⚛️ quantum physics

Boundary exceptional point driven chiral switch of topological edge states

This paper theoretically demonstrates that dynamically encircling boundary-induced exceptional points in a finite non-Hermitian Su-Schrieffer-Heeger chain enables a robust, chiral transfer of topological edge states between opposite boundaries with perfect fidelity, all while leaving the bulk energy spectrum unaffected.

Original authors: Bikashkali Midya, Barsa Rani Sahu

Published 2026-10-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Bikashkali Midya, Barsa Rani Sahu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of physics, waves are the messengers of energy, whether they are ripples on a pond, sound traveling through air, or light moving through a fiber optic cable. For decades, scientists have discovered that under certain conditions, these waves can become trapped in specific patterns that are incredibly difficult to disrupt. This phenomenon, known as topology, ensures that waves can flow along the edges of a material without scattering back, much like a train staying on a dedicated track regardless of the terrain it passes over. This principle, called bulk-boundary correspondence, means that the internal structure of a material dictates what happens at its surface. Recently, physicists have begun exploring what happens when these systems are not perfectly closed but instead interact with their environment, gaining or losing energy. This introduces a new layer of complexity where the rules of standard physics bend, allowing for strange behaviors like the merging of distinct wave states into a single, indistinguishable point.

A team of researchers at the Indian Institute of Science Education and Research in Berhampur has now uncovered a way to harness these interactions to move information across a material with perfect precision, using only the edges of the system. They focused on a specific arrangement of atoms or sites, known as a Su-Schrieffer-Heeger chain, which acts like a one-dimensional highway for waves. In their theoretical model, the middle of this highway is perfectly stable and unchanging, but the two ends are equipped with special controls that can either add energy to the system or remove it. By carefully tuning these boundary controls, the researchers found they could create a unique point where two distinct wave states, normally living at opposite ends of the chain, merge together. This merging point is called an exceptional point, a special condition where the system loses its ability to distinguish between the two states.

The researchers discovered that by moving these boundary controls in a specific circular pattern around this merging point, they could force a wave to jump from one end of the chain to the other. The direction in which they moved the controls determined the outcome. If they moved the controls in a counter-clockwise circle, a wave starting at the left edge would travel across the entire length of the chain and settle perfectly at the right edge. If they moved the controls in a clockwise circle, the wave would simply return to where it started. This process is remarkably robust; the wave does not get lost or scattered into the middle of the chain. Instead, it remains strictly confined to the topological edge states, moving cleanly from one side to the other without ever disturbing the stable interior of the material.

What makes this finding particularly significant is that it solves a long-standing challenge in the field. Previous attempts to move waves across large systems often required changing the properties of the entire material, which disrupted the very stability that made the transport possible. In contrast, this new method keeps the middle of the chain completely untouched and static. The researchers showed that by applying non-Hermitian potentials—essentially controlled gain and loss—only at the very first and very last sites of the chain, they could achieve high-fidelity transfer across arbitrarily large distances. They derived an exact mathematical formula for the conditions needed to create the merging point, proving that this effect works regardless of how many sites are in the chain. This means the method is scalable and does not break down as the system grows larger.

The team demonstrated that this chiral switching mechanism relies on a non-adiabatic jump, a rapid transition that occurs when the system is driven quickly enough to bypass the usual slow evolution. As the boundary parameters are swept around the exceptional point, the wave state is forced to choose a path based on the direction of the sweep. One direction amplifies the wave at the destination while suppressing it at the start, effectively pumping the energy across. The other direction reverses this process, returning the wave to its origin. This behavior was confirmed through detailed simulations of chains with sixteen sites, where the researchers tracked the movement of the wave population over time. The results showed that the transfer is nearly perfect, with the wave ending up entirely on the opposite side or back at the start, depending solely on the orientation of the control loop.

This work suggests a new way to build switches for future technologies that rely on the flow of light or electricity. Because the method requires only localized changes at the boundaries, it could be implemented in various experimental platforms, such as photonic lattices made of light-guiding materials, electrical circuits designed to mimic quantum behavior, or arrays of superconducting qubits. In these systems, engineers could use active amplifiers or controlled loss channels at the ends of a device to steer signals without touching the core of the system. The researchers note that while their work is currently theoretical, the principles are clear and the design is ready for experimental verification. By proving that boundary control alone is sufficient to drive complex, large-scale topological transport, this study opens the door to more efficient and reliable ways of moving information in advanced physical systems.

