🔬 mesoscale physics

Spin resolved spectral topology and re-entrant localization in a non Hermitian quasiperiodic SSH chain

This study reveals that in a non-Hermitian quasiperiodic Su-Schrieffer-Heeger chain with Rashba spin-orbit coupling, increasing non-Hermiticity induces a re-entrant localization transition and a concurrent real-complex-real spectral evolution, where finite spin-dependent hopping splits the spectral loops into four distinct spin-resolved branches with unique topological winding numbers.

Original authors: Hemant K Sharma

Published 2026-05-15

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CC BY 4.0

Original authors: Hemant K Sharma

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 long, narrow hallway made of tiles. In a normal hallway, the tiles are all the same, and if you drop a ball, it rolls freely from one end to the other. This is like a "perfect" crystal in physics.

Now, imagine a hallway where the tiles are arranged in a pattern that never quite repeats itself—a "quasiperiodic" hallway. In this hallway, the floor gets bumpy in a specific, rhythmic way. If you drop a ball here, it might get stuck in one spot (localization) or it might still roll freely (delocalization), depending on how bumpy the floor is.

This paper explores what happens when we add three special ingredients to this hallway:

Spin: Imagine the balls are actually tiny magnets that can point "Up" or "Down."
Spin-Dependent Hopping: Imagine the floor is slippery for "Up" magnets but sticky for "Down" magnets, or vice versa.
Non-Hermitian Magic: Imagine the hallway has invisible "wind" or "leaks" that can either push the balls forward or absorb them, making the physics slightly different from our everyday world (this is the "non-Hermitian" part).

Here is what the researchers discovered, using simple analogies:

1. The "Re-entrant" Rollercoaster

Usually, if you make a system more chaotic or "leaky" (increasing the non-Hermitian parameter), things get stuck. You expect the balls to get trapped and stay there.

But in this paper, the researchers found a surprise: The balls get stuck, and then they get unstuck again.

Phase 1 (Free): At first, the balls roll freely.
Phase 2 (Stuck): As they turn up the "leakiness," the balls get trapped in specific spots.
Phase 3 (Free Again): If they turn the "leakiness" up even more, the balls suddenly start rolling freely again!

They call this a "re-entrant" transition. It's like a door that locks when you push it, but if you push it really hard, it unlocks and swings open again.

2. The Map of the Hallway (Spectral Topology)

The researchers didn't just watch the balls; they looked at a "map" of all the possible energy states the balls could have. In this weird physics world, this map isn't just a flat line; it's a 3D shape that can twist into loops.

Without the "Spin-Dependent" trick: The map forms two big loops. It's like two separate race tracks that look almost identical.
With the "Spin-Dependent" trick: This is the big discovery. When they added the rule that "Up" and "Down" magnets feel the floor differently, those two big loops split apart.
- Suddenly, there are four separate loops instead of two.
- The "Up" magnets take one set of tracks, and the "Down" magnets take another.
- This splitting creates a new kind of "topological" structure (a fancy way of saying the shape of the map has changed in a fundamental way).

3. The Connection Between Getting Stuck and the Map Shape

The most important finding is that these two things happen at the exact same time:

When the balls start getting stuck (localized), the map forms loops.
When the balls get unstuck (delocalized), the loops collapse back into a flat line.

It's as if the act of the balls getting trapped is what creates the loops on the map. When they are free to roam, the loops disappear.

4. The "Spin-Selective" Effect

Because the "Up" and "Down" magnets experience the floor differently, they don't get stuck in the same way.

Sometimes, the "Down" magnets get very tightly trapped, while the "Up" magnets are still rolling around freely.
This creates a situation where the hallway acts like a filter, sorting the magnets by which way they are pointing.

Summary

The paper describes a strange, bumpy hallway where:

Making the hallway "leaky" causes a three-stage dance: Free $\rightarrow$ Stuck $\rightarrow$ Free again.
Adding a rule that treats "Up" and "Down" magnets differently splits the energy map from two loops into four.
The act of the particles getting stuck is directly linked to the shape of this energy map.

The researchers suggest this model could be built in real life using ultra-cold atoms (atoms cooled to near absolute zero) or light circuits (lasers in glass fibers), where scientists can control these "leaky" and "spin-dependent" effects to create new types of switches or filters.

Technical Summary: Spin-resolved spectral topology and re-entrant localization in a non-Hermitian quasiperiodic SSH chain

Problem Statement
Quasiperiodic systems serve as a critical intermediate platform between perfectly periodic crystals and disordered media, exhibiting sharp localization transitions distinct from the Anderson localization found in random systems. While the interplay between the Su–Schrieffer–Heeger (SSH) model, quasiperiodicity, and non-Hermiticity has been explored, the specific combined role of Rashba spin–orbit interaction (RSOI) and spin-dependent hopping in determining the localization and spectral topology of non-Hermitian SSH systems remains largely unexplored. Specifically, the systematic investigation of how spin-dependent hopping influences the formation of spin-resolved spectral loops and the splitting of winding numbers is lacking.

