Altermagnetic $\alpha$-MnTe with Néel vector along the $y$-axis exhibits a finite anomalous Hall conductivity (AHC) and weak ferromagnetism along the $z$-axis. As already demonstrated in the bulk, there is the breaking of the C$_6$ symmetry by the in-plane Néel vector, leaving a C$_2$-type magnetic symmetry. The surface of $\alpha$-MnTe breaks the C$_2$, leaving only a time-reversed mirror symmetry with respect to the $x=0$ plane. Therefore, we demonstrate that on the surface, the interplay between breaking of the crystal symmetry and Néel vector orientation produces a reduction of the space group from hexagonal P6$_3$/mmc to orthorhombic Amm2. As a result, the surface exhibits not only a polar distortion along the $z$-axis, but also a polar distortion and a weak ferrimagnetism along the $y$-axis. To describe the surface of MnTe in an accessible way, we simplify the problem and examine the effect of the in-plane electric field in bulk MnTe. Moreover, as a doped ionic semiconductor, the properties of MnTe can be influenced by lattice polarization under an applied electric field. We investigate the interplay between the intrinsic anomalous Hall effect and lattice polarization, showing that polarization effects can substantially affect the AHC. Since the electric field breaks inversion symmetry, this contribution from the lattice polarization coexists with the non-linear anomalous Hall effect, highlighting the rich transport phenomenology of altermagnets.

Altermagnetic magnons in crystalline materials exhibit momentum-dependent splitting whose nodal structure and chiral character are governed by the point-group symmetry of the magnetic sublattice rotation. Here, we demonstrate the first synthetic realization of altermagnetic magnonics in a continuum platform composed of antiferromagnetically coupled ferromagnetic films with alternating in-plane exchange anisotropies, showing that the key signatures of altermagnetic magnonics emerge beyond the crystalline setting. Solving the linearized Landau-Lifshitz equation within a dipole-exchange framework, we show that this architecture reproduces the characteristic momentum-dependent splitting, nodal directions, and anisotropic isofrequency contours of A-type altermagnets. Long-range dipolar interactions qualitatively reconstruct this exchange-driven spectrum by lifting the nominal nodal degeneracy, hybridizing opposite-chirality modes, and producing a finite, thickness-dependent wave-vector splitting along directions that are nodal in the exchange-only limit. Extending the bilayer to finite multilayers reveals that synthetic altermagnetism undergoes a parity-dependent reconstruction that separates surface and bulk altermagnetic excitations. These results establish altermagnetic magnon phenomenology as an engineerable collective response of dipole-exchange multilayers beyond microscopic crystal symmetries.

Half-metallic ferromagnets exhibit a gap in the density of states for one spin projection while remaining gapless for the opposite spin. We show that in helimagnets an unusual half-metallic state can exist, where the spin projection that experiences the gap is determined by the direction of the wave vector. This state originates from the nontrivial topology of the band structure, specifically from the dispersion forming a multi-sheeted covering over the Brillouin zone. We present two-dimensional tight-binding models for $p$-wave and $f$-wave half metals. These structures can be realized in crystalline and van der Waals systems. The complex band structure of the unusual $p$-wave half metal in nanostructures is also discussed. In a quantum well, a standing wave with a helical spin structure is formed, and a persistent spin current exists.

Combinatorial memory is a class of memory in which information is encoded in the set of paths through a structured mesh. In this work, we introduce a systematic encoding framework, referred to as the Color-Rule-Function (CRF) approach, for representing information in combinatorial memory. The method consists of four key steps: selecting a sequence of paths in the mesh, assigning values (e.g., colors) to each cell, defining a set of rules based on the values encountered along each path, and constructing a Boolean function that determines the state of each path.. The coding procedure is illustrated by several examples. The design space scales of the CRF scale fundamentally faster compared to conventional memory. This apparent advantage arises from the use of rule-based and functional representations but is accompanied by increased hardware complexity. A possible hardware realization of the CRF framework is discussed. Importantly, the hardware overhead can be substantially reduced through the use of customized modules. The examples of the customized design are described in the text. The combination of CRF coding with customized module design may lead to a practical advantage in data storage density. According to the estimates, the data storage density may exceed Exabit per centimeter squared. A key problem that requires further investigation is related to the minimum Hamming distance between an arbitrary target bit sequence and the closest sequence realizable within the CRF framework under fixed hardware constraints.

