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Krylov complexity, mode-resolved complexity and entanglement entropy across phase transitions in the non-Hermitian extended Su-Schrieffer-Heeger model

This paper investigates phase transitions in a non-Hermitian extended Su-Schrieffer-Heeger model by demonstrating that Krylov spread complexity and entanglement entropy effectively signal topological changes and dynamical phases under different evolution protocols, while momentum-resolved complexity further elucidates the underlying characteristic modes driving these transitions.

Original authors: Ling-Feng Zhang, Wing Chi Yu

Published 2026-07-07

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Ling-Feng Zhang, Wing Chi Yu

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 vast, intricate dance floor where thousands of tiny dancers (quantum particles) move in perfect synchronization. Physicists usually study how these dancers move when the music is perfectly balanced and predictable. But what happens when the music gets a bit "glitchy"—when some dancers lose energy while others gain it, and the rules of the dance become slightly unbalanced? This is the world of non-Hermitian physics, and this paper explores a specific, complex version of that dance floor called the extended Su-Schrieffer-Heeger (SSH) model.

Here is a breakdown of what the researchers did, using simple analogies:

1. The Dance Floor Setup

The researchers looked at a chain of dancers arranged in pairs. In the "normal" version of this dance, the pairs are linked in a simple pattern. In this extended version, the dancers can also reach out and grab hands with neighbors two steps away, creating a much more complex web of connections.

They introduced a "glitch" into the system: an imaginary chemical potential. Think of this as a rule where one side of the dance floor is constantly losing energy (like a dancer getting tired), while the other side is gaining energy (like a dancer getting a sugar rush). This makes the system "non-Hermitian"—it's not a closed, perfect loop anymore; energy is flowing in and out.

2. The "Magic Points" (Exceptional Points)

In the normal, balanced dance, the dancers change their formation (a phase transition) at specific moments when the music stops briefly (a "gap closing").

The researchers found that when they added the "glitch" (the non-Hermitian part), something fascinating happened at these transition moments. The "gap" didn't just close; it split into pairs of "magic points" called Exceptional Points.

The Analogy: Imagine a road that usually splits into two lanes. In this glitchy world, at the split, the two lanes don't just separate; they twist around each other and merge into a single, strange loop before splitting again. The number of these loops is directly related to how complex the dance formation was before the glitch started.

3. Measuring the "Confusion" (Complexity)

To understand how the dancers react to these changes, the researchers used two main tools to measure "complexity" (how hard it is to describe or prepare the dance):

Tool A: The "Instant Snapshot" (Unitary Evolution)
They asked: "If we magically teleport the dancers from a simple starting pose to the final, complex pose, how 'spread out' does the information become?"
- Result: This method was like a high-precision camera. It clearly snapped a picture of the transition. Whenever the dancers changed their formation, the "complexity" jumped or spiked. It was a very reliable way to spot the phase transitions.
Tool B: The "Long-Term Video" (Non-Unitary Evolution)
They asked: "If we let the dancers perform this glitchy dance for a long time, what does the average complexity look like?"
- Result: This was like watching a blurry, long-exposure video. It could still see the major transitions, but it also showed some small, confusing bumps and dips within a single phase. It was a bit noisier than the instant snapshot.

4. Zooming In: The "Solo Dancers" (Mode-Resolved Complexity)

The researchers realized that looking at the whole dance floor at once was like looking at a crowd from a helicopter—you see the big picture but miss the details. So, they invented a new tool: Mode-Resolved Complexity.

The Analogy: Instead of watching the whole crowd, they put on special glasses that let them watch one specific dancer (or a specific rhythm) at a time.
What they found: They could see exactly which dancers were causing the confusion. They tracked how these "critical dancers" moved and changed as the music parameters shifted.
The Fidelity Test: They used a "similarity score" (fidelity) to see how much the behavior of one dancer resembled another. When the system hit a transition, this score dropped sharply, acting like a perfect alarm bell for the phase change.

5. The "Settling Down" Phase (Saturation)

Finally, they looked at what happens when the dancers have been dancing for a very long time in the "glitchy" regime (where the energy is purely imaginary).

The Analogy: Imagine a spinning top that eventually slows down and stops. The researchers measured how long it took for the dance to "settle" into a steady, boring rhythm.
The Discovery: They found that the time it took to settle down depended on the "slowest" dancer in the group—the one who was barely moving.
- In the original, simple dance model, the "slowest dancer" was always the same person, no matter how you changed the music.
- In this new, extended model: The "slowest dancer" changes identity as you tweak the music. The person who is the slowest at one moment might be replaced by a different dancer a moment later. This creates a continuous, smooth shift in how the system settles, rather than a sudden jump.

6. The "Entanglement" Check

They also measured Entanglement Entropy, which is a way of measuring how "connected" the two halves of the dance floor are.

The Result: The entanglement entropy behaved almost exactly like the "settling time" of the complexity. It slowed down and sped up in the same patterns.
Why it matters: Since measuring entanglement is often easier in real-world experiments than measuring "complexity," this finding suggests that scientists can use entanglement as a simpler, more accessible way to detect these complex phase transitions and dynamical phases.

Summary

In short, this paper shows that when you add a specific type of "glitch" to a complex quantum dance floor:

New "magic points" appear at the edges of phase changes.
Complexity measures act like sensitive detectors for these changes, with "instant snapshots" being clearer than "long videos."
Zooming in on individual rhythms reveals exactly how the system transforms.
The settling time of the system is controlled by a "slowest dancer" that changes its identity as you tune the system, creating a unique signature for this extended model.
Entanglement can be used as a practical, real-world tool to spot these same behaviors.

