Majorana Edge States in Atomic Wires Coupled by Pair Hopping

We present evidence for Majorana edge states in a number conserving theory describing a system of spinless fermions on two wires that are coupled by pair hopping. Our analysis is based on a combination of a qualitative low energy approach and numerical techniques using the density matrix renormalization group. In addition, we discuss an experimental realization of pair-hopping interactions in cold atom gases confined in optical lattices.

Figure 1

(a) Ladder Hamiltonian: Atoms in the ground states $a$ and $b$ confined in the upper resp. lower wire can tunnel individually along the $x$ direction, and can hop in pairs between the wires. (b) Atomic scheme for ${}^{40}\mathrm{K}$ atoms: the two states $|F=9/2$, $m_F=9/2⟩$ and $|7/2$, $7/2⟩$ serve as $\uparrow$ and $\downarrow$ spin states in the microscopic implementation, while laser beams of different polarizations realize the spin dependent potential (see 34). (c) Typical lattice structure which realizes arrays of two-leg ladders: the circled areas correspond to the plaquette in panel (a). Black arrows denote counterpropagating beams on the $xy$ plane with ${ϵ}_z$ polarization, while red (blue) arrows describe counterpropagating beams in the $yz(xz)$ plane with ${ϵ}_x({ϵ}_y)$ polarization with a $\pi /4$ angle with respect to $yz(xz)$.

Figure 2

(a)–(d): Implementation of the pair hopping: (a) Ladder setup as a combination of two wires with opposite energy offsets (b). The dashed box denotes a single plaquette, with site indices in parenthesis $(\cdots )$; $t_{am}$, $t_{bm}$ are the tunneling amplitudes from the $a$ and $b$ wire to the central sites, respectively. Atoms in the center [(C)] of the plaquette interact with strength $U$ (shaded area). (b) The single wire is realized as a bipartite lattice of $\uparrow$ and $\downarrow$ fermions with Raman-assisted tunneling (Rabi frequency $\Omega$). (c),(d) Energy offsets along the diagonal of the plaquette in (c) for the $\downarrow$ resp, d) $\uparrow$ species, and corresponding virtual processes indicating pair tunneling (see text). The energy offsets $-{ϵ}_1$ and ${ϵ}_2$ inhibit single particle hopping between the wires, while the pair hopping respects energy conservation. (e) Time evolution of the state $a_{1,a\uparrow }^{†}{a}_{2,a\downarrow }^{†}|0⟩$ according to the microscopic dynamics of ${\tilde{H}}_{\square }$ (see text): the blue (red) curve indicate the population $p_{a,b}(t)=⟨n_{a/b,1}{n}_{a/b,2}⟩$ as a function of time.

Figure 3

(a) Closing of the energy gaps with the system size ($L=12$, 24, 36) for $W=-1.8$ and $n=1/3$. For OBCs, the gap $\Delta E_1$ closes exponentially (left panel), in contrast to the polynomial closing in the case of PBCs (right panel, open blue triangles). The energy gap $\Delta E_2$ closes polynomially independent of the boundary conditions (right panel, red diamonds for the OBC and closed triangles for PBC). Note that $\Delta E_{2,\mathrm{PBC}}=\Delta E_{1,\mathrm{PBC}}$. (b) Nonlocal fermionic correlations $G_{lj}$ on the upper wire for $L=24$. (c) and (d) Entanglement spectrum for the system of the size $L=24$ shows double degeneracy for both the OBC (c) and PBC (d).

Figure 4

Effects of imperfections on the topological order ($L=24$, $n=1/3$, OBC). (a),(b) Effects of static disorder: The nonlocal correlations (a) and the degeneracy of the ES (b) indicated the topological state in the presence of disorder with $V_r=0.1t$. (c) Effect of disorder in the pair hopping. The ES remains degenerate. (d) Ground state degeneracy in the presence of an interwire single-particle hopping $H_{\perp }=\sum _i{t}_y{a}_i^{†}{b}_i+\mathrm{H}.\mathrm{c}.$