Topological metal-insulator transitions in one-dimensional non-Hermitian quasicrystals: beyond PT-symmetry

This paper investigates a non-PT-symmetric one-dimensional non-Hermitian quasicrystal model, demonstrating that it generally supports triple phase transitions involving localization, topology, and degeneracy breaking, while also exhibiting distinct localization-delocalization transitions without topological or degeneracy-breaking changes, thereby extending the understanding of topological metal-insulator transitions beyond PT-symmetric systems.

The Gist

Imagine a long, one-dimensional train track where the train cars (electrons) can hop from one station to the next. In a normal, "perfect" world, the stations are evenly spaced, and the train moves freely. In a "disordered" world, the stations are scattered randomly, and the train gets stuck in one spot (this is called localization).

This paper explores a strange, "in-between" world called a quasicrystal. Here, the stations aren't random, but they aren't perfectly repeating either. They follow a complex, rhythmic pattern (like the Fibonacci sequence) that creates long-range order without ever repeating exactly.

Now, add a twist: this world is non-Hermitian. In physics terms, this means the system isn't perfectly balanced; it has "gain" (energy coming in) and "loss" (energy going out), like a train track with some sections that boost the train's speed and others that act as brakes.

Here is what the researchers discovered, explained through simple analogies:

1. The "Ghost Wind" and the "Traffic Jam"

In these special non-Hermitian systems, there is a phenomenon called the Non-Hermitian Skin Effect (NHSE). Imagine a strong, invisible wind blowing down the track. This wind pushes all the passengers (electrons) to pile up at one end of the train, even if the train is moving. This is the "Skin Effect."

Usually, scientists studied these systems only when they had a special balance called PT-symmetry (Parity-Time symmetry). Think of PT-symmetry as a perfect mirror: for every "boost" on the left, there is an equal "brake" on the right. When this balance exists, the system behaves in a very specific, predictable way.

The Paper's Big Discovery:
The authors asked: What happens if we break that perfect mirror? What if the "boosts" and "brakes" are slightly out of sync? They created a model where the real and imaginary parts of the potential (the boost and brake) are shifted by a "phase angle" (a timing delay).

2. The "Triple-Decker" Transition

When they tweaked this timing (the phase shift), they found that the system could undergo a Triple Phase Transition. Imagine a traffic light that changes three things at once:

1. Localization: The train goes from moving freely to getting stuck in a traffic jam.
2. Topology: The "shape" of the track's energy changes, creating a loop that can't be untangled (like a knot).
3. Degeneracy Breaking: In the "stuck" state, two identical train cars that were previously twins (having the exact same energy) suddenly become different individuals.

In most of the parameter space, these three things happen simultaneously. It's as if the moment the traffic jam forms, the track twists into a knot, and the twins separate. This is driven by that "ghost wind" (NHSE) pushing things around.

3. The "Pure Traffic Jam" (The Surprise)

The most interesting finding is that this "Triple-Decker" behavior isn't the only thing that happens.

The researchers found specific settings (when the timing shift is exactly zero or a full circle) where the "ghost wind" disappears. In these specific cases:

• The train still gets stuck in a traffic jam (Localization).
• But, the track does not twist into a knot (No Topology).
• And, the twins stay identical (No Degeneracy Breaking).

This is like a traffic jam that looks exactly like the ones in normal, boring, "Hermitian" physics. It's a "pure" localization transition that doesn't rely on the weird non-Hermitian skin effects.

4. The "Four-Decker" Special Case

There was one special setting (when the timing shift is exactly 90 degrees) where the system regained its perfect mirror balance (PT-symmetry). Here, a fourth thing happened: the energy levels of the train cars suddenly shifted from being real numbers to complex numbers (a "Real-Complex" transition). This created a "Quartet" transition, adding one more layer of complexity to the triple-decker.

Summary

The paper shows that non-Hermitian quasicrystals are more versatile than previously thought.

• Most of the time: You get a complex "Triple-Decker" transition where getting stuck, twisting the track, and separating twins all happen at once, driven by the non-Hermitian "skin effect."
• Sometimes: You can dial the system to a setting where you get a "Pure" traffic jam, just like in normal physics, without the extra weirdness.

Essentially, the authors expanded our understanding of how these systems work, showing that you don't always need the "perfect mirror" (PT-symmetry) to get interesting physics, and that you can actually "turn off" the weird non-Hermitian effects to get a standard localization transition if you tune the phase shift correctly.

