Emergence of Hermitian topology from non-Hermitian knots

This paper demonstrates that a first-order knot transition in the complex eigenvalue spectrum of a non-Hermitian Hamiltonian can emerge from a topological phase transition in an underlying Hermitian model defined by the non-Hermitian system's singular values, a process characterized by discrete jumps in eigenvalues rather than exceptional points.

The Gist

Imagine you are a detective trying to solve a mystery involving two different types of maps: a Standard Map (Hermitian) and a Weird, Twisted Map (Non-Hermitian).

Usually, these two maps tell very different stories. The Standard Map is like a calm, flat landscape where everything is predictable and real. The Twisted Map is like a rollercoaster in a dream; it has loops, knots, and strange "dead ends" where things merge together.

In this paper, the researchers (Gaurav, Ranjan, and Bhabani) discovered a hidden secret: The knots in the Twisted Map are actually just reflections of the hills and valleys in the Standard Map.

Here is the story of their discovery, broken down into simple concepts:

1. The Two Maps

• The Standard Map (Hermitian): Think of this as a standard topological map of a city. It has "roads" (energy levels) that are always real numbers. Sometimes, you can change a parameter (like a traffic light) to switch the city from a "loop-free" layout to a "looped" layout. This is called a Topological Phase Transition.
• The Twisted Map (Non-Hermitian): This map is wilder. Its roads can be complex numbers (imagine roads that go "sideways" into a different dimension). Because of this, the roads can twist into knots (like a pretzel or a figure-eight). Usually, these knots only untie or retie when the roads crash into each other at a specific point called an Exceptional Point (EP).

2. The "Shadow" Connection

The researchers asked a clever question: What if the "hills and valleys" (singular values) of the Twisted Map are actually just the "roads" of the Standard Map?

They took a Standard Map (specifically a famous model called the SSH model, which describes how electrons hop between atoms in a chain) and forced the Twisted Map to copy its hills and valleys.

The Surprise:
They found that whenever the Standard Map changed its shape (a topological transition), the Twisted Map immediately changed its knot shape too!

• When the Standard Map went from a "flat" phase to a "looped" phase, the Twisted Map went from an Unlinked knot (two separate rings) to an Unknot (a single loop).
• When the Standard Map went from one loop to two loops, the Twisted Map went from a single loop to a Hopf Link (two rings interlocked like a chain).

3. The "Magic Jump" (The Big Discovery)

Here is the most exciting part. Usually, for a knot to change shape in the Twisted Map, the roads have to crash and merge (an Exceptional Point). It's like a car crash that forces the traffic to reroute.

But in this case, there was no crash.

Instead, the roads simply jumped.

• Imagine you are driving on a bridge. Suddenly, without any warning or crash, the bridge teleports you to a different height.
• The researchers call this a "First-Order Knot Transition."
• At the exact moment the Standard Map changes its topology, the Twisted Map's roads make a sudden, discrete jump. The knot unties or reties instantly, without the roads ever touching or merging.

4. The One-Way Street

The researchers also found an interesting rule about this connection:

• If the Standard Map changes, the Twisted Map changes. (The shadow follows the object).
• But, if the Twisted Map changes, the Standard Map doesn't have to. (The shadow can move on its own sometimes).

They found that the Twisted Map can sometimes get into knots and untie them at other points where the Standard Map is perfectly calm and doing nothing. So, seeing a knot change doesn't always mean the Standard Map is changing, but if the Standard Map changes, the knot will change.

The Big Picture Analogy

Imagine you are holding a shadow puppet (the Standard Map) against a wall. The shadow is simple and flat.
Behind you, there is a complex, twisting wire sculpture (the Twisted Map).

Usually, the wire sculpture does its own thing. But the researchers found a way to build the wire sculpture so that its "thickness" matches the shadow puppet perfectly.

• When you move your hand to change the shadow from a "circle" to a "square," the wire sculpture behind you instantly snaps from a simple loop to a complex knot.
• It doesn't happen because the wires are melting or crashing; they just teleport into a new shape the moment the shadow changes.

Why Does This Matter?

This discovery is a new bridge between two worlds of physics.

1. It helps us understand the weird world of Non-Hermitian physics (which is used in lasers, optics, and open quantum systems) by using the simpler, well-understood rules of standard physics.
2. It suggests new ways to build materials. If we want a material that has a specific "knot" in its energy levels (which could protect it from errors), we can just design a standard material with a specific shape, and the "knot" will naturally appear in the complex version.

In short: The complex knots of the weird world are just the shadows of the simple hills of the normal world, and they change shape with a magical jump, not a crash.

Technical Summary

1. Problem Statement

Non-Hermitian (NH) systems exhibit unique topological phenomena not found in Hermitian systems, such as Exceptional Points (EPs) where eigenvalues and eigenvectors coalesce, and complex eigenvalue spectra that form knot structures (braiding). A central question in the field is understanding the relationship between the topology of NH systems and their underlying Hermitian counterparts.

Specifically, the authors address the following problem:

• Singular values of an NH Hamiltonian are always real and can be interpreted as the eigenvalues of a Hermitian Hamiltonian.
• If a Hermitian Hamiltonian undergoes a topological phase transition (characterized by a change in winding number), does this transition induce a corresponding change in the knot topology of the complex eigenvalues of the underlying NH Hamiltonian?
• Crucially, does this NH knot transition occur via the standard mechanism (an EP) or through a different mechanism?

