⚛️ quantum physics

Exceptional by Design: Long-Range Hopping as a Knob for Exceptional Point Control

This paper investigates a generalized non-Hermitian Rice-Mele model with balanced gain/loss and next-nearest-neighbor hopping, revealing that while long-range hopping leaves periodic boundary exceptional points unchanged, it significantly alters open boundary spectra by shifting and generating new exceptional points, inducing size-dependent degeneracies, and enabling a topological phase diagram with distinct edge state sectors despite the absence of the non-Hermitian skin effect.

Original authors: Carolina Martinez-Strasser, Dario Bercioux, Nico Leumer

Published 2026-06-24

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CC BY 4.0

Original authors: Carolina Martinez-Strasser, Dario Bercioux, Nico Leumer

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

Imagine a musical instrument, like a guitar, but instead of just strings, it has two sets of strings (sublattices) that can be tuned to either gain energy (like a microphone boosting a sound) or lose energy (like a sound being absorbed). This is the world of non-Hermitian physics: a realm where energy isn't perfectly conserved, and things get a bit "messy" compared to the tidy, predictable world of standard physics.

In this paper, the authors are tuning a specific mathematical model of such a system (a "Rice-Mele" model) to find a very special, rare state called an Exceptional Point (EP).

Here is the breakdown of their discovery using everyday analogies:

1. The "Sweet Spot" (Exceptional Points)

In normal physics, if you tune two strings to the same pitch, they just sound the same. But in this "messy" non-Hermitian world, there are special "sweet spots" called Exceptional Points.

The Analogy: Imagine two dancers spinning. Usually, if they spin at the same speed, they are just two separate dancers. At an Exceptional Point, they don't just match speeds; they physically merge into a single, confused dancer who can't spin properly anymore. The system becomes "defective."
Why it matters: The paper notes that these points are famous for being super-sensitive. If you nudge the system slightly near this point, the reaction is huge (like a magnifying glass). This is often called "enhanced sensing."

2. The New "Knob": Long-Range Hopping

The authors introduced a new control knob to their system: Next-Nearest-Neighbor (NNN) hopping.

The Analogy: Imagine a row of houses (atoms).
- Normal hopping: You can only walk from your house to your immediate neighbor's.
- Long-range hopping (The new knob): You can now skip a house and walk directly to the second neighbor's house.
The authors wanted to see if this "skipping" ability changed where those special "sweet spots" (EPs) were located.

3. The Big Surprise: It Depends on the Walls

The most important finding is that the effect of this "skipping" knob depends entirely on whether the system has walls or not.

Scenario A: No Walls (Periodic Boundary Conditions)
- Imagine the row of houses is actually a giant circle. You can walk forever without hitting a wall.
- The Result: The "skipping" knob does nothing to the location of the sweet spots. It's like adding a shortcut in a circular track; the runners (energy levels) still meet at the exact same time and place. The sweet spots stay on the same lines and circles regardless of how much you "skip."
Scenario B: With Walls (Open Boundary Conditions)
- Imagine the row of houses is a straight line with a wall at the start and end.
- The Result: The "skipping" knob changes everything.
  1. It shifts the location of the existing sweet spots.
  2. It creates brand new sweet spots that didn't exist before.
  3. There is a specific "magic ratio" (when the skipping strength equals twice the wall-to-wall hopping) where the system undergoes a dramatic change, closing a gap in its energy spectrum.

4. The "Skin Effect" Mystery

In many similar non-Hermitian systems, energy tends to pile up at the edges, like water collecting in a corner. This is called the Non-Hermitian Skin Effect (NHSE).

The Finding: The authors checked their system and found no skin effect. The energy didn't pile up at the edges. The "bulk" (the middle) and the "boundary" (the edges) behaved in a standard, predictable way. This is rare and important because it means the system is easier to understand and predict.

5. The Edge States (The Ghosts at the Door)

Even without the "skin effect," the system still has "edge states"—special energy modes that live only at the ends of the line.

The Analogy: Think of a long hallway. Most sounds travel through the middle (bulk states), but at certain settings, a whisper gets trapped right at the front door and another at the back door, unable to move into the hallway.
The authors mapped out a "phase diagram" (a map of settings) showing exactly when you get zero, one, or two of these trapped whispers. They confirmed that the math predicting these whispers (using a "winding number" concept) matched perfectly with what they saw in the computer simulations.

Summary

The paper is essentially a study of a complex, energy-leaking system. The authors found that:

Exceptional Points (the super-sensitive sweet spots) exist in this system.
Adding a "long-range skip" (hopping over a neighbor) has zero effect on these spots if the system is a loop (no walls).
However, if the system has walls, that same "skip" completely reshapes the map of where these spots are, creating new ones and shifting old ones.
The system behaves "normally" regarding edge states (no weird energy piling up), making it a clean example of how topology works in these messy, non-Hermitian systems.

The authors conclude that by tuning this "long-range hopping" knob, scientists could potentially move these sensitive sweet spots around to make better sensors, provided they can build systems with physical walls (open boundaries).

