Observability of a sharp Majorana transition in a few-body model

We propose experimentally observable signatures of topological Majorana quasiparticles in the few-body limit of the interacting cold-atom model of Iemini et al. [F. Iemini, L. Mazza, L. Fallani, P. Zoller, R. Fazio, and M. Dalmonte, Phys. Rev. Lett. 118, 200404 (2017)]. In this limit, the total on-site density and single-body correlations change smoothly with the model parameters, while the calculated mutual information of opposite ends of the lattice indicates a sharp transition of the system to a topological ground state. Furthermore, local-density and -parity measurements provide an experimentally viable path for observing the ground-state Majorana quasiparticles in ultracold atoms. Our results lay out a promising future for utilizing few-body systems as a testing ground for Majorana physics.

Figure 1

The lowest-energy eigenvalues of the even-parity sector, shifted so that the center of the low-lying energies is approximately zero by subtracting a cubic fit of the average of the first four energies. A cubic fit is the lowest degree polynomial fit that gives an accurate approximation to the mean low-energy spectrum as a function of $W/T$ when the problem was solved using exact diagonalization. The fit $Q_3(W/T)$ is given by $0.0016{(W/T)}^3-0.1303{(W/T)}^2+0.8999(W/T)-64.131$. During the crossover from the nontopological to the topological phase, a zoo of states is formed and we can see the four lowest-energy states separate, with the ground state becoming approximately (doubly) degenerate. We use the same color scheme for all plots and focus on the four lowest-energy states as these are the states that interact with the lowest band in the quasiadiabatic few-body limit. Inset: There is an avoided crossing between the lowest-energy state and the second excited state. When $U=0=U_{\pm }$, the Hamiltonian may be written in a four-block-diagonal basis, and there is a nonavoided crossing with the first excited state, which lies in another symmetry sector (but becomes coupled when on-site interactions are included).

Figure 2

Single-species correlators $\langle a_{\uparrow ,+,1}^{†}{a}_{\uparrow ,+,j}\rangle$ are shown for the four lowest-energy states for $(L,N,T,U,{\alpha }_R,b)=(7,7,-1,0,4,4)$ (refer to Fig. 1 for the legend). (a) In the trivial phase ($W=10T$), all low-energy states exhibit a staggered decay with no rebounding of single-particle correlations. (b) In the topological phase ($W=16T$) the degenerate ground-state manifold exhibits exponential decay into the bulk and rebounds on the other end of the chain while the trivial states exhibit correlations that do not rebound. As $W$ is increased in magnitude, the bulk continues to lose correlation with the edges.

Figure 3

Top: Sample lattice with division into three regions $A, B$, and $C$. Bottom: Mutual information between the first and last sites ($L_A=1=L_C$) for the four lowest-energy states as $W/T$ is varied (refer to Fig. 1 for the legend). The degenerate ground-state manifold has topological properties after the transition region $W/T\in [11,14]$, while the first-excited-state manifold has no edge-edge correlation beyond a product state.

Figure 4

Mutual information of the left and right edges for (a) $L_A=2=L_C$ and (b) $L_A=3=L_C$. As the widths of the edges for the mutual information calculations are increased, one can see the evolution towards bulk behavior as short-ranged entanglement provides the majority of the entanglement between the two halves with finite-size effects causing the cusplike features in the transition region.

Figure 5

(a) In the trivial regime of the model ($W=10T$ in this plot), all low-energy states have very low densities at one edge of the lattice for each particle species (refer to Fig. 1 for the legend). (b) In the topological regime ($W=16T$), the densities of the topological ground-state manifold have comparable values on both ends of the lattice, while the nontopological excited states maintain near-zero edge densities at one end of the lattice.

Figure 6

Local ground-state parity measurements of the left-hand side $\langle P_{+,L_A}\rangle$ plotted for $L_A=1,2,3$ in the even-parity sector. With just a single site, the local-parity measure shows a small but noticeable transition, while the local parity of the first three sites provides a very strong evidence of the phase transition.