🔬 mesoscale physics

ScatterWorks: A Python package for building and solving scattering network models

ScatterWorks is an open-source Python package that facilitates the flexible construction and efficient solution of scattering network models for studying non-interacting transport and wave-propagation phenomena in disordered and topological systems, featuring composable operations, sparse matrix support, and visualization utilities.

Original authors: R. Johanna Zijderveld, Hélène Spring, Anton R. Akhmerov

Published 2026-08-31

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: R. Johanna Zijderveld, Hélène Spring, Anton R. Akhmerov

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 a vast, tangled web of pathways where tiny waves of energy travel, bouncing off junctions and weaving through a landscape of disorder. This is the world of network models, a powerful way physicists describe how electricity or light moves through materials that are not perfectly uniform. Instead of tracking every single atom, scientists map these systems as a series of connections and junctions. At each junction, the waves split and recombine according to specific rules, much like water flowing through a complex system of pipes and valves. This approach has become essential for understanding strange states of matter, such as those found in twisted layers of carbon atoms or materials where electrons get trapped in place by disorder. However, building these digital maps has traditionally been a laborious task, often requiring researchers to write custom code for every new shape or pattern they wanted to study. This made it difficult to compare different systems or to explore the most complex geometries where the most interesting physics often hides.

To solve this problem, a team of researchers at Delft University of Technology has created a new open-source software tool called ScatterWorks. Think of this package as a universal construction kit for these wave networks. Before this tool existed, changing the shape of a network or the way its parts were connected often meant rewriting the core logic of a simulation from scratch. ScatterWorks changes that by allowing scientists to build complex networks using simple, reusable building blocks. A researcher can start with a small, basic unit, much like a single tile, and then use the software to copy, paste, and rearrange these tiles into vast, intricate structures. The software handles the heavy lifting of connecting the pieces, ensuring that the rules for how waves scatter at every junction remain consistent, whether the network is a simple loop or a massive grid with millions of connections.

The power of this tool lies in its ability to handle two very different types of problems with equal ease. First, it can work with symbolic mathematics, allowing researchers to keep variables like reflection and transmission rates as abstract symbols rather than fixed numbers. This lets them derive exact, analytical formulas for how a system behaves, essentially solving the puzzle on paper but with the help of a computer to manage the complexity. The authors demonstrated this by building a model of a Fabry–Perot interferometer, a device where light bounces between two mirrors. Using the software, they were able to reconstruct the known mathematical formula for how light transmits through this device, proving the tool works for theoretical derivations.

Second, the package is built to handle massive numerical simulations where exact formulas are impossible to find. Here, the software assigns specific numbers to the scattering rules and calculates how waves move through a disordered landscape. The team tested this by simulating the Chalker–Coddington model, a famous network used to study the transition between insulating and conducting states in the quantum Hall effect. In these simulations, the researchers introduced random disorder and watched how the conductance, or the ability of the material to carry current, changed as they tweaked the scattering rules. The software successfully reproduced the expected behavior: at a specific critical point, the conductance peaked at a value of approximately 0.5 times the fundamental quantum unit of conductance, a result that matches decades of theoretical predictions.

What makes ScatterWorks particularly effective is its use of a specialized mathematical technique that allows it to solve these problems without running out of memory, even when the networks become enormous. By treating the connections as sparse, meaning most of the possible links are empty, the software can focus its computing power only on the active pathways. This efficiency allows researchers to study systems that are far larger than previously possible, opening the door to more precise measurements of how disorder affects transport. The package also includes visual tools that let scientists see the network they have built, hovering over nodes to check labels and connections, which is crucial for debugging complex geometries.

The results presented in the paper show that this tool is not just a convenience but a significant step forward in making network models accessible and robust. The authors have verified their code against known analytical solutions and established numerical benchmarks, ensuring that the results are reliable. By providing a unified environment where one can switch between symbolic derivation and large-scale simulation, ScatterWorks removes the barrier of technical complexity that has kept many researchers from using these powerful models. As the software continues to evolve, it promises to make the study of wave transport in disordered and topological systems a more routine and collaborative part of modern physics, allowing scientists to focus on the physics rather than the mechanics of building their models.

Technical Summary: ScatterWorks

Problem Statement
Network models provide an efficient framework for studying non-interacting transport and wave-propagation phenomena in disordered and topological systems, such as the Chalker–Coddington model for the Quantum Hall transition, moiré systems, and Floquet problems. Despite their versatility, these models are less widely adopted than continuum or tight-binding approaches within parts of the condensed matter community. This underutilization stems from a lack of general-purpose tools; researchers often rely on private, problem-specific codes that are difficult to repurpose. Furthermore, standard transfer matrix methods can suffer from numerical instabilities due to exponentially growing and decaying modes. Changes in geometry, boundary conditions, or disorder protocols often require reworking core setup logic rather than simply adjusting parameters. There is a need for a unified, open-source environment that facilitates the flexible construction and robust solution of both symbolic and numerical network models, particularly for large systems where dense linear algebra methods become computationally limiting.

Methodology
The paper introduces ScatterWorks, an open-source Python package designed to build and solve scattering network models. The methodology is centered on a compact network representation and a modular workflow:

Network Representation: The package uses a Network object defined by a canonical list of directed links (canonical_links). Each link connects a source node to a sink node and includes optional shift vectors to handle periodic boundary conditions and tiling. This representation separates the topological definition of the network from the assignment of scattering matrices.
Composable Operations: The package provides a set of operations to manipulate network geometry without altering the underlying canonical link order, thereby preserving the basis for scattering matrices. Key operations include:
- tile(): Replicates a unit cell to create larger periodic or finite networks.
- union(): Combines disjoint networks.
- cut(): Selects specific subsets of links or nodes.
- relink(): Reconnects link endpoints by modifying a link_map array, allowing for complex geometry changes without redefining the canonical basis.
Scattering Equation Compilation: Local scattering matrices are assigned to nodes based on labels (e.g., node families, sublattices) or explicit indexing. These local matrices are assembled into a global scattering operator $S$ . The package supports both symbolic matrices (using SymPy) for analytical derivations and numerical matrices (dense or sparse) for simulations.
Solvers and Observables: To compute transport observables, the package distinguishes between bulk links and lead links (incoming and outgoing). It eliminates bulk links to derive an effective scattering matrix ( $S_{eff}$ $S_{e f f}$ ) between leads.
- For general cases, solve_scattering_equations computes the effective matrix.
- For large sparse systems, schur_solve utilizes the Schur complement method (via the MUMPS library) to efficiently handle the inversion of the bulk sub-block $(I - S_{bb})^{-1}$ . This approach avoids the numerical instabilities of transfer matrices and scales better for large networks.
- Transport observables, such as the dimensionless conductance $g$ , are calculated using the Landauer formula applied to the transmission block of $S_{eff}$ .
Visualization: The package includes plotting utilities that render network geometries with label-aware hover and static labels, aiding in the debugging of complex network constructions.

Key Contributions

Unified Framework: ScatterWorks offers a single interface for constructing both symbolic and numerical network models, bridging the gap between analytical derivations and large-scale numerical simulations.
Flexible Geometry Construction: The use of composable operations (tile, cut, relink, union) allows users to define complex unit cells, ribbons, and closed geometries without rewriting core logic for every new geometry.
Sparse Solvers: By implementing Schur-based solvers for sparse scattering matrices, the package enables the study of larger network sizes than is practical with dense methods, addressing a key bottleneck in network model simulations.
Symbolic Capabilities: The ability to assign symbolic scattering matrices allows for the recovery of analytical expressions directly from the network model code.
Label-Based Workflow: The use of pandas DataFrames for node and link labels facilitates the assignment of scattering matrices and the selection of leads, streamlining the setup for custom geometries.

Results
The paper demonstrates the package's capabilities through two primary examples:

Symbolic Fabry–Perot Interferometer: The package was used to assemble a small network with symbolic scattering matrices representing mirrors and phase nodes. The resulting effective scattering matrix was derived analytically, recovering the standard Fabry–Perot transmission and reflection structure, including the round-trip resonance denominator.
Numerical Chalker–Coddington Model: A numerical implementation of the Chalker–Coddington network was constructed to study the Quantum Hall phase transition.
- The package successfully reproduced the quasienergy band structure, identifying the critical point at $\theta = \pi/4$ where eigenphases cross.
- Conductance calculations for various system sizes ( $L$ ) near the critical point showed the expected peak value of approximately $0.5 e^2/h$ .
- Runtime analysis confirmed that the Schur-based solver exhibits the expected $O(L^3)$ scaling with system size, validating its efficiency for large networks compared to dense methods.

Significance
The paper positions ScatterWorks as a practical tool to lower the barrier to entry for network model research. Its significance lies in providing a flexible, open-source environment that maintains performance through sparse formulations. By decoupling network geometry construction from scattering matrix assignment and solution, the package allows researchers to explore complex geometries and disorder protocols without re-engineering their code. The inclusion of plotting utilities and label-aware rendering further supports the debugging and reproducibility of large, custom network setups. The authors note that the package is designed to be extensible, with future work aimed at improving the user interface and expanding functionality to other settings such as Floquet-driven physics and random walk studies.

Technical Summary: ScatterWorks

More like this