Hidden Zeeman Field in Odd-Parity Magnets: An Ideal Platform for Topological Superconductivity

arXiv:2603.15147 · cond-mat.supr-con · Submitted 2026-03-16 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai, and with me are Mira and Lev, guest researcher.

Mira: Today's paper: "Hidden Zeeman Field in Odd-Parity Magnets".

Kai: Odd-parity magnets (OPMs) are characterized by time-reversal-preserving non-relativistic spin splitting (NSS), but they intrinsically break time-reversal symmetry (T) due to their underlying magnetic order.

Mira: First, who's behind it and why it matters.

Title and authors: Kai: So, to get started, let's talk about the title and authors of this paper, "Hidden Zeeman Field in Odd-Parity Magnets: An Ideal Platform for Topological Superconductivity." It sounds quite specific and points toward a very targeted area of condensed matter research.

Mira: It does sound very focused; the title immediately tells us that they aren't just looking at odd-parity magnets generally, but specifically linking them to topological superconductivity through this hidden field concept.

Lev: As a quantum error correction researcher, I’m curious if this hidden field concept is something that can be modeled simply enough to actually translate into a practical error correction protocol later on.

Kai: That's fair; the complexity comes from the underlying physics, but the authors are establishing that OPMs host this hidden Zeeman field rooted in their intrinsic time-reversal symmetry breaking, which is a big conceptual step forward.

Mira: I think that’s where it gets interesting; they’re moving past just looking at the NSS and insisting that because of the T-breaking inherent to magnetism, this hidden field must be there and fundamentally reshapes the band structure.

Lev: If it reshapes things fundamentally, then we need to know if those resulting energy scales are large enough to matter for any hardware application. I'm thinking about what a realistic superconducting state looks like when you factor in these intrinsic fields.

Kai: Well, they show that this hidden field is tied to the magnetic order, which they reveal through an analytical f-wave magnet model where the magnetism generates an emergent gauge field manifesting as real-space spin loop current order.

Mira: That emergent gauge field concept is really powerful because it provides a physical picture of how the spin dynamics are coupled to the charge degrees of freedom in a way we hadn't seen before in this context.

Lev: Coupling magnetism to charge, that’s usually where things get messy for experimentalists; we need to see if that coupling translates into something measurable, perhaps through transport properties or specific spectroscopic signatures.

Kai: They show that the large NSS energy scale, which is on the eV scale, enables conventional superconductivity to coexist robustly with this hidden Zeeman field reaching hundreds of meV in Zeeman splitting.

Mira: That coexistence is the core claim; it’s not just that they exist separately, but that they are robustly stable together within this unique band structure created by the odd-parity magnets.

Lev: Coexistence is good for stability, but how does it affect the gap structure? Does it make the superconducting gap smaller or larger in a way that's useful? I need to know what we’re actually looking at experimentally.

Kai: The paper suggests that this unique band structure establishes OPMs as an ideal platform for topological superconductors, supporting large topological regions, which means the topology isn't confined to tiny pockets.

Mira: Supporting large topological regions is a big deal because it suggests that the material has the potential to host larger domains of topological matter compared to systems where topology is highly localized.

Lev: If we can support larger domains, it opens up possibilities for scaling up superconducting devices, which is what error correction ultimately aims to do.

Kai: So, what we’re seeing here is that this paper introduces the idea that OPMs aren't just interesting because of NSS; they are intrinsically linked to this hidden Zeeman field which dictates a new type of topological state.

Mira: Exactly; it shifts the focus from merely observing the spin splitting to understanding how that intrinsic symmetry breaking dictates the material’s entire superconducting phase diagram.

Lev: And for error correction, if we can engineer systems with large, robust topological regions, we have a better chance at creating fault-tolerant qubits.

The paper's summary: Kai: Now that we've covered the basics of the paper "Hidden Zeeman Field in Odd-Parity Magnets: An Ideal Platform for Topological Superconductivity," the authors summarize their main findings by emphasizing that OPMs universally host a hidden Zeeman field arising from intrinsic time-reversal symmetry breaking.

Mira: And they emphasize that this discovery is paired with the coexisting large odd-parity NSS, which together establish OPMs as an ideal platform for realizing topological superconductors.

Lev: From my perspective, what I’m focusing on is how these two features interact—how the hidden Zeeman field and the NSS actually cooperate to create the unique band structure they describe.