Methodology
The authors investigate a generalized non-Hermitian SSH–Aubry–Andr´e–Harper (AAH) lattice defined by a Hamiltonian incorporating:

Dimerized Hopping: Intracell ( $v$ ) and intercell ( $w$ ) hopping amplitudes.
Quasiperiodic Potential: A complex onsite potential with strength $\lambda$ and a phase $\phi = \theta + ih$ , where the imaginary component $h$ controls the degree of non-Hermiticity (breaking Hermiticity while preserving PT symmetry when $\theta=0$ ).
Spin-Orbit and Spin-Dependent Terms: The model includes Rashba spin–orbit coupling terms (spin-conserving $\alpha_1, \alpha_2$ and spin-flip $\alpha_3, \alpha_4$ ) and a spin-dependent hopping term $g_h$ .

The study employs exact diagonalization of the Hamiltonian for finite system sizes ( $L$ ). The analysis relies on three primary metrics:

Localization: Calculated via the average inverse participation ratio (IPR) and normalized participation ratio (NPR), including spin-resolved IPRs ( $\text{IPR}_\uparrow, \text{IPR}_\downarrow$ ).
Spectral Nature: Characterized by the fraction of eigenvalues with non-zero imaginary parts ( $\rho$ ), distinguishing between real, mixed, and fully complex spectra.
Topological Character: Quantified by spectral winding numbers ( $w$ ) defined in the complex-energy plane. The authors define multiple winding numbers ( $w_1, w_2, w_3, w_4$ ) to account for disconnected spectral contours arising from spin splitting, in addition to a total winding number ( $w_{tot}$ ).

Key Results
The study reveals a complex interplay between non-Hermiticity, quasiperiodicity, and spin degrees of freedom, characterized by the following findings:

Re-entrant Localization Transition:
As the non-Hermitian parameter $h$ increases, the system undergoes a sequence of phase transitions: Extended $\to$ Intermediate (coexistence of localized and extended states) $\to$ Localized $\to$ Intermediate $\to$ Extended. This re-entrant behavior ( $E \to I \to L \to I \to E$ ) is driven by the non-Hermitian modulation. The inclusion of spin-dependent hopping and spin-flip terms suppresses the fully localized phase and broadens the coexistence region.
Real-Complex-Real Spectral Transition:
The spectral transition mirrors the localization sequence. The system transitions from a purely real spectrum ( $R$ ) to a mixed phase ( $M$ ) with coexisting real and complex eigenvalues, then to a fully complex phase ( $C$ ), and finally re-enters a predominantly real spectrum at large $h$ ( $R \to M \to C \to M \to R$ ). A direct correspondence is established: localized regimes coincide with complex eigenvalues, while extended phases correspond to real spectra.
Spin-Resolved Spectral Topology and Splitting:
- Without Spin-Dependent Hopping ( $g_h = 0$ ): The complex-energy spectrum forms two nearly spin-degenerate loops characterized by winding numbers $w = \pm 2$ .
- With Finite Spin-Dependent Hopping ( $g_h \neq 0$ ): The spin degeneracy is lifted. Each of the two spectral loops splits into two independent spin-resolved branches, resulting in four disconnected spectral contours. These contours carry distinct winding numbers (e.g., $-1, 0, 1, 2$), creating multiple winding sectors.
- Spin-Selective Localization: The spin-resolved IPR analysis shows that spin-up and spin-down sectors evolve differently, exhibiting distinct localization strengths and spectral structures, particularly near the outer spectral branches.
Re-entrant Topological Sequence:
The topological transition follows the sequence $w = 0 \to w \neq 0 \to w = 0$ . The emergence of non-zero winding numbers coincides with the formation of complex spectral loops and enhanced localization. As $h$ increases further, the loops collapse back onto the real axis, winding numbers vanish, and the system returns to a topologically trivial extended phase.

Significance and Claims
The paper claims to demonstrate a direct correspondence between localization, spectral topology, and spin-resolved spectral splitting in non-Hermitian quasiperiodic systems. The central contribution is the revelation that spin-dependent hopping fundamentally reshapes the spectral topology, transforming degenerate loops into multiple spin-resolved branches with distinct winding sectors.

The authors assert that their results highlight the nontrivial role of spin-dependent hopping in generating unconventional spectral topology and spin-selective localization. They propose that the presented model provides a platform for exploring controllable non-Hermitian topological phases and spin-resolved spectral engineering. The paper suggests potential experimental realization in ultracold atoms (using optical lattices, Raman-assisted tunneling, and synthetic spin–orbit coupling), photonic lattices, topolectric circuits, and other synthetic quantum systems where complex onsite modulation and spin-dependent couplings can be independently controlled.