Euler band topology in superfluids and superconductors

This paper establishes a unified framework for Euler band topology in superfluids and superconductors by demonstrating that specific three-dimensional topological phases in symmetry classes DIII and CI, including the B phase of superfluid Helium-3, possess nontrivial Euler classes that link to known invariants and predict unique phenomena such as linked nodal lines and higher-order topology.

The Gist

Imagine you are holding a piece of fabric. In the world of physics, electrons moving through materials (like superconductors or superfluids) are often described as if they are flowing on this fabric. Usually, this fabric is smooth and predictable. But sometimes, the fabric gets twisted, knotted, or woven in such a complex way that it creates "holes" or "loops" that can't be untangled without ripping the fabric apart. This is called topology.

For a long time, physicists knew about two main ways to twist this fabric: one that breaks time (like a movie playing backward) and one that doesn't. But recently, scientists discovered a third, stranger way to twist the fabric called Euler band topology. It's like a knot that only exists if you have a specific kind of mirror symmetry in your room.

This paper by Shingo Kobayashi, Manabu Sato, and Akira Furusaki is like a new instruction manual for finding these special knots in superconductors (materials that conduct electricity with zero resistance) and superfluids (liquids that flow with zero friction).

Here is the breakdown of their discovery using simple analogies:

1. The "Magic Mirror" Symmetry

To find these special Euler knots, you need a very specific setup. Imagine a room with a mirror on the floor and a mirror on the wall. If you walk into the room, your reflection is flipped both horizontally and vertically. In physics, this is called Space-Time Inversion Symmetry.

The authors looked at materials that have this "double-mirror" symmetry. They found that in these materials, the electrons form a special kind of "real" pattern (like a pattern you could draw with a pencil, not a ghostly mathematical one). This pattern is protected by a number called the Euler Class.

• The Analogy: Think of the Euler Class as a "knot count." If the count is zero, the fabric is flat. If the count is non-zero (like 1 or 2), the fabric has a permanent, unbreakable knot.

2. The Helium-3 Mystery (The "Spinning Top")

The paper uses Superfluid Helium-3 (a super-cold liquid version of helium) as a prime example.

• The Old View: Physicists knew Helium-3 had a "winding number" (a measure of how many times the electron spins wrap around). If this number was odd, it was a topological superfluid.
• The New View: The authors proved that this "winding number" is actually just a different way of counting the Euler knot.
• The Twist: Usually, if you put a magnet near a superconductor, it breaks the symmetry and destroys the magic. But because Helium-3 has this special "Euler knot," it is incredibly tough. Even if you apply a magnetic field (as long as it's in a specific direction), the knot stays tied. The material remains a topological superfluid.

Why does this matter? It explains why Helium-3 behaves strangely in magnetic fields, showing off "Majorana particles" (ghostly particles that are their own antiparticles) on its surface. The Euler knot is the reason these particles are so stubborn and hard to destroy.

3. The "Linked Rings" in Superconductors

The paper also looked at a different type of material called Class CI superconductors.

• The Scenario: Imagine a 3D ball of dough (the material). Inside, there are lines where the superconducting power vanishes (called "nodal lines").
• The Discovery: If the Euler knot count is non-zero, these lines of "nothingness" inside the material don't just float randomly. They link together like two rings in a chain.
• The Analogy: Imagine two hula hoops floating inside a giant jelly. If the Euler number is special, the hoops are locked together. You can't pull them apart without breaking the jelly.
• The Proof: The authors built a computer model of a multi-orbital superconductor (a material with complex electron orbits). When they added a tiny bit of "twist" (a perturbation), the single ring of "nothingness" split into two smaller rings, but they remained linked. This linking structure is a direct fingerprint of the Euler topology.

4. Why Should You Care?

This isn't just abstract math. It provides a unified language for understanding weird quantum phenomena.

