Localization due to topological stochastic disorder in active networks

An active network is a prototype model in nonequilibrium statistical mechanics. It can represent, for example, a system with particles that have a self-propulsion mechanism. Each node of the network specifies a possible location of a particle and its orientation. The orientation (which is formally like a spin degree of freedom) determines the self-propulsion direction. The bonds represent the possibility to make transitions: to hop between locations or to switch the orientation. In systems of experimental interest (Janus particles), the self-propulsion is induced by illumination. An emergent aspect is the topological stochastic disorder (TSD). It is implied by the nonuniformity of the illumination. In technical terms the TSD reflects the local nonzero circulations (affinities) of the stochastic transitions. This type of disorder, unlike a nonhomogeneous magnetic field, is non-Hermitian and can lead to the emergence of a complex relaxation spectrum. It is therefore dramatically distinct from the conservative Anderson-type or Sinai-type disorder. We discuss the consequences of having TSD. In particular we illuminate three different routes to underdamped relaxation and show that localization plays a major role in the analysis. Implications of the bulk-edge correspondence principle are addressed too.

Figure 1

Model geometry. (a) Janus particle in a one-dimensional random environment. The nonuniform illumination (arrows) induce self-propulsion in the direction of the head (white). The orientation of the head is called below polarization. (b) Lattice modeling of the system. Each node represents a possible location $n$ and possible polarization $s=\uparrow ,\downarrow$. One tile of the lattice is plotted. The system is composed of $N$ tiles with periodic boundary conditions. The transition rates along the vertical bonds are $w=1$ in both directions, representing random flips of polarization. The horizontal bonds are biased: The stochastic field there $\mathcal{E}$ is written as the sum of a drift $f_n$ and a self-propulsion term ${\varphi }_n$. The latter reflects that we are dealing with an active network. (c) Illustration of a representative segment of the lattice. The black sites are those that serve as sinks for the stochastic flow in the presence of strong disorder. They support the floor-level relaxation modes.

Figure 2

Simulation of polarization as a function of time. The average polarization $D_n$ at each location $n$ is color coded and imaged as a function of time $t$. In all the panels the polarization of the initial perpetration is modulated with wave number $k=2\pi /5$, namely, $p_{n,s}\propto 1\pm cos\left(kn\right)$. Only 50 sites are displayed. (a) Overdamped oscillations for $\varphi =0.5$ are observed since the self-propulsion is below the critical threshold. (b) For $\varphi =2$, which is above the threshold, one observes underdamped relaxation due to topological symmetry breaking. (c) Once disorder is added, the nonuniform NESS pattern takes over almost immediately. The parameters here are as in Fig. 3. (d) In the latter case we provide an image of the time derivative ${\dot{D}}_n$, hence one can resolve how the underdamped relaxation is blurred.

Figure 3

Representative relaxation spectrum for finite propulsion ($\overline{\varphi }=2$) and TSD (${\sigma }_{\varphi }=1$) with $n=500$. The eigenvalues are presented in the complex plane. Each associated eigenmode is characterized by various measures: The real and $\text{Im}\left[\lambda \right]>0$ points are color coded by the participation number $M$, while the conjugate $\text{Im}\left[\lambda \right]<0$ points are color coded by the effective propulsion $B$. See the text for definitions. The solid line illustrates the spectrum of the nondisordered system (13).

Figure 4

Representative relaxation spectra. The presentation is the same as in Fig. 3 but with the conjugate eigenvalues excluded. (a) Same propulsion as in Fig. 3 but with CSD (${\sigma }_f=1$) instead of TSD. (b) Same TSD as in Fig. 3, but the average propulsion is zero. (c) Same as (b) but with weak drift ($\overline{f}=0.02$). (d) Same as (b) but with stronger drift ($\overline{f}=0.8$).

Figure 5

Bloch spectrum ${\lambda }_{k,s}$. The two bands are $q=1$ (blue, lower half-plane) and $q=2$ (red, upper half-plane). (a)–(c) $f=0$ and (a) $\varphi =0.98$, (b) $\varphi =1.2$, and (c) $\varphi =1.6$. Note that ${\varphi }_c\approx 0.96$. Equation (13) has been used. For presentation the horizontal pieces of the band have been shifted off the real axis. (d) Same $\varphi$ as in (a) but with an added $f=0.01$ that lifts the $\pm k$ degeneracy. The black arrows show the direction in which ${\lambda }_k$ is changing as $k\in [0,2\pi ]$ is increased from zero and eventually returns.

Figure 6

Diagonalization procedure. We go from the site representation $|n,s\rangle$ of Fig. 1 to the mode representation $|\alpha ,\pm \rangle$, which is illustrated on the left. The diagonal arrows represent anti-Hermitian couplings due to self-propulsion. The thick double-sided arrows represent the Hermitian hopping elements between cells. The disorder affects all those couplings and also adds vertical hopping elements. The unperturbed diagonal energies are $\lambda =2$ and $\lambda =4$. With hopping we get two bands $[0,4]$ and $[2,6]$ that are illustrated by the solid line on the right. Neglecting the interband couplings, the spectrum is still real, represented by the blue vertical segments. The complex spectrum appears due to band mixing, as explained in the text.

