Atiyah-Hirzebruch spectral sequence for topological insulators and superconductors: $E_2$ pages for 1651 magnetic space groups

This paper computes the $E_2$ pages of the momentum-space and real-space Atiyah-Hirzebruch spectral sequences for topological crystalline insulators and superconductors across 1651 magnetic space groups in up to three dimensions, enabling the determination of $K$-groups for approximately 59% of these symmetry settings under a physically reasonable assumption.

The Gist

The Big Picture: Mapping the "Shape" of Matter

Imagine you are an architect trying to design a building that has a very specific, unchangeable "soul." In the world of quantum physics, this "soul" is called a topological phase. Materials like topological insulators and superconductors are special because their electrons are arranged in a way that makes them robust; you can't easily change their state without breaking the material entirely.

The authors of this paper, Ken Shiozaki and Seishiro Ono, are like master cartographers. Their goal was to draw a complete map of all possible "souls" (topological phases) that electrons can have when they are living inside a crystal with magnetic properties.

The Challenge: Too Many Possibilities

There are 1,651 different types of magnetic crystals (called Magnetic Space Groups). Each type has a unique set of rules for how the atoms are arranged and how they interact with magnetism.

For each of these 1,651 crystal types, the electrons can form different "shapes" or phases. The scientists wanted to list every single possible shape for every single crystal type. This is a massive puzzle because the math involved is incredibly complex, like trying to solve a billion-piece jigsaw puzzle where the pieces keep changing shape.

The Tool: The "AHSS" (A Mathematical Ladder)

To solve this, the authors used a powerful mathematical tool called the Atiyah-Hirzebruch Spectral Sequence (AHSS).

Think of the AHSS as a multi-story construction ladder:

• The Ground Floor (E1 Page): This is where you start. You look at the smallest building blocks of the crystal (the atoms and their immediate neighbors) and ask, "What shapes can form right here?"
• The Second Floor (E2 Page): This is the main focus of the paper. You take the answers from the ground floor and see how they fit together as you move up to larger sections of the crystal. This step gives you a very good approximation of the final shape.
• The Top Floors (E3, E4, etc.): These are the final, perfect details. However, calculating these floors is extremely difficult and often impossible to do systematically for every single crystal type.

The authors realized that while they couldn't always reach the very top floor (the perfect answer), they could calculate the second floor (the E2 page) very efficiently for all 1,651 crystal types.

The Strategy: Two Different Maps

Here is the clever trick the authors used to get the most accurate results possible without doing the impossible math:

1. The Momentum Map: They looked at the electrons from the perspective of their movement (momentum space). This is like looking at a city from a helicopter to see the flow of traffic.
2. The Real-Space Map: They looked at the electrons from the perspective of their physical location (real space). This is like walking through the city street by street to see the buildings.

In physics, these two maps must describe the same reality. They are two sides of the same coin.

The authors calculated the "second floor" (E2 page) for both maps for all 1,651 crystal types. Then, they compared the two maps.

• If the helicopter view and the street view gave different answers, they knew the answer wasn't final yet.
• If the two views agreed, they knew they had found the true "soul" of the material.

The Results: Solving 59% of the Puzzle

By cross-referencing these two maps, the authors were able to definitively determine the topological "soul" for about 59% of the magnetic crystal types they studied.

For the remaining 41%, the two maps didn't give a single, unique answer. This means there are still a few possibilities left for those specific crystals, and the "higher floors" of the mathematical ladder (E3 and E4) would be needed to solve them. However, the authors provided a list of all possible candidates for those cases, narrowing the search significantly.

Summary in a Nutshell

• The Goal: To catalog every possible stable state of electrons in 1,651 different magnetic crystals.
• The Method: They used a mathematical "ladder" (AHSS) to build up the answer step-by-step. They focused on the second step (E2 page) because it's calculable for everything.
• The Hack: They calculated this step from two different angles (movement vs. location) and compared them. Where the angles matched, they found the exact answer.
• The Outcome: They successfully identified the exact topological classification for 59% of the cases and provided a shortlist of possibilities for the rest.

The paper essentially provides a massive, pre-computed database (available online) that other scientists can use to instantly know the topological properties of these materials without having to do the heavy math themselves.

Technical Summary

Problem Statement
The classification of topological crystalline insulators (TCIs) and superconductors (TCSs) protected by crystalline symmetries is a central problem in condensed matter physics. While the classification of topological phases in free fermion systems is theoretically described by K-theory, explicitly computing the K-groups for all possible symmetry settings in three dimensions is computationally challenging. Specifically, the classification involves determining the K-groups for 1651 magnetic space groups (MSGs), 528 magnetic layer groups, and 393 magnetic rod groups. The primary difficulty lies in the fact that while the $E_2$-page of the Atiyah-Hirzebruch spectral sequence (AHSS) provides a first approximation of the K-group, the full classification requires computing higher-order differentials ($d_r$ for $r \geq 2$) and solving extension problems, which are generally difficult to determine systematically for arbitrary symmetry classes.