• Robustness: It explains why some quantum states survive even when you mess with them (like adding a magnet).
• New Materials: It gives scientists a checklist to find new materials. If a material has the right symmetry and a specific "knot count," it might host these exotic particles. The authors suggest materials like UTe2 and KFe2As2 are likely candidates.
• Future Tech: These "Majorana" particles are the holy grail for quantum computers because they are naturally protected from errors. Understanding the Euler knot helps us design better, more stable quantum computers.

The Bottom Line

The authors have shown that the mysterious "Euler Class"—a concept previously thought to be only for electrons in normal metals—is actually the hidden key to understanding the most robust and exotic states of superconductors and superfluids.

They essentially said: "If you see a superconductor that acts weirdly in a magnetic field or has linked rings of zero-energy, don't panic. It's just wearing an Euler knot. And that knot makes it super strong."

This discovery connects the dots between different types of quantum weirdness, offering a single map to navigate the complex world of topological materials.

Technical Summary

1. Problem Statement

Real band topology, characterized by invariants such as the Euler class and the second Stiefel–Whitney (SW) class, typically arises in systems with space-time inversion symmetry ($I_{ST}$) that squares to the identity. While this topology has been extensively studied in normal-state electronic systems (e.g., twisted bilayer graphene, topological semimetals) and artificial platforms, its emergence and characterization in superconductors and superfluids remain less understood.

Most existing studies on superconducting Euler topology rely on specific models or assume nontrivial Euler topology in the normal-state Hamiltonian. A critical open question is how Euler band topology emerges in a general setting within the Bogoliubov-de Gennes (BdG) framework, particularly in symmetry classes DIII and CI, and how it relates to established topological invariants like the 3D winding number. Furthermore, the robustness of these phases against time-reversal symmetry breaking (TRSB) perturbations (e.g., magnetic fields) requires a unified theoretical framework.

2. Methodology

The authors develop a unified framework for analyzing the band topology of BdG Hamiltonians possessing space-time inversion symmetry ($I_{ST}$).

• Symmetry Analysis: They focus on Altland-Zirnbauer symmetry classes DIII (spin-triplet, $s=1$) and CI (spin-singlet, $s=0$).
  • For Class DIII, $I_{ST}$ is defined as the product of two-fold rotation ($C_{2z}$) and time-reversal ($T$), i.e., $I_{ST} = C_{2z}T$.
  • For Class CI, $I_{ST}$ can be either $C_{2z}T$ or the product of spatial inversion ($P$) and time-reversal ($PT$).
• Mathematical Formulation:
  • The BdG Hamiltonian is transformed into an off-diagonal form using a chiral basis.
  • The topology is analyzed via the homotopy of the unitary matrix $Q(k) \in U(N)/O(N)$ derived from the singular value decomposition of the off-diagonal block.
  • The Euler class ($e_2$) is defined as the flux integral of the real Berry curvature over an $I_{ST}$-invariant 2D plane (e.g., $k_z=0$ or $\pi$) in the 3D Brillouin zone (BZ).
• Derivation of Formulas:
  • The authors derive simplified formulas for the Euler class under $n$-fold rotational symmetry ($C_{nz}$).
  • They utilize the Stokes' theorem and properties of the sewing matrix (constructed from eigenstates at time-reversal invariant momenta, TRIMs) to express the Euler class in terms of Pfaffians and rotation eigenvalues.
  • They establish explicit algebraic relationships between the Euler class and stable topological invariants (3D winding number $w_3$, 2D/3D $\mathbb{Z}_2$ invariants).

