Non-Hermitian physics describes systems that exchange energy with their surroundings—think of a set of coupled oscillators that leak energy, or photonic circuits with gain and loss baked in. In such worlds, the usual picture of bulk bands and smooth edge states can break down. Enter the geometry-dependent skin effect, a startling phenomenon where the very shape of a lattice decides where and how edge states pile up. A new paper from researchers at Tsinghua University and collaborators in Hong Kong Shenzhen charts a path through this geometric murkiness by building a two-dimensional (2D) theory that keeps the edge from becoming a mere afterthought.
The study, led by Chenyang Wang, Jinghui Pi, Qinxin Liu, Yaohua Li, and Yong-Chun Liu, anchors the ideas at the State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University. The Chinese University of Hong Kong Shenzhen Research Institute also contributed. The work offers what the authors call a geometry-dependent non-Bloch band theory for 2D lattices, centered on a construct they term the Strip Generalized Brillouin Zone (SGBZ). The upshot is both conceptual and practical: a universal framework to understand when and how geometry will reshape boundary behavior in non-Hermitian materials, and a concrete, testable condition for when geometry will matter in 2D.
Geometry as a control knob, not a curiosity
In ordinary Hermitian systems, a crystal’s boundary behavior tends to be predictable: bulk bands predict edge states, and the spectrum under open boundary conditions (OBC) converges to what you’d expect from the periodic-boundary case. In non-Hermitian lattices—systems with gain and loss—this harmony often evaporates. A famous twist is the non-Hermitian skin effect (NHSE), where a large fraction of eigenstates under OBC localize at the boundary, and the spectrum under OBC can look nothing like the spectrum under periodic boundary conditions (PBC). To tame this, physicists developed the non-Bloch band theory in 1D: you promote Bloch wavevectors into complex momenta, and the spectrum under OBC becomes a curve in the complex plane known as the Generalized Brillouin Zone (GBZ). The 1D story is beautiful and powerful because the GBZ turns a boundary pathology into a tractable, even topological, story.
But in 2D and higher dimensions, geometry steps in like a stubborn sculptor. The energy spectrum and the way modes localize under OBC can depend on the lattice’s shape, not just its internal couplings. That geometry-dependence—GDSE for geometry-dependent skin effect—has haunted attempts to generalize 1D non-Bloch theory. The new paper identifies, isolates, and then exploits this geometry dependence in 2D, showing that the problem is not just harder; it is fundamentally shaped by how you let the geometry grow toward infinity.
Takeaway: geometry isn’t a side note in non-Hermitian 2D systems; it is a central character. You can tune where the edge sits by choosing how you stretch or truncate the lattice in different directions, which means engineers can, in principle, sculpt boundary behavior with geometry alone.
SGBZ: A geometry-aware map of 2D non-Hermitian bands
The core technical advance is the Strip Generalized Brillouin Zone (SGBZ). To reach it, the authors invent and follow a two-step, quasi-1D limiting procedure. First, they pick a major axis along which the strip extends to infinity. They treat the lattice as an infinite chain in that direction but keep a finite width across the minor axis. In this setup, the complex momentum along the major axis, β1, meets a 1D GBZ constraint, yielding what they call the quasi-1D major-axis GBZ or QMGBZ. Next, they push the strip’s width to infinity, which projects the minor-axis physics into a parametric family of minor-axis GBZs—PMGBZs. The limit of the QMGBZs as width grows gives the SGBZ, the geometry-aware boundary map that governs the strip geometry in 2D.
What matters isn’t the presence of a single GBZ, but the competition between multiple, inequivalent SGBZs that can arise when you choose different strips to probe the same 2D lattice. If the SGBZs for two strips are incompatible, the boundary states and even the OBC spectrum can differ between the two geometries. That mismatch is the heart of GDSE: geometry can tilt the whole boundary physics in a 2D non-Hermitian lattice.
To formalize when this GDSE appears, the authors introduce a crucial sufficient condition: a phenomenon they call non-Bloch dynamical degeneracy splitting, or non-Bloch DDS. In short, if a continuous set of complex momenta (the driver of exponential boundary localization in the 2D context) collapses to a discrete set when geometry changes—if the degeneracy that underpins the OBC modes splits in the non-Bloch sense—GDSE is triggered. If you imagine a crowd of possible boundary modes represented as a continuous band of momenta, a geometry-induced splitting thins that crowd into discrete personalities; boundary behavior becomes geometry-dependent in a robust way. This DDS picture generalizes the 1D Bloch-case DDS and is the theory’s hinge for predicting when GDSE occurs in 2D.
Another big idea is the connection to the Amoeba formulation, a mathematical approach to geometry-agnostic generalized Brillouin zones in higher dimensions. The SGBZ analysis reveals that the Amoeba spectrum—the broad geometry-agnostic map of where energies lie—can be seen as a union of all possible SGBZ spectra. In other words, Amoeba provides a big-picture view, while SGBZ zooms in on geometry-specific slices. For the 2D Hatano-Nelson model used in the paper, the authors show that the Amoeba spectrum corresponds to the x- and y-strip SGBZ spectra, linking these two powerful ideas into a cohesive story.
Amoebas, holes, and the practical meaning of GDSE
One of the paper’s striking visual metaphors is the Amoeba: a geometric shape in a logarithmic space that encodes where solutions to the lattice’s characteristic polynomial live. In this language, a “hole” in the Amoeba—an excluded region in the log-space—corresponds to a central circle in the PMGBZ world. The authors show that holes (central holes) in Amoeba spectra imply the existence of central circles in PMGBZ space, and these, in turn, are intimately tied to whether the GDSE appears. If a reference energy lands inside an Amoeba hole, the corresponding SGBZ landscape may force a central circle, signaling a GDSE-prone setting. If the Amoeba has no hole for that energy, GDSE can still arise, but the relationship becomes subtler and depends on how PMGBZ points intersect with base manifolds in the SGBZ construction.
The 2D Hatano-Nelson model serves as the main worked example. By tweaking the complex couplings along different lattice directions, the authors show three distinct strip geometries—two along the standard x- and y-axes and a diagonal [11] direction—that generate different SGBZs. In simulations, spectra computed by plugging the SGBZ into the non-Bloch band theory agree with numerically exact OBC spectra for the parallelogram regions aligned with the respective strips. When the strips’ major axes conflict, the OBC spectrum shifts with the aspect ratio, a smoking gun for GDSE. The upshot is tangible: change how you lay out the sample, and the spectrum changes in a predictable, geometry-driven way.
The work also maps out how a universal condition can determine when GDSE is inevitable. If you can find a pair of SGBZs that cannot be simultaneously satisfied by a single geometry, you’re in the land of GDSE. Conversely, if all SGBZs line up, boundary behavior is robust against how you extend or truncate the lattice. This is more than a mathematical curiosity: it offers a practical rubric for designing devices whose boundary behavior you can dial up or down with geometry alone.
From theory to design: what this means for the future of non-Hermitian materials
Why should we care about a geometry-sensitive boundary state in a non-Hermitian lattice? Because it reframes how we think about building devices that exploit edge modes, lasing, sensing, or unidirectional transport. If the edge behavior can be steered by geometry, engineers gain a new, tangible control knob. In photonics, phononics, or cold-atom analogs, 2D lattices are common; GDSE implies you might tailor where light or sound concentrates simply by shaping the lattice’s footprint. The SGBZ framework is a map for navigating that design space—allowing researchers to predict, before fabrication, how an actual sample’s shape will sculpt its spectral and spatial edge properties.
Beyond devices, the work deepens a philosophical thread in non-Hermitian physics: the bulk-boundary relationship is more nuanced in higher dimensions and cannot be divorced from geometry. The proud promise of non-Bloch band theory—retrodicted from 1D into higher dimensions—gets a rigorous, geometry-aware extension in 2D. The SGBZ is not a mere mathematical gadget; it’s the language by which geometry speaks to spectra, localization, and the fate of boundary states.
As a practical roadmap, the authors show how to check whether GDSE will appear using a single geometry’s SGBZ, thanks to their major-axis transformation analysis. In other words, you don’t need to chase every possible strip to know whether the geometry will bite you. That kind of sufficiency result is exactly what engineers and experimentalists crave when venturing into new, non-Hermitian material platforms.
What to watch for next
The paper opens multiple avenues. One is experimental validation across platforms that realize 2D non-Hermitian lattices: photonic lattices with engineered loss and gain, acoustic metamaterials, or superconducting circuits with controlled non-Hermiticity. A second is extending the framework to Floquet (driven) systems, where the time-dependent modifications of gain and loss might create even richer GDSE-like phenomena. A third thread is the deeper mathematical tie to Amoeba theory and how central holes map to boundary phenomena across more complicated lattice geometries.
All of this points to a future where geometry isn’t just the backdrop for physics to play out; it’s an active design parameter. The dream is to encode robust, geometry-tuned edge behavior into devices that do not rely on delicate fine-tuning of couplings but instead exploit how you physically shape the lattice. The SGBZ would then be the designer’s notebook, translating a patient, geometric tweak into a predictable spectral outcome.
Author note: This article distills a research program from a paper by Chenyang Wang, Jinghui Pi, Qinxin Liu, Yaohua Li, and Yong-Chun Liu, affiliated with the State Key Laboratory of Low-Dimensional Quantum Physics at Tsinghua University and collaborators at The Chinese University of Hong Kong Shenzhen Research Institute. The lead authors and the senior author Yong-Chun Liu play central roles in articulating the geometry-dependent non-Bloch framework and its implications for 2D non-Hermitian physics.
Why it matters now: As experimental platforms push into higher dimensions with tunable non-Hermiticity, a geometry-aware theory is not just elegant — it is essential. The Strip GBZ approach provides the precise vocabulary to discuss, predict, and harness the edge in 2D non-Hermitian systems, turning a curious oddity into a toolkit for tomorrow’s devices.
Bottom line: The edge isn’t just where the physics ends; in these 2D non-Hermitian systems, the edge is where geometry begins. The Strip GBZ framework turns that idea into a concrete map, revealing when geometry will steer the spectrum and when it won’t, and linking the new world of GDSE to the broader mathematical landscape of Amoebas. It’s a bold step toward a future where engineers design with geometry as a primary ingredient rather than a stubborn constraint.
