The quest for precise defect-field-driven microstructure-functionality manipulation represents a frontier in the design of next-generation ferroic materials. In ferroelastics, ferroelectrics, and ferromagnetics alike, nanoscale domain configurations-such as strain glass, relaxor, and spin glass states-govern critical functional properties, including colossal caloric effects, giant piezoelectricity, and ultralow hysteresis. However, a fragmented understanding for ferroic nanostructure-functionality relations persist across these material classes, largely because existing theoretical and computational descriptions lack a unified treatment of the local fields generated by defects. Based on phase-field modeling, a particularly powerful route to bridge microstructure and functionality, this review synthesizes cross-disciplinary research to establish a universal physical paradigm: defect-induced local suppression fields act as the common microscopic origin that disrupts long-range domain percolation, pins domain walls, and stabilizes functional nanodomains. We demonstrate a cohesive phase-field framework uniquely capable of such targeted microstructure-functionality manipulation, which integrates conventional energy descriptions with quantitatively derived local defect fields, obtained from atomistic simulations and experimental strain mapping, to predictively simulate and rationally tailor complex domain patterns. This review not only unifies the physics governing nanodomain formation and glassy transitions but also hopes to provide a predictive computational toolkit for high-performance applications in sensing, actuation, energy conversion, and solid-state cooling.

The work is devoted to a fundamental study of the magnetic properties of dilute magnetic semiconductors (DMSs). Cubic Sn 1-x-y Si x Mn y Te (A 4 B 6 ) and hexagonal layered In 1-x Mn x Se (A 3 B 6 ) were studied. Analysis of the temperature dependencies of dynamic magnetic susceptibility by fitting with Gaussian curves revealed four universal types of cluster magnetic subsystems (CMSs) in both DMSs, regardless of their crystal structure and the main mechanism of exchange interaction (RKKY in A 4 B 6 or Kramers in A 3 B 6 ). It is assumed that CMSs are formed by dynamic magnetic clusters (DMCs), whose structure and dimensionality (0D, 1D, 2D, and 3D) are determined by localized vibrational modes (LVMs) and the maximum range of thermal vibrations of Mn ions, which gives them the character of quasistationary formations similar to standing waves. Each CMS is characterized by a set of five critical temperatures (T F , T C , Θ, T K , β), of which T K and β are introduced for the first time as analogues of T C and Θ in the region of spin and cluster glasses. This approach made it possible to reveal a symmetry between two types of magnetic disorder: "overheated" paramagnetic (PM) and "supercooled" spin-glass and cluster-glass. The overlap of Gaussian curves corresponding to different CMSs explains the coexistence of spin glass (SG), cluster glass (CG), and ferromagnetic (FM) phases, which we previously observed in Ge 1-x-y Sn x Mn y Te (A 4 B 6 ). The effect of annealing on the rearrangement of magnetic subsystems in In 1-x Mn x Se is shown. The concept of integral magnetic susceptibility (IMS) is introduced as a useful tool for the quantitative study of CMSs dynamics. The idea of the possibility of co-using ordered (FM) and disordered magnetic phases (SG and CG) in new functional materials for spintronics is put forward.

A continuous nonlinear optimization perspective on the Spin Glass Problem

Glasses have complex energy landscapes and exhibit nonequilibrium aging dynamics. Here, we propose a generalized trap model for activated aging based on a key static property of the energy landscape: the distribution of energy barriers. Our theory predicts that, upon cooling, weak ergodicity breaking (WEB) in quenching dynamics occurs before strong ergodicity breaking in equilibrium dynamics. Furthermore, the theory indicates that the characteristic size of activation clusters can be deduced from the logarithmic decay of the time-correlation function. We rigorously test the model's assumptions and predictions using the simplest spin glass model, the random energy model. The predicted aging behavior is also universally observed in paradigmatic structural glasses, including the Weeks-Chandler-Andersen (WCA) model and amorphous silica. Applying our framework to the WCA model allows us to extract a static length from the nonequilibrium dynamics, extending its observable growth range from a mere factor of 2 to 3 to a full order of magnitude and providing supportive evidence for the random first-order transition scenario. Last, we propose a unified ergodic-WEB phase diagram for aging dynamics in general glassy systems.

