🔬 mesoscale physics

Floquet multiple exceptional points with higher-order skin effect

This paper investigates periodically driven open quantum systems in microcavities, demonstrating that a quenched driving protocol enables the controlled creation and annihilation of multiple Floquet exceptional points characterized by quantized winding numbers, while simultaneously inducing a higher-order skin effect with modes localized at both edges and corners.

Original authors: Gaurab Kumar Dash, Subhendu Kumar Patra, Diptiman Sen, Manisha Thakurathi

Published 2026-02-20

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CC BY 4.0

Original authors: Gaurab Kumar Dash, Subhendu Kumar Patra, Diptiman Sen, Manisha Thakurathi

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 you have a tiny, magical playground made of light trapped inside a glass box (a microcavity). In this playground, light waves can spin clockwise or counter-clockwise. Usually, physics tells us that if you leave things alone, the light behaves predictably. But this paper explores what happens when we shake this playground rhythmically and introduce a bit of "magic" where light can be created (gain) or destroyed (loss) in specific spots.

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

1. The "Ghost" Meeting Points (Exceptional Points)

In normal physics, if two roads cross, you can usually tell them apart. But in this "non-Hermitian" world (where energy isn't perfectly conserved because of the gain and loss), something strange happens. Sometimes, two different paths of light don't just cross; they merge into a single, ghostly point where they become indistinguishable.

The scientists call these Exceptional Points (EPs). Think of it like two rivers flowing together until they become one single, wider river. At this exact spot, the rules of the game change completely. The paper shows that by shaking the system (periodic driving), they can create multiple of these merging points, make them appear, and make them disappear again, just by tuning the speed of the shake.

2. The "Shake" That Creates Monsters (Floquet Engineering)

The researchers didn't just sit still; they applied a rhythmic "kick" to the system, like shaking a jar of marbles. In physics, this is called Floquet engineering.

The Analogy: Imagine a swing. If you push it at just the right rhythm, it goes higher and higher. If you push it at a weird rhythm, it might do something totally unexpected, like spinning in a circle instead of going back and forth.
The Result: By shaking their light-box, they created a new "effective" world where these ghostly meeting points (EPs) pop up in places they never existed before. They found they could control the "birth" and "death" of these points simply by changing how long they waited between shakes.

3. The "Scream" of the System (Fidelity Susceptibility)

How do you know when these ghostly points appear? You can't see them with a regular microscope. The scientists invented a special "detector" called Fidelity Susceptibility.

The Analogy: Imagine a room full of people whispering. If you change the temperature slightly, most people might not notice. But if you hit a specific "tipping point," the whole room suddenly screams in unison.
The Discovery: When the system hits one of these ghostly meeting points, this detector goes crazy. It spikes to infinity (a sharp peak). The paper shows that if you add up all these "screams" across the system, you get a massive explosion in the data exactly when the number of ghost points changes. It's like a seismograph detecting an earthquake.

4. The "Skin" Effect (Light Clinging to the Edges)

Usually, if you have a crowd of people in a room, they spread out evenly. But in this non-Hermitian world, the light waves get scared and run to the walls. This is called the Skin Effect.

The Analogy: Imagine a party where everyone suddenly decides to stand only on the walls of the room, leaving the middle empty.
The Twist: The paper discovered a Higher-Order Skin Effect. It's not just the walls (edges) that get crowded; the corners get crowded too! The light waves pile up specifically at the four corners of the box. It's as if the light waves are huddled in the corners of the room, terrified of the open space in the middle.

5. Why Does This Matter?

This isn't just a cool math trick.

Controlling Light: It gives scientists a new way to trap and manipulate light at the microscopic level.
New Materials: It helps us understand how to build materials that behave in weird, useful ways when they are being driven or shaken.
The Future: By understanding how to make these "ghost points" appear and disappear, we might be able to build super-sensitive sensors or new types of lasers that are impossible to build with current technology.

In a nutshell: The authors found a way to use rhythmic shaking to create and destroy invisible "merging points" in a system of light. They proved that when these points appear, the system "screams" (detectable by their new math tool), and the light waves get scared and hide in the corners of the box. This opens up a new playground for engineers to design light-based devices with superpowers.

1. Problem Statement

The paper addresses the complex interplay between non-Hermitian (NH) physics and periodic driving (Floquet engineering) in open quantum systems. While exceptional points (EPs) and the non-Hermitian skin effect (NHSE) are well-studied in static systems, their combined behavior in periodically driven systems remains largely unexplored. Specifically, the authors aim to:

Investigate the emergence of Floquet Exceptional Points (FEPs) in driven NH systems.
Understand the mechanism of pair-production and annihilation of these FEPs via tuning driving parameters.
Detect and characterize these singularities using bi-orthogonal fidelity susceptibility.
Explore the emergence of higher-order skin effects (localization at corners and edges) in driven NH systems, which differ from conventional topological insulators.

