Topological states in a microscopic model of interacting fermions

We present a microscopic model of interacting fermions where the ground state degeneracy is topologically protected. The model is based on a double-wire setup with local interactions in a particle number conserving setting. A compelling property of this model is the exact solvability for its ground states and low energy excitations. We demonstrate the appearance of topologically protected edge states and derive their braiding properties on a microscopic level. We find the non-Abelian statistics of Ising anyons, which can be interpreted as Majorana-like edge states.

Figure 1

Setup. (a) We consider a double chain (two-leg ladder) of spinless fermions with upper (lower) chain denoted as $a \left(b\right)$. (b) The number-conserving Hamiltonian is given by intrachain terms $A_i^x \left(x=a,b\right)$ and interchain couplings $B_j$.

Figure 2

Ground states. (a) For every filling $N$ with parity $P={(-1)}^N$, there are two degenerate zero-energy ground states for open boundary conditions, characterized by their (upper) subchain parity $\alpha =P_a$. (b) The chosen fermion gauge leads to the simple description of the ground states given in the text. (c) Behavior of the spectrum in the low-energy sector of symmetry subspaces classified by the total parity $P$ and the subchain parity $\alpha$. Both open (OBC) and periodic (PBC) boundary conditions are shown; zero-energy states are drawn bold.

Figure 3

Ground state properties. (a) Intrachain single-particle correlation $\langle a_i^{†}{a}_j\rangle$ (Green's function) as a function of the distance $|i-j|$ for various fillings $\rho$ and a chain of length $L=30$. The revival for $|i-j|\sim L$ indicates exponentially localized edge states (gray region). (b) Overlap of the ground states for time-reversal invariant (TRI) and breaking (TRB) perturbations of $H$ in dependence on the position $i,j$ of the subchain-parity violating single-particle hopping (blue: $\rho =0.5$ TRI; red: $\rho =0.5$ TRB; gray: $\rho =0.25$ TRB).

Figure 4

Entanglement. (a) Two branches ($\beta =\pm 1$: red/blue) of the entanglement spectrum for a chain of length $L=20$ and splitting $S=10$ with fillings $N=10,20,30$ (diamonds, circles, squares, respectively). The half-filling branch is highlighted gray. Physically, the index $n$ describes the subsystem filling while $\beta$ describes the subsystem subchain parity. This illustrates the twofold degeneracy of the entanglement spectrum. (b) The entanglement entropy $S^{\text{ent}}$ as a function of subsystem size $S$ for various fillings $\rho$. It obeys an area law with logarithmic corrections.

Figure 5

Braiding. (a) Setup of four open chains $L_i,i=1,2,3,4$ (black) with controllable weak single-particle couplings between the inner four end points. The partner chains (gray) are not involved in the braiding and can be disregarded. (b) The dynamics takes place in the eight-dimensional Hilbert space spanned by the subchain-parity eigenstates with fixed total parity $\alpha ={\alpha }_1{\alpha }_2{\alpha }_3{\alpha }_3=-1$. The colors denote the subchain parities ${\alpha }_i$ of the four black chains. (c) Spectrum of the weak coupling Hamiltonian during the braiding procedure depicted below the plot. A black arrow indicates single-particle hopping. There are four degenerate zero-energy ground states. The deviation from zero energy (perfect adiabaticity) due to the finite time evolution is shown in the inset $\left(\sim {10}^{-8}\right)$. (d) Time evolution for the initial zero-energy state $|00;0\rangle$. Shown are the (moduli of the) overlaps with $|00;0\rangle$ and $|11;0\rangle$ (see inset).