Technical Summary: Krylov Complexity, Mode-Resolved Complexity, and Entanglement Entropy across Phase Transitions in the Non-Hermitian Extended Su-Schrieffer-Heeger Model

Problem Statement
This study investigates phase transitions within the extended Su-Schrieffer-Heeger (SSH) model, incorporating next-nearest-neighbor (NNN) hoppings and an imaginary staggered chemical potential. While the original SSH model is well-understood, its extended non-Hermitian version exhibits a richer phase diagram with multiple topological phases and complex non-equilibrium dynamics. The authors aim to characterize these phases and their transitions using information-theoretic tools, specifically Krylov spread complexity and entanglement entropy. A key motivation is to understand how exceptional points (EPs) emerge from gap-closing momenta in the Hermitian limit under weak non-Hermiticity and to determine whether complexity measures can effectively signal these transitions and distinguish dynamical phases in regimes with purely imaginary spectra.

Methodology
The authors analyze the model using two distinct dynamical protocols, both admitting analytical solutions due to the system's mapping to the $su(2) \times su(2)$ Lie algebra:

Unitary Protocol: Preparing the non-Hermitian ground state via unitary evolution from a reference state (the lowest-weight state of the algebra) at time $t=1$ .
Non-Unitary Protocol: Evolving the system under the non-Hermitian Hamiltonian for long times from the same reference state.

The primary observables are:

Spread Complexity ( $C_\Omega$ and $\bar{C}$ ): Calculated for both the unitary ground state and the long-time averaged non-unitary evolution.
Entanglement Entropy ( $S$ ): Computed for the long-time evolution to compare with spread complexity.
Mode-Resolved Complexity: To resolve the detailed structure of transitions, the authors introduce momentum-resolved complexity and associated fidelity measures ( $F_{C_\Omega}(t_c)$ and $\tilde{F}_{C_\Omega}(k)$ ) to track the evolution of specific modes and identify critical momenta.
Saturation Analysis: In the regime of purely imaginary spectra, the authors derive analytical scaling laws for the saturation time ( $t^*$ ) of spread complexity and entanglement entropy, relating them to the slowest decay mode of the spectrum.

Key Contributions and Results

Emergence of Exceptional Points: In the presence of small non-Hermiticity, the authors demonstrate that exceptional points emerge in pairs from the gap-closing momenta found at the topological phase boundaries of the Hermitian limit. The number of emergent EPs is twice the number of original gap-closing points, which corresponds to the absolute difference in winding numbers between adjacent phases. As non-Hermiticity increases, these EPs expand and eventually coalesce or vanish.
Complexity as a Phase Transition Detector:
- Unitary Protocol: The spread complexity ( $C_\Omega$ ) and its derivative serve as robust indicators of topological phase transitions in both Hermitian and non-Hermitian regimes. Discontinuities or divergences in the derivative align with phase boundaries.
- Non-Unitary Protocol: The long-time spread complexity ( $\bar{C}$ ) and entanglement entropy also detect transitions. However, the authors observe minor peaks or valleys within phases for $\bar{C}$ and $S$ that do not correspond to topological transitions. Consequently, the authors suggest that the spread complexity from the unitary protocol is a cleaner indicator of phase transitions than the long-time non-unitary measures.
Mode-Resolved Analysis: The introduction of mode-resolved complexity reveals that transitions are driven by specific critical modes.
- Fidelity measures based on the distribution of mode-resolved complexity accurately capture phase boundaries.
- In the Hermitian limit, fidelity across momentum detects gap-closing momenta.
- In the non-Hermitian case, it identifies characteristic momenta where gap-closing regions begin or end.
- This approach decodes how gapless modes evolve as non-Hermiticity strength increases, showing that modes with opposite momenta contribute anti-symmetrically in the non-Hermitian regime.
Dynamical Phases and Saturation Behavior:
- In the regime where the spectrum is purely imaginary, distinct dynamical phases are identified based on the saturation behavior of spread complexity and entanglement entropy.
- The authors derive an analytical scaling for the saturation time $t^*$ , showing it is inversely proportional to the gap of the slowest decay mode ( $\Gamma(k^*)$ ).
- Unlike the conventional SSH model where the critical mode $k^*$ is fixed within a phase, the extended model exhibits a continuous variation of $k^*$ with the driving parameter.
- The entanglement entropy exhibits a saturation behavior identical to the spread complexity, suggesting it is a viable, experimentally accessible proxy for identifying these dynamical phases.

Significance
The paper claims that Krylov spread complexity, particularly when derived from unitary evolution, provides a robust and sensitive tool for characterizing topological phase transitions in non-Hermitian systems, outperforming long-time non-unitary measures in terms of signal clarity. The introduction of mode-resolved complexity offers a granular view of transition mechanisms, linking macroscopic complexity changes to specific momentum modes and exceptional points. Furthermore, the demonstration that entanglement entropy mirrors the saturation dynamics of spread complexity in purely imaginary spectral regimes suggests that entanglement entropy can serve as a more experimentally accessible probe for dynamical phases in non-Hermitian systems. The work extends the understanding of non-Hermitian physics beyond the standard SSH model, highlighting the intricate interplay between NNN hoppings, topology, and dissipation.