2. Methodology

The authors employ a formalism based on Singular Value Decomposition (SVD) to construct a bridge between Hermitian and Non-Hermitian systems.

• Construction of NH Hamiltonians:

  • They start with a translational invariant Hermitian Hamiltonian $H(k, \omega)$ (where $k$ is lattice momentum and $\omega$ is a tunable parameter).
  • They ensure the spectrum of $H$ is real and positive by adding a constant shift $\mu I$.
  • Using SVD, they decompose $H$ as $H = A A^\dagger$, where $A$ is the underlying NH Hamiltonian.
  • $A$ is constructed as $A = U \Sigma V^\dagger$, where $\Sigma$ contains the square roots of the eigenvalues of $H$, $U$ is derived from the eigenvectors of $H$, and $V$ is an arbitrary unitary matrix representing gauge freedom.
  • The authors primarily investigate cases where $V$ is $k$-independent and traceless to isolate the topological effects of $H$.

• Topological Invariants:

  • Hermitian System: The topological phase is characterized by the standard winding number $\nu_H$ calculated from the eigenvectors of $H$.
  • Non-Hermitian System: The knot topology of the complex eigenvalues of $A$ is characterized by a winding number $\nu$ defined in the complex plane:
    $$\nu = \int_0^{2\pi} \frac{dk}{2\pi i} \frac{d}{dk} \ln \det \left( A - \frac{1}{2}\text{Tr}(A) \right)$$
    (Note: The reference energy is adjusted to the spectral center).

• Models Studied:

  • Model I: A standard Su-Schrieffer-Heeger (SSH) model. It transitions from winding number $\nu=0$ to $\nu=1$ at $\omega=1$.
  • Model II: An extended SSH model with next-nearest-neighbor hopping. It transitions from $\nu=1$ to $\nu=2$ at $\omega=1$.

3. Key Contributions and Results

A. Emergence of Knot Transitions from Hermitian Transitions

The primary finding is a direct correlation: when the Hermitian Hamiltonian $H$ undergoes a topological phase transition (gap closing at specific $k$ points), the complex eigenvalues of the constructed NH Hamiltonian $A$ undergo a knot transition (change in knot topology, e.g., from unlink to unknot, or unknot to Hopf-link).

• This transition occurs at the exact same parameter value ($\omega=1$) where the Hermitian gap closes.
• The result is robust across different choices of the unitary matrix $V$, provided $V$ is $k$-independent. If $V$ is $k$-dependent, the specific knot types may vary, but the transition point remains tied to the Hermitian gap closing.

B. Discovery of "First-Order Knot Transitions"

The most significant theoretical contribution is the identification of a novel type of topological transition in NH systems:

• Standard NH Transitions: Usually accompanied by an Exceptional Point (EP), where eigenvalues coalesce and the matrix becomes non-diagonalizable.
• This Work's Finding: The knot transition induced by the Hermitian topological transition does not involve an EP.
  • Instead, the real and imaginary parts of the NH eigenvalues exhibit a discrete jump (discontinuity) at the transition point.
  • The authors term this a "First-Order Knot Transition."
  • At the transition point ($\omega=1$), the NH matrix $A$ remains normal (diagonalizable), and no EP exists, yet the knot topology changes abruptly due to the spectral discontinuity.

C. Asymmetry of the Correspondence

The study establishes an asymmetric relationship:

• Hermitian $\to$ NH: A topological transition in the Hermitian system always induces a knot transition in the corresponding NH system.
• NH $\to$ Hermitian: A knot transition in the NH system does not necessarily imply a topological transition in the Hermitian system.
  • The authors identify additional transition points in the NH spectrum (e.g., at $\omega \approx 34$ for Model I and $\omega \approx 67$ for Model II) where the knot topology changes.
  • These NH transitions are accompanied by Exceptional Points (EPs) but occur without any gap closing in the Hermitian spectrum. Thus, they represent purely non-Hermitian phenomena with no Hermitian counterpart.

4. Significance and Implications

• New Classification of NH Transitions: The paper introduces a new class of topological transitions in NH systems that are distinct from EP-driven transitions. These "first-order" transitions are driven by spectral discontinuities rather than eigenvalue coalescence.
• Bridging Hermitian and NH Physics: It provides a constructive framework to generate NH models with specific knot topologies by starting from well-understood Hermitian topological phases. This suggests that signatures of Hermitian topology can persist in the complex eigenvalue structure of NH systems.
• Experimental Relevance: The authors suggest that these transitions could be probed in photonic platforms, where NH systems are easily realized. The distinct signature (discontinuous jump vs. EP coalescence) offers a clear experimental observable to distinguish between Hermitian-induced knot transitions and intrinsic NH transitions.
• Theoretical Insight: The work clarifies that while NH band theory is incomplete without homotopy theory, the singular values (and their link to Hermitian eigenvalues) provide a robust "shadow" of Hermitian topology that dictates the global knot structure of the NH spectrum, independent of local EPs.

Conclusion

The paper demonstrates that the topological phase transitions of a Hermitian Hamiltonian are imprinted onto the knot topology of the complex eigenvalues of a constructed Non-Hermitian Hamiltonian. This imprint manifests as a unique "first-order knot transition" characterized by spectral discontinuities rather than Exceptional Points. This finding establishes a robust, albeit asymmetric, link between Hermitian topology and Non-Hermitian knot theory, offering new avenues for designing and understanding topological phases in open quantum systems.