3. Key Contributions

1. Unified Framework for BdG Systems: The paper extends the theory of Euler band topology from normal-state insulators to superconductors and superfluids, specifically addressing classes DIII and CI.
2. Derivation of Euler Class Formulas:
   • For Class DIII ($C_{2z}T$ symmetry): Derived a formula relating the Euler class $e_2$ to the product of Pfaffians of the sewing matrix at TRIMs.
   • For Class CI ($C_{4z}T$ symmetry): Derived a formula relating the Euler class to rotation eigenvalues and Pfaffians, accounting for orbital doublets.
3. Connection to Winding Numbers:
   • Proved that for Class DIII, the 3D winding number $w_3$ is equivalent to the Euler class modulo 2: $(-1)^{w_3} = (-1)^{\bar{e}_{2z}}$, where $\bar{e}_{2z}$ is the difference in Euler classes between $k_z=\pi$ and $k_z=0$.
   • For Class CI, established a relation between the Euler class and the 3D winding number (which is an even integer in this class), showing $(-1)^{\bar{e}_{2z}/2} = (-1)^{w_3/2}$.
4. Identification of "Euler Superconductors": Defined a new class of topological phases where the nontrivial Euler class persists even when time-reversal symmetry is broken, provided $I_{ST}$ symmetry is preserved.

4. Key Results

A. Class DIII Systems (e.g., Superfluid $^3$He-B)

• Topological Characterization: The 3D topological phase with an odd winding number ($w_3 = 1$) possesses a nontrivial Euler class ($|\bar{e}_{2z}| = 1 \mod 2$).
• Robustness under TRSB: The authors demonstrate that the B phase of superfluid Helium-3 ($^3$He-B) in a magnetic field is an Euler superfluid.
  • While a magnetic field breaks $T$ symmetry (making $w_3$ ill-defined), if the field is perpendicular to the rotation axis, $C_{2z}T$ symmetry is preserved.
  • The nontrivial Euler class ensures the persistence of gapless surface states on surfaces parallel to the magnetic field.
• Physical Implications: This Euler topology provides a unified explanation for:
  • Majorana Ising spin susceptibility: The robustness of surface states against magnetic fields.
  • Higher-order topology: The existence of Majorana hinge states, linked to the nontrivial second SW class (since $e_2 = w_2 \mod 2$).

B. Class CI Systems (Spin-Singlet Superconductors)

• Nodal Line Linking: In centrosymmetric Class CI superconductors with a nontrivial Euler class, the topology dictates the existence of superconducting nodal lines that link with degeneracy lines of the normal-state Hamiltonian.
• Model Demonstration: Using a tight-binding model of a multi-orbital $s_{\pm}$-wave superconductor:
  • When inversion symmetry is present, a nontrivial Euler class ($\bar{e}_{2z} = 2 \mod 4$) implies the existence of a nodal loop inside the 2D sphere in the BZ.
  • The model exhibits a linking structure between the superconducting nodal loop and the normal-state band degeneracy line.
  • Upon breaking $C_{4z}$ to $C_{2z}$ (via perturbation), the nodal loop splits into two smaller loops, preserving the total Euler charge, while the linking structure remains robust.

5. Significance

• Theoretical Unification: The work connects disparate phenomena in superconductivity—such as the Majorana Ising response in $^3$He-B, higher-order topology, and nodal line linking—under the single umbrella of Euler band topology.
• Robustness Insight: It clarifies that topological protection in these systems does not solely rely on time-reversal symmetry but can be maintained by the combined $C_{2z}T$ (or $PT$) symmetry, which is crucial for understanding materials in magnetic fields.
• Material Candidates: The paper identifies potential candidate materials for Euler superconductivity, including UTe$_2$ (a heavy-fermion superconductor) and KFe$_2$As$_2$ (an iron-based superconductor), which satisfy the required symmetry conditions.
• Experimental Predictions: The findings suggest new experimental probes, such as spin susceptibility measurements, scanning tunneling microscopy (STM) for hinge states, and transport measurements, to detect Euler topology in superconductors.

In summary, this paper establishes that the Euler class is a fundamental topological invariant for superconductors with space-time inversion symmetry, offering a robust mechanism for protecting surface and hinge states against time-reversal symmetry breaking perturbations.