⚛️ quantum physics

Transparent Domain Walls through Information Convex Sets

This paper develops an entanglement-bootstrap framework using information convex sets to characterize transparent domain walls in $(2+1)$ -dimensional topologically ordered states directly from ground state wavefunctions, revealing how these walls modify ground state degeneracy, transmute anyons, and generate non-factorizable symmetries without requiring categorical defect data as input.

Original authors: Jintae Kim, Amanda Gatto Lamas, Jacopo Gliozzi, Bowen Shi, Taylor L. Hughes, Jong Yeon Lee

Published 2026-09-24

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CC BY 4.0

Original authors: Jintae Kim, Amanda Gatto Lamas, Jacopo Gliozzi, Bowen Shi, Taylor L. Hughes, Jong Yeon Lee

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

In the strange, silent world of quantum materials, there exists a state of matter where the rules of everyday physics seem to dissolve. Imagine a sheet of material where the electrons do not behave like individual particles, but rather as a single, unified entity that stretches across the entire object. This is a topological phase, a state defined not by the arrangement of atoms, but by the global shape of the quantum information woven through the system. In these materials, particles known as anyons can emerge. Unlike the electrons or protons we know, anyons carry a unique kind of memory; if you move one around another, the system remembers the path, changing its state in a way that depends only on the topology of the journey, not the speed or the details of the route. These materials are famous for their robustness, hosting a ground state degeneracy, meaning they can exist in several distinct, stable configurations simultaneously, a property that makes them prime candidates for building fault-tolerant quantum computers.

However, these materials can also host invisible boundaries called transparent domain walls. Unlike a normal wall that separates two different rooms, a transparent wall is a topological feature that can be moved anywhere without changing the local physics. It is invisible to any probe that looks at a small, local patch of the material. Yet, despite being locally undetectable, these walls have a profound global effect: they can swap the identities of anyons as they pass through, effectively shuffling the quantum information of the entire system. For years, physicists have understood these walls through abstract mathematical theories that assumed the existence of the wall as a starting point. The challenge remained: could we see these invisible walls and understand their effects directly from the quantum wavefunction of the material itself, without assuming we already knew what the wall looked like?

A team of researchers has now answered this question by developing a new method to extract the hidden structure of these materials directly from their ground state. They focused on a specific type of quantum material defined on a torus, a shape like a doughnut, which allows for loops that cannot be shrunk to a point. The researchers used a mathematical tool called an information convex set, which acts like a lens to focus on the possible quantum states that can exist within a specific region of the material. By analyzing these sets on both small, local rings and large, non-contractible rings that wrap around the doughnut shape, they were able to map out the invisible walls. Their approach does not require any prior knowledge of the wall's structure or a microscopic model of the material; it derives the wall's properties purely from the entanglement patterns inherent in the quantum state.

The team discovered that these transparent walls leave a distinct fingerprint in the way the material's quantum information is distributed. When they examined the large rings that cross a transparent wall, they found that the number of distinct quantum states available in that region changes in a predictable way. Specifically, the wall acts as a filter, allowing only certain types of anyons to pass through unchanged while transforming others. This filtering effect reduces the number of independent quantum states that can exist on the ring, a reduction that the researchers could measure directly. They found that the amount of "topological entanglement entropy," a measure of how deeply the quantum information is woven together, changes depending on whether the ring crosses a wall or runs parallel to it. If the ring crosses a wall, the entanglement entropy reveals the wall's presence by showing a specific reduction in the system's complexity. If the ring runs parallel to the wall, the entropy remains unchanged, but a different measure, which they call entanglement asymmetry, reveals the wall's influence on the global structure of the system.

Perhaps the most striking finding is how these invisible walls reshape the symmetries of the material. In a standard topological phase without walls, the symmetries that connect different ground states can be thought of as independent operations running along the two main directions of the doughnut shape. The researchers showed that when transparent walls are present, this independence breaks down. The symmetries become entangled, requiring operations that run along both directions simultaneously to be understood. They demonstrated that certain transformations of the quantum state cannot be broken down into separate actions on the two loops; instead, they must be performed as a single, unified process that winds around the entire system. This means that the invisible walls fundamentally alter the way the material's quantum states are connected, creating a new kind of symmetry that is supported jointly on the two fundamental cycles of the torus.

To prove their framework works, the researchers applied their method to three specific lattice models of quantum materials, including a variation of the toric code and a model known as Wen's plaquette model. In these simulations, they successfully reconstructed the properties of the transparent walls directly from the ground state wavefunctions. They confirmed that the walls could change the ground state degeneracy, reducing the number of stable configurations the system can hold. They also verified that the walls could permute the anyons, swapping their identities as they moved around the loops. In one specific case involving a model with odd dimensions, they showed that the presence of the walls prevented the symmetry operators from being separated into independent loops, exactly as their theory predicted. This provided a concrete, numerical demonstration that their method could identify and characterize these elusive features without any external input.

The work offers a powerful new way to diagnose topological phases, moving beyond the need for abstract mathematical descriptions or specific microscopic models. By treating the quantum wavefunction as a source of truth, the researchers have shown that the global structure of a material, including its invisible boundaries, can be read directly from the local patterns of entanglement. This approach not only clarifies how transparent domain walls modify the ground state degeneracy and anyon transport but also reveals a deeper layer of symmetry in topological matter that was previously hidden. The ability to detect these walls and understand their impact on quantum symmetries brings us closer to a complete picture of how topological order behaves in complex, real-world scenarios, potentially guiding the design of more robust quantum technologies.