2. Methodology

The authors employ a theoretical framework combining Floquet theory with non-Hermitian Hamiltonian dynamics.

System Model: They propose a two-dimensional, two-band model realized in an array of microcavity resonators. The system consists of clockwise and anticlockwise modes.
Driving Protocol: A periodically quenched driving protocol is introduced. The system evolves under two alternating Hamiltonians ( $H_1$ $H_{1}$ and $H_2$ $H_{2}$ ) over a period $2T$ $2 T$ :
- $H_1$ : A Hermitian Hamiltonian representing unitary evolution.
- $H_2$ : A non-Hermitian Hamiltonian introducing on-site gain and loss (imaginary potential).
Analytical Derivation:
- They derive the Floquet effective Hamiltonian ( $H_F$ ) using the time-evolution operator $U_{2T} = e^{-iH_2 T} e^{-iH_1 T}$ .
- They analyze the quasi-energy spectrum $\epsilon(k)$ to identify band degeneracies.
Topological Characterization:
- Winding Numbers: They calculate integer-quantized winding numbers around zero and $\pi$ quasi-energies to characterize the topological nature of FEPs.
- Bi-orthogonal Fidelity Susceptibility (BFS/BFFS): They introduce a Bi-orthogonal Floquet Fidelity Susceptibility ( $\chi_F$ ). Unlike Hermitian systems, this utilizes both left and right eigenvectors to account for non-orthogonality. The divergence of $\chi_F$ serves as a precise marker for EPs.
Skin Effect Analysis: They examine the system under Open Boundary Conditions (OBC) to identify skin modes. They utilize a weighted Inverse Participation Ratio (IPR) to distinguish between corner-localized and edge-localized modes.

3. Key Contributions

Discovery of Multiple FEPs: The study demonstrates that a periodically quenched drive can generate multiple pairs of FEPs within the Floquet Brillouin zone (FBZ), a phenomenon not present in the static counterpart.
Control of FEP Dynamics: The authors show that the creation and annihilation of FEPs can be precisely controlled by fine-tuning the driving time period ( $T$ ) and system parameters (gain/loss strength $\xi$ ).
Novel Detection Tool: They establish Bi-orthogonal Floquet Fidelity Susceptibility as a robust tool for detecting FEPs. They prove that the momentum-summed susceptibility diverges sharply exactly when the number of FEPs changes.
Higher-Order Skin Effect in Driven Systems: The paper reveals a higher-order skin effect where skin modes localize simultaneously at corners and edges. This is distinct from standard NHSE (edge-only) and is protected by spatio-inversion symmetry.
Type-II FEPs: They identify a new class of FEPs (Type-II) that emerge solely due to the driving period, lacking direct counterparts in static Hermitian or non-Hermitian diabolic point systems.

4. Key Results

Static Limit: The static model ( $\xi \neq 0$ ) hosts EPs characterized by integer winding numbers ( $W = \pm 1$ ). The bi-orthogonal fidelity susceptibility diverges at these points.
Floquet Dynamics:
- The driven system hosts FEPs at quasi-energies 0 and $\pi/2T$ .
- As the driving period $T$ increases, Type-II FEPs appear. The number of FEPs ( $N_0$ and $N_\pi$ ) changes discretely, leading to a sharp divergence in the summed fidelity susceptibility (Fig. 4b).
- Topological Invariants: The FEPs are topologically protected, evidenced by robust integer winding numbers around the zero and $\pi$ quasi-energy gaps.
Spectral Features:
- Under OBC, the system exhibits Floquet edge states around zero energy and Dirac-like dispersion (surface Dirac cones) around $\pi$ .
- The complex eigenspectrum shows Floquet point gaps that are absent in the periodic boundary condition (PBC) spectrum.
Higher-Order Skin Effect:
- The system exhibits a higher-order skin effect where eigenstates localize at both corners and edges.
- The corner modes are four-fold degenerate and appear in the point gaps of the quasi-energy spectrum.
- This effect is robust and protected by spatio-inversion symmetry ( $P$ ).

5. Significance and Implications

Experimental Realizability: The proposed model is directly realizable in microcavity resonator arrays, where gain and loss can be engineered via asymmetric scattering or external pumping.
Floquet Engineering: The work provides a blueprint for engineering topological singularities (FEPs) in driven dissipative systems, offering a new degree of freedom (time-period) to manipulate light-matter interactions.
Fundamental Physics: It bridges the gap between non-Hermitian topology and Floquet theory, revealing that periodic driving can induce higher-order topological phenomena (corner modes) that are impossible in static Hermitian systems.
Applications: These findings have significant potential for manipulating light at the microscale, potentially leading to advanced photonic devices with enhanced sensitivity (due to EPs) and robust transport properties (due to skin effects).

In conclusion, the paper establishes a comprehensive framework for understanding and controlling the rich non-equilibrium physics of driven non-Hermitian systems, highlighting the unique ability of periodic driving to generate multiple exceptional points and higher-order skin effects.