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Weak topological phases in the presence of interactions

This paper mathematically computes the groups of weak symmetry-protected topological phases in dimensions zero through three for all tenfold-way symmetry types by comparing homotopical free and interacting classifications using Atiyah's Real KR-theory and Freed-Hopkins' invertible field theory, thereby predicting their stability under short-range interactions and identifying potential intrinsically-interacting phases.

Original authors: Omar Antolín Camarena, Arun Debray, Cameron Krulewski, Natalia Pacheco-Tallaj, Daniel Sheinbaum, Luuk Stehouwer

Published 2026-07-16

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CC BY 4.0

Original authors: Omar Antolín Camarena, Arun Debray, Cameron Krulewski, Natalia Pacheco-Tallaj, Daniel Sheinbaum, Luuk Stehouwer

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 is built from tiny, invisible Lego bricks called atoms. When you stack these bricks into a crystal, they form a rigid, repeating pattern. For decades, scientists have known that the way these bricks are arranged dictates how electricity flows through them. But in the last few years, a new kind of "magic" has been discovered in these crystals. It turns out that some materials aren't just defined by their ingredients, but by the secret, invisible knots in the way their electrons dance. These are called topological phases. Think of a coffee mug and a donut: to a topologist, they are the same because you can stretch one into the other without tearing. Similarly, some materials have a "knot" in their electronic structure that can't be untied, no matter how you squish or stretch the material, as long as you don't break the rules of symmetry.

Now, here is the tricky part. In the real world, electrons don't just dance alone; they bump into each other. They interact. For a long time, physicists had a perfect map for describing these materials when the electrons ignored each other (the "free" case). But once you let them interact, the map gets messy, and the old rules often break down. The big question for scientists has been: Do these magical, knot-protected states survive when the electrons start fighting with each other? Or do the interactions rip the knots apart? This paper dives deep into that question, specifically looking at a special type of knot called a "weak" phase, which relies on the crystal's repeating pattern to exist.

The authors of this paper, a team of mathematicians and physicists, have built a sophisticated mathematical bridge to connect the "free" world (where electrons ignore each other) with the "interacting" world (where they don't). They didn't just guess; they used advanced tools from a field called homotopy theory and a concept called T-duality, which is like a magical mirror that swaps the crystal's physical layout with its momentum map. By comparing the two sides of this mirror, they could predict exactly which of these "weak" topological phases are sturdy enough to survive the chaos of electron interactions, and which ones will crumble.

Here is what they found. They looked at ten different "families" of materials (known as the tenfold way), each with different rules for how time and charge behave. For each family, they checked the materials in one, two, and three dimensions. Their results are a mix of good news and surprising new discoveries.

First, they confirmed that some phases are incredibly tough. For example, in a specific family of materials called Class BDI, they found that three specific "counting" numbers (which tell you how many layers of a certain type exist) remain perfectly stable even when interactions are turned on. This matches what other scientists had guessed based on physical experiments. Similarly, in Class A, a famous property called the Hall conductivity (which measures how electricity flows sideways) was proven to be stable under interactions. This gives a solid mathematical backing to previous physical theories.

However, the paper also reveals that not everything survives. In several cases, the "free" map predicted a phase that simply vanishes once interactions are introduced. The authors calculated exactly how many of these "free" phases are unstable. For instance, in Class AII (which includes the famous topological insulators), they showed that while the main "strong" knot is safe, the "weak" knots formed by stacking layers are also safe in dimensions up to three. This means if you build a 3D topological insulator by stacking 2D layers, those layers will keep their special properties even if the electrons start interacting.

But the most exciting part is what they found that no one knew before. The authors discovered entirely new types of phases that only exist because of interactions. These are called "interaction-enabled" phases. They are like a new kind of knot that cannot be tied if the electrons are lonely; it only forms when they start bumping into each other. Specifically, in Class CI in four dimensions (three space, one time), they predicted a group of four distinct new phases that have never been seen in the literature. They also found similar hidden gems in other classes, like Class AIII and Class CII.

The paper is very careful to state that these are mathematical predictions based on a specific set of assumptions (the "low-energy effective field theory" ansatz). They haven't built these materials in a lab yet, but they have provided a rigorous mathematical proof that these phases should exist and be stable. They also ruled out the idea that these new phases could be explained by the old "free" models; these new phases are genuinely new, born from the complexity of interactions.

In short, this paper acts as a guidebook for the future of quantum materials. It tells experimentalists which materials to trust will keep their magic when the temperature drops and interactions kick in, and it points them toward a hidden landscape of new, interaction-born phases that are waiting to be discovered. It confirms that some of our best ideas about topological matter are rock-solid, while simultaneously opening the door to a whole new world of quantum states that we didn't even know were possible.

Problem Statement
Symmetry-protected topological (SPT) phases of matter are traditionally classified using homotopical techniques. For non-interacting (free) fermion systems, this classification is well-established via K-theory groups, famously codified in the "tenfold way" periodic table. However, the introduction of short-range interactions generally alters these classifications, rendering the free-fermion K-theory framework insufficient. While homotopical methods for interacting SPTs have been developed using the framework of topological quantum field theory (TQFT) and bordism groups (specifically the Freed-Hopkins program), these methods have largely focused on "strong" phases—those protected solely by internal symmetries without spatial symmetries.