Technical Summary

Technical Summary: Topological metal-insulator transitions in one-dimensional non-Hermitian quasicrystals: beyond PT-symmetry

Problem Statement
One-dimensional non-Hermitian quasicrystals (NHQCs) with parity-time ($PT$) symmetry are known to exhibit simultaneous localization-delocalization, topological, and $PT$-symmetry-breaking transitions (often termed "triple phase transitions"). However, existing literature largely restricts investigations to $PT$-symmetric systems. The behavior of NHQCs in more general non-Hermitian settings, specifically those lacking $PT$ symmetry, remains largely unexplored. This work addresses the gap in understanding how the absence of $PT$ symmetry impacts topological metal-insulator transitions and the nature of localization in such systems.

Methodology
The authors propose a one-dimensional tight-binding model of a non-Hermitian quasicrystal where the onsite potential $V_n$ consists of real and imaginary parts modulated by cosine functions with a relative phase shift $\phi$:
$$V_n = v_R \cos(2\pi\alpha n) + i v_I \cos(2\pi\alpha n + \phi)$$
Here, $\alpha$ is an irrational number ensuring quasiperiodicity. The model generally breaks $PT$ symmetry, except at specific values $\phi = m\pi + \pi/2$.

To analyze the system, the authors employ a duality transformation to map the real-space Hamiltonian to momentum space. In the dual picture, the non-Hermiticity manifests as asymmetric nearest-neighbor couplings ($|J_+| \neq |J_-|$), while the onsite potential becomes Hermitian. This duality allows for the analysis of Non-Hermitian Skin Effects (NHSEs) and Anderson localization interplay.

The study utilizes numerical simulations with periodic boundary conditions on a system size $L = F_{15} = 610$ (Fibonacci approximation). Key characterization metrics include:

• Inverse Participation Ratio (IPR) and Fractal Dimension (FD) to distinguish between extended and localized phases.
• Winding Number ($\omega$) calculated via twisted boundary conditions to characterize spectral topology.
• Degeneracy Rate ($\eta$) to track spectral degeneracy breaking.
• Maximum Imaginary Part of Eigenenergies to identify $PT$-symmetry breaking.

Key Contributions and Results

1. Triple Phase Transitions in General NHQCs:
   The authors demonstrate that in most parameter regions (general $\phi$), the system supports a simultaneous "triple phase transition" involving:

   • Localization-Delocalization: A transition between extended and localized states.
   • Topological Transition: A change in the spectral winding number ($\omega$) from 0 (trivial) to $\pm 1$ (non-trivial).
   • Degeneracy-Breaking Transition: A transition from a spectrum with nearly twofold degeneracy to a non-degenerate spectrum.
     These three transitions coincide along the same phase boundary defined by $v = \sqrt{2/(1 + |\sin \phi|)}$. The topological and degeneracy-breaking transitions are shown to originate from NHSEs in momentum space (asymmetric couplings).

2. Distinct Types of Localization:
   The paper identifies two distinct mechanisms for localization in 1D NHQCs:

   • Type I (NHSE-driven): Occurs when asymmetric couplings are present. This type is accompanied by topological transitions and degeneracy breaking.
   • Type II (Hermitian-analogous): Occurs at specific phase shifts $\phi = m\pi$ where asymmetric couplings vanish ($J_+ = J_-$). In this case, the system undergoes a localization-delocalization transition without activating topological phase transitions or degeneracy-breaking transitions. This behavior is analogous to localization in Hermitian quasicrystals.

3. Special Symmetry Points:

   • At $\phi = m\pi + \pi/2$ ($PT$-symmetric): The system exhibits a "quartet phase transition," encompassing the triple transitions mentioned above plus a $PT$-symmetry-breaking transition (real-to-complex spectral transition).
   • At $\phi = m\pi$ (Time-reversal symmetric, no NHSE): Only the localization-delocalization transition persists. The winding number vanishes, and the degeneracy rate remains near unity, confirming the absence of non-Hermitian topological and degeneracy-breaking features.

Significance
The authors claim that this work extends the understanding of topological metal-insulator transitions from $PT$-symmetric systems to a broader class of non-Hermitian settings. By decoupling the localization transition from topological and degeneracy-breaking transitions, the study reveals that non-Hermitian systems can host distinct types of localization behavior. Specifically, it highlights that localization in NHQCs is not always tied to non-Hermitian skin effects or topology; under specific conditions (vanishing asymmetric couplings), it reverts to a mechanism analogous to the Hermitian case. The relative phase shift $\phi$ is identified as a tunable parameter that can control the nature of these phase transitions.