Kai: They detail that in their representative f-wave magnet model H1, after applying unitary transformations, they arrive at a block-diagonal Hamiltonian

H1'': (seven) which reveals three distinct bands: h0(k), h1(k), and h2(k).

Mira: That specific mathematical structure is what allows them to analyze the spin expectation value sx, y, z at any wave vector k, and it shows a f-wave pattern in sz(k) within those bands.

Lev: If they can map out these distinct bands, that’s useful for figuring out which states are the relevant ones for our error correction efforts; we need to know the energy spectrum of those states.

Kai: They also show that the energy scales of the NSS and the HZF are directly set by hopping amplitude t and exchange coupling J, which in magnetic materials can reach eV and hundreds of meV scales respectively.

Mira: That direct scaling is a very concrete result because it tells us precisely how to tune these properties by changing fundamental material parameters like t or J.

Lev: Knowing the specific scaling helps us set expectations for what kind of experimental manipulation we might need to perform on the material to achieve those target energy scales.

Kai: They further show that the HZF manifests identically in both frames, lifting Kramers degeneracy at time-reversal invariant momenta, which is the explicit T-breaking part of their argument.

Mira: That manifestation across frames confirms that this intrinsic field is a fundamental property of the magnetic order and not just an artifact of our choice of frame.

Lev: For us, it means we have a consistent physical picture to work with; inconsistency in how we define the T-breaking could lead to wildly different predictions, which is something you have to be careful about.

Kai: So, what they are doing is connecting the magnetic structure directly to these topological features through this hidden Zeeman field and its resulting band structure.

Mira: They’re showing that OPMs are fundamentally different from what we thought before because of this unified mechanism involving the NSS and the HZF.

The paper's improvements: Kai: So, shifting to the improvements suggested by the authors in "Hidden Zeeman Field in Odd-Parity Magnets: An Ideal Platform for Topological Superconductivity," they are proposing that OPMs offer decisive advantages over conventional approaches relying on external magnetic fields.

Mira: They argue that this is because OPMs intrinsically host both odd-parity NSS and a hidden Zeeman field, and the hidden Zeeman field can reach hundreds of meV in real materials, which far exceeds what laboratory fields can achieve.

Lev: That intrinsic scale is what makes it compelling for hardware development; it suggests that we don't need to fight against external fields to get that level of protection against disorder.

Kai: This large intrinsic magnetic gap, enabled by the eV-scale NSS, coexisting with conventional superconductivity robustly due to this hidden Zeeman field reaching hundreds of meV, dramatically expands the topological regime in parameter space.

Mira: Expanding the topological regime in parameter space means there are many more material parameters where these states can exist than previously thought, which is a major mathematical gain.

Lev: A larger parameter space is always better for research because it gives us more freedom to search for stable phases without being constrained by external field limitations.

Kai: Furthermore, this intrinsic gap enhances the robustness of Majorana boundary modes against disorder significantly when compared to conventional approaches that rely on laboratory fields, which is a key feature they highlight.

Mira: That increased robustness against disorder is exactly what we need for building reliable topological states; it means the material itself provides a stronger protection than an external magnetic field would offer.

Lev: If the material provides that intrinsic protection, we can focus our experimental efforts on controlling other aspects of the system rather than just fighting environmental noise.

Kai: They also show that depending on whether you look at Type-I, Type-II, or Type-III OPMs, you can engineer different topological features like unidirectional Majorana edge states or Majorana Kramers pairs.

Mira: That classification is what makes the paper practical for experimentalists; it tells them exactly which magnetic texture they need to target to achieve the desired topological outcome.

Lev: Having that roadmap is essential; it means we know which specific material configuration to look for when trying to build a device, rather than just blindly sweeping parameter space.

Kai: And they point out that Type-III OPMs can be engineered to host both Majorana corner states and spin-group-protected Majorana Kramers pairs.

Mira: That’s a very specific prediction because it connects the type of magnetic order directly to the resulting topological features, which is a big step in connecting microscopic magnetism to macroscopic topology.

Lev: That level of specificity is what makes any potential material synthesis viable; you can design for a specific topological outcome based on the magnetic structure you expect to see.

Conclusion: Kai: So, wrapping up the discussion on "Hidden Zeeman Field in Odd-Parity Magnets: An Ideal Platform for Topological Superconductivity," we've seen how this paper establishes OPMs as a robust environment for topological superconductivity thanks to the hidden Zeeman field.

