Papers

Next to the directed percolation (DP) universality class, parity conserving directed percolation (pcDP; also called parity conserving branching annihilating random walks, pcBARW) is the second-most important model with an absorbing state transition. Its distinction from ordinary DP is that particle number is conserved modulo 2, which implies that there are two distinct sectors in systems with a finite initial number of particles: Realizations with even and odd particle numbers show different scaling behaviors, and systems in the odd sector cannot die. An intriguing feature of pcDP it is that some of its critical exponents seem to be very simple rational numbers. The most prominent is the one describing the average number of particles (or active sites) in the even sector, which is asymptotically constant. In contrast, the dynamical critical exponent (which is the same in both sectors) seems not close to any simple rational. Finally, the order parameter exponent $\beta$ (which is also the same in both sectors) is, according to the most precise previous simulations, rather close to 1, but incompatible with it. We present high statistics simulations which clarify this situation, and which indicate several other intriguing properties of pcPD clusters. In particular, we find that all exponents which were close to rationals are even closer, and $\beta = 1.000$ with the error in the next digit.

The fuzzy-sphere regularization is an emerging numerical and theoretical technique for studying conformal field theories (CFTs). In this paper, we apply it to the $O(N)$ vector model, one of the most prominent theories for critical behavior in three space-time dimensions. We construct a model that realizes the $O(N)$ Wilson-Fisher and free-scalar CFTs for general $N$. For $N=2,3,4$, we present numerical evidence including the operator spectra and correlation functions in agreement with conformal symmetry and conformal bootstrap results.

Motivated by recent breakthrough studies of wave hyperbolicity in extremely anisotropic natural materials and artificial composites, we investigate the radiation pattern of a localized emitter in a hyperbolic medium. Since the emission of a point source is associated with the Fourier transform of the iso-frequency contours of a medium, we derive and analyze the properties of the Fourier transform of hyperbolic dispersion, which sheds light into the emission properties in the presence of hyperbolic bands. Our analysis leads to a generalized form of Huygens' principle for hyperbolic waves, connecting to the emergence of negative refraction and focusing with hyperbolic media. We also highlight the occurrence of aliasing artifacts in polariton imaging. More broadly, our findings provide analytical tools to model polariton propagation in materials with extreme anisotropy, and may be applied to several other physical platforms featuring hyperbolic responses, from astrophysics to seismology.

Two-dimensional (2D) materials are attractive platforms for realizing high-performance quantum bits (qubits). However, radio-frequency (RF) charge detection, which is a key technique for qubits readout, remains challenging in such systems. We demonstrate RF reflectometry with impedance matching for high-resistance quantum dot devices based on bilayer graphene and molybdenum disulfide. By integrating a tunable strontium titanate (SrTiO3) varactor into a resonant circuit, we achieve nearly perfect impedance matching, enabling sensitive charge detection. The demodulated RF signal clearly shows Coulomb oscillations, and the SrTiO3 varactor exhibits robustness against both magnetic fields and voltage noise on the varactor. Our results establish SrTiO3 varactors as effective tunable matching components for RF reflectometry in high-resistance 2D material quantum devices, providing a foundation for high-speed qubits readout.

Using a rotationally invariant formulation of spin-lattice coupling, we derive a rigorous definition of the chemical potential for magnon-polaron quasiparticles in collinear ferromagnets (FMs) and antiferromagnets (AFMs), valid when magnetoelastic scattering equilibrates magnons and acoustic phonons on timescales much shorter than those associated with quasiparticle-nonconserving relaxation processes. While our microscopic framework applies to generic magnon-phonon interactions, here we focus on high-symmetry crystals where the two transverse acoustic modes form a degenerate doublet. This doublet can combine into circularly polarized phonons, making the chiral selectivity of the coupling manifest: the FM magnon mode hybridizes only with the co-rotating phonon, whereas in collinear AFMs each magnon branch of opposite handedness couples to the phonon of the same chirality. We show that, in both FM and AFM systems, the nonequilibrium magnon-polaron gas is governed by a single chemical potential conjugate to the conserved axial angular momentum. In FMs, the two hybrid branches in the co-rotating sector share this chemical potential, weighted by their magnonic fractions; in AFMs, the four magnon-polaron branches split into two chiral sectors that carry opposite angular momenta and couple with opposite sign to the same chemical potential. Building on this microscopic thermodynamic framework, we formulate a Boltzmann transport theory for magnon-polarons and derive compact expressions for angular-momentum and heat currents that interpolate continuously to the decoupled regime and reproduce the phenomenological magnon-polaron transport framework underlying previous spin Seebeck analyses.

