Papers

Braiding statistics of $p$- and $q$-dimensional topological excitations is conventionally defined in $p+q+2$ spatial dimensions. We find a novel statistical process $W_N(X,Y)=(Y^{-1}X^{-1})^N(YX)^N$ for two order-$N$ excitations in $p+q+1$ dimensions, detecting the Bockstein response $A\smile \beta(B)$. This new statistics and fermionic loop statistics exhaust all loop statistics in three dimensions whose fusion rules form an Abelian group $G$, classified by $H^5(B^2G,U(1))$. Surprisingly, conventional three-loop braiding goes beyond this classification, so it must have non-Abelian fusion rules. We suggest viewing three-loop braiding as particle-loop braiding together with exotic fusion rules between loops and point-like defects. We also try to clarify the relationship between statistics and symmetry anomaly.

Synthetic quantum matter provides a highly tunable route to fractional quantum Hall physics beyond the constraints of conventional electronic materials. However, previous theoretical studies have mostly focused on their ground state properties. It remains unclear to what extent such platforms could reveal key excitation properties of fractional quantum Hall states. Here, we study charge-neutral collective excitations in a non-abelian lattice fractional quantum Hall state realized in the bosonic Harper-Hofstadter model at unity filling factior, realizing a Moore-Read ground state. Combining full exact diagonalization, band-projected exact diagonalization, and matrix-product-state simulations, we demonstrate the existence of a long-lived chiral graviton mode, probed by chiral 3-body correlators, for the first time on lattice non-Abelian states. The graviton signal is topological sector-independent and could be observed via geometric quenches in small open droplets directly relevant to current cold-atom experiments, while other neutral modes, such as the magnetoroton and neutral fermion, are less resolved at presently achievable volumes.

Ferroelectric nematic liquid crystals (FNLCs) are polar fluids in which spontaneous polarization coexists with nematic orientational order, giving rise to unusual dielectric and electromechanical responses. However, the collective modes underlying their giant dielectric response remain unclear. Here, we show that this response originates from the superposition of two distinct relaxation modes rather than a single process. Dielectric spectroscopy reveals that the low-frequency mode exhibits soft-mode-like behavior associated with short-axis molecular rotation, whereas the high-frequency mode corresponds to a Goldstone-like phase displacement of an effective transverse polarization component rotating around the director. These assignments are supported by systematic analyses of temperature, electric-field, cell-thickness, and alignment-layer dependences. Our results demonstrate that the giant dielectric response of ferroelectric nematics reflects multiple collective polarization dynamics with different symmetries and restoring forces, providing a framework for interpreting dielectric spectra in polar nematic fluids.

Conventionally, the local spin magnetization in itinerant magnets is determined from the equilibrium local spin density. Here, we propose a thermodynamic approach in which the local spin magnetization is defined from the response of the system to an infinitesimal external magnetic field. The predictions of the two theories are identical for collinear magnets, but differ qualitatively and quantitatively for non-collinear magnets. In the present thermodynamic approach, the spin coherences determine an alternative distribution of local spin magnetization due to the field-induced deformation of the energy eigenstates. This effect is captured by a Berry-curvature-like contribution reminiscent of orbital magnetization and has several distinct observable consequences. We explore the differences between the conventional and thermodynamic approaches in several test cases.

Rising energy demand from data-center and AI applications has renewed interest in reversible computation, where logic need not dissipate heat at every step if information is uncomputed. Implementations have so far been classical: adiabatic CMOS recovers part of the switching energy but still moves thousands of $k_BT$ per logic node at room temperature. Here we propose classical reversible logic implemented by coherent spin dynamics in a spin quantum-dot array, with inputs and outputs in classical basis states and no algorithmic use of superposition. The same spin stores, transports, and computes, with unitary rotation replacing irreversible switching. The universal building block is an iToffoli gate driven by DC voltage pulses and exchange between hole spins in Ge/SiGe quantum dots. Simulations with realistic model parameters reproduce the Toffoli truth table and yield a testable error landscape. Because shuttling transports the bit without measurement, logic and data movement remain unitary until readout. Millivolt pulses on femtofarad gates with superconducting lines dissipate only dielectric loss and have no thermodynamic floor. With the loss parameters assumed for a demonstrated device the gate energy is $\sim 10^{2}\,k_BT\ln 2$ at 4 K; at a design point within reach of existing devices it can fall below the Landauer scale $k_BT\ln 2$, five to seven orders of magnitude less than a CMOS Toffoli. The same semiconductor hardware therefore serves both purposes, supporting quantum algorithms when superposition is used and classical reversible logic otherwise.

We investigate the estimation of bath-induced coherence in a finite quantum system interacting with thermal reservoirs. Enhancement of coherence estimation is transient and the estimation precision totally disappears at the steady state despite the system retaining finite coherence. By analyzing the full Liouvillian eigenspectrum, we demonstrate that the optimal sensing window emerges from the competition between identifiable contributory modes' temporal relaxation and statistical importance. Neither is the linear inverse scaling of Liouvillian gap with transient optimal time a signature of unimodal contribution to optimal sensing, nor is the existence of multimodal dynamics a signature of nonlinear scaling. The inverse Liouvillian gap does not obey any general scaling with the optimal sensing time of coherence and we prove our numerical results analytically using a general Markovian framework. We further show that coupling the finite system to a quantum cavity and maintaining a thermal bias, transforms the transient metrological optimization into a sustained steady-state resource.

We formulate a symmetry-based hierarchy of strain-derived conjugate fields for ferroaxial order, and demonstrate strain-selected ferroaxial domain control using first-principles calculations. Since ferroaxial order is even under both spatial inversion and time reversal, ordinary electric and magnetic fields cannot serve as universal linear conjugate fields. Homogeneous strain, however, can generate symmetry-allowed piezoaxial fields whose leading order is determined by the parent point group and by the chosen ferroaxial-axis component. For basal-plane strain, the leading field is linear in orthorhombic systems, quadratic in tetragonal systems, and cubic in trigonal and hexagonal systems. Cubic parent groups further split into two classes: cubic-I groups, $23$ and $m\bar{3}$, allow linear full-strain fields for selected axes, whereas cubic-II groups, $432$, $\bar{4}3m$, and $m\bar{3}m$, forbid linear fields and require quadratic or cubic strain combinations depending on the selected axis. In trigonal systems, the basal-plane deviatoric strain with signed amplitude $\varepsilon_{\rm u}$ and principal-axis angle $\theta$ gives the single-axis field $h\propto\varepsilon_{\rm u}^3\sin6\theta$. First-principles calculations for the trigonal ferroaxial compound Na$_2$BaMg(PO$_4$)$_2$ verify both the predicted angular dependence and cubic strain scaling of the ferroaxial domain splitting, and fixed-strain atomic relaxations show strain-selected evolution from the para-axial structure. These results establish static homogeneous strain as a symmetry-allowed conjugate field for ferroaxial order and suggest a route to ferroaxial domain control through strain-field cooling.

The Fredrickson-Andersen model with hyperparameter $K=1$ is a severely constrained kinetic lattice spin system, such that any site is temporarily blocked from changing its packing state (empty or occupied) if there is one or more occupied nearest neighbors. Starting from a completely random initial configuration with a fraction $\rho$ of sites being occupied, some of the sites may be permanently frozen to their initial state under this severe kinetic constraint. The remaining sites can switch states at least occasionally, and they form the unfrozen subsystem associated with the given initial configuration. In the present work we investigate thermodynamic phase transitions in such unfrozen subsystems of the two-dimensional square lattice and the three-dimensional cubic lattice by extensive numerical simulations. We demonstrate that the giant connected component of the unfrozen subsystem collapses at certain critical value $\rho_{c}$ of initial packing density, with $\rho_c = 0.2475$ for the square lattice and $\rho_c = 0.2809$ for the cubic lattice. This phase transition belongs to the same universality class of the conventional site percolation. We also observe that the ground states (densest packing configurations) experience a continuous crystal-to-glass phase transition at the critical value $\rho^* = 0.1423$ of initial packing density for the cubic lattice. For the two-dimensional square lattice we argue that long-range crystalline order is destroyed in the ground states as long as the initial packing density $\rho$ is positive.