Mira: The central theme is that this intrinsic T-breaking mechanism provides a strong, internally generated protection against disorder that is superior to external magnetic fields, especially when coupled with the large odd-parity NSS.

Lev: I think what’s most important for scaling up is realizing that we have a material-based source of stability for these topological states that doesn't require constantly tuning external fields.

Kai: This discovery opens new avenues for exploring Majorana physics in magnetic materials by showing how to leverage the intrinsic properties of OPMs, particularly their large intrinsic gap.

Mira: It really broadens the parameter space where conventional superconductivity can coexist with this hidden Zeeman field, which is a substantial mathematical gain for theorists.

Lev: For our error correction work, having a material-based source of stability that doesn't require constantly tuning external fields is what we need when scaling up any quantum state.

Kai: So, "Hidden Zeeman Field in Odd-Parity Magnets: An Ideal Platform for Topological Superconductivity" has fundamentally changed how we view OPMs and their potential for topological applications.

Mira: It’s a significant result because it unifies the NSS and the HZF into a single physical picture of magnetism driving superconductivity.

Lev: We've seen that this material-based source of stability is what we need when scaling up any quantum state.

High Magnetic Field Laboratory, HFIPS, Chinese Academy of Sciences · Department of Physics, Hong Kong University of Science and Technology

cond-mat.supr-con

Submitted: 2026-03-16

Updated: 2026-09-24

Comments: 24 pages, 8 figures

Journal ref: Phys. Rev. Lett. 137, 136003 (2026)

DOI: 10.1103/j9m4-386b

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 86/100

The gist: Odd-parity magnets (OPMs) are characterized by time-reversal-preserving non-relativistic spin splitting (NSS), but they intrinsically break time-reversal symmetry (T) due to their underlying magnetic

Key concepts

Odd-Parity Magnets (OPMs): These materials are characterized by time-reversal-preserving non-relativistic spin splitting, but they intrinsically break time-reversal symmetry due to their underlying magnetic order.
Hidden Zeeman Field (HZF): This field arises from the intrinsic time-reversal symmetry breaking in OPMs and is tied to the magnetic order. It manifests as an emergent gauge field that reshapes the band structure of the material.
Topological Superconductivity: The paper establishes OPMs as an ideal platform for topological superconductors because they host both odd-parity spin splitting and a hidden Zeeman field, which supports large topological regions.
Coexistence: The core claim is that conventional superconductivity can robustly coexist with the hidden Zeeman field reaching hundreds of meV in Zeeman splitting, establishing a unique band structure.

Terminology

Robustness — 26× in this paper · explained in 5 episodes
Degeneracy — 21× in this paper · explained in 3 episodes
Symmetry Breaking — 4× in this paper · explained in 3 episodes
Altermagnets — 2× in this paper · explained in 1 episode
Andreev Bound States — 1× in this paper · explained in 1 episode
Anyons — 1× in this paper · explained in 2 episodes
Core — 1× in this paper · explained in 2 episodes
Hubbard Model — 1× in this paper · explained in 3 episodes
Majorana Zero Modes (MZMs) — 1× in this paper · explained in 3 episodes

Summary

Odd-parity magnets (OPMs) are characterized by time-reversal-preserving non-relativistic spin splitting (NSS), but they intrinsically break time-reversal symmetry (T) due to their underlying magnetic order. This work reveals that OPMs universally host a hidden Zeeman field (HZF) rooted in this T-breaking, which fundamentally reshapes their band structure.

The paper demonstrates that the large NSS (eV scale) enables conventional superconductivity to coexist robustly with the hidden Zeeman field, with Zeeman splitting reaching hundreds of meV. This unique band structure establishes OPMs as an ideal platform for topological superconductors (TSCs), supporting large topological regions. Based on OPMs, the authors engineer a series of TSCs hosting distinct Majorana boundary modes, including unidirectional Majorana edge states. The work corrects a fundamental misconception about OPMs and establishes them as a versatile platform for field-free and robust TSCs.

The mechanism is revealed through an analytical f-wave magnet model (H1). In this model, the magnetic order generates an emergent gauge field, manifesting as a real-space spin loop current order. Crucially, the large NSS (eV scale) pins the spin, enabling conventional superconductivity to robustly coexist with the strong HZF (with Zeeman splitting reaching hundreds of meV).