We demonstrate that Shannon's information entropy and the thermodynamic entropy of Boltzmann and Gibbs are quantitatively equivalent for real condensed-matter systems. By interpreting atomic configurations as information sources, we compute entropy directly from the compressibility of molecular-dynamics trajectories, without physical partitioning or empirical modeling. A custom lossy-compression algorithm measures the minimum number of bits required to describe a microstate at finite precision, and this bit count maps exactly to thermodynamic entropy through the Shannon-Boltzmann relation. The method reproduces benchmark entropies for metals, semiconductors, oxides, and refractory ceramics in both solid and liquid phases, establishing information as the fundamental quantity underlying thermodynamic disorder. This equivalence unifies information theory and statistical mechanics, providing a general and computationally efficient framework for determining entropies and free energies directly from atomic data.

Development of spintronic and quantum computing devices increases demand for efficient, energy saving method of spin manipulation at molecular scale. Polyoxovanadate molecular magnets being susceptible to both electric and magnetic fields may serve here as a good base material. In this paper two isostructural anions [V$_{12}$As$_8$O$_{40}$(HCO$_2$)]$^{n-}$ (with $n=3,5$) featuring two different mixed-valence states with itinerant and localized valence electrons are studied. The impact of the electric field on their magnetic properties is investigated by means of two complementary methods informed by magnetic measurements: effective Hamiltonian calculations and density functional theory. It is demonstrated that the magnetoelectric effect in theses molecules is induced mostly by relocation of itinerant electrons, is highly anisotropic, depends on the valence state and can be detected even at room temperature. These findings can pave the way to practical applications in which an electric field control over spin state is required.

We present an experimental method to determine the refractive index of $Al_{x}Ga_{1-x}As$ (x = 0.0 - 0.5) from 300 K to 4 K across the 500 - 1100 nm wavelength range. The values are extracted from spectroscopically observed microcavity resonances in thin $Al_{x}Ga_{1-x}As$ membranes embedded between fully and partially reflective gold mirrors. Refined Varshni and Paessler models are used to describe temperature-dependent bandgap shifts and material composition. By tracking resonance shifts and benchmarking against finite-difference time-domain simulations, we derive the dispersive optical response with high precision. This yields a quantitatively improved analytical expression for the refractive index of $Al_{x}Ga_{1-x}As$ matching the experimental results with a coefficient of determination as high as $R^2=0.993$, enabling accurate modeling near the band edge at cryogenic temperatures. The method is straightforward and broadly applicable to other semiconductor systems, offering a valuable tool for the design of micro photonic devices such as quantum light sources.

Quantum entanglement offers powerful opportunities for enhancing measurement sensitivity beyond classical limits, with optical atomic clocks serving as a leading platform for such advances. This chapter introduces the principles of entanglement-enhanced quantum metrology and explores their applications to timekeeping. We review the theoretical framework of quantum phase estimation, comparing frequentist and Bayesian approaches, and discuss paradigmatic entangled states such as spin-squeezed and GHZ states. Particular emphasis is placed on the challenges posed by decoherence, which constrain the practical advantages that can be realized in large-scale devices. The discussion then turns to frequency estimation in atomic clocks, highlighting how experimental constraints shape the translation of abstract quantum limits into real performance gains. Finally, we outline emerging directions of contemporary quantum metrology. Together, these developments underscore the increasingly close interplay between quantum information processing and precision metrology.