We introduce a spectral approach to characterizing the three-dimensional Edwards-Anderson spin glass. By analyzing the eigenvalue statistics of overlap matrices constructed from two-dimensional cross sections, we identify a crossover from the Wigner semicircle law at high temperatures toward a Gaussian distribution, which is consistently attained near the spin-glass critical point. Visible for different distributions of the random coupling, the Gaussian distribution can potentially serve as a robust spectral indicator of criticality. Remarkably, the spectral density is well described by Tsallis statistics, with the entropic index q evolving from q=-1 (semicircle, T=∞) to q=1 (Gaussian) at T_{c}, revealing a statistical structure inside the paramagnetic phase. We find q≤1 within numerical precision. While the local level statistics remain consistent with Gaussian orthogonal ensemble statistics, reflecting standard level repulsion, the temperature dependence appears mainly in the global spectral density. Our results present spectral statistics as a computationally efficient complement to multireplica correlator methods and provide a new perspective on cooperative and critical phenomena in disordered systems.

Random lasers (RLs) differ fundamentally from conventional lasers by replacing engineered optical cavities with disorder-mediated multiple scattering feedback, enabling cavity-free emission through photon interactions in disordered gain media. RLs have demonstrated many application potentials in the domains of optics and optoelectronics because of their low coherence, multidirectionality, and multiple wavelengths. However, the principle and procedure of RL generation, as well as the method for achieving low-threshold emission, still present difficulties. This study demonstrates the replica symmetry breaking (RSB) in coherent RL systems utilizing Enteromorpha prolifera (EP), a marine bio-scatterer exhibiting multiscale photonic architecture. The transition from the photon paramagnetic phase to the spin glass phase is seen with the increase of the pump energy in this random liquid phase system. The multiscale structure of EP enables efficient dye adsorption while enhancing photon scattering to amplify optical feedback, thereby achieving stable coherent random lasing with a low threshold. Power Fourier transform analysis was employed to calculate the effective optical cavity length of the RL system, depicting the coherent lasing dynamics and determining the free spectral range. The bio-enabled RL system demonstrated remarkable spatial coherence control with good speckle-free imaging potentials through quantitative analysis of speckle contrast and derived mode numbers, revealing pump-energy-dependent correlations that elucidate intermodal interaction dynamics. These findings advance the fundamental understanding of disordered photonic systems while establishing a sustainable framework for engineering bio-derived RL sources with tailored coherence properties.

The Ga-rich region of the Fe–Ni–Ga ternarysystemwas investigated, by exploring a line of compositions FexNi3–xGa4, with 0.5 ≤ x ≤ 2.5. The single-phasecubic material was found only at the composition Fe1.5Ni1.5Ga4 and its immediate vicinity, representinga new phase in the Fe–Ni–Ga diagram. The homogeneityrange of this phase was estimated by additionally exploring a setof compositions FexNiyGaz around the central compositionFe1.5Ni1.5Ga4. The structural modelwas constructed based on the structure of the binary Ni3Ga4 parent phase, which crystallizes in the cubic Ia3̅d space group. We have consideredthat by substituting Fe for Ni, the Ia3̅d structure is preserved, with the Fe and Ni being statisticallydistributed at their 48g Wyckoff site. The possibilityof a symmetry-reduced chiral structural model I4132 driven by chemical ordering of Fe and Ni cannot be entirelyruled out on the basis of the crystallographic study. The magneticstudy of the Fe1.5Ni1.5Ga4 phasehas revealed that the material forms a spin glass phase below thespin freezing temperature Tf ≈9 K. Since the spin glass ordering of the Fe and Ni magnetic momentsis compatible with their random distribution, the magnetic study supportsthe disordered cubic Ia3̅d model.