The hidden Zeeman field arises directly from intrinsic T-breaking and manifests as the lifted Kramers degeneracy at time-reversal invariant momenta. In a representative model H1, after a local unitary transformation that rotates all moments to the x-direction, the exchange coupling term becomes a uniform Zeeman field, capturing broken T. Moreover, an emergent gauge field appears: "using Q · a1 = Q · a2 = Q · a3 = 2π/3 (mod 2π), each hopping term acquires a uniform phase factor e−iπσ̃z /3. Remarkably, these complex hoppings generate a loop flux on each triangular plaquette [Fig. 1(c)], which corresponds to a spin loop current order in the real space although the charge current on each bond vanishes (see Appendix B for details)."

The band structures of OPMs are analyzed by mapping them to a two-band model, h0(k), which allows one to obtain the analytical expression for the spin expectation value sx,y,z(k). The study shows that the energy scales of the NSS and the HZF are set by the hopping amplitude t and the exchange coupling J, respectively, which in magnetic materials can reach the eV and hundreds of meV scales. Furthermore, the HZF manifests identically in both the original lattice frame and the rotated frame: it lifts Kramers degeneracy at time-reversal invariant momenta, accounting for the explicit T-breaking.

Compatibility between OPMs and superconductivity is investigated by performing self-consistent mean-field calculations for models H1,2 with s-wave pairing. In these systems, the generated gauge field pins the spin along the z-direction, while the HZF lies in the xy-plane. This orthogonal configuration is reminiscent of Ising superconductors. The results show that superconductivity coexists with coplanar magnetic order over a wide range of J, and Tc decreases continuously to zero, characteristic of a second-order transition.

Engineering TSCs based on OPMs involves diagonalizing models like HBdG in a nanowire geometry. This yields energy spectra exhibiting Majorana flat bands coexisting with gapless bulk states. When the chemical potential lies within this gap and an s-wave pairing is introduced, the corresponding 1D slice enters a TSC phase. The study shows that the energy spectrum of H̃BdG closes when fz (k) = 0 and J = ∆ + (f0 − µ) [54]. This condition is met at points along the Γ-M̃1,2,3 lines where fz(k) vanishes, producing six pairs of nodal heliconets.

Furthermore, the work demonstrates that "the coexistence of gapless bulk states and dispersive edge states lifts constraints on the propagation direction of edge states, allowing scattering into the bulk. Consequently, edge states on opposite boundaries can propagate either in the same or opposite directions. When they flow in the same direction, unidirectional edge states emerge [59]."

The paper concludes that OPMs offer decisive advantages over conventional approaches relying on external magnetic fields because "OPMs intrinsically host both odd-parity NSS and a HZF. The latter can reach hundreds of meV in real materials, for instance, ∼130 meV in the p-wave magnet EuIn2 As2 [54, 61], far exceeding the scale achievable with laboratory fields. This large intrinsic magnetic gap can coexist robustly with superconductivity due to the eV-scale NSS, dramatically expanding the topological regime in parameter space and significantly enhancing the robustness of Majorana boundary modes against disorder."

In summary, "we have uncovered a HZF universally present in OPMs, arising directly from their intrinsic time-reversal breaking. This discovery, together with the coexisting large odd-parity NSS, establishes OPMs as an ideal platform for realizing robust and field-free TSCs." The work resolves a fundamental misconception in the description of OPMs and opens new avenues for exploring Majorana physics in magnetic materials.

The study also classifies OPMs into three types based on their spin textures: Type-I (collinear), Type-II (coplanar), and Type-III (non-coplanar). The authors show that the systematic absence of Kramers pairs at time-reversal momenta in these bands signals the presence of a hidden Zeeman field in OPMs, which opens a magnetic energy gap. This gap is significant: "This large intrinsic gap can coexist robustly with superconductivity due to the eV-scale NSS, dramatically expanding the topological regime in parameter space and significantly enhancing the robustness of Majorana boundary modes against disorder. The study also shows that Majorana Kramers pairs can be realized in a magnetic domain-wall geometry and that for Type-III OPMs, both Majorana corner states and spin-group-protected Majorana Kramers pairs can be engineered." The material EuIn2 As2 is highlighted as an example where the intrinsic gap is approximately 130 meV.

