Thermal instability of protected end states in a one-dimensional topological insulator

We have studied the dynamical thermal effects on the protected end states of a topological insulator (TI) when it is considered as an open quantum system in interaction with a noisy environment at a certain temperature $T$. As a result, we find that protected end states in a TI become unstable and decay with time. Very remarkably, the interaction with the thermal environment (fermion-boson) respects chiral symmetry, which is the symmetry responsible for the protection (robustness) of the end states in this TI when it is isolated from the environment. Therefore, this mechanism makes end states unstable while preserving their protecting symmetry. Our results have immediate practical implications in recently proposed simulations of TIs using cold atoms in optical lattices. Accordingly, we have computed lifetimes of topological end states for these physical implementations that are useful to make those experiments realistic.

Figure 1

The TI system is pictured in (b) as a ladder of hopping spinless fermions with amplitudes $K$ (horizontal and diagonal) and $M$ (vertical). The fermions are coupled to a magnetic gauge field on the lattice that is perpendicular to the plaquettes. This is a lattice gauge theory known as the Creutz Ladder (CL).[17, 18] The wavy lines at each site indicate interaction with thermal bosonic baths. In (a), the interaction vertex of fermions with the bosonic bath is shown.

Figure 2

Fidelities for the state of the CL with the initial Fermi sea. Here we have taken $\theta =\pi /2$, $T=1$ (in units of $K$, Eq. (2)), and ${\omega }_c=3/\left(2K\right)$ (of the order of the mean distance between the two bands). We see the fragility of the topological phase $(m<1)$ in comparison with the rest of the cases $(m>1)$. The main plot corresponds to the eight-site CL, and the inset depicts the initial decay rate in the thermodynamic limit (see main text).

Figure 3

Asymptotic occupation of the fermions for different values of $m$. The temperature is set to $T=1$ (in units of $K$, Eq. (2)). On the left side we have represented some examples which present topological order, on the right side the system is out of the topological phase. Note that on the left side the point 4 is fixed whereas the point 5 is fixed on the right side. For the sake of comparison we have depicted also the Fermi-Dirac distribution which corresponds to the thermal state.

Figure 4

Instability of the topological end states against thermal noise in an eight-site CL (the six-site CL provides the same results). Here, $\theta =\pi /2$ and the bath temperature is $T=0.04$ (in units of $K$, Eq. (2)), that is 1$\%$ of the CL gap. The $x$ axis represents the longitudinal position on the CL. Thus, at $t=0$ all fermions are localized at the left end, and as time increases they tend to delocalize along the whole chain. The inset plot depicts the probability that the fermions are in the end for several values of the magnetic flux $\theta$. The decay rate of the end state is smoother at $\theta =0$ and $\theta =\pi /2$ as the distance between bands increases for these values. Therefore, it is clear that the topological end states are unstable for any value of the magnetic flux. Note that for the decay of a right-end state the result is symmetric because of the chiral symmetry of the interaction (4).