PbTiO$_3$ is a ferroelectric perovskite semiconductor with favourable electronic and optical properties, making it suitable for a wide range of applications, including photo-catalysis and (opto)electronic devices. Despite its relevance, an accurate ab-initio description of the optical absorption spectrum and of the impact of ferroelectric distortion on the excitonic properties is still lacking. We combine $G_0W_0$ and Bethe-Salpeter Equation calculations to investigate the electronic and optical properties of PbTiO$_3$, tracking the evolution of its excitonic spectrum along the transition from the cubic paraelectric to the tetragonal ferroelectric phase. As the polar distortion increases, the first absorption peak of the cubic phase splits into two distinct features due to symmetry breaking, which partially lifts the degeneracy of the underlying excitonic state. Crucially, the distortion further introduces an in-plane/out-of-plane anisotropy in the spectra and controls the energy separation between the resulting excitonic branches. These findings highlight the potential for tuning the optical absorption properties of PbTiO$_3$ via the application of an external electric field.

We demonstrate that the recent observations of spatial oscillations in the energy position of the superconducting coherence peaks in the cuprate, transition metal dichalcogenide, iron-based and heavy-fermion superconductors are consistent with the possible presence of a pair-, charge- or spin-density-wave phase. We show that for all three cases, the spatial oscillations of the superconducting order parameter, $\Delta({\bf r})$ can be imaged via the critical Josephson current, $I_c({\bf r})$ measured in Josephson scanning tunneling spectroscopy experiments. Finally, we show that the spatial oscillations of the density waves and of $\Delta({\bf r})$ exhibit relative phase shifts of $\Delta \phi = 0$ or $\pi$ for a charge-density wave, and $\Delta \phi =\pi/2$ for the spin-density wave.

The metallic kagome compound Sc$_3$Mn$_3$Al$_7$Si$_5$ has attracted attention as a candidate platform where geometric frustration and itinerant electrons may cooperate to stabilize a quantum-disordered magnetic ground state. Here, we combine bulk thermodynamic probes, low-noise FIB-device transport, and comprehensive $^{55}$Mn Nuclear Magnetic Resonance (NMR) measurements to elucidate the low-temperature spin dynamics of this system. The bulk data reveal strongly reduced magnetic entropy, a negative magnetoresistance arising from spin scattering, and field-dependent transport indicates the spin fluctuations, while showing no signatures of long-range magnetic order. NMR provides a direct local view of the correlated Mn moments: the nuclear spin-spin relaxation $T_2$ exhibits a pronounced low-temperature enhancement driven by an indirect internuclear coupling through electronic spin fluctuations, whose temperature and distance dependence point to partially gapped low-energy spin excitations. The spin-lattice relaxation rate $T_1^{-1}$ displays a Hebel-Slichter-like coherence peak near \SI{10}{K}, coincident with the resistivity crossover and a subtle heat-capacity anomaly, indicating the formation of short-range spin-singlet correlations. Together, our results demonstrate that Sc$_3$Mn$_3$Al$_7$Si$_5$ hosts an unconventional correlated state dominated by frustrated, gapped spin dynamics, placing it among the rare metallic kagome systems proximate to a quantum spin liquid.

