quantum wire have been studied [42–45]. The realistic experiments in ultracold atoms are not on strictly 1D systems, which motivates our present study on the existence and properties of MFs in higher di- mensional ultracold atom systems [46]. The necessity of studying the physics beyond 1D systems also arises from the failure of mean field theory in 1D where there is no long range ordering due to Mermin-Wagner theorem. The inclusion of a weak tunneling in the transverse direc- tions in a quasi-1D system could effectively suppress the quantum fluctuations and stabilize the mean field super- fluid order [47]. However, the presence of such transverse tunneling terms, even if treated as a perturbation, may pairwise couple the MFs and create a gap in the low energy spectrum. It follows that, unless the number of chains in the transverse directions is odd (which is dif- ficult to control experimentally), the system of coupled chains may not support any MFs at all. Because if this, whether or not MFs exist in weakly coupled quasi-1D (with finite number of chains), 2D, and 3D cold atom systems with artificial SOC and Zeeman fields has re- mained an important open question both theoretically and experimentally. In this paper we show that multiple localized zero en- ergy MFs still exist in a multi-chain system in a wide range of parameter space. In the limit of infinite number of chains in the transverse directions, the MFs form a zero energy flat band that can be probed experimentally. Our work is based on a chiral symmetric analysis of a multi-chain system which is not applicable when there is a nonzero in-plane Zeeman field [48–51]. Thus, in ad- dition to the existence of the MFs we also explore their topological robustness against an additional (in-plane) Zeeman field which may give rise to spatially inhomoge- neous order parameters in the resultant FF phase [52, 53]. In contrast to the previous studies, we have considered here strictly 1D SOC which has been realized in ultracold Fermi gases recently [35, 36]. Model system.— We first consider quasi-1D optical lat- tices aligned along the x direction. The tight-binding Hamiltonian in the mean field approximation can be writ- ten as, H tb = H 0 + H ⊥ + H Δ (1) The first term H 0 = −t ∑ iσ (c † i,σ c i+ˆ ex,σ + H.c.) − µ ∑ i,σ n i,σ + α 2 ∑ i (c † i-ˆ ex,↓ c i↑ − c † i+ˆ ex,↓ c i↑ + H.c.) − V z ∑ i (c † i↑ c i↑ − c † i↓ c i↓ ) − V y ∑ i (−ic † i↑ c i↓ + ic † i↓ c i↑ ) is the Hamiltonian of the parallel chains along x direction where c † i,σ is the fermionic operator creating a particle with spin σ at site i =(i x ,i y ,i z ). t is the tunneling strength along x direction, µ is the chemical potential, α is the 1D SOC strength, V z and V y are the out-of-plane