Majorana Quasiparticles Protected by ${\mathbb{Z}}_2$ Angular Momentum Conservation

We show how angular momentum conservation can stabilize a symmetry-protected quasitopological phase of matter supporting Majorana quasiparticles as edge modes in one-dimensional cold atom gases. We investigate a number-conserving four-species Hubbard model in the presence of spin-orbit coupling. The latter reduces the global spin symmetry to an angular momentum parity symmetry, which provides an extremely robust protection mechanism that does not rely on any coupling to additional reservoirs. The emergence of Majorana edge modes is elucidated using field theory techniques, and corroborated by density-matrix-renormalization-group simulations. Our results pave the way toward the observation of Majorana edge modes with alkaline-earth-like fermions in optical lattices, where all basic ingredients for our recipe—spin-orbit coupling and strong interorbital interactions—have been experimentally realized over the last two years.

Figure 1

Schematics of the orbital Hubbard model in the presence of spin-orbit coupling as realized with alkaline-earth-like atoms. (a)–(b) The model we consider in Eq. (1) describes tunneling ($H_t$), spin-orbit-coupling ($H_{\mathrm{so}}$), and spin-exchange processes ($H_W$). In cold atom settings, the spin degree of freedom is represented by different Zeeman states with nuclear spin $m_F$, $m_F+1$, while the orbital degree of freedom is encoded in different electronic states, ${}^1{S}_0$ and ${}^3{P}_0$. In these systems, $H_W$ and $H_{\mathrm{so}}$ are described by the gray and red arrows, respectively. (c) In the quasitopological phase of the model, the entanglement spectrum displays a characteristic twofold degeneracy: eigenvalues of the reduced density matrix with the same number of particles come in pairs with opposite parities (see text).

Figure 2

DMRG analysis of the topological properties of the ground state for model $H$, at fixed parity, with parameters $W=-8$, ${\alpha }_R=b=4$, $U=0$, at distinct fillings $n=1/4$ and $n=1/6$. (a) Algebraic scaling of the gap computed at fixed parity, compatible with $\sim L^{-1}$. (b) Exponential scaling of the gap between the distinct parity sectors. (c)–(d) Single particle correlations $G(j,\ell )=⟨c_{j,\uparrow ,1}^{†}{c}_{\ell ,\uparrow ,1}⟩$ at the bulk (c) and at the edges (d), for a system with $L=48$ sites.

Figure 3

DMRG analysis for the adiabatic continuity of model $H$ to an exactly solvable model. (a) Parity gap “${\alpha }_{\mathrm{sp}}$” along the adiabatic continuity path $(\beta =0,\lambda =9/10)\to (\beta =1,\lambda =1)$. Inset illustrates the Hamiltonian parameters varied along the path. (b) Single particle edge correlations along the adiabatic path—similar behavior follows for $n=1/6$. In all plots we consider a system with $L=48$ sites and use $m=140$ number of kept states in the DMRG simulations.