Vapor-pressure-mismatched materials, such as transition metal chalcogenides, have emerged as key electronic, photonic, and quantum materials. Hybrid pulsed laser deposition (hPLD) has become a preferred method for epitaxial or textured growth of these materials; however, unintentional carbon (C) incorporation remains a persistent concern, particularly when using organic chalcogen precursors as safer alternatives to toxic hydrides. The mechanisms governing C incorporation and its impact on film growth and properties in hPLD remain poorly understood. Here, we investigate the influence of C-containing side products generated from organosulfur precursor pyrolysis on ZnS, BaTiS$_3$, and TiS$_2$ thin films grown by hPLD using tert-butyl disulfide (TBDS). Structural characterization via X-ray diffraction and atomic force microscopy, combined with secondary ion mass spectrometry, is used to systematically examine the effects of growth temperature and TBDS partial pressure on film morphology, crystallinity, and C incorporation. Optimal growth temperatures of 400{\deg}C, 500{\deg}C, and 700{\deg}C are identified for ZnS, TiS$_2$, and BaTiS$_3$, respectively. Growth above or below these temperatures leads to increased C incorporation at both the interface and within the film, correlating with degraded texture. In contrast, highly textured films exhibit minimal C content, comparable to films grown without TBDS. For TiS$_2$, C incorporation depends strongly on TBDS pressure, with 10$^{-1}$ Pa identified as the optimal pressure for minimizing contamination. At higher pressures, loss of preferential texture is observed, likely due to C graphitization poisoning the interface and bulk. These results provide new insight into process-induced C impurities in hPLD-grown chalcogenide thin films and have important implications for sulfide-based thin film technologies.

We present a thermodynamically consistent theoretical framework for lyotropic liquid crystals (LCs) based on the GENERIC (General Equation for the Non-Equilibrium Reversible-Irreversible Coupling) formalism. This formalism ensures conservation of energy and production of entropy, while coupling concentration, momentum balance, and liquid crystalline order. Starting from a binary nematic-isotropic mixture, we derive a theory for these key variables, which is then extended to multi-component systems. The binary equations are solved numerically using a Julia-based solver that relies on an upwind finite-difference scheme, enabling stable and efficient simulations capable of handling multiple time scales while satisfying fundamental mathematical constraints. The results of simulations are consistent with experimental observations of topological core defects in chromonic LCs, as well as flow-driven droplet shape transitions under Couette and Poiseuille flows. This work provides a platform for simulations of multi-component lyotropic LCs that can be extended to systems with multiple interfaces, active materials, and materials subject to external fields.

Vanadium dioxide (VO$_2$) exhibits hysteresis in resistance while undergoing a thermally driven insulator-metal transition (IMT). Understanding the nonequilibrium effects in resistance is of great interest, as VO$_2$ is a strong candidate for brain-inspired computing, which is more energy efficient for AI tasks compared to traditional computing. Accurate models of the connection between microscopic and macroscopic transport properties and microscopic imaging of VO$_2$ will allow us to better utilize VO$_2$ in future applications. However, predictions of macroscopic resistance of VO$_2$ that quantitatively match observations using spatially resolved data have not yet been achieved. Here, we demonstrate an accurate prediction of the macroscopic resistance of VO$_2$ throughout the entire temperature range of interest, by developing a multiscale resistor network model incorporating the assumption of fractal sub-pixel structure of the optical data, where the configuration of insulating and metallic domains within each pixel are drawn from the random field Ising model near criticality. This strongly indicates that the observed fractal, power law structure of metallic and insulating domains extends down to much smaller length scales than the current record for experimental resolution of this system, and that the two-dimensional random field Ising model near criticality is a suitable model for describing the metal and insulator patches of VO$_2$ down to scales that approach the unit cell.

We investigate the spin-1/2 $J_1-J_2$ triangular-lattice Heisenberg antiferromagnet in a magnetic field by combining large-scale density matrix renormalization group (DMRG) simulations with self-consistent spin-wave theory. The resulting field-coupling phase diagram reveals that quantum fluctuations stabilize coplanar order across the entire parameter range, giving rise to a characteristic sequence of magnetization plateaux. Near the quantum-spin-liquid window $0.06 \lesssim J_2/J_1 \lesssim 0.16$, which extends to magnetic field $B \sim J_1$, we identify overlapping $m = 1/3$ and $m = 1/2$ plateaux - a distinctive hallmark of the system's proximity to the low-field spin-liquid regime. The excellent quantitative agreement between DMRG and self-consistent one-loop spin-wave calculations demonstrates that semiclassical approaches can reliably capture and parameterize the plateau phases of triangular quantum antiferromagnets.

We study fusion rings, or symmetry topological field theories (SymTFTs), which lie outside the non-negative integer matrix representation (NIM-rep), by combining knowledge from generalized symmetry and that from pseudo-Hermitian systems. By applying the Galois shuffle operation to the SymTFTs, we reconstruct fusion rings that correspond to nonlocal CFTs constructed from the corresponding local nonunitary CFTs by applying the similarity transformations. The resultant SymTFTs are outside of NIM-rep, whereas they are ring isomorphic to the NIM-rep of the corresponding local nonunitary CFTs. We study the consequences of this correspondence between the nonlocal unitary model and local nonunitary models. We demonstrate the correspondence between their classifications of massive or massless renormalization group flows and the discrepancies between their boundary or domain wall phenomena. Our work reveals a new connection between ring isomorphism and similarity transformations, providing the fundamental implications of ring-theoretic ideas in the context of symmetry in physics.

The tunable mechanical response of knitted fabrics underpins applications ranging from soft robotics and artificial muscles to morphing electromagnetic field sensors. Elasticity in fabrics emerges from the bending of yarn in the knitted structure; however, properties beyond elasticity are relatively unexplored. Here, we demonstrate that knitted fabrics subjected to cyclic uniaxial stress exhibit significant hysteresis and the remarkable ability to "remember" their response to previous deformations – reminiscent of classical return point memory in magnetic systems. The hysteretic behavior deviates from the two standard models of hysteresis that usually apply to solid-state materials, viscoelasticity and plasticity. Thus, we develop a phenomenological extension of the Preisach model of hysteresis which well replicates our data, and discuss implications of these results on the underlying mechanisms of memory in knitted fabrics.

We present general four-loop template $\beta$-functions and anomalous field dimensions for renormalisable scalar-fermion theories in three dimensions. By imposing $\mathcal{N}=1$ and $\mathcal{N}=2$ supersymmetry, we obtain relations between the template RGE coefficients, valid in any renormalisation scheme. Directly in $d=3$, we identify a new theory with a non-trivial IR fixed point that is under perturbative control in a large-$N$ limit. We provide up-to-date numerical results for all required massive tadpole master integrals up to four loops and complement them with analytic expressions where available.

Quantum computers and simulators offer unparalleled capabilities of probing quantum many-body states, by obtaining snapshots of the many-body wave function via collective projective measurements. The probability distribution obtained by such snapshots (which are fundamentally limited to a negligible fraction of the Hilbert space) is of fundamental importance to determine the power of quantum computations. However, its relation to many-body collective properties is poorly understood. Here, we develop a theoretical framework to link quantum phases of matter to their snapshots, based on a combination of data complexity and network theory analyses. The first step in our scheme consists of applying Occam's razor principle to quantum sampling: given snapshots of a wave function, we identify a minimal-complexity measurement basis by analyzing the information compressibility of snapshots over different measurement bases. The second step consists of analyzing arbitrary correlations using network theory, building a wave-function network from the minimal-complexity basis data. This approach allows us to stochastically classify the output of quantum computers and simulations, with no assumptions on the underlying dynamics, and in a fully interpretable manner. We apply this method to quantum states of matter in one-dimensional translational invariant systems, where such classification is exhaustive, and where it reveals an interesting interplay between algorithmic and computational complexity for many-body states. Our framework is of immediate experimental relevance, and can be further extended both in terms of more advanced network mathematics, including discrete homology, as well as in terms of applications to physical phenomena, such as time-dependent dynamics and gauge theories.

The Kibble-Zurek mechanism quantifies defect formation during adiabatic passage across a continuous phase transition, providing key insights into universality in quantum many-body systems. We explore counting statistics of defects in adiabatic passage experiments on long 1D Rydberg atom chains. The experiments reveal an anomaly in the defect number distribution at long ramp times, challenging the hypothesis of defect formation through independent domain mergers. Numerical simulations confirm the anomaly and suggest its link to non-critical coarsening dynamics, which we suppress in prepare-and-hold experiments. Our results highlight the ability of quantum simulators to uncover unexpected correlated quantum phenomena.

