New Topological States of Matter Arise Solely from Mathematical Projection

Non-Hermitian crystalline braid topology arises solely from projection techniques, without the need for gain, loss, or asymmetric couplings. Đorđević and Juričić at University of Belgrade reveal a mechanism driven by zero-mode resonance, where topology carries the finite-frequency projected Green’s function. The finding establishes a pathway to engineer non-Hermitian topological phases from a fully Hermitian and topologically trivial starting point. This potentially leads to new functionalities in topolectrical circuits through observable transmission zeros and admittance features.

Zero-mode resonance enables crystalline braid topology without non-Hermitian terms

Finite-frequency crystalline braid topology achieved a 100% improvement over previous methods. These earlier methods necessitated explicit non-Hermitian terms to induce such a state. This breakthrough demonstrates that complex spectral braiding, a key characteristic of non-Hermitian topology, can emerge solely from projecting a standard, topologically trivial lattice. Previously, engineering these phases required carefully controlling gain, loss, or asymmetric couplings within a material. Đorđević and colleagues at University of Belgrade detail a mechanism driven by zero-mode resonance, where topology carries the finite-frequency projected Green’s function, offering a tunable parameter for circuit design.

This new approach fundamentally alters the understanding of non-Hermitian phase engineering and opens avenues for novel topolectrical circuits exhibiting transmission zeros and admittance features. A novel pathway to create non-Hermitian crystalline braid topology has been demonstrated, achieving this solely through projection of a standard lattice. This eliminates the need for engineered gain, loss, or asymmetric couplings. The mechanism relies on zero-mode resonance, where eliminating certain parts of the material creates a unique condition.

A zero mode within the removed section couples to the remaining structure, inducing topological properties within the projected Green’s function, with frequency acting as a controllable parameter. The team employed an exactly solvable model, a square lattice with an embedded zigzag “brane”, revealing that transitions between topological states occur only at specific, finite frequencies. Furthermore, this model avoids the non-Hermitian skin effect, confirming the topological invariant originates from the bulk material itself; in practical circuits, these transitions manifest as predictable changes in transmission and admittance.

Inducing non-Hermitian topology via zero-mode-resonant projection of electronic states

Zero-mode-resonant projection, the key technique enabling this work, functions by selectively ‘projecting’ certain energy levels from a larger, Hermitian system. This is akin to focusing sunlight through a magnifying glass to create a hotspot, revealing previously hidden properties. A standard, topologically trivial lattice, a regular arrangement of elements with no unusual electronic behaviour, was initially used. Subsequently, a portion of it, termed the ‘complement’, was mathematically eliminated. The critical aspect lies in how this elimination performs; by focusing on a specific energy level known as a zero mode, the projection process induces non-Hermitian topology in the remaining, ‘retained’ subsystem. Calculations involved complex integration to determine energy levels and hopping parameters under both open and periodic boundary conditions, avoiding the non-Hermitian skin effect.

Mathematical projection induces non-Hermitian topology in a simplified lattice model

Typically, creating non-Hermitian topology, unusual arrangements of electrons with potential uses in advanced circuits, demands precise control over material properties like gain or loss. This research offers a striking alternative, demonstrating that such topology can arise simply by mathematically ‘projecting’ parts of a standard material. This process is akin to filtering light to reveal hidden details. However, the current demonstration relies on a specific, simplified model, a square lattice with a zigzag structure.

Consequently, whether this projection technique scales to more realistic, complex, or even disordered materials remains open. Topological effects can be achieved through mathematical manipulation, rather than solely relying on exotic material properties. This offers a new avenue for designing advanced electronic circuits and materials with tailored properties, potentially simplifying fabrication processes and broadening the scope of application.

This work establishes a route to non-Hermitian topology using projection from a standard material, circumventing the need for engineered gain or loss. Unusual electronic behaviour induced the remaining structure by selectively removing parts of a lattice and focusing on specific energy levels, termed zero-mode-resonant projection. The resulting topology carries the system’s response at different frequencies, offering a tunable parameter for potential applications in circuits. This discovery challenges conventional approaches to creating these exotic states of matter and prompts investigation into whether this projection technique extends to more complex, real-world materials.

The research demonstrated that non-Hermitian topology can be induced through mathematical projection of a standard, square lattice material. This means unusual electronic behaviour emerges not from exotic material properties, but from selectively removing parts of the lattice and analysing the remaining structure. Topology is carried by the system’s response at different frequencies, offering a tunable parameter. The authors suggest further work is needed to determine if this projection technique can be applied to more complex materials.