Methodology
The authors employ the Atiyah-Hirzebruch spectral sequence (AHSS) as a systematic computational framework to approximate the K-groups for topological phases. The methodology is divided into two dual approaches:

1. Momentum-Space AHSS: Based on the topology of band structures in the Brillouin zone (BZ), described by K-cohomology.
2. Real-Space AHSS: Based on the "crystalline topological liquid" picture, where topological phases are viewed as patches of lower-dimensional topological phases localized on cells, described by K-homology.

The authors implement a specific cell decomposition of space (both momentum and real space) with a crucial constraint: if a group action fixes a cell setwise, it must fix the cell pointwise. This condition simplifies the calculation of the $E_1$-page. The computation proceeds as follows:

• $E_1$-page Calculation: The authors compute the $E_1$-page by decomposing the space into $p$-cells (0-cells, 1-cells, 2-cells, and 3-cells). For each cell, they determine the little group and the irreducible representations (irreps) of the symmetry group acting on that cell. The classification of gapped Hamiltonians on these cells is determined by the Altland-Zirnbauer (AZ) classes and Wigner criteria, yielding $\mathbb{Z}$ or $\mathbb{Z}_2$ classifications with specific matrix dimensions for the generating Dirac Hamiltonians.
• $E_2$-page Calculation: The first differential $d_1$ is computed by analyzing how irreps on $p$-cells decompose into irreps on adjacent $(p+1)$-cells (in momentum space) or $(p-1)$-cells (in real space). This involves calculating induced representations and character overlaps, accounting for factor systems and symmetry actions (unitary/antiunitary).
• Constraint on K-groups: Since computing higher differentials ($d_2, d_3$) and extension problems is generally intractable for all symmetry settings, the authors propose a constraint method. They enumerate all possible higher differentials and extension scenarios compatible with the computed $E_2$-pages. By assuming that higher differentials do not reduce the rank of the K-group (a physically reasonable assumption based on the nature of band inversions and gapless point creation), they generate sets of candidate K-groups from both the momentum-space and real-space AHSS. The true K-group must lie in the intersection of these two sets.

Key Contributions

• Systematic Computation of $E_2$-pages: The paper provides a detailed algorithm and implementation for computing the $E_2$-pages of the AHSS for free fermionic insulators and superconductors in one, two, and three spatial dimensions.
• Comprehensive Symmetry Coverage: The authors compute results for 1651 magnetic space groups, 528 magnetic layer groups, and 393 magnetic rod groups, covering all one-dimensional representations of pairing symmetries for superconductors.
• Constraint Technique: A novel technique is introduced to constrain the possible K-groups by comparing the candidate sets derived independently from momentum-space and real-space AHSS. This intersection method significantly reduces the ambiguity in the classification.
• Algorithmic Implementation: The paper details the mathematical machinery required for these computations, including the handling of factor systems, the construction of cell decompositions (fundamental domains), and the computation of extension groups (Baer sums) and higher differentials.

Results

• Determination of K-groups: By applying the intersection constraint method, the authors successfully determined the K-groups for approximately 59% of the symmetry settings in three dimensions (specifically for the degree $n=0$, which corresponds to the classification of gapped topological phases).
• Higher Dimensions: For other degrees ($n \neq 0$) and lower dimensions (1D and 2D), the percentage of uniquely determined K-groups is generally higher, reaching over 90% for many 1D and 2D magnetic point groups.
• Data Availability: All computed $E_2$-pages and the resulting sets of candidate K-groups are made available via a provided URL and Supplemental Material.
• Specific Examples: The paper illustrates the method with detailed examples, such as spinful class D superconductors with MSG $P2_11'$, showing how the intersection of momentum and real space candidates narrows down the possible K-groups (e.g., to $\mathbb{Z} \oplus \mathbb{Z}_4^{\oplus 2} \oplus \mathbb{Z}_2^{\oplus 2}$).

Significance
The paper claims that while the full classification of topological crystalline phases requires solving higher differentials and extension problems—which remains an open challenge for many symmetry classes—the proposed method provides a robust and systematic way to constrain the possible K-groups. By leveraging the duality between momentum-space and real-space descriptions, the authors demonstrate that a significant portion (approx. 59%) of the classification problem in three dimensions can be resolved without needing the explicit form of higher differentials. This work establishes a comprehensive database of $E_2$-pages and candidate K-groups, serving as a foundational resource for the classification of topological phases in magnetic and non-magnetic crystalline systems. The authors note that their approach is limited to free fermion systems and that generalizing the comparative approach to interacting or bosonic systems remains an open problem.