🔬 condensed matter

Loop currents in Haldane's model and in time-reversal-breaking superconductors

This paper theoretically demonstrates that the integrated optical spectral weight of the frequency-dependent ac Hall conductivity serves as a direct experimental probe for quantifying steady loop currents in both Haldane's model and time-reversal-breaking chiral superconductors on a honeycomb lattice.

Original authors: Azzam Alzahrani, Victor M. Yakovenko

Published 2026-08-25✓ Author reviewed ⓘ

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CC BY 4.0

Original authors: Azzam Alzahrani, Victor M. Yakovenko

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 by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

For centuries, the magnetism of everyday objects like refrigerator magnets or compass needles has been understood as a property of tiny atomic spins, little internal arrows that align to create a magnetic field. But there is another, more subtle way for matter to generate magnetism, one that does not rely on these spins at all. Instead, imagine electrons not as spinning tops, but as travelers moving in continuous, unending circles between atoms within a crystal. These are called loop currents. Unlike the electric current in a wire that stops when you flip a switch, these microscopic loops can persist indefinitely, creating a magnetic field without any external power source. While the existence of such currents has been proposed in theory for decades, proving they are actually present in real materials has been difficult because they are invisible to the naked eye and often too small to measure directly. The challenge for physicists has been to find a way to see these hidden loops without disturbing them, essentially looking for their shadow rather than the object itself.

Two researchers at the University of Maryland, Azzam S. Alzahrani and Victor M. Yakovenko, have developed a theoretical guide for spotting these elusive currents using light. They focused on two specific types of materials: one is a theoretical model of a special insulator known as Haldane's model, and the other is a type of superconductor that breaks a fundamental symmetry of nature called time-reversal symmetry. In simple terms, time-reversal symmetry means that the laws of physics look the same whether time is moving forward or backward. When a material breaks this symmetry, it behaves differently depending on the direction of time, a condition often linked to magnetism. The researchers calculated exactly how these persistent loop currents would interact with light, specifically light that oscillates at different frequencies. They discovered that the presence of these currents leaves a distinct fingerprint in the way the material conducts electricity when hit by light, a property known as the optical Hall conductivity.

The core of their work involves connecting the invisible flow of electrons to a measurable signal. In the first part of their study, they examined Haldane's model, a grid of atoms arranged in a honeycomb pattern. They calculated the average strength of the loop currents flowing between the atoms and found that these currents depend on specific details of how electrons jump from one atom to another. Crucially, they showed that these currents are directly linked to the total amount of light energy the material absorbs at different frequencies. If you were to shine light on such a material and measure how much it absorbs while also measuring its electrical response, the total "weight" of that absorption would tell you exactly how strong the loop currents are. This provides a clear path for experimentalists: instead of trying to measure the tiny currents directly, they can measure the light absorption, which is a much more accessible task.

In the second part of their study, the researchers turned their attention to a chiral superconductor, a material that conducts electricity with zero resistance but also breaks time-reversal symmetry due to the unique way its electrons pair up. This type of superconductor was previously studied by other scientists, who noticed a sharp, sudden spike in the material's response to light, but the reason for this spike was unclear. Alzahrani and Yakovenko demonstrated that this spike is not a random occurrence but a direct result of electrons jumping between different energy levels within the material, a process triggered specifically by the time-reversal-breaking nature of the superconductor. They found that the frequency of this light spike is exactly twice the energy difference between the electrons' current state and a specific point in the material's energy structure. Just as in the first model, the strength of this light spike is directly proportional to the magnitude of the loop currents circulating inside the material's unit cells.

This connection between the optical spike and the loop currents is significant because it offers a way to verify the existence of these currents in real superconductors. The researchers noted that while scientists have already detected a related effect called the polar Kerr effect in some superconductors using specialized equipment, the signal is often very weak at high frequencies. Their work suggests that the strongest and clearest signal would be found at lower frequencies, specifically at the frequency of the absorption peak they identified. By tuning their instruments to this specific frequency, experimentalists could potentially see a much stronger signal, making it easier to confirm the presence of these mysterious loop currents. The study does not claim to have discovered these currents in a new material, but rather provides the theoretical map and the specific tools needed to find them. It clarifies that the optical properties of these materials are not just abstract numbers but are direct reflections of the steady, circulating currents flowing within them, offering a new window into the magnetic secrets of quantum materials.