The paper's key findings are:

OPMs universally host a hidden Zeeman field (HZF) arising from intrinsic T-breaking, which reshapes the band structure.
The large NSS (eV scale) enables conventional superconductivity to coexist robustly with the HZF (hundreds of meV).
This unique structure makes OPMs ideal platforms for robust and field-free TSCs, supporting large topological regions.
Distinct types of OPMs can be used to engineer different TSC features, including unidirectional Majorana edge states and Majorana Kramers pairs.
The HZF provides a robust protection against disorder compared to conventional approaches relying on external magnetic fields, as its scale far exceeds laboratory field capabilities.

Relevant quotes:

Hidden Zeeman Field in Odd-Parity Magnets: An Ideal Platform for Topological Superconductivity (Page 1)

"Through an analytical f-wave magnet model, we show that NSS microscopically originates from an emergent gauge field, manifesting as a real-space spin loop current order. Crucially, the large NSS (eV scale) enables conventional superconductivity to coexist robustly with the hidden Zeeman field, with Zeeman splitting reaches hundreds of meV." (Page 1)

This unique band structure renders OPMs ideal platforms for TSCs, supporting large topological regions. (Page 1)

The HZF arises directly from intrinsic T-breaking and manifests as the lifted Kramers degeneracy at time-reversal invariant momenta [Figs. 1(e) and 1(f)]. (Page 1)

The energy scales of the NSS and the HZF are set by the hopping amplitude t and the exchange coupling J, respectively, which in magnetic materials can reach the eV and hundreds of meV scales. (Page 2)

"This large intrinsic gap can coexist robustly with superconductivity due to the eV-scale NSS, dramatically expanding the topological regime in parameter space and significantly enhancing the robustness of Majorana boundary modes against disorder." (Page 5)

"In summary, we have uncovered a HZF universally present in OPMs, arising directly from their intrinsic time-reversal breaking. This discovery, together with the coexisting large odd-parity NSS, establishes OPMs as an ideal platform for realizing robust and field-free TSCs." (Page 5)

"OPMs are magnetic systems characterized by a NSS that is odd under spatial inversion, satisfying S(k) = −S(−k), where S(k) = (sx (k), sy (k), sz (k)). Based on the dimensionality of S(k), OPMs are classified into three distinct types [24]." (Page 14)

"This selective lifting of degeneracy in H2,3 follows from the symmetry T̃1 = T τ, with T̃12 = −1 at kx = 0. The systematic absence of Kramers pairs at time-reversal momenta in these bands signals the presence of a hidden Zeeman field in OPMs, which opens a magnetic energy gap." (Page 8)

"The large intrinsic gap can coexist robustly with superconductivity due to the eV-scale NSS, dramatically expanding the topological regime in parameter space and significantly enhancing the robustness of Majorana boundary modes against disorder." (Page 5)

"The material EuIn2 As2 is a candidate p-wave magnet [24], which crystallizes in a hexagonal structure and hosts a helical spin configuration, as shown in Fig. S5(a). This compound belongs to the spin space group 1 P 6001 63 /2100 m1 m6001 c(1, 1, 3100m [24] and respects the effective time-reversal symmetry T̃ ۱ = T σz and a spingroup symmetry g2 = [31 τ], which constrain the spin expectation values to satisfy sx,y (k) = sx,y (−k) and sz (k) = −sz (−k)." (Page 18)

(Note: The provided text contains significant repetition in the extracted quotes, which is a characteristic of an arXiv preprint's formatting, but all unique scientific points have been included.)

The material EuIn2 As2 is a candidate p-wave magnet [24], which crystallizes in a hexagonal structure and hosts a helical spin configuration, as shown in Fig. S5(a). This compound belongs to the spin space group 1 P 6001 63 /2100 m1 m6001 c(1, 1, 3100m [24] and respects the effective time-reversal symmetry T̃ ۱ = T σz and a spingroup symmetry g2 = [31 τ], which constrain the spin expectation values to satisfy sx,y (k) = sx,y (−k) and sz (k) = −sz (−k).

Improvements for AI systems

Based on the provided scientific paper, here are specific improvements for AI systems that could be derived from its physical insights:

AI System Improvement Recommendations:

Acknowledge and Model Intrinsic Symmetry Breaking:
Implement Hidden Field Detection Algorithms (HZF):
Develop Topological Phase Prediction Models for Magnetic Materials (OPMs):
Engineer Robust Majorana Edge State/Corner State Simulators:

Specific Capabilities of the Improved AI System:

Detailed Improvements and Specific Outputs: