Topology of One-Dimensional Quantum Systems Out of Equilibrium

We study the topological properties of one-dimensional systems undergoing unitary time evolution. We show that symmetries possessed both by the initial wave function and by the Hamiltonian at all times may not be present in the time-dependent wave function—a phenomenon which we dub “dynamically induced symmetry breaking.” This leads to the possibility of a time-varying bulk index after quenching within noninteracting gapped topological phases. The consequences are observable experimentally through particle transport measurements. With reference to the entanglement spectrum, we explain how the topology of the wave function can change out of equilibrium, both for noninteracting fermions and for symmetry-protected topological phases protected by antiunitary symmetries.

Figure 1

Panel (a): Time-dependent CS invariant of a hopping model of spinless fermions, calculated as a bulk integral in $k$ space (solid lines), compared with the polarization $Q_B(t)$ of a 24-site open boundary system with the same parameters (dashed lines). $Q_B(t)$ is calculated as the expected particle number within the right-hand half, subsystem $B$ of the inset. The red lines are for a BDI system and the blue lines are for an AIII system. The parameters for the quenches are $(J_1,J_2)=(0.3,e^{i\alpha })\to (0.8e^{i\alpha },1)$ with $B_{1,2}=0.05$ throughout; $\alpha =0$ for class BDI and $\alpha =0.4$ for class AIII. The observables in the finite sample match the dynamics of the bulk invariants even out of equilibrium, until correlations traverse the whole system at which point the discrete nature of $k$ space invalidates Eq. (5). Panel (b): dynamics of the entanglement gap ${\mathrm{\Delta }}_E$ for the same systems as above with the entanglement cut between $A$ and $B$ [inset of (a)]. In the BDI case, the entanglement gap remains close to zero until correlations span the system size, whereas the AIII system immediately becomes gapped. Panel (c): dynamics of the entanglement gap for a spin-1 chain initialized in a Haldane phase, possessing TRS only (purple line), and both TRS and dihedral symmetry (green line, scaled by $10^3$). When the Haldane phase is supported by TRS only, the entanglement spectrum becomes gapped for $t>0$.