Random-walk topological transition revealed via electron counting

The appearance of topological effects in systems exhibiting a nontrivial topological band structure strongly relies on the coherent wave nature of the equations of motion. Here, we reveal topological dynamics in a classical stochastic random walk version of the Su-Schrieffer-Heeger model with no relation to coherent wave dynamics. We explain that the commonly used topological invariant in the momentum space translates into an invariant in a counting-field space. This invariant gives rise to clear signatures of the topological phase in an associated escape time distribution.

Figure 1

(a) Stochastic random walk on a lattice with SSH coupling geometry. The state of the system $\left|n\right.〉$ changes randomly as a function of time with alternating hopping probabilities according to Eq. (1). (b) This random walk describes the dynamics of a SET connected to two Markovian leads with properly adjusted chemical potentials, which are coupled to the dot with strengths ${\mathrm{\Gamma }}_{\mathrm{R}}$ and ${\mathrm{\Gamma }}_{\mathrm{L}}$. By counting the number of particles tunneled into the right reservoir $m$ and monitoring the dot occupation $d\in \{0,1\}$, one can infer the state $n=2m-d$. (c) Illustration of the TI in the SSH model, which is equivalent to a winding of the curve $\left[l_x\left(\chi \right),l_y\left(\chi \right)\right]$ (depending on ${\mathrm{\Gamma }}_{\mathrm{R}/\mathrm{L}}$) around the origin. (d) Stochastic trajectory of ${\left|n\right.〉}_t$ as a function of time in units of $2\overline{\gamma }={\mathrm{\Gamma }}_{\mathrm{R}}+{\mathrm{\Gamma }}_{\mathrm{L}}$. At time $t_{\mathrm{e}}$ the state escapes from the region $\left\{1,...,N\right\}$, resembling the quantum SSH model with an open boundary condition.

Figure 2

Comparison of the spectrum of ${\mathbf{L}}_{\mathrm{SSH}}$ in Eq. (4) with open boundary conditions and the exponent spectrum ${\beta }_i$ of the integrated ETD according to Eq. (6). In (a), (b), and (c) we depict the results for the chain lengths $N=10, N=18$, and $N=17$, respectively. The orange (solid) lines depict the spectrum of the matrix ${\mathbf{L}}_{\mathrm{SSH}}$. The top panels depict the exact results obtained by a diagonalization. The dotted exponent spectrum depicts the levels with nonvanishing coefficients $A_j$ [see Eq. (6)]. The coefficients $A_j$ are represented by the blue regions, whose width is proportional to $A_j$. The bottom panels depict the exponent spectrum obtained by a simulation of random trajectories. To construct the integrated ETD we used $i_{\max }={10}^5$ random trajectories, which have been fitted with $K=3$.

Figure 3

(a) depicts the reconstructed escape time distribution (ETD) $P_{\mathrm{e}}\left(t\right)$ (dots), the reconstructed integrated ETD $P_{\mathrm{e},\mathrm{int}}\left(t\right)$ (crosses), and the corresponding fitted curves with solid and dashed lines for $\alpha =-0.5\overline{\gamma }$. (b) Cumulants ${\kappa }_m$ of the ETD for $m=1$ (solid), $m=2$ (dashed), and $m=3$ (dotted) as a function of $\alpha$. The depicted and higher cumulants do not reveal any signature of topology as, e.g., a nonanalyticity at $\alpha =0$.