We develop a rigorous topological theory of anomalies on the lattice, which are obstructions to gauging global symmetries and the existence of trivial symmetric states. We also construct $\Omega$-spectra of a class of invertible states and quantum cellular automata, which allows us to classify both anomalies and symmetry protected topological phases up to blend equivalence.

We investigate the dynamics of strongly disordered spin chains in the presence of random local measurements. By studying the transverse-field Ising model with a site-dependent random longitudinal field and an effective $l$-bit many-body localized Hamiltonian, we show that the prethermal and MBL regimes are unstable to local measurements along any direction. Any non-zero measurement density induces a volume-law entangled phase with a subsequent phase transition into an area-law state as the measurement rate is further increased. The critical measurement rate $p_c$, where the transition occurs, is exponentially small in the strength of disorder $W$ and the average overlap between the measurement operator and the local integrals of motion $O$ as $p_c \sim \exp[-\alpha W/(1-O^2)]$. In the measurement-induced volume-law phase, the saturation time scales as $t_s \sim L $, contrasting the exponentially slow saturation $t_s \sim e^{aL}$ in the prethermal and MBL regimes at $p = 0$.

Cellular renewing active matter - assemblies of proliferating and apoptotic cells - underlies tissue homeostasis, morphogenesis, and clonal competition. Previous work in one-dimensional periodic systems identified a fitness advantage associated with rapid dead-cell clearance, an "opportunistic" competition mechanism. Extending this framework, we study two-dimensional cellular aggregates and show that dimensionality modifies the interplay between competition mechanisms for clones with different clearance rates: in 2D, opportunistic and homeostatic-pressure-based competition jointly shape clonal selection, to varying degrees. We then introduce an explicit circular confinement to probe how boundaries modulate this interplay. While opportunistic competition persists, distinct timescale-dependent behaviors emerge through weakened homeostatic-pressure-based competition near boundaries. Structural analysis reveals that confinement promotes tangential alignment and spatially heterogeneous homeostatic pressure, thereby reshaping competitive outcomes at tissue edges. Our study connects newly discovered competition mechanisms with more realistic biological contexts, highlighting how dimensionality and spatial constraints influence tissue structures and modulate competition in heterogeneous cell populations, with implications for tumor growth dynamics and tissue development.

We investigate the properties of CuVOF$_4$(H$_2$O)$_6$$\cdot$H$_2$O, in which two different spin species, Cu(II) and V(IV), form antiferromagnetic spin-1/2 dimers with weak interdimer coupling provided via hydrogen bonding. Using radio-frequency susceptometry and electron-spin resonance (ESR), we show how the temperature-magnetic field spin-dimer phase diagram evolves as a function of applied hydrostatic pressure and correlate this with pressure-induced changes to the crystal structure. These results, coupled with pressure-tuned DFT calculations, confirm the prior prediction that the primary exchange interaction is mediated via an unusual mechanism in which the V(IV) ions provide considerable spin density to the oxygen that joins the two spins in each dimer and which lies along the Jahn-Teller axis of the Cu(II) ion. In addition, the dissimilarity in the spins that make up each dimer unit leads to a non-linear field dependence of the electronic energy levels as detected in the ESR measurements.

We investigate the impact of an anisotropic surface tension on the late-stage dilute phase separation dynamics, revisiting the seminal Lifshitz-Slyozov (LS) theory, which traditionally relies on the assumption of isotropic surface tension. Using a perturbative treatment for weak anisotropy, we demonstrate that although the characteristic $t^{1/3}$ drop growth law remains unchanged, the anisotropy causes a significant breakdown of morphological universality. Specifically, we calculate explicitly a one-parameter family of nonequilibrium drop shapes that depend on the scaled drop size. These shapes are close to the equilibrium Wulff shape, but the smaller drops are more spherical and the larger drops have an enhanced anisotropy in comparison to the Wulff shape. We also demonstrate that the the drop size distribution is modified from the isotropic LS distribution at second order in the anisotropy strength.