Probabilistic computing has gained attention for solving combinatorial optimization problems (COPs), mainly using the Ising model, which may not be suitable for complex COPs. Instead, this work proposes a multi-state probabilistic computing system based on the Potts model using stochastic threshold switching floating-body metal-oxide-semiconductor field-effect transistors (FB-MOSFETs) as the multi-state probabilistic bits (p-bits) to solve challenging COPs. The system employs drain voltage sharing and a one-hot sampling method to achieve controllable probabilistic behavior and scalable annealing. Experimental validations on spin glass and max-4-cut problems demonstrate that the system efficiently samples a tunable Boltzmann distribution while converging faster than traditional methods. Comparative analyses further highlight superior energy efficiency and decreased time-to-solution, underscoring the potential of multi-state probabilistic computing for large-scale, complex COPs using only MOSFET devices.

Linear-time classical approximate optimization of cubic-lattice classical spin glasses

Networks of Hebbian networks: more is different.

The 2024 Nobel Prize in Physics was awarded for pioneering contributions at the intersection of artificial neural networks (ANNs) and spin-glass physics, underscoring the profound connections between these fields. The topological similarities between ANNs and Ising-type models, such as the Sherrington–Kirkpatrick model, reveal shared structures that bridge statistical physics and machine learning. In this perspective, we explore how concepts and methods from statistical physics, particularly those related to glassy and disordered systems like spin glasses, are applied to the study and development of ANNs. We discuss the key differences, common features, and deep interconnections between spin glasses and neural networks while highlighting future directions for this interdisciplinary research. Special attention is given to the synergy between spin-glass studies and neural network advancements and the challenges that remain in statistical physics for ANNs. Finally, we examine the transformative role that quantum computing could play in addressing these challenges and propelling this research frontier forward.

Temperature chaos may lead to many thermodynamic states in spin glasses

We propose a practical hybrid decoding scheme for the parity-encoding architecture. This architecture was first introduced by N. Sourlas [Phys. Rev. Lett. , 070601 (2005)] as a computational technique for tackling hard optimization problems, especially those modeled by spin systems such as the Ising model and spin glasses, and reinvented by W. Lechner, P. Hauke, and P. Zoller [Sci. Adv. , e1500838 (2015)] to develop quantum annealing devices. We study the specific model, called the SLHZ model, aiming to achieve a near-term quantum annealing device implemented solely through geometrically local spin interactions. Taking account of the close connection between the SLHZ model and a classical low-density parity-check (LDPC) code, two approaches can be chosen for the decoding: (1) finding the ground state of a spin Hamiltonian derived from the SLHZ model, which can be achieved via stochastic decoders such as a quantum annealer or a classical Monte Carlo sampler; or (2) using deterministic decoding techniques for the classical LDPC code, such as belief propagation and a bit-flip decoder. The proposed hybrid approach combines the two approaches by applying bit-flip decoding to the readout of the stochastic decoder based on the SLHZ model. We present simulations demonstrating that this approach can reveal the latent potential of the SLHZ model, realizing the soft-annealing concept proposed by Sourlas.