These lecture notes are intended as a coherent introduction to conformal field theory in general, and composite operators in particular, through a semiclassical framework for computing scaling dimensions, with emphasis on operators of the form $\phi^n$. In doing so, they aim to fill a gap in the literature and to help decode some of the relevant concepts. The physical idea is that at large $n$ an (heavy) operator creates a highly occupied state. Through the state-operator correspondence, this state lives on the cylinder $\mathbb{R}\times S^{d-1}$, and its scaling dimension is the corresponding energy of the theory on the cylinder. The notes are organized as a self-contained route from conformal symmetry to semiclassical dynamics. Part I reviews the conformal group, primary operators, radial quantization, the state-operator correspondence, and operator mixing. Part II builds the semiclassical framework, first in the free scalar theory, where the dimension of $\phi^n$ is recovered in three independent ways, and then through the double-scaling limit, the action variable, and Bohr-Sommerfeld quantization. Part III develops the general machinery of periodic saddles, Floquet theory, fluctuation determinants, the Gel'fand-Yaglom method, and the Gutzwiller trace formula. Part IV applies the framework to the $O(N)$ $\phi^4$ theory in $d=4-\epsilon$ at the Wilson-Fisher fixed point, deriving the classical elliptic solution, the Lamé fluctuation spectrum, the zero modes, and the one-loop contribution to the large-$n$ scaling dimensions. Beyond the explicit computation, the notes emphasize the role of composite operators as probes of collective sectors of quantum field theory, with extensions to gauge theories, conformal windows, and asymptotically safe field theories.

Identifying topological invariants are endowed with profound physical connotations in the fields of condensed matter physics. However, they are often limited by the failure of standard theorems and high computational costs. Traditional machine learning methods typically treat this problem as a black-box regression, which fails to learn the underlying mathematical structures of lattices. To this end, we propose a straightforward yet powerful multimodal model that fundamentally improves topological invariants discovery through two key methodological innovations. First, instead of feeding raw data into a standard network, our model introduces a dual-track alignment mechanism. This mechanism treats eigenvalues as context sequences and eigenvectors as matrix structures, enabling the model to naturally capture the interrelated algebraic and geometric properties of topological states. Second, to resolve the common problem where deep learning models make numerical mistakes in exact mathematical calculations, we use a tool-integrated reasoning paradigm. Within this paradigm, the neural network acts as a logical controller to calculate discrete topological indices. This design eliminates the uncertainty of deep learning, ensuring zero-error calculations from continuous physical data. We demonstrate that our model can accurately reconstruct complex three-dimensional generalized Brillouin zones and achieve 97% accuracy in calculating non-Bloch Chern numbers in long-range transition systems. By combining the flexible logic of multimodal artificial intelligence with the rigorous precision of symbolic computation, this work provides a reliable and highly versatile tool for exploring exotic topological phases.

We study discrete time-crystalline (DTC) phases in one-dimensional spin-1/2 chains with power-law-graded Ising interactions under periodic Floquet driving. By generalizing Stark localization to power-law-graded Ising interaction profiles, we identify robust period-doubled dynamics across a wide range of interaction exponents, stabilized by the interplay between coherent driving and spatially varying coupling. Within the DTC phase, the energy stored in the system, interpreted as a quantum battery, increases superlinearly with system size, although no scaling advantage persists in normalized power. Beyond energy storage, we demonstrate that the DTC phase supports enhanced quantum sensing. The quantum Fisher information associated with estimating timing deviations in the drive scales superextensively with system size, surpassing the Heisenberg limit. The degree of quantum advantage can be tuned by varying the interaction exponent, though DTC behavior remains robust throughout. Our results position power-law-graded Ising interacting Floquet systems as robust platforms for storing quantum energy and achieving metrological enhancement.

Recently, it has been revealed that a variety of novel phenomena emerge in hyperbolic spaces, while non-Hermitian physics has significantly enriched the landscape of condensed matter physics. Building on these developments, we construct a geodesic-based method to study the non-Hermitian skin effect (NHSE) in non-reciprocal hyperbolic lattices. Additionally, we develop a geodesic-periodic boundary condition (geodesic-PBC), akin to the Euclidean periodic boundary condition (PBC), that complements its open boundary condition. Importantly, we find that the non-reciprocal directionality within a hyperbolic regular polygon and the geodesic-based boundary determine the spectral sensitivity, and hence, the NHSE. Unlike in Euclidean models, however, we must utilize boundary localization to distinguish non-trivial skin modes from their trivial boundary counterpart due to the extensive boundary volume of hyperbolic lattices. We also relate the spatial density profile with the finite-size scaling of hyperbolic lattices. These aspects highlight the profound impact of hyperbolic geometry on non-Hermitian systems and offer new perspectives on the intricate relationship between the geometric characteristics of hyperbolic lattices and non-Hermitian physics.

We propose a quantum bosonic qubit model on a fcc lattice that realizes the canonical source structure of mimetic dark matter as a defect of a rank-two lattice Gauss law. The standard contribution from general relativity is implemented similarly to previous works in the literature, while the mimetic sector modifies the constraint equations through additional source terms. Different theories such as mimetic dark matter, vector mimetic dark matter, and tensor-vector-scalar models are implemented on the lattice. In all these cases, a generalized Gauss law incorporates an additional Gauss-law (topological) defect depending on the type of generalization, but always fitting into the structure of the defects from the general relativity contribution. The mimetic constraint is treated in its full ADM form, retaining the normal derivative of the scalar field, and the known ghost and gradient instabilities of the minimal continuum mimetic theory are summarized. The lattice construction is therefore presented as a formal realization of the canonical source structure rather than as a complete cosmological model.