Topologically Protected Long Edge Coherence Times in Symmetry-Broken Phases

We argue that symmetry-broken phases proximate in phase space to symmetry-protected topological phases can exhibit dynamical signatures of topological physics. This dynamical, symmetry-protected “topological” regime is characterized by anomalously long edge coherence times due to the topological decoration of quasiparticle excitations, even if the underlying zero-temperature ground state is in a nontopological, symmetry-broken state. The dramatic enhancement of coherence can even persist at infinite temperature due to prethermalization. We find exponentially long edge coherence times that are stable to symmetry-preserving perturbations and not the result of integrability.

Figure 1

Sketch of the dominant processes that tunnel between the two ferromagnetic ground states. Domain walls (DW) are represented by blue bars, and their decorated counterparts (DW*) are red and carry a ${\mathbb{Z}}_2$ charge. Under periodic boundary conditions (PBC), the two types of domain walls are equivalent. With open boundary conditions (OBCs), however, the decorated domain walls cannot be annihilated at the edges without breaking the symmetry, so will “bounce off” instead. Decorated domain walls are therefore unable to flip the edge spin without breaking the symmetry.

Figure 2

Autocorrelation of the edge spin at zero temperature computed with exact diagonalization (ED) for 14 spins and OBC. The parameters are $J=5.2$, $B=1.424$, and $V(g_1,\dots ,g_5)$ is chosen to break integrability completely [40]. Inset: sketch of the phase diagram for Eq. (1) as a function of $x$ and $J$. Phases are described in the text. The location of the dots corresponds to the data by color.

Figure 3

($\mathbf{T}=\mathbf{0}$) Comparison of the coherence time (data) with its analytical prediction (lines). Data are computed on 14 spins via ED with parameters $(J,B)=(5.2,1.27)$. The symbols were obtained with $V=0$, whereas the dashed lines were obtained with $V\ne 0$. ($\mathbf{T}=\infty$) Comparison of coherence times for OBC and PBC at infinite temperature on 14 spins with $(J,B)=(1.57,9.03)$ and $V$ chosen so that the model is not integrable–see Figs. 4 and 4. Numerical details are given in the Supplemental Material [40].

Figure 4

(a) Autocorrelation $C_{\infty }(t)$ at $x=1$ and $T=\infty$ under OBC and varying system size. $C_{\infty }(t)$ remains close to one for a time $\tau$ until it drops to its thermal value of 0, and $\tau$ increases exponentially with system size until its saturation. (b) The same autocorrelation $C_{\infty }(t)$ under various conditions on 14 sites. “Edge” is same as in the main panel, “bulk” corresponds to ${\sigma }_{L/4}^z$, “PBC” corresponds to periodic boundary conditions, and “No Sym” corresponds to a system where the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry was broken explicitly with edge perturbations ${\sigma }_0^x{\tau }_1^z$ and ${\sigma }_0^y{\tau }_1^z$. (c) Histogram of the differences in adjacent energy levels showing the nonintegrability of the model and (d) normalized density of states in the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ even/even sector on 16 spins. (e) Coherence time for $x=1$. Here $J=1.57$ and $B=8.42$ in (a)–(d). Numerical details are given in the Supplemental Material [40].