The specific problem addressed in this paper is the classification of weak SPT phases in the presence of interactions. Weak phases are protected by lattice translation symmetries and can often be constructed by stacking lower-dimensional strong phases. The authors aim to mathematically predict the stability of these weak phases under short-range interactions and to identify "interaction-enabled" phases (intrinsically interacting phases that have no free-fermion analog) for all ten Altland-Zirnbauer symmetry classes in spatial dimensions 0 through 3.

Methodology
The paper employs a homotopical approach, comparing the classification of free phases (via K-theory) with interacting phases (via Anderson-dualized bordism groups of invertible field theories). The core methodology involves extending the Freed-Hopkins "free-to-interacting map" to the context of discrete translation-invariant systems.

Mathematical Frameworks:
- Free Phases: Modeled using K-theory of the Brillouin zone torus ( $T^d$ ). For real symmetry classes, this involves KR-theory or KO-theory; for complex classes, complex K-theory.
- Interacting Phases: Modeled using the group of deformation classes of reflection-positive invertible field theories (IFTs). Following Freed and Hopkins, these are classified by the Anderson dual of the bordism group $\Omega^{H}_{d+2}(Y)$ , where $H$ is the spacetime symmetry group associated with the tenfold way class and $Y$ is the spatial manifold.
- Weak Phases Ansatz: The authors propose that weak interacting phases are classified by the generalized cohomology group $\mathfrak{G}^{d+2}_{H(s)}(T^d)$ , where $T^d$ is the spatial (unit cell) torus. This builds on the "crystalline equivalence principle" and Freed-Hopkins' ansatz for families of field theories.
Key Technical Tools:
- T-Duality: The authors utilize T-duality isomorphisms to relate the K-theory of the Brillouin torus (momentum space) to the K-theory of the spatial torus. This allows them to recast free weak phases, traditionally defined over the Brillouin zone, into a framework compatible with the spatial torus used for interacting phases.
- Binomial Formula for Torus Cohomology: They employ a splitting formula (James splitting) for the generalized cohomology of a torus. This decomposes the cohomology of $T^d$ into a direct sum of cohomologies of spheres (cells), effectively expressing weak phases as a combination of strong phases from lower dimensions.
- Free-to-Interacting Map: The central object of study is the homomorphism $F2I: \text{Free} \to \text{Interacting}$ . The kernel of this map identifies free phases that become trivial when interactions are turned on (unstable phases), while the cokernel identifies interaction-enabled phases.
Computational Strategy:
By combining T-duality with the binomial splitting, the authors derive that the weak free-to-interacting map in dimension $d$ is a direct sum of strong free-to-interacting maps in dimensions $0$ through $d$ . Specifically, $F2I^d_{\text{weak}} \cong \bigoplus_{k=0}^d \binom{d}{k} F2I^{d-k}_{\text{strong}}$ . This reduces the problem of computing weak classifications to computing the known strong classifications in lower dimensions.

Key Contributions and Results
The paper provides a complete mathematical computation of the weak free-to-interacting maps for all ten symmetry classes in spatial dimensions 1, 2, and 3. The results are summarized in a series of corollaries (3.4 through 3.33) detailing the kernel (unstable free phases) and cokernel (interaction-enabled phases) for each case.

Stability of Weak Phases: The authors confirm that many weak phases remain stable under interactions. For instance, in Class AII (time-reversal invariant topological insulators), the weak phases in dimensions up to 3 are robust to interactions, consistent with physical expectations.
Interaction-Enabled Phases: The study identifies specific instances of intrinsically interacting weak phases.
- In Class CI (3+1 dimensions), the authors predict a $\mathbb{Z}_2^4$ group of interaction-enabled weak phases. These phases do not appear in the free classification and are noted as a new prediction not previously found in the literature.
- In Class A (3+1 dimensions), the Hall conductivity (a weak invariant) is shown to be stable, providing a mathematical derivation of previous physical results.
- In Class BDI (3+1 dimensions), the three $\mathbb{Z}$ -valued indices of free weak phases are found to remain nontrivial under interactions, agreeing with recent physical studies.
Unstable Phases: The computations identify free phases that become trivial upon adding interactions. For example, in Class BDI, the $\mathbb{Z}$ -valued strong phases in 1D become $\mathbb{Z}_8$ in the interacting limit; this reduction propagates to higher dimensions, creating a kernel in the weak classification.

Significance and Claims
The paper claims to provide a rigorous mathematical framework for understanding the fate of weak topological phases under interactions. Its significance lies in:

Unification: It unifies the classification of free and interacting phases for weak SPTs by extending the Freed-Hopkins formalism to include spatial translation symmetry via T-duality and torus cohomology.
Predictive Power: It offers precise mathematical predictions regarding the stability of weak phases and the existence of interaction-enabled phases, which can guide future theoretical and experimental investigations.
Validation: The results corroborate existing physical literature (e.g., for Class AII and BDI) while offering new predictions (e.g., for Class CI) that have not yet been explored in the physics literature.
Methodological Advancement: It demonstrates that the complex problem of interacting weak phases can be reduced to the computation of lower-dimensional strong phases, leveraging the binomial structure of torus cohomology.

The authors explicitly state that their work is an introduction to free-to-interacting maps for weak phases and a full computation for the tenfold way, noting that ongoing work will address more general crystalline symmetries. They do not propose specific experimental protocols but rather provide the theoretical classification necessary to interpret potential experimental observations of weak phases in interacting systems.

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