Technical Summary: Boundary Exceptional Point Driven Chiral Switch of Topological Edge States

Problem Statement
The integration of non-Hermitian physics with topological phases has revealed unique phenomena such as exceptional points (EPs), where eigenvalues and eigenvectors coalesce. Dynamically encircling these EPs can induce asymmetric, chiral state switching. However, existing implementations of non-Hermitian chiral state transfer have been largely restricted to two-level systems or small-scale configurations. Scaling this mechanism to large, multi-site lattices has proven difficult; previous attempts relied on global bulk modulations (e.g., staggered non-Hermiticity) that disrupt the extended bulk energy spectrum. Consequently, establishing a high-fidelity state transfer process across an extended lattice using strictly localized boundary controls, while ensuring the bulk topology and energy spectrum remain entirely unperturbed, remains an open challenge.

Methodology
The authors theoretically investigate a finite Su-Schrieffer-Heeger (SSH) lattice in its topologically non-trivial phase ( $v < w$ ), where $v$ and $w$ represent intra-cell and inter-cell hopping amplitudes, respectively. The bulk of the chain is strictly Hermitian. Non-Hermiticity is introduced exclusively at the boundaries (sites $j=1$ and $j=2N$ ) via time-dependent complex potentials $\Gamma(t) = \delta(t) + i\gamma(t)$ and $-\Gamma(t)$ , representing gain, loss, and detuning.

The study employs the following analytical and numerical approaches:

Analytical Derivation: The authors solve the eigenvalue problem for the $2N \times 2N$ Hamiltonian matrix to derive exact analytical coordinates for boundary-induced exceptional points (EPs) within the topological gap. They also utilize a reduced two-dimensional subspace approximation spanned by decoupled left ( $|L\rangle$ ) and right ( $|R\rangle$ ) edge states to validate the full-system results.
Dynamic Encircling Protocol: A closed parameter loop is defined in the complex $\Gamma = (\delta, \gamma)$ plane. The loop is designed to encircle both conjugate EPs simultaneously, starting and ending in a Hermitian regime ( $\gamma=0$ ) with large real detuning ( $|\delta| \gg |\kappa_{edge}|$ ) to ensure edge states are decoupled and strictly localized.
Numerical Simulation: The time-dependent Schrödinger equation is solved numerically for a lattice of $2N=16$ sites to track the dynamical evolution of state populations and the trajectory of the dynamical energy on Riemann surfaces.

Key Contributions and Results

Exact Analytical Coordinates for Boundary EPs: A primary theoretical achievement is the derivation of an exact analytical expression for the boundary-induced EPs in arbitrarily large systems ( $N \gg 1$ ). The coordinates are given by $\Gamma_{EP} = \pm i v^N / w^{N-1}$ (with $\delta_{EP}=0$ ). This overcomes the mathematical intractability typically associated with finding EPs in high-dimensional systems.
Bulk-Preserving Chiral Switching: The study demonstrates that dynamically encircling these boundary EPs facilitates a robust, chiral edge state transfer. Depending on the chirality (orientation) of the encircling loop:
- A counter-clockwise (CCW) loop transfers a state localized at one boundary to the opposite boundary with perfect fidelity.
- A clockwise (CW) loop returns the state to its starting position via a double-switching process.
Non-Adiabatic Jump Mechanism: The state transfer is driven by non-adiabatic transitions as the system traverses complex energy Riemann surfaces. The loop chirality dictates whether the state vector jumps to a dominant gain sheet (facilitating transfer) or undergoes a double jump (facilitating return).
Bulk Spectrum Integrity: Crucially, because non-Hermiticity is strictly confined to the boundaries, the bulk energy bands and bulk topological invariants remain entirely unperturbed. The edge states remain confined within the topological gap throughout the evolution, avoiding the transient merging into bulk states required by traditional Thouless pumping.
Scalability: Numerical analysis indicates that while the minimum protocol duration ( $T_{min}$ ) required for high-fidelity transfer scales linearly with system size, the mechanism remains effective for large-scale lattices, confirming the scalability of the boundary-driven approach.

Significance
The paper establishes a minimal framework for a large-scale topological chiral switch that is experimentally verifiable. By confining non-Hermitian driving to the boundaries, the authors provide a solution to the problem of scaling non-Hermitian state transfer without disrupting the bulk spectrum. The work offers clear, scalable design principles for controlling topological edge states through boundary non-Hermitian driving. The authors suggest that this mechanism could be realized in existing experimental platforms, including active non-Hermitian photonic lattices, topolectrical circuits, and programmable superconducting qubit arrays, where boundary potentials can be precisely tuned in real-time. The findings represent a fundamental shift from traditional bulk-modulation-based topological transport to a boundary-controlled paradigm.

Technical Summary: Boundary Exceptional Point Driven Chiral Switch of Topological Edge States

More like this