Technical Summary: Optical Manifestations of Loop Currents in Haldane's Model and Time-Reversal-Breaking Superconductors

Problem Statement
The paper addresses the theoretical challenge of detecting persistent loop currents in periodic crystals. While such currents are known to break time-reversal symmetry (TRS) and can generate an anomalous Hall effect (AHE), experimental evidence for them remains scarce and debated. A primary difficulty lies in distinguishing these steady (dc) loop currents from other magnetic phenomena. The authors posit that while loop currents are zero-frequency (dc) phenomena, their presence manifests in the frequency-dependent optical properties of a material, specifically within the optical Hall conductivity. The study aims to establish a quantitative theoretical link between the expectation value of loop currents and measurable optical spectral weights in two distinct systems: Haldane's model and chiral time-reversal-breaking superconductors on a honeycomb lattice.

Methodology
The authors employ a tight-binding theoretical framework on a two-dimensional honeycomb lattice to analyze two specific models:

Haldane's Model: A Chern insulator where TRS is broken by complex next-nearest-neighbor (NNN) tunneling amplitudes ( $t_{nnn} = t'_{nnn} + i t''_{nnn}$ ) induced by a periodic magnetic flux. The authors derive the Hamiltonian in momentum space, calculate the Green's function, and explicitly compute the expectation value of the loop current operator. They relate this to the optical Hall spectral weight ( $W_{Hall}$ ) by evaluating the commutator of current operators and integrating the imaginary part of the ac Hall conductivity ( $\sigma_H(\omega)$ ).
Chiral Superconductor: A spin-singlet superconductor with $d_{x^2-y^2} \pm i d_{xy}$ pairing on the honeycomb lattice. In this model, the normal state preserves TRS ( $t''_{nnn}=0$ ), but TRS is broken by the chiral superconducting pairing. The authors utilize the Bogoliubov-de Gennes (BdG) formalism to construct the Hamiltonian and Green's function. They generalize the current operator to the Nambu basis to calculate the loop current expectation value induced by the superconducting pairing. They further analyze the ac Hall conductivity to identify the origin of optical absorption peaks.

Throughout both sections, the authors utilize magneto-optical sum rules to relate the integrated optical spectral weight to the expectation value of the loop current commutator.

Key Contributions and Results

Explicit Calculation of Loop Currents in Haldane's Model:
- The paper provides the first explicit calculation of the expectation value of loop currents ( $I_{lc}$ ) for Haldane's model in terms of model parameters.
- The authors demonstrate that a non-zero $I_{lc}$ requires both the real ( $t'_{nnn}$ ) and imaginary ( $t''_{nnn}$ ) parts of the NNN tunneling amplitude to be non-zero. This contrasts with the dc quantum Hall effect, which requires only the imaginary part.
- A direct proportionality is established between the loop current expectation value and the optical Hall spectral weight: $I_{lc} \propto W_{Hall}$ . This implies that measuring the integrated optical spectral weight of the ac Hall conductivity provides a direct method to determine the magnitude of steady loop currents.
Optical Manifestations in Chiral Superconductors:
- The study investigates a chiral superconductor where loop currents are induced by the pairing mechanism rather than an external magnetic flux.
- The authors identify a sharp optical absorption peak in the ac Hall conductivity. They demonstrate that this peak originates from electronic excitations between the lower and upper Dirac bands near the $K$ and $K'$ points in the Brillouin zone.
- The frequency of this peak is determined to be twice the energy difference between the Fermi level and the Dirac point, a value typically much larger than the superconducting gap.
- Similar to Haldane's model, the optical spectral weight of this peak is directly related to the magnitude of the loop currents induced in the unit cells by the chiral pairing.
Theoretical Framework for Optical Detection:
- The paper establishes that the antisymmetric velocity kernel, $K_{\alpha\beta}(k)$ , plays a central role in describing both the dc and ac Hall responses. It is shown to be a generalization of the Berry curvature tensor, which applies strictly to the dc limit ( $\omega=0$ ).
- The results confirm that the strongest signal in the Hall channel for optical detection occurs at the frequency of the absorption peak corresponding to interband transitions.

Significance and Claims
The paper claims that its theoretical study contributes to the understanding of spectral properties required to interpret experimental observations of the polar Kerr effect (PKE) and magneto-optical Kerr effect (MOKE) in superconductors. By linking the integrated optical spectral weight of the ac Hall conductivity directly to the magnitude of loop currents, the authors propose that optical tools can be employed to experimentally study the presence and magnitude of these currents.

The authors note that while PKE signals are weak at high optical frequencies (e.g., 820 nm), they are expected to be stronger at lower frequencies (terahertz and sub-terahertz ranges) closer to the relevant electronic energy scales. The identification of the specific frequency of the absorption peak (twice the energy difference to the Dirac point) and its direct relation to loop currents provides a clear theoretical target for such low-frequency experimental measurements. The work does not propose new experimental setups but rather offers the necessary theoretical interpretation for existing and future measurements in the terahertz regime.

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