Figure 7

Spectral distribution of the eigenvalues. (a) Fractions of eigenvalues per spectral window versus the disorder strength ${\sigma }_{\varphi }$, with $\overline{\varphi }=\overline{f}={\sigma }_f=0$. (b) Fraction of complex eigenvalues (out of all eigenvalues) divided into the different spectral windows. In some realizations of the system there are residual complex eigenvalues in the first window ($\text{Re}\left[\lambda \right]<2$), with a small imaginary part. It is not clear whether these are numerical issues or not. The threshold for complexity here is $\text{Im}\left[\lambda \right]>{10}^{-4}$. The plots are for rings of $N=500$ cells, with ${\varphi }_n={\alpha }_n{\sigma }_{\varphi }$, where ${\alpha }_n\in [-1,1]$ are random numbers. Each point corresponds to a different ${\sigma }_{\varphi }$, while the ${\alpha }_n$ are not modified.

Figure 8

Fraction of complex eigenvalues. (a) Fraction of complex eigenvalues relative to the number of eigenvalues in the second window ($\text{Re}\left[\lambda \right]\in [2,4]$) for the same ring as in Fig. 7 and for nonzero ${\sigma }_f$ too. In the latter case the fraction becomes smaller. (b) Total fraction of complex eigenvalues for a simple ring. Each point in the plots is averaged over 100 realizations. The solid and dashed lines are for $\sigma =3$ and $\sigma =6$, respectively. The different lines (from bottom to top at $B>4$) are for rings of length $L=5,10,20,30,60$.

Figure 9

Participating sites for each eigenmode. (a) Participation number $M$ versus $\text{Re}\left[\lambda \right]$ for ${\sigma }_{\varphi }=1$ and ${\sigma }_f=0.1$, with $\overline{\varphi }=\overline{f}=0$. (b) Same as (a) but with the disorder parameters swapped: ${\sigma }_f=1$ and ${\sigma }_{\varphi }=0.1$. (c) Various occupations volumes versus ${\sigma }_{\varphi }$. In (a) and (b) the points are color coded by $\text{Im}\left[\lambda \right]$. In (c) each point is an average over all the eigenstates within $2<\text{Re}\left[\lambda \right]<4$. The inset shows that the inverse localization length has a roughly quadratic dependence on the disorder strength, as in the Anderson model.

Figure 10

(a) Effective circulation $B$ of the eigenstates versus their $\sqrt{L^{*}}/L$, calculated for TSD with ${\sigma }_{\varphi }=2$ (red) and ${\sigma }_{\varphi }=4$ (blue), as well as $\overline{\varphi }=\overline{f}={\sigma }_f=0$. The solid and dashed lines are given by Eq. (23). (b) We define $\tilde{B}={\sum }_n{P}_n\left|{\mathcal{B}}_n\right|$ and verify that it agrees with the estimate $\sim (L^{*}/L){\sigma }_{\varphi }$. Both panels refer to the same set of eigenstates, namely, those that reside in the spectral window $2<\text{Re}\left[\lambda \right]<4$.

Figure 11

Floor-level occupation. (a) Floor-level occupation $Q$ versus $\text{Re}\left[\lambda \right]$ for ${\sigma }_{\varphi }=8$, with $\overline{\varphi }=\overline{f}={\sigma }_f=0$. (b) The fraction of floor-level eigenstates drops down from 25% as $\overline{\varphi }$ is increased. Floor-level eigenstates are defined as those that have $Q>0.7$. This is compared with $F$ from Eq. (25) and with the number of states with $\text{Re}\left[\lambda \right]<2$ (${\sigma }_{\varphi }=8$). (c) Fraction of floor-level eigenstates versus ${\sigma }_f$ and ${\sigma }_{\varphi }$. The average propulsion is $\overline{\varphi }=2$.

Figure 12

The NESS for a system that is composed of two regions. The probabilities on the upper and lower chains are plotted as dashed blue and solid red lines. (a) The two regions are with opposite $f$. The sink interface is located at $n=10$, while the other interface is at $n=60$. (b) The $[10,60]$ region is with $\varphi$ only and the other region is with $f$ only. (c) The two regions are with opposite $\varphi$. Strong localization near the interface is observed only in case (a), where the topological index is nonzero. The insets of (a) and (b) show close-ups.

Figure 13

(a) Image of $|A_{ij}{A}_{ji}^{*}{|}^{1/2}$ in the $|\alpha ,\pm \rangle$ basis. The indices are sorted by ${\lambda }_i^{\left(0\right)}$. The propulsion is $\overline{\varphi }=2, {\sigma }_{\varphi }=0.2$, and $\overline{f}={\sigma }_f=0$. States that have similar $k$ content are strongly coupled. If $\overline{\varphi }=0$ (not displayed) all the couplings becomes comparable as implied by Eq. (C4). (b) Here $\overline{\varphi }=0$ and ${\sigma }_{\varphi }=1$. The elements are divided by the eigenvalue differences $|{\lambda }_i^{\left(0\right)}-{\lambda }_j^{\left(0\right)}|$. Large values implies strong coupling that induces strong mixing. Note that in both panels diagonals are excluded. The color scale in (a) is arbitrary, while that of (b) is absolute.