Disorder-induced topological phase transition in a driven Majorana chain

We study a periodically driven one-dimensional Kitaev model in the presence of disorder. In the clean limit our model exhibits four topological phases corresponding to the existence or nonexistence of edge modes at zero and $\pi$ quasienergy. When potential disorder is added, the system parameters get renormalized and the system may exhibit a topological phase transition. When starting from the Majorana $\pi$ mode (MPM) phase, which hosts only edge Majoranas with quasienergy $\pi$, disorder induces a transition into a neighboring phase with both $\pi$ and zero modes on the edges. We characterize the disordered system using (i) exact diagonalization, (ii) Arnoldi mapping onto an effective tight-binding chain, and (iii) topological entanglement entropy.

Figure 1

(a) A typical autocorrelation function $A_{\infty }\left(nT\right)$ as a function of ${log}_{10}\left(n\right)$. The autocorrelation is calculated using both the matrix $W$ (red) as well as through exact diagonalization (blue). The two methods agree and therefore the blue symbols are not visible. The results are presented for a single disorder realization scaled such that the standard deviation is $0,0.3,\text{and}0.6$ (from top to bottom). (b) The nearest neighbor hopping ${\beta }_m$ as a function of the Krylov site index $m$, in the effective Krylov-space chain, given by the first diagonal next to the main one of the matrix $iln\left(W\right)$. (c) The first few elements of $iln\left(W\right)$ represented by color. Darker colors corresponds to larger elements. In the above calculation we have used the model in Eq. (1) with parameters $w=\mathrm{\Delta }=1$ and $\overline{\mu }=0.6$. $\sigma$ is the standard deviation of the disorder, measured relative to the average chemical potential $\overline{\mu }$.

Figure 2

(a) A typical autocorrelation function $A_{\infty }\left(nT\right)$ as a function of ${log}_{10}\left(n\right)$, calculated using exact diagonalization of a chain of $L=200$ sites, shown for varying disorder strength. (b) The same data as on the left, but averaged over 10 disorder realizations. The model parameters used are the same as in Fig. 1.

Figure 3

Arnoldi spectrum, i.e., spectrum of $iln\left(W\right)$, and wave functions, for a single disorder realization. (a) The spectrum of the Arnoldi chain plotted in order of increasing quasienergy magnitude (b) Probability distribution for the state at zero energy. The first 20 Arnoldi sites are shown. (c) Probability distribution for the state at energy $\pi$. Besides the range of disorder strengths, the parameters are the same as in Fig. 1. In some cases, there is a state near zero energy in the Arnoldi spectrum which is not localized at the edge of the sample. In that case it is not plotted in the middle panel.

Figure 4

Evolution of the spectrum as the disorder strength is increased. As the disorder is increased the quasienergy gap around $\epsilon =\pi$ closes but the edge MPMs remain localized. Moreover, the gap around $\epsilon =0$ also closes and a MZM emerges. The Majorana edge modes can be seen in Fig. 5.

Figure 5

Exact diagonalization spectrum and wavefunctions for a single disorder realization with increasing disorder strength (top to bottom). (a) Spectrum, (b) zero modes, and (c) $\pi$ modes. In both the middle and right columns, the black curves are states at energies indistinguishable from $\epsilon T=0$ or $\epsilon T=\pi$. The purple curves are states with energies in the vicinity of zero or $\pi$.

Figure 6

The topological entanglement entropy and inverse localization length. (a) Partitions used to calculate the topological entanglement entropy. [(b) and (c)] The topological entanglement entropy as a function of the disorder strength for a single disorder realization (b), and averaged over 100 disorder profiles (c). In both cases, a chain of 200 sites was used. (d) The spatial decay rates ${\mathrm{\Gamma }}_0$ and ${\mathrm{\Gamma }}_{\pi }$ (inverse localization length), in units of inverse lattice constant, corresponding to the 0 and $\pi$ modes respectively. The light (orange) purple lines are $\left({\mathrm{\Gamma }}_{\pi }\right){\mathrm{\Gamma }}_0$, calculated using Eq. (28) [(29)] averaged over 100 disorder realizations with error bars corresponding to the standard deviation. The average of these 100 realizations exactly coincide with Eqs. (31) and (30), ploted in darker shades, where the average is done by integrating a Gaussian distribution for $\delta \mu$. Note that the condition for an edge mode is $\mathrm{\Gamma }<0$ as the opposite regime represents a non-normalizable function. In both (c) and (d), a phase transition is observed around $\sigma /\overline{\mu }\approx 0.6$, as indicated by the vertical dashed line.

Figure 7

The phase diagram for a clean system based on the criteria (28) and (29). Here blue is the trivial phase, light blue is the phase with a MZM, yellow has a MPM and red is a $\mathrm{MZM}+\mathrm{MPM}$ phase. The grid lines indicate the parameters chosen for much of this paper, $\mu =0.6$ and $\mathrm{\Delta }Tmod2\pi$ with $\mathrm{\Delta }=1$ and $T=8.25$.

Figure 8

The phase diagram for a disordered system with $\overline{\mu }=0.6$ and $T=8.25$ based on the criteria (30) and (31) where $\delta {\mu }_i$ is taken from a Gaussian distribution with variance ${\sigma }^2$. Here blue is the trivial phase, yellow is the phase with a MPM and red is a $\mathrm{MZM}+\mathrm{MPM}$ phase. The grid line indicates $\mathrm{\Delta }Tmod2\pi$ with $\mathrm{\Delta }=1$ and $T=8.25$, as chosen for much of this paper.

Figure 9

(a) The phase diagram for a disordered system with $\overline{\mu }=0.6$ and $T=8.25$ based on the criteria (30) and (31) for a uniform distribution in the range $[-\mathrm{\Sigma }/2,\mathrm{\Sigma }/2]$. Here blue is the trivial phase, light blue is the phase with a MZM, yellow has a MPM, and red is the $\mathrm{MZM}+\mathrm{MPM}$ phase. (b) The topological entanglement entropy, $S^{\mathrm{topo}}$, calculated for the same parameters as on the left panel. The chain length is $L=100$ and the results are averaged over 1000 disorder realizations. The grid line in both figures indicates $\mathrm{\Delta }Tmod2\pi$ with $\mathrm{\Delta }=1$ and $T=8.25$, as chosen for much of this paper.