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Theory of the intrinsic optical anomalous Hall responses in time-reversal symmetry breaking superconductors

This paper establishes a unified quantum-geometric theory for the intrinsic optical anomalous Hall effect in time-reversal symmetry breaking superconductors, distinguishing the microscopic requirements for unitary and nonunitary pairing states and providing a framework to identify their specific pairing symmetries.

Original authors: Jia-Long Zhang, Wen Huang

Published 2026-10-22

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CC BY 4.0

Original authors: Jia-Long Zhang, Wen Huang

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

Superconductors are materials that conduct electricity with zero resistance, a property that emerges when electrons pair up and move in perfect unison. For decades, physicists have been trying to figure out exactly how these pairs form, as the answer reveals the fundamental nature of the material. A particularly intriguing possibility is that these electron pairs break a fundamental rule of nature called time-reversal symmetry. In simple terms, this means the behavior of the pairs would look different if time were played backward, much like a spinning top that only spins one way. When this happens, the material can generate a Hall effect, a phenomenon where an electric current flowing through the material is pushed sideways, creating a voltage perpendicular to the flow. Detecting this sideways push in superconductors is a powerful way to identify these exotic states, but for a long time, it was unclear exactly how the microscopic machinery of the superconductor creates this effect, especially in materials that are free of impurities.

A team of researchers has now developed a unified theory to explain how this intrinsic optical Hall effect arises in clean, multi-band superconductors. They focused on materials that possess a specific type of symmetry where the structure looks the same if flipped inside out. By analyzing the quantum mechanics of these systems, the researchers discovered that the key to generating this sideways voltage lies in the "quantum geometry" of the electron pairs. This concept describes how the electron waves are shaped and connected across different energy levels within the material. The study reveals that the effect is driven by electrons tunneling, or jumping, between these different energy levels. This tunneling is facilitated by a specific property of the material's structure that acts like a bridge between the levels, allowing the electrons to move in a way that breaks the usual symmetry and creates the Hall response.

The researchers found that the mechanism depends heavily on the type of superconducting state. They distinguished between two main categories: unitary states, where the time-reversal symmetry is broken by the orbital motion of the electron pairs, and nonunitary states, where the breaking is linked to the spin polarization of the electrons. A crucial finding is that for unitary states to produce this Hall effect, the electron pairs must form across different energy bands, a process known as interband pairing. Without this cross-band connection, the effect simply cannot happen in these specific materials. In contrast, nonunitary states are more flexible; they can generate the Hall effect even if the pairs form only within a single energy band, provided the material has a specific interaction between the electron's spin and its motion. This distinction provides a clear rule for scientists to follow when trying to identify the nature of a superconductor.

To understand these processes, the team broke down the complex quantum interactions into two distinct types of microscopic events. The first involves single electrons tunneling between energy levels, while the second involves entire Cooper pairs tunneling together. Both processes rely on the same geometric feature of the material to function. The researchers also constructed a simplified model that treats the superconductor as a system of electrons with modified properties, effectively removing the complexity of the paired holes to see the core mechanism. This alternative approach confirmed their earlier findings, showing that the pairing interaction itself endows the electrons with a new, effective curvature in their quantum landscape, which is what drives the Hall response.

Perhaps the most practical outcome of this work is a new way to distinguish between these different types of superconducting states using light. The researchers proved that low-frequency Hall resonances, which are specific peaks in the response to light, can only appear near the energy gap of nonunitary superconductors. In unitary states, these low-frequency peaks are strictly forbidden by symmetry. This means that by shining light at frequencies just above the superconducting gap and looking for these specific resonances, experimentalists can tell whether a material is in a unitary or nonunitary state. This offers a precise spectroscopic tool to settle debates about the nature of mysterious superconductors, such as the controversial material Sr2RuO4, where the exact pairing mechanism has been a subject of intense investigation for years. The work establishes quantum geometry as the fundamental organizing principle behind these optical responses, providing a clear guide for future experiments to map the hidden symmetries of quantum matter.

Technical Summary: Theory of Intrinsic Optical Anomalous Hall Responses in Time-Reversal Symmetry Breaking Superconductors

Problem Statement
Determining the symmetry of Cooper pairing in superconductors remains a central challenge in condensed matter physics. While time-reversal symmetry breaking (TRSB) is a distinctive feature of certain pairing states (e.g., chiral $p_x + ip_y$ ), the microscopic mechanisms enabling the optical anomalous Hall effect (AHE) in clean, multiband superconductors are not fully elucidated. Previous interpretations of the polar Kerr effect as evidence for chiral superconductivity face theoretical hurdles: in clean single-band systems, Galilean invariance forbids the optical Hall effect because the chiral relative motion of Cooper pairs decouples from their center-of-mass motion. Although multiband effects have been proposed to bypass this, a unified organizing principle for the intrinsic Hall response in multiband TRSB superconductors has been lacking. Key unresolved questions include the necessity of interband Cooper pairing for different TRSB states, the specific microscopic processes (single-electron vs. Cooper-pair tunneling) driving the response, and the existence of low-frequency Hall resonances near the superconducting gap.