Technical Summary: Transparent Domain Walls through Information Convex Sets

Problem Statement
In $(2+1)$ -dimensional topologically ordered many-body states, gapped domain walls typically act as interfaces where anyons may tunnel, condense, or become confined. A special class of these, known as transparent domain walls, presents a unique diagnostic challenge. Unlike standard defects, transparent domain walls are intrinsically topological and "invisible" to local topological probes; their position can be topologically deformed away from any local region. Consequently, standard local diagnostics (such as those based on disk intersections) fail to detect them. However, these walls are not physically trivial: they can permute anyon types when transported across noncontractible loops, modify the ground state degeneracy (GSD), and alter topological entanglement entropy (TEE).

The central question addressed by this work is: What information associated with transparent domain walls can be derived directly from ground state wavefunctions at fixed points of Abelian topological phases, without assuming a microscopic realization or an a priori categorical description of the defect structure?

Methodology: Entanglement Bootstrap and Information Convex Sets
The authors employ the entanglement bootstrap framework, which reconstructs emergent topological structures from local entropic conditions and the consistency of reduced density matrices. The core mathematical tool is the Information Convex Set (ICS), denoted $\Sigma(X)$ , defined as the set of density matrices on a subregion $X$ that are locally indistinguishable from a reference state $\sigma$ (satisfying axioms A0 and A1) and satisfy specific information-theoretic constraints.

The methodology distinguishes between two types of regions:

Local Annuli: Contained within a disk. The extreme points of the ICS on a local annulus correspond to anyon types.
Noncontractible Annuli: Annuli whose boundaries are noncontractible loops on the torus. The authors argue that while local annuli cannot distinguish different transparent domain walls, noncontractible annuli can.

The framework analyzes the interplay between the ICS on local annuli and noncontractible annuli. Specifically, it utilizes:

Transport Automorphisms: Moving a local annulus along a noncontractible loop induces an automorphism $\phi$ on the set of anyon types. A transparent domain wall is defined as a topological feature that induces a nontrivial automorphism when an annulus crosses it.
Fusion Rules on Noncontractible Annuli: The authors derive fusion rules governing the action of anyons on the extreme points of the ICS on noncontractible annuli.
Minimum Entropy States (MESs): These are extreme points of the ICS on the full torus whose reductions to a noncontractible annulus are extreme points of the ICS on that annulus. The authors establish a one-to-one correspondence between MESs and transport-invariant extreme points of the noncontractible annulus ICS.

Key Contributions and Results

Characterization of Transparent Domain Walls via ICS:
- The number of extreme points of the ICS on a noncontractible annulus equals the number of anyon types invariant under transport along the annulus's direction. A reduction in this number (compared to the total anyon types) signals the presence of a transparent domain wall crossing the annulus.
- Quantum Dimensions: For Abelian phases, the quantum dimensions $d_\alpha$ of extreme points on a noncontractible annulus can exceed unity, even if all anyons have unit quantum dimension. These dimensions encode the effect of crossing transparent domain walls.
Topological Entanglement Entropy (TEE):
- The maximal TEE on a noncontractible annulus is derived as $\gamma_{LW}^{\max} = \log(D/d_\alpha)$ , where $D$ is the total quantum dimension and $d_\alpha$ is the quantum dimension of an extreme point.
- Unlike the standard TEE, this maximal value is sensitive to the specific transparent domain walls crossing the chosen annulus. Consequently, $\gamma_{LW, x}^{\max}$ and $\gamma_{LW, y}^{\max}$ can differ depending on the orientation of the annulus relative to the walls.
Entanglement Asymmetry and Ground State Degeneracy:
- The authors define entanglement asymmetry $\Delta S_X$ as the increase in entropy under averaging over the effective symmetry action on the noncontractible annulus.
- The maximal entanglement asymmetry is found to be $\Delta S_X^{\max} = \log(\text{GSD})$ .
- Crucially, this value is identical for both annulus orientations ( $x$ and $y$ ). It reflects the combined effect of transparent domain walls along both fundamental cycles on the global ground state space, whereas TEE probes the net effect of walls crossing a specific annulus.
Symmetry Structure and Obstructions:
- In the presence of transparent domain walls, the authors identify symmetries supported jointly on two fundamental cycles that cannot be decomposed into a product of two independent 1-form symmetry operators (one per cycle).
- These "non-factorizable" symmetries arise from anyon transmutation induced by the walls. The action of these symmetries on MESs is clarified through anyon tunneling and fusion rules, providing a constructive description of how symmetries connect distinct ground states.
Applications to Lattice Models:
The framework is applied to three specific models:
- Wen's Plaquette Model: Demonstrates cases where transparent domain walls arise due to odd lattice dimensions, leading to reduced GSD and non-factorizable symmetries (e.g., a symmetry requiring transport along both $x$ and $y$ cycles).
- Anisotropic Dipolar Toric Code: Shows how the GSD and TEE depend on the greatest common divisor of the lattice size and the anyon charge $N$ .
- Rank-2 Toric Code: Applied to demonstrate the framework's utility in higher-rank gauge theories.

Significance
The paper claims to provide a direct derivation of topological defect structures (specifically transparent domain walls) from ground state wavefunctions, bypassing the need for input from categorical defect data or microscopic models. By utilizing the entanglement bootstrap, the work establishes that:

Transparent domain walls are detectable via the structure of ICSs on noncontractible annuli.
The distinction between maximal TEE (sensitive to wall crossings) and maximal entanglement asymmetry (sensitive to the global wall configuration and GSD) offers a refined diagnostic toolkit for topological phases.
The existence of symmetries that cannot be decomposed into independent 1-form symmetries challenges the standard spontaneous 1-form symmetry breaking picture in the presence of transparent domain walls, necessitating a more complex understanding of the ground state degeneracy.

The work concludes that these entanglement-based diagnostics are robust and applicable to a broad class of topological lattice models, including those exhibiting UV/IR mixing.

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