Entanglement topological invariants for one-dimensional topological superconductors

Entanglement provides characterizing features of true topological order in two-dimensional systems. We show how entanglement of disconnected partitions defines topological invariants for one-dimensional topological superconductors. These order parameters quantitatively capture the entanglement that is possible to distill from the ground-state manifold and are thus quantized to 0 or $log2$. Their robust quantization property is inferred from the underlying lattice gauge theory description of topological superconductors and is corroborated via exact solutions and numerical simulations. Transitions between topologically trivial and nontrivial phases are accompanied by scaling behavior, a hallmark of genuine order parameters, captured by entanglement critical exponents. These order parameters are experimentally measurable utilizing state-of-the-art techniques.

Figure 1

Partitions associated with the entanglement topological order parameter and the phase diagram of the interacting Kitaev chain. (a) schematics of the partitions $A$ (shaded, green) and $B$ (shaded, blue) considered here. Each site of the chain hosts a spinless fermion degree of freedom, $a_j$, that can be decomposed into two Majorana fermions $c_{2j}$ and $c_{2j+1}$. The orange circle magnifies the cut across partitions: Deep in the topological phase, neighboring Majorana fermions belonging to different physical sites are coupled (dashed line). The partition cut takes place exactly between the two coupled Majorana fermions. (b) disconnected von Neumann entropy $S^D$ as a function of $\mu /t,U/t$, at fixed $\mathrm{\Delta }=1$. Black lines are from Ref. [23]. The color plot is obtained via interpolation on a 5 x 7 grid. Clearer evidence of the quantization of $S^D$ in a phase and the sharpness of the transition requires a lot of points, as we highlight on lines I and II in Sec. 3b.

Figure 2

Schematics of the correspondence between the Kitaev chain and ${\mathbb{Z}}_2$ lattice gauge theories. (a) Hilbert space structure and gauge-invariant building blocks. (b) Three examples of the mapping between states in the fermionic (left), gauge theory (center), and string representation (right). (c) String representation of ${|\mathrm{\Psi }\rangle }_{+}$.

Figure 3

Color plot of the disconnected $n$ entropies, $S_n^D$, of the Kitaev wire without interaction, $U=0$, for (a) $n=1$ and (b) $n=2$. The results are obtained with the free-fermion technique. The $y$ axis represents the chemical potential, $\mu /t$, while the $x$ axis the superconducting amplitute, $\left|\mathrm{\Delta }\right|/t$. A grid with $16\times 30$ points is considered. The two theoretically expected phase transition lines occur for $\left|\mathrm{\Delta }\right|=0$ and $\mu /t=2$, respectively. Using the definitions of Fig. 1, $L=50,L_A=L_B=12$, and $L_D=32$. The diagrams coincide with the results obtained in Ref. [31].

Figure 4

Finite-size scaling properties of $S^D$ in (a) normal scale, (b) log scale, for a chain with $L_A=L_B=(L_{A\cap B}+L_{A\cup B})/2$, and $U=0,\mathrm{\Delta }/t=1$. In the topologically trivial phase, $S^D$ quickly vanishes. In contrast, in the topological phase $\left(\mu =1.0,1.5\right)$, $S^D$ increases as a function of system size, and approaches its thermodynamic value exponentially fast when increasing $L_A$, as shown in (b).

Figure 5

Finite size scaling of $S^D$ (in units of ${log}_2)$ along lines (I) and (II) of Fig. 1 as a function of $L_D$ and $\mu$ or $U$ using DMRG. In all plots, $L_A=L_B=12$. (a) $S^{D}L$ as a function of $\mu$ for different sizes: The critical value ${\mu }_c$ is the intersection of all curves; we obtain ${\mu }_c=1.978$. (b) Scaling of $\lambda \left(x\right)$ for different system sizes: Curve collapse. The collapse is best realized for $a=b=1$, values that also minimize the square root of the residual sum of squares. (c) $S^{D}L$ as a function of $U$ for different sizes: The critical value is here $U_c\left(12\right)=-0.314$. (d) The collapse is again best realized for $a=b=1$. Simulations with more sites (especially using free fermion techniques) only confirm these results.

Figure 6

Time evolution of $S^D$ after a quantum quench from (a) ${\mathrm{\Delta }}_0=0.5$ to $\mathrm{\Delta }=1.5$, (b) from ${\mu }_0=1.0$ to $\mu =3.0$, and (c) from ${\mu }_0=3.0$ to $\mu =1.0$ with $U=0,\mu =0,L=8L_A$. Finite-size scaling of $t_c$ or $t_c^{\prime }$ for two values of (d) $\mathrm{\Delta }$, (e) ${\mu }_0$, and (f) $\mu$ of the quenched Hamiltonian. In all cases, the width of the plateau diverges linearly with system size, as expected for topological invariants. The threshold lines of $S^D$ according to the definition of $t_c$ are depicted as a dashed line.

Figure 7

Scaling in system size of the mean value of SD for $t=\mathrm{\Delta }=\mu =1,U=0,L_A=L_B=(L_{A\cap B}+L_{A\cup B})/2$ and for three amplitudes of disorder: $W=1$ and 3 in the topological phase and $W=12$ for the trivial disordered phase (200 realizations of the disorder for each point). The inset provides a logarithmic scale for the axis of $S^D$: The scaling is exponential. The standard deviation is smaller than ${10}^{-6}$ for each point. The same study using Rényi-2 entanglement entropies leads to the same results quantitatively. The phase transition is expected at $W_c\sim 11$ for these parameters [45, 46].

Figure 8

The Kitaev chain with $L$ sites and open boundary conditions. Each site $i$ (denoted by a dashed circle) can be occupied by one spinless fermion and can be decomposed into two Majorana fermions (denoted by black dots) in $2i-1$ and $2i$. Associating the Majoranas $2i$ and $2i+1$ allows the construction of a quasilocal fermionic basis denoted with tildes. The ground states in the topological stereotypical regime will see its neighboring Majorana fermions pairing up, so, in the tilded basis, each site is unoccupied. Only the two Majorana on the edges do not need to pair up.

Figure 9

A physical cut can only be done between sites, here, on the link $c$, partitioning the chain into two subsets: $A$ and $B$.

Figure 10

A partition of the chain into $n$ consecutive connected subsets $A_1,A_2$,..., $A_n$.

Figure 11

Scheme of the partitions. Two “tricks” are used to compute only standard bipartition of the chain: (i) a circular permutation of the partition used for $S({\rho }_A)$, $S\left({\rho }_{A\cup B}\right)$, and $S\left({\rho }_{A\cap B}\right)$ and (ii) reindexing the sites, thus making the Hamiltonian long range, but making the subset $B$ connected and used for $S\left({\rho }_B\right)$.