Technical Summary: Exceptional by Design: Long-Range Hopping as a Knob for Exceptional Point Control

Problem Statement
Exceptional points (EPs) are spectral degeneracies unique to non-Hermitian (nH) systems where eigenvalues and eigenvectors coalesce, rendering the Hamiltonian defective. While EPs offer potential for enhanced sensing and exhibit unique topological properties (such as eigenstate permutation upon adiabatic encircling), their existence typically requires fine-tuning of system parameters. Furthermore, the interplay between non-Hermitian gain/loss, long-range hopping, and boundary conditions in topological models remains complex. Specifically, the role of next-nearest-neighbor (nnn) hopping in controlling EPs and topological edge states within a generalized Rice-Mele (RM) model, particularly under open boundary conditions (OBC) where the non-Hermitian skin effect (NHSE) is absent, requires systematic investigation.

Methodology
The authors investigate a generalized non-Hermitian Rice-Mele model defined on a 1D tight-binding lattice with two sublattices ( $A$ and $B$ ). The model includes:

Intra-cell ( $t$ ) and inter-cell ( $t'$ ) hopping.
Next-nearest-neighbor (nnn) hopping ( $m$ ) within sublattices.
Balanced gain and loss ( $\pm i\gamma$ ) on the two sublattices, introducing $PT$-symmetric non-Hermiticity.
Onsite energy differences ( $\epsilon_A, \epsilon_B$ ).

The study employs a combination of analytical and numerical techniques:

Symmetry Analysis: The model is classified within the 38-fold non-Hermitian symmetry scheme (Kawabata et al.), identifying preserved symmetries (specifically TRS II) and constraints required for others.
Bulk Analysis: The bulk Hamiltonian $H(k)$ is diagonalized to derive dispersion relations and identify bulk EPs via the vanishing of the discriminant of the secular equation.
Boundary Condition Comparison: The system is analyzed under both Periodic Boundary Conditions (PBC) and Open Boundary Conditions (OBC) for finite system sizes ( $N$ ).
Numerical Identification: EPs are located numerically by computing the condition number $K(U)$ of the eigenvector matrix (divergence indicates an EP) and confirmed via Jordan decomposition.
Topological Characterization: A winding number framework (based on Ref. [19]) is applied to the bulk Hamiltonian to predict edge states in the absence of standard symmetries. The existence and localization of these states are verified using the Directional Inverse Participation Ratio (dIPR).

Key Contributions and Results

Control of Exceptional Points via Long-Range Hopping:
- Under PBC: The nnn hopping term $m$ enters the bulk Hamiltonian only as an identity contribution ( $-2m \cos k \cdot \mathbb{1}_2$ ). Consequently, $m$ does not influence the existence or location of bulk EPs in parameter space (which lie on lines and ellipses defined by $t, t', \gamma$ ). It only shifts the energy of the degeneracy point.
- Under OBC: The independence is broken. The nnn hopping $m$ actively shifts the energy of existing EPs and, crucially, generates new EPs. The parameter constraints for EPs under OBC become transcendental and dependent on $N$ . A specific condition, $t' = 2m$ , is identified as signaling a topological gap closing in the OBC spectrum, distinct from the PBC case.
Order and Degeneracy of EPs:
- The system hosts exclusively second-order EPs (EP2) under both PBC and OBC.
- However, at specific parameter points (e.g., intersections of EP lines), multiple simultaneous second-order EPs occur. This yields degenerate configurations where the total degeneracy grows with the system size $N$ .
Topological Phase Diagram and Edge States:
- Using a winding number invariant defined for nH systems without symmetry protection, the authors map a phase diagram revealing sectors with zero, one, or two edge states ( $W=0, 1, 2$ ).
- The presence of the staggered potential ( $\epsilon_- \neq 0$ ) is essential for the $W=1$ sector (single edge mode).
- The bulk-boundary correspondence is confirmed: topological edge states emerge under OBC for non-trivial winding numbers but are absent in PBC spectra.
- Absence of NHSE: The Generalized Brillouin Zone (GBZ) coincides with the ordinary Brillouin Zone, and the non-Hermitian skin effect is absent in this model.
Spectral Properties:
- Edge state eigenvalues exhibit a fixed imaginary part ( $\text{Im}[E] = \pm \gamma$ ) tied to the gain/loss amplitude, while their real part drifts continuously with the hopping ratio $t/t'$ .
- Phase transitions between topological sectors occur without the closing of the bulk spectral gap, consistent with the "topology-without-gap-closing" mechanism.

Significance and Claims
The paper claims to demonstrate that long-range hopping ( $m$ ) serves as a specific "knob" for controlling exceptional points, but this control is highly dependent on boundary conditions. While $m$ is inert regarding the location of bulk EPs in parameter space under PBC, it fundamentally alters the EP landscape under OBC by generating new degeneracies and shifting existing ones.

The authors emphasize that their findings provide a mechanism to re-localize EPs in parameter space for OBC systems. This is significant because it suggests that effective nnn hopping (engineered, for instance, via gate potentials in quantum ladder setups) could be used to tune systems toward EPs for enhanced sensing applications, where small perturbations yield magnified responses. The work also reinforces the validity of winding number frameworks for predicting edge states in nH systems lacking standard symmetries, confirming that bulk-boundary correspondence holds even in the absence of the NHSE. The study remains modest regarding experimental realization, noting that while atomic systems typically couple nearest and next-nearest neighbors simultaneously, engineered quantum ladder setups offer a pathway to control effective $m$ independently.