Methodology
The authors develop a unified theoretical framework for TRSB multiband superconductors possessing inversion symmetry. The study employs three complementary approaches:

Diagrammatic Analysis: Using linear response theory, the optical Hall conductivity is expressed via the transverse current-current correlator. The authors decompose the one-loop Feynman diagrams into contributions from normal and anomalous Gor'kov Green's functions. This allows for the classification of microscopic processes into two distinct classes: single-electron tunneling and Cooper-pair tunneling.
Effective Electronic Theory: By projecting out the hole degrees of freedom from the Bogoliubov-de Gennes (BdG) Hamiltonian, the authors construct an effective electron-only Hamiltonian. This approach incorporates pairing-induced corrections to reveal how superconductivity endows electrons with an effective Berry curvature.
Symmetry Analysis: The authors utilize particle-hole and inversion symmetries to derive selection rules for optical transitions, specifically investigating the conditions under which low-frequency Hall resonances can exist near the superconducting gap edge.

The analysis is conducted in the Bloch-band basis, fully accounting for spin-orbit coupling (SOC) and distinguishing between unitary TRSB states (where TRSB originates in the orbital part of the wavefunction, with no spontaneous spin polarization) and nonunitary TRSB states (where TRSB is associated with finite spin polarization).

Key Contributions and Results

Quantum Geometry as the Unifying Principle: The study establishes that the intrinsic optical Hall response is fundamentally a quantum-geometric phenomenon. The response is driven by interband velocity, which is intimately tied to the interband Berry connection between distinct Bloch bands. This interband velocity facilitates coherent interband charge transfer, violating Galilean invariance in the superconducting state.
Microscopic Mechanisms: The diagrammatic analysis identifies two classes of processes:
- Class A (Single-electron tunneling): Involves virtual interband tunneling of a single electron.
- Class B (Cooper-pair tunneling): Involves the transfer of Cooper pairs (or scattering between intra- and interband pairs).
  Both classes require at least one interband tunneling event driven by the interband velocity.
Distinction Between Unitary and Nonunitary States:
- Unitary TRSB States: The authors rigorously prove that interband Cooper pairing is a prerequisite for generating an intrinsic Hall response. In the absence of interband pairing, the Hall response vanishes even if the state is chiral. The response arises from the interplay of intraband and interband pairing mediated by interband velocity.
- Nonunitary TRSB States: Interband pairing is not absolutely required. In states with only intraband pairing, a finite Hall response can arise via single-electron tunneling processes (Class A1), provided that spin-orbit coupling is present. This mechanism is analogous to the Hall effect in spin-orbit-coupled ferromagnets, where the breaking of perfect cancellation between spin species leads to a residual charge Hall conductivity.
Low-Frequency Hall Resonances: A critical finding is the symmetry selection rule for low-frequency resonances. The authors demonstrate that intrinsic optical Hall resonances near the superconducting gap edge ( $\omega \approx 2\Delta$ ) are allowed in nonunitary states but forbidden in unitary TRSB states. In unitary chiral states, the lowest-lying Bogoliubov bands are degenerate and related by symmetries that cause optical transition contributions to cancel exactly. Consequently, the Hall resonance in unitary states occurs at much higher frequencies (normal-state interband separation), whereas nonunitary states exhibit a resonance near the gap edge.

Significance
This work provides a definitive theoretical guide for interpreting spectroscopic techniques, particularly the polar Kerr effect, in the search for TRSB pairing states in quantum materials. By establishing quantum geometry as the organizing principle, the paper clarifies why certain TRSB states (like specific chiral pairings) may not exhibit a Kerr signal in clean limits unless interband pairing is present.

The distinction between unitary and nonunitary mechanisms offers a pathway to experimentally differentiate candidate pairing states. Specifically, the prediction that low-frequency Hall resonances exist only in nonunitary states suggests that infrared or terahertz spectroscopy on pristine samples could distinguish between these pairing symmetries. This is particularly relevant for controversial materials like $\text{Sr}_2\text{RuO}_4$ , where identifying the precise nature of the TRSB pairing remains an open challenge. The study does not propose new experimental setups but rather refines the theoretical interpretation of existing and future measurements within the constraints of symmetry and quantum geometry.

Technical Summary: Theory of Intrinsic Optical Anomalous Hall Responses in Time-Reversal Symmetry Breaking Superconductors

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