🔬 mesoscale physics

High second Chern number induced by long-range hopping in a four-dimensional Dirac model

This paper demonstrates that introducing long-range hopping into a four-dimensional Dirac model enables the realization of topological phases with high second Chern numbers and facilitates the transformation of trivial insulators into topological ones, thereby establishing long-range hopping as a powerful tool for engineering unconventional 4D topological states.

Original authors: Zheng-Rong Liu, Xiang Liu, Rui Chen, Bin Zhou

Published 2026-08-11

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CC BY 4.0

Original authors: Zheng-Rong Liu, Xiang Liu, Rui Chen, Bin Zhou

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 the universe as a giant, multi-layered video game. In the games we play every day, characters move through three dimensions: left-right, up-down, and forward-backward. But in the hidden world of quantum physics, scientists suspect there are secret levels with even more dimensions—four, five, or more—that we can't see but can mathematically explore. These extra dimensions aren't just for sci-fi; they hold the keys to understanding "topological phases of matter." Think of these phases like different types of dough. You can twist a piece of dough into a ball, a donut, or a pretzel. No matter how much you squish or stretch it, a ball stays a ball (it has no holes), and a donut stays a donut (it has one hole). You can't turn a ball into a donut without tearing it. In physics, these "shapes" of electron waves are called topological invariants. They are incredibly stable, meaning if you build a material based on a "donut" shape, it will conduct electricity perfectly on its edges without getting messy or losing energy, even if the material is a bit dirty or imperfect.

For a long time, scientists have been studying these shapes in our familiar 3D world. But recently, they've started looking at "4D" systems. In this four-dimensional world, the "shape" of the material is measured by something called the "second Chern number." You can think of this number as a scorecard for how complex and twisted the electron waves are. A higher score means a more complex, robust, and potentially useful state of matter. Until now, the standard model used to study these 4D worlds was like a basic, pre-made puzzle kit. It was simple and reliable, but it only allowed for a few specific scores: 0, 1, 3, or -1. It was as if the puzzle kit only had pieces that could make a ball or a simple donut, but nothing more intricate. Scientists wondered: Is that all there is? Can we twist the 4D dough into something far more complex, or are we stuck with just those few shapes?

This is where the new research comes in. The authors, a team from Hubei University, decided to see if they could break the limits of that basic puzzle kit. They asked: What happens if we let the electrons in this 4D world "hop" further than their immediate neighbors? In the standard model, electrons only jump to the spot right next to them. But in this new study, the researchers introduced "long-range hopping," allowing electrons to skip over a few spots to land further away. It's like changing a game of hopscotch where you can only jump to the next square, to a version where you can leap three or four squares ahead.

The results were surprising and exciting. By adding these long-distance jumps, the team found they could transform a boring, empty 4D space (a "trivial insulator" with a score of 0) into a highly complex topological state. In their computer simulations, they discovered they could reach a second Chern number of -6. That's a huge jump from the previous limit of 3! They didn't stop there. When they started with a 4D material that already had a topological shape, the long-range hopping allowed them to twist it even further, creating new phases with scores of -6 and even -7. These are shapes that were impossible to make with the old, simple rules.

The team also checked if these new shapes were real by looking at the "edges" of their 4D world. In physics, there's a rule called "bulk-boundary correspondence," which basically says that if the inside of the material is twisted in a certain way, the surface must show a specific number of "gapless" paths where electricity can flow freely. The simulations showed exactly this: when the score was -6, six distinct paths appeared on the surface; when the score was -7, seven paths appeared. This confirmed that the complex shapes they created were mathematically sound and stable.

While this work is currently a theoretical exploration using mathematical models and simulations, it opens a door to a much richer world of 4D physics. The researchers suggest that because these high-score states are so complex, they might lead to stronger and more unique electrical responses than we've seen before. They also point out that while we can't build a real 4D room, we can build "artificial" 4D worlds using things like circuits, sound waves, or light. Since we can already build circuits that mimic long-range hopping, this paper suggests that we might soon be able to build these exotic, high-scoring 4D materials in a lab, turning a mathematical curiosity into a physical reality.

