The unique control features of topological stochastic and quantum systems

Abstract

Topological phases support edge states that can be robust to material deformations and other perturbations. While well-studied in quantum systems, topological phases have also been observed in stochastic and biochemical systems, yet it remains unclear which of their properties remain similar or different from those in quantum systems. In this paper, we derive analytical expressions for the spectral properties of simple quantum and stochastic models on the same lattice to rigorously characterize these complex systems. Intriguingly, we find that non-reciprocity moves states away from the steady-state in stochastic systems while clustering states at zero-energy in quantum systems. In contrast, making the system more topological does the opposite: it clusters more states around the steady-state in stochastic systems but moves states away from the zero-energy state in quantum systems. These results provide control parameters for selection and modulation of different purposes while quantifying the size of gap which protects the longest-lived states. Lastly, we discover a mode unique to stochastic systems that we dub the topologically emerging state, which persists across different models and dimensions, including in the presence of non-equilibrium currents.

Figures (6)

• Figure 1: Properties of 1D Hatano-Nelson non--hermitian quantum and stochastic lattices. (a) A uniform lattice with open boundary conditions and non-reciprocal transition rates, parameterized by the forward rate $\gamma_{\mathrm{in}}$ and the backward rate $\gamma_{\mathrm{in}}^{\prime}$. The strength of non-reciprocity is characterized by the ratio $c = \frac{\gamma_{\mathrm{in}}}{\gamma_{\mathrm{in}} + \gamma'_{\mathrm{in}}}$. Here $N$ is the total number of sites in the chain and $j$ labels a lattice site. (b) The stochastic cluster for system size $N$ overlaps with the quantum cluster of size $N{-}1$. In the stochastic system, as the system size $N$ increases toward the thermodynamic limit, the spectral gap between the cluster and the steady state saturates to a constant value. Red circles denote the spectrum of $\mathcal{A}$. Triangles denote the spectrum of $\mathcal{W}$, with blue corresponding to the spectral cluster, purple to the steady state, and green to the topologically emerging state. The last state is explored in Sec. \ref{['sec:pushed']}. The stochastic gap is defined as the distance between the steady-state eigenvalue and the second-largest eigenvalue. Here $c = 0.8$. (c) We show the stochastic and quantum spectrum as a function of $c$. As the non-reciprocity parameter $c$ increases, both quantum and stochastic clusters collapse to a single point, while the stochastic steady-state eigenvalue remains fixed. Here $N = 6$. (d) Stochastic eigenvectors exhibit an exponential envelope with sinusoidal modulation. Stochastic eigenvector profiles at $c = 0.8$ for $N = 6$.
• Figure 2: Properties of 1D SSH non--hermitian quantum and stochastic lattices. (a) A 1D lattice with non-reciprocal transition rates, parameterized by the alternating forward internal rate $\gamma_{\mathrm{in}}$, forward external rate $\gamma_{\mathrm{ex}}$, backward internal rate $\gamma_{\mathrm{in}}^{\prime}$, and backward external rate $\gamma_{\mathrm{ex}}^{\prime}$. The strength of non-reciprocity is characterized by the ratio $\frac{\gamma_{\mathrm{in}}}{\gamma_{\mathrm{in}}^{\prime}} = \frac{c}{1-c}$. The yellow shading indicates one unit cell. (b) The stochastic cluster of system size $N$ (integer number of unit cells) overlaps with the quantum cluster of size $N{-}1$ (non-integer number of unit cells). Red circles show the spectrum of $\mathcal{A}$; triangles show the spectrum of $\mathcal{W}$, with blue for the cluster, purple for the steady state, and green for the topologically emerging state. Here, $\frac{\gamma_{\mathrm{in}}}{\gamma_{\mathrm{ex}}^{\prime}} = 8$ and $\frac{\gamma_{\mathrm{ex}}}{\gamma_{\mathrm{in}}^{\prime}} = 4$. (c) The topologically emerging state lies closer to the steady state in the stochastic spectrum within the topological regime, despite being the middle mode of the spectrum. As the non-reciprocity parameter $c$ increases, quantum clusters converge to one point, and stochastic clusters converge to two separate points, while the stochastic steady state remains fixed. Here $c$ is varied at fixed ratio $r = 0.8$, and $N = 6$. (d) The parameter $r$ is varied at fixed non-reciprocity $c = 0.8$. The gray shading indicates the topological phase, and the white shading indicates the trivial phase. The diagonal green line tracks the topologically emerging eigenmode, which approaches the steady state linearly as $r \to 1$. Here $N = 6$.
• Figure 3: The step-like spatial profile of TES persisted in 1D SSH model. Left: Eigenvector profiles at $c = 0.8$, $r = 0.8$ for $N = 6$. Right: Eigenvector profiles at $c = 0.999$, $r = 0.8$ for $N = 6$. Same color scheme as Fig. \ref{['fig:lattice']}(d).
• Figure 4: Properties of 2D SSH non--hermitian quantum and stochastic lattices. (a) A 2D lattice with non-reciprocal transition rates, parameterized by alternating forward rates $\gamma_{\mathrm{in}}$, $\gamma_{\mathrm{ex}}$ and backward rates $\gamma_{\mathrm{in}}^{\prime}$, $\gamma_{\mathrm{ex}}^{\prime}$. The strength of non-reciprocity is characterized by $c$, and the relative strength of internal and external rates is characterized by $r$. The yellow shading indicates one unit cell. A pair of numbers $(i,j)$ indicates the position of the cell. (b) We show the real and imaginary spectra of the quantum and stochastic systems, where gray shading indicates the topological phase and the white shading indicates the trivial phase. In the strongly topological regime, the topologically emerging state (green line) and other states cause the stochastic spectrum (blue lines) to contain a significantly larger number of states near the steady state (purple line), as compared to the the quantum spectrum (red lines) near zero energy. These points of comparison are indicated with circles in both the stochastic and quantum systems, in the strong topological limit. Here, $c = 0.8$, $N_i = 2$, and $N_j = 3$.
• Figure 5: At extreme non-reciprocity, the eigenvector of the cluster completely overlaps with that of the TES. Eigenvector profiles at $c = 0.99$, $r = 0.5$ for $N = 6$. Same color scheme as Fig. \ref{['fig:lattice']}(d).