The observation of the Anomalous Hall Effect (AHE) in collinear antiferromagnets, despite their vanishing net magnetization, has been a subject of significant interest. The existence of a pseudovector that is responsible for AHE in collinear antiferromagnets is tightly connected with  the existence of the Dzyaloshinskii's invariant in the system. While all such invariants in collinear antiferromagnets have been understood for all symmetry classes, the microscopic mechanism of AHE in collinear antiferromagnets is still under research.

In this work we provide a microscopic basis for AHE in all relevant types of collinear and compensated antiferromagnets known to exhibit this effect. We show that such antiferromagnets are collinear weak ferromagnets and collinear ferrimagnets. Although, it is typical to think of weak ferromagnets as antiferromagnets with canted Néel order, in reality weak ferromagnets can have a collinear Néel order with the pseudovector consistent with the Dzyaloshinskii's invariant carried by either conducting fermions or magnons. In addition, we uncover a distinct class of ferrimagnets in which the direction of the pseudovector derived from the Dzyaloshinskii's invariant is parallel with the direction of the Néel order. The 

We show that main ingredients behind the AHE in collinear antiferromagnets are shown to be the spin-orbit coupling and spin-splitting of conducting fermions, both derived from the symmetries that allowed for the Dzyaloshinskii's invariant in the system.

There are more weak ferromagnets in Nature than the collinear antiferromagnets without the Dzyaloshinskii's invariant. Typical examples of weak ferromagnets are RuO$_2$, CrSb, CoF$_2$, NiF$_2$, $\alpha$-Fe$_2$O$_3$, MnTe, LuFeO$_3$, MnF$_2$, CoCO$_3$ and many more. Some weak ferromagnets are metallic and some insulating.

Figure 1: A minimal models of a collinear N\'{e}el ordered d-wave antiferromagnet on a square lattice, ferrimagnet on a square lattice and weak ferromagnet on rutile lattice, illustrating the magnetic (±m) and non-magnetic (green, purple and cyan) sites. This lattice geometry, along with specific symmetry breaking, underpins the emergence of AHE.

V. P. Golubinskii  and V. A. Zyuzin
JETP Letters 123, issue 5 (2026)

In quantum theory, both quantum mechanics and quantum field theory, symmetries of the action or Hamiltonian play a huge role. Symmetries can be either continuous and local, like gauge symmetry in quantum field theory, or discrete and global. In the case of quantum field theory, phase transitions occur when the parameters of the theory (temperature) change, violating the original symmetries. The properties of an effective theory in different phases turn out to be fundamentally different. The simplest example illustrating this is the 2D Ising model, in which a second-order phase transition occurs. In the high-temperature phase, the average magnetization is zero, $Z_2$ symmetry is preserved, while in the low-temperature phase, the $Z_2$ symmetry is broken and the average magnetization is nonzero.

In this paper, we study discrete quantum gravity defined on a 4D irregular lattice (simplicial complex). This theory is a mathematical model of the following physical idea: at extremely small scales, space-time exhibits granularity.

The model includes gravitational degrees of freedom as well as a Dirac fields $\Psi_{\cal V}$ and $\Psi^{\dagger}_{\cal V}$. The bare action of the theory has a global discrete $Z_4$ symmetry. The fermionic part of this action, which is bilinear relative to Dirac fields, can be rewritten in terms of effective fermionic variables $\Phi_{\cal V}$ and $\Phi^{\dagger}_{\cal V}$. These are linear combinations of the Dirac field and its conjugate at each vertex of the lattice. It can be said that the effective Dirac variables are analogous to the quasiparticle operators in superconductivity theory, constructed using Bogolyubov transformations.

A peculiarity of the construction of effective fermion operators in the lattice theory of gravity is the following fact: "particle" operators $\Phi_{\cal V}$ are invariant under $Z_4$ transformations, but their Hermitian conjugate operators $\Phi^{\dagger}_{\cal V}$ are transformed according to the simplest non-trivial representation of the group $Z_4$.

In the model under study, high-temperature expansion is correct at the highest temperatures.

As in the Ising model, it is established in this way that at the highest temperatures a $Z_4$-symmetrical phase occurs. As the temperature decreases, the $Z_4$ symmetry is broken down to its $Z_2$ symmetry subgroup. The order parameter here is the contribution to the lattice action, which transforms into the Hilbert-Einstein action in the long-wave limit. This contribution preserves $Z_2$-symmetry, but it breaks the $Z_4$-symmetry.

As the temperature decreases further, another phase transition occurs, breaking the $Z_2$-symmetry. The corresponding order parameter is a tetrad, the mean of which becomes nonzero. Spacetime emerges.

S.N. Vergeles
JETP Letters 123, issue 3 (2026)

In recent years, there has been a surge of interest in developing new approaches to improve the efficiency of existing computational methods or create fundamentally new paradigms. A particularly promising way is the shift from digital computing schemes to analog systems. One such physical system capable of emulating the structure of artificial neural networks are the diffractive neural networks. In this architecture, computation occurs passively and at the speed of light as a coherent wavefront propagates through spatially engineered diffractive layers, performing predetermined operations. However, successfully offloading computations onto an analog physical platform presents a significant challenge: it requires precise and accurate mathematical modeling that faithfully accounts for all the intricacies of the physical implementation.  Any discrepancy between the numerical model and the real-world system can lead to computational errors and degraded performance.

In this work, we directly address this challenge. We experimentally validate the correctness of our numerical modeling framework for a Fourier-diffractive neural network, in particular, we check the fidelity of using the fast Fourier transform to calculate the propagation of light in free space and its interaction with the lens and directly demonstrate the legitimacy of using the pixel of the phase mask as a weighting factor of the neural layer. Furthermore, we perform a comprehensive numerical investigation into how the exact geometry of the optical system influences the final accuracy of the computations. This study provides essential insights and design rules for bridging the gap between theoretical models and robust, high-fidelity physical mplementations of optical neural networks.

Konovalova A., Popkova A., Baluyan T., Fedyanin A.
JETP Letters 123, issue 2 (2026)