Technical Summary: High Second Chern Number Induced by Long-Range Hopping in a Four-Dimensional Dirac Model

Problem Statement
Four-dimensional (4D) topological systems, particularly the 4D Dirac model, serve as a paradigmatic platform for exploring topological phenomena characterized by the second Chern number ( $C_2$ ). However, in the standard minimal 4D Dirac model, the topological nontrivial phases are restricted to a limited set of second Chern numbers, specifically $C_2 = \pm 1$ and $\pm 3$ . While recent studies have shown that external mechanisms like magnetic fields or periodic driving can induce even second Chern numbers, the potential of intrinsic lattice modifications—specifically long-range hopping—to expand the topological phase space of higher-dimensional systems remains unexplored. The central question addressed is whether introducing long-range hopping can overcome the limitations of the minimal model to generate unexplored topological phases with high second Chern numbers.

Methodology
The authors investigate a 4D Dirac model modified by the inclusion of next-nearest-neighbor (NNN) and next-next-nearest-neighbor (NNNN) hopping terms. The Hamiltonian $H(k)$ is constructed using Dirac matrices $\Gamma_j$ and a momentum-dependent mass term $M(k)$ that incorporates:

Nearest-neighbor hopping ( $t_0$ ).
NNN hopping ( $t_1$ ), introducing terms proportional to products of two cosines (e.g., $\cos k_x \cos k_y$ ).
NNNN hopping ( $t_2$ ), introducing terms proportional to products of three cosines.

The energy spectrum is derived by diagonalizing this Hamiltonian. To characterize the topological properties, the authors calculate the second Chern number ( $C_2$ ) over the 4D Brillouin zone using the non-Abelian Berry curvature. They employ an adaptive mesh refinement method to ensure accurate evaluation of $C_2$ , particularly near topological phase transition points where the bulk gap closes. The study analyzes the system under open boundary conditions along one spatial direction to verify the bulk-boundary correspondence by observing the emergence of gapless boundary states.

Key Contributions and Results
The paper demonstrates that long-range hopping acts as a powerful mechanism for engineering 4D topological states with high second Chern numbers, significantly expanding the phase space beyond the minimal model.

Inducing Topological Phases in Trivial Insulators: Starting from a 4D trivial insulator (where the Dirac mass $m=5$ yields $C_2=0$ ), the introduction of NNN hopping ( $t_1$ ) drives a topological phase transition. As $t_1$ increases, the bulk gap closes and reopens, transforming the system into a topological insulator with a high second Chern number of $C_2 = -6$ .
Rich Phase Diagram with Combined Hopping: When both NNN ( $t_1$ ) and NNNN ( $t_2$ ) hoppings are present, the system exhibits a complex phase diagram. The authors identify topological phases with various second Chern numbers, including $C_2 = -1, -2, 4,$ and $-6$. Notably, even with only NNNN hopping ( $t_1=0$ ), the system can transition from a trivial insulator to a phase with $C_2 = -1$ .
Tuning Existing Topological Insulators: For initial states that are already topological (e.g., $m=1$ with $C_2=3$ or $m=3$ with $C_2=-1$ ), long-range hopping induces transitions into new phases with even higher magnitudes of $C_2$ . Specifically, the study reveals phases with $C_2 = -6$ and $C_2 = -7$ , values that are absent in the 4D minimal Dirac model.
Verification of Bulk-Boundary Correspondence: The theoretical predictions are verified by calculating energy spectra under open boundary conditions. The results show that the number of gapless 3D boundary modes emerging in the bulk gap exactly matches the magnitude of the second Chern number ( $|C_2|$ ). For instance, the $C_2 = -7$ phase exhibits seven gapless boundary modes.

Significance and Claims
The paper claims that long-range hopping provides a versatile and intrinsic route to generate and control 4D topological phases with high second Chern numbers, overcoming the constraints of minimal models. The authors assert that these high $C_2$ phases are significant because the second Chern number directly determines the quantized nonlinear electromagnetic response coefficient in four dimensions; thus, these new phases are expected to exhibit enhanced nonlinear transport responses compared to those in the minimal model.

Regarding experimental realization, the authors note that while genuine 4D space is inaccessible in conventional materials, synthetic dimensions in artificial quantum systems (such as ultracold atoms, photonic lattices, acoustic systems, and electric circuits) have enabled the investigation of 4D topological phases. They specifically suggest that long-range hopping can be realized in artificial metamaterials like electric circuits, acoustic crystals, and superconducting quantum circuits, making the proposed 4D model with long-range hopping a feasible target for experimental implementation in these platforms.

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