Growth and shape formation in crowded multicellular assemblies arise from the interplay of chemical gradients, single-cell expansion and mechanical interactions, making it essential to understand how these processes jointly shape collective organization. Using a particle-based model that resolves nutrient fields as well as cellular orientations and their inheritance, we investigate how orientational order emerges within expanding fronts whose morphology is set by nutrient limitation. We identify a transition in nematic order controlled by front morphology, with orientational memory influencing alignment only on one side of this transition. Under strong inheritance, orientational order varies non-monotonically: both thin active layers (fingering morphologies) and thick active layers (flat fronts) produce strong alignment, whereas intermediate cases are less ordered. Analysis of velocities, reorientation statistics, and stress anisotropies shows that this behavior reflects a shift from inheritance-dominated to mechanically driven alignment that overrides lineage memory. The resulting differences in front speed produce a fitness advantage of orientational memory only in the diffusion-limited, memory-dominated regime. These findings elucidate how nutrient supply, mechanical interactions, and single-cell expansion together shape self-organization during growth.

Probabilistic error cancellation is an attempt to reverse the effect of dissipative noise channels on quantum computers by applying unphysical channels after the execution of a quantum algorithm on noisy hardware. We investigate on general grounds the properties of such unphysical quantum channels by considering a random matrix ensemble modeling noisy quantum algorithms. We show that the complex spectra of denoiser channels inherit their structure from random Lindbladians. Additional structure imposed by the locality of noise channels of the quantum computer emerges in terms of a hierarchy of timescales.

We investigate the two-dimensional classical Heisenberg ($N=3$) model with long-range (LR) interactions decaying as $1/r^{2+\sigma}$, focusing on the marginal case $\sigma=2$. Large-scale Monte Carlo simulations up to linear size $L=8192$ reveal a finite-temperature phase transition, in striking contrast to the short-range case where the system is asymptotically free. An adiabatic renormalization-group analysis shows that the LR interaction provides a positive contribution to the stiffness flow and gives rise to a weak continuous transition through non-topological spin-wave fluctuations. We show that the low-temperature phase exhibits a novel logarithmic quasi-long-range order (log-QLRO), with correlations decaying as $C(r)\sim(\ln r)^{-\eta_\ell}$ and vanishing magnetization in the strict thermodynamic limit. High-precision data confirm this picture, including the predicted identity $2\pi\kappa_R\eta_\ell=N-1$, with $\kappa_R$ the renormalized spin stiffness. The decay, with $\eta_\ell\propto T$, is slower than any power law, making the phase practically indistinguishable from true long-range order in realistic systems. Our results establish a novel log-QLRO phase and a continuous transition beyond the Ginzburg-Landau-Wilson paradigm in the two-dimensional Heisenberg model at $\sigma=2$.

We investigate the nonequilibrium current response in diffusive superconductor-normal-metal-superconductor junctions subjected to a low-frequency AC voltage. Using a kinetic description based on the adiabatic motion of Andreev bound states, we derive a general expression for the admittance of a junction under a DC phase bias, formulated entirely in terms of the phase-dependent density of states induced by the proximity effect. A numerical solution of the full nonlinear Usadel equations that describe the dynamics of the junction is presented. The obtained results for the admittance and the Josephson current-phase relation apply to two-dimensional planar junctions with Rashba spin-orbit coupling and an in-plane Zeeman field, as well as to Josephson junctions formed with topological insulator surface states as the normal layer. The frequency dependence of the admittance captures the crossover between the hydrodynamic and collisionless regimes, distinguished by the relation between the drive frequency and the inelastic relaxation rate in the normal region.