Classification of topological insulators and superconductors out of equilibrium

We establish the existence of a topological classification of many-particle quantum systems undergoing unitary time evolution. The classification naturally inherits phenomenology familiar from equilibrium—it is robust against disorder and interactions, and exhibits a nonequilibrium bulk-boundary correspondence, which connects bulk topological properties to the entanglement spectrum. We explicitly construct a nonequilibrium classification of noninteracting fermionic systems with nonspatial symmetries in all spatial dimensions (the ‘tenfold way'), which differs from its equilibrium counterpart. Direct physical consequences of our classification are discussed, including important ramifications for the use of topological zero-energy bound states in quantum information technologies.

Figure 1

Illustration of topological restrictions on unitary dynamics. The various topological phases are denoted ‘0,’ ‘$+1$,’ and ‘$-1$.’ In each case (a)–(d), some phases are compatible with the symmetry constraints on the initial state, and others are not; the inaccessible phases are marked with hatches. Diagrams (a), (b), and (d) feature initial states ${\rho }_1$ and ${\rho }_2$ that are separated by an equilibrium topological phase transition (horizontal line), while in (c) the initial states are forced to be in the same equilibrium phase by the initial symmetries. In diagrams (a) and (c), the phase boundaries persist even after dynamically induced symmetry breaking—it is not possible for a state with one topology to evolve into a state with different topology. In (b) and (d), the topological distinction between the two phases is broken out of equilibrium, i.e., the initial state topology is lost under dynamics. In cases (b)–(d), the nonequilibrium classification is trivial due to initial state restrictions and/or the breakdown of equilibrium phase boundaries due to symmetry breaking.

Figure 2

The physical Brillouin zone (BZ) as the equator of a higher dimensional momentum space $\mathrm{\Sigma }\left(\text{BZ}\right)$ parametrized by $(\vec{k},\theta )$. At the poles $\theta =\pm \pi$, the BZ is contracted to a point, representing the $\vec{k}$-independent reference state. We also identify the two poles, ensuring periodicity in $\theta$.

Figure 3

Dimensional extension of a chiral symmetric insulator after unitary time evolution. The higher dimensional BZ, which was compact at the poles $\theta =\pm \pi$, becomes open for $t>0$.

Figure 4

Geometry of the entanglement cut for 2D systems with periodic boundary conditions in the $x$ direction and open boundary conditions with a large system size in the $y$ direction. The dashed line represents the divide between regions $A$ and $B$.

Figure 5

Dynamics of the single-particle modes of the entanglement spectrum (eigenvalues ${\zeta }_n$ of the reduced single-particle density matrix $C_{i,j}=\langle \mathrm{\Psi }\left(t\right)|{\widehat{\psi }}_i^{†}{\widehat{\psi }}_j|\mathrm{\Psi }\left(t\right)\rangle$, where $i,j$ belong to the spatial region $A$) for the Haldane model (top) and the Kane-Mele model (bottom). In both systems, we start with a topologically nontrivial initial state at $t=0$ (left) and then time evolve under a different Hamiltonian by a time $t=2J_1^{-1}$. The entanglement spectrum of the time-evolved state is plotted (right). In the Haldane model, where topology is preserved, the edge state remains gapless after time evolution. However, in the Kane-Mele model, the gapless modes are TRS protected and topology is destroyed out of equilibrium, hence the edge state becomes gapped at finite times.

Figure 6

Decoherence of Majorana qubit memories due to temporal noise, as witnessed by the recovery fidelity [Eq. (18)]. We compare three systems in $d=1$ [with Hamiltonians given in Eq. (17)], in symmetry classes DIII and BDI—these have entries ${\mathbb{Z}}_2\to 0$ and $\mathbb{Z}\to {\mathbb{Z}}_2$, respectively, in the nonequilibrium classification (Table 2). Accordingly, the topology of models (17a) and (17c) is unstable out of equilibrium. This is reflected in the fidelity of storage in the associated Majorana modes when local Lorentzian (TRS) noise is present: The fidelity for the stable model (17b) saturates at a constant value, indicating the preservation of the qubit, whereas the initial state information stored in the unstable models decays away completely, indicating that there is no measurement which can be made to extract the initial qubit state.