Multiple re-entrant topological windows induced by generalized Bernoulli disorder

1. Problem Statement

The paper addresses the interplay between disorder and topology in one-dimensional (1D) chiral lattices, specifically the Su-Schrieffer-Heeger (SSH) model. While the Topological Anderson Insulator (TAI) effect—where disorder drives a trivial system into a topological phase—is well-established, most previous studies focused on:

• Continuous random disorder.
• Deterministic or quasiperiodic modulations.

There is a gap in understanding how multivalued discrete random distributions (specifically generalized Bernoulli disorder) influence topological phase diagrams. The authors aim to determine if such structured disorder can generate multiple disconnected re-entrant topological windows (regions where the system transitions from trivial $\to$ topological $\to$ trivial $\to$ topological, etc.) and how the statistical parameters of the disorder (values and probabilities) control these windows.

2. Methodology

Model System

The authors study a 1D SSH chain with:

• Uniform interdimer hopping: $t_2$ (set to 1 as the energy unit).
• Disordered intradimer hopping: $t_1$ is modified by a random variable $\xi_i$.
• Hamiltonian:
  $$\hat{H} = -\sum_i \left[ (t_1 - \xi_i)\hat{c}^\dagger_{i,A}\hat{c}_{i,B} + t_2\hat{c}^\dagger_{i,B}\hat{c}_{i+1,A} + \text{H.c.} \right]$$
• Disorder Distribution: The random variable $\xi_i$ follows a generalized Bernoulli distribution with $M$ discrete values $\xi^{(1)}, \dots, \xi^{(M)}$ occurring with probabilities $p_1, \dots, p_M$.

Topological Characterization

Since translational symmetry is broken, standard momentum-space invariants cannot be used. The authors employ:

1. Reflection-Matrix Topological Quantum Number ($Q$): Based on the zero-energy scattering matrix. $Q=1$ indicates a nontrivial phase (zero-energy boundary modes), while $Q=0$ indicates a trivial phase.
2. Disorder-Averaged Real-Space Winding Number: Used as a complementary check to confirm phase boundaries.
3. Mean Chiral Displacement ($C(t)$): A dynamical probe defined as the time-averaged expectation value of the chiral symmetry operator $\Gamma$ and position operator $X$. This is proposed as an experimental observable.

Analytical Approach

The phase boundaries are derived analytically using the inverse localization length ($\gamma$) of the zero-energy modes. The transition occurs when $\gamma = 0$. For a generalized Bernoulli distribution, this yields a weighted geometric-mean condition:
$$\left| \prod_{j=1}^M (-t_1 + \xi^{(j)})^{p_j} \right| = 1$$
This condition replaces the simple arithmetic average often used in other disorder models, reflecting the multiplicative nature of the recursion relations in the SSH model.

3. Key Contributions & Results

A. Generation of Multiple Re-entrant Windows

• Binary Disorder ($M=2$): When the disorder has two values (e.g., $\xi^{(1)}=\lambda, \xi^{(2)}=2\lambda$), varying the disorder strength $\lambda$ at fixed $t_1$ (in the regime $t_1 > 1$) induces two disconnected topological windows. The system transitions: Trivial $\to$ Topological $\to$ Trivial $\to$ Topological $\to$ Trivial.
• Multivalued Disorder ($M>2$): For distributions with $M$ values (e.g., $M=3, 4, 5$), the number of disconnected topological windows increases. In the large-$t_1$ regime, the number of windows matches the number of disorder components ($M$).

B. Control via Statistical Parameters

The authors demonstrate that the number and widths of these windows are systematically tunable:

• Probabilities ($p_j$): Changing the probability weights of the disorder values shifts the phase boundaries unequally. For example, in the binary case, increasing $p_1$ shrinks the first window while broadening the second.
• Disorder Values ($\xi^{(j)}$): The relative spacing between disorder values (e.g., $\xi^{(2)}/\xi^{(1)}$) controls the compression or expansion of the topological intervals. Increasing the separation between values generally compresses the windows.

C. Analytical-Numerical Agreement

The phase boundaries derived from the analytical condition (Eq. 3 in the paper) show excellent agreement with numerical simulations of the disorder-averaged topological quantum number $Q$ and the real-space winding number.

D. Decoupling of Topology and Localization

The study analyzes the localization properties using the fractal dimension (via the Inverse Participation Ratio).

• Result: Re-entrant extended regimes (delocalized states) appear in the parameter space, but their locations do not coincide with the topological windows.
• Significance: This proves that the re-entrant topological behavior and re-entrant localization behavior arise from distinct mechanisms. Topology is governed by the geometric mean of couplings, while localization is influenced by effective bond breaking/homogenization.

E. Dynamical Detection and Implementation

• Dynamical Probe: The time-averaged mean chiral displacement $\langle C \rangle$ successfully tracks the topological transitions. It approaches $\approx 1$ in topological windows and $\approx 0$ in trivial regions, resolving the multiple windows dynamically.
• Photonic Implementation: The authors propose a feasible implementation using photonic waveguide lattices. By introducing an auxiliary waveguide and adiabatically eliminating it, they can engineer the effective disordered intradimer hopping ($t_1 - \xi_i$) via detuning and coupling strengths, while keeping interdimer coupling ($t_2$) fixed.

4. Significance

1. New Mechanism for Topological Engineering: The paper establishes that multivalued discrete disorder is a powerful tool for engineering complex topological phase diagrams. Unlike continuous disorder which typically destroys topology, structured discrete disorder can create multiple, tunable "islands" of topology.
2. Analytical Tractability: The derivation of the phase boundary via the weighted geometric mean provides a simple, analytically solvable framework for predicting disorder-induced topological transitions in chiral systems.
3. Experimental Relevance: The proposal for photonic waveguide implementation offers a direct path to experimentally observe these multiple re-entrant windows. The use of mean chiral displacement as a dynamical probe bridges the gap between theoretical topological invariants and measurable physical quantities.
4. Fundamental Insight: The work clarifies that topology and localization in disordered 1D chiral systems are not strictly coupled; one can have re-entrant topology without re-entrant delocalization, challenging simplified views of the Anderson transition in topological contexts.

In summary, this work reveals that the statistical structure of discrete disorder (specifically generalized Bernoulli distributions) acts as a "knob" to systematically organize and multiply re-entrant topological windows in 1D chiral lattices, offering new avenues for robust topological device design.