Emergent Supersymmetry from Strongly Interacting Majorana Zero Modes

We show that a strongly interacting chain of Majorana zero modes exhibits a supersymmetric quantum critical point corresponding to the $c=\frac{7}{10}$ tricritical Ising model, which separates a critical phase in the Ising universality class from a supersymmetric massive phase. We verify our predictions with numerical density-matrix-renormalization-group computations and determine the consequences for tunneling experiments.

Figure 1

(a) The model with nearest-neighbor hopping and interactions between four nearest Majoranas. (b) The phase diagram of Hamiltonian (1) for $g=1$ as a function of $t$.

Figure 2

The symmetry broken states at strong coupling. The thick lines indicate pairs of Majorana operators combined to form Dirac operators. The empty circles symbolize that the resulting Dirac levels are empty. For both of the states shown, there is another degenerate state, where are all Dirac levels are filled.

Figure 3

Top: The detection of the critical point through one of the universal ratios. Bottom: The values of four universal gap ratios at the tricritical point $t/g=0.00405$ as a function of $L$ (for systems with $2L$ Majoranas). The numerical results (data points) show excellent agreement with the CFT predictions (black lines at $7/2,7/24,3/8$, and $35/8$). The DMRG truncation errors are insignificant (up to 1000 states were kept in the computations).

Figure 4

The scaling behavior of the fermionic Green’s function (obtained with DMRG keeping 800 states) for the Ising and the TCI CFTs observed in our microscopic model.