The double perovskite compound Nd₂NiMnO₆ has garnered considerable attention due to its intriguing magnetic properties, particularly the exchange bias effect, which is of great relevance for spintronic and magnetic storage applications. This review provides a comprehensive and comparative analysis of the exchange bias phenomenon observed in various structural forms of Nd₂NiMnO₆ including bulk, nanoparticles, and thin films, emphasizing the distinctions in their origin, magnitude, and tunability. The discussion systematically explores the influence of structural ordering, antisite disorders, interfacial coupling, magnetic anisotropy, grain size, and strain effects on exchange bias behavior. In bulk Nd₂NiMnO₆, exchange bias arises mainly from ferromagnetic/antiferromagnetic interfacial interactions originating due to antisite disorder induced antiphase boundaries. In NdSrNiMnO₆ nanoparticles, exchange bias is described by a core–shell model, where antiferromagnetic cores are surrounded by spin glass shells, and uncompensated spins at the interface contribute to the observed bias. In Nd₂NiMnO₆ thin films, grown on mismatched substrate (LaAlO₃), the exchange bias effect is primarily governed by strain induced antisite disorders leading to ferromagnetic/spin glass interfacial coupling. By correlating these mechanisms across different morphologies, this review highlights how antisite disorder and microstructural engineering critically dictate the exchange bias behavior in Nd₂NiMnO₆ systems. The insights presented here not only deepen the understanding of exchange bias in double perovskite oxides but also offer guidelines for tailoring their magnetic properties for advanced spintronic and multifunctional device applications.

Quantum phase transitions: from ordered magnets to spin glasses.

Spin glass models and market volatility: A multiple threshold nonlinear autoregressive distributed lag approach

Energy landscapes of spin glasses on the dice lattice

Decorated clusters and geometrical frustration in cluster spin glass: A random graph approach

Polaritonic chemistry has garnered increasing attention in recent years due to pioneering experimental results, which show that site- and bond-selective chemistry at room temperature is achievable through strong collective coupling to field fluctuations in optical cavities. Despite these notable experimental strides, the underlying theoretical mechanisms remain unclear. In this focus review, we highlight a fundamental theoretical link between the seemingly unrelated fields of polaritonic chemistry and spin glasses, exploring its profound implications for the theoretical framework of polaritonic chemistry. Specifically, we present a mapping of the dressed many-molecules electronic-structure problem under collective vibrational strong coupling to the analytically solvable spherical Sherrington-Kirkpatrick (SSK) model of a spin glass. This mapping uncovers a collectively induced spin glass phase of the intermolecular electron correlations, which could provide the long sought-after seed for significant local chemical modifications in polaritonic chemistry. Overall, the qualitative predictions made from the SSK solution (e.g., dispersion effects, phase transitions, differently modified bulk and rare event properties, heating, etc.) agree well with available experimental observations. Our connection not only demonstrates the relevance of moving beyond the dilute gas approximation, where the Fermionic nature of the electrons becomes an essential ingredient, but it also paves the way for novel computational strategies to quantify the subtle chemical characteristics of the cavity-induced spin glass phase. Moreover, our mapping provides a versatile framework to incorporate, adapt, and explore a wide range of spin glass concepts within polaritonic chemistry. Ultimately, the connection also offers fresh insights into the applicability of spin glass theory beyond condensed matter systems suggesting novel theoretical directions such as spin glasses with explicitly time-dependent (random) interactions.

The analytical solution to the out-of-equilibrium dynamics of mean-field spin glasses has profoundly shaped our understanding of glassy dynamics, which take place in many diverse physical systems. In particular, the idea that during the aging dynamics, the evolution becomes slower and slower but keeps wandering in an unbounded space (a manifold of marginal states), thus forgetting any previously found configuration, has been one of the key hypotheses to achieve an analytical solution. This hypothesis, called weak ergodicity breaking, has recently been questioned by numerical simulations and attempts to solve the dynamical mean-field equations (DMFEs). In this Letter, we introduce a new integration scheme for solving DMFEs that allows us to reach very large integration times, t=O(10^{6}), in the solution of the spherical 3+4-spin model, quenched from close to the mode coupling temperature down to zero temperature. Thanks to this new solution, we can provide solid evidence for strong ergodicity breaking in the out-of-equilibrium dynamics on mixed p-spin glass models. Our solution to the DMFE shows that the out-of-equilibrium dynamics undergo aging, but in a restricted space: the initial condition is never forgotten, and the dynamics take place closer and closer to configurations reached at later times. During this new restricted aging dynamics, the fluctuation-dissipation relation is richer than expected.