We estimate the impact of Polyakov loop (PL) contributions on electroweak phase transitions (PTs). We show that the PL, which is unavoidable in thermal gauge field theory, tends to tame thermal contributions, thereby softening electroweak PTs and affecting bubble dynamics, nucleation, and the related gravitational-wave spectrum. Including this non-perturbative contribution in perturbative approaches results in a thermal effective potential that disfavours first-order PTs over either second-order PTs or smooth cross-overs. This feature is universal for both fermionic and bosonic contributions to the effective potential.

Using theory, simulation, and experiment, we introduce a new class of active particle which we term doubly chiral active Brownian particles (dcABPs), which show robust topologically protected transport along boundaries without backscattering at corners. Their double chirality stems from the coexistence of an intrinsic angular velocity, which can cause rotation independently of translation, and a translation-rotation coupling inducing cross-alignment to the instantaneous velocity, which causes rotation only concomitantly with translation. A mechanically detailed model shows that the latter effect can arise from an asymmetric friction distribution in the direction perpendicular to the self-propulsion direction. We show that topologically protected modes emerge when the two sources of chirality have opposite sign and the intrinsic rotation is weaker than the translation-rotation coupling. In the deterministic limit, we characterize the emergence of these modes not only along straight boundaries, but also along curved boundaries and during interparticle interactions. We provide a proof-of-principle experimental realization by building a doubly chiral vibrobot. While setting the work into context, we moreover show that the topologically protected boundary-induced transport of dcABPs stands in contrast to the edge currents observed for simple chiral ABPs, which we demonstrate are not associated with boundary-induced transport, as well as to those observed for chiral active rods or self-aligning chiral ABPs, which we show to be associated with boundary-induced transport but to backscatter at corners, implying lack of topological protection.

We derive thermodynamic and kinetic uncertainty relations valid for arbitrary currents in coherent, strongly coupled, linear systems out of equilibrium. Exploiting properties of the transport statistics, in particular fluctuation theorems, we identify the relevant entropy production and activity that determine the cost of precision at the level of individual scattering events. The resulting bounds include higher-order fluctuations and remain valid far from equilibrium. We illustrate our results in normal and superconducting hybrid structures, and show that their predictiveness and validity range exceeds existing formulations.

Kelvin probe force microscopy (KPFM) probes local surface-potential variations through the electrostatic force between a conductive tip and a surface, which depends on the potential difference and the tip-surface capacitance gradient (CG). In heterodyne KPFM, the oscillating tip is usually treated by combining a bias-modulated electric field with a first-order truncated Taylor-series expansion of the CG. Although convenient, this treatment is limited to a poorly defined small-amplitude regime and leaves the convergence of the series unresolved. Here, we establish a rigorous spectral description of the CG dynamics and of the resulting electrostatic force beyond this approximation. We formulate a non-truncated Taylor-series description of the CG and prove its convergence for a realistic Hudlet-based capacitance model in both monomodal and bimodal motion. In the monomodal case, we show the equivalence between Fourier-series and Taylor-series descriptions, derive explicit expressions for the dominant Fourier coefficients, and introduce order-truncation criteria that replace the usual qualitative notion of a small-amplitude regime. We then extend the formalism to bimodal motion and derive the effective CG coefficients governing the static, first-eigenmode, and second-eigenmode components of the electrostatic interaction. Numerical simulations confirm the convergence of the Taylor-based coefficients toward the Fourier coefficients and support the truncation-regime hierarchy in both configurations. This work establishes the formal basis for describing electrostatic force components and AFM observables in open-loop heterodyne experiments and provides a general framework for CG dynamics in multimode force microscopy involving nonlinear electromechanical coupling and frequency conversion.

We predict that intercalating bilayer graphene with nonmagnetic WSSe or magnetic MnSSe Janus monolayers induces a layer-selective switch of the in-plane Rashba spin texture, resulting in opposite spin current directions in the top and bottom graphene layers. First-principles calculations reveal that both Janus monolayers decouple the two graphene layers while simultaneously inducing opposite signs of the proximity-induced Rashba spin-orbit coupling in each. Tight-binding modeling of the proximitized layers, combined with Rashba-Edelstein charge-to-spin conversion calculations, confirms that the spin current direction can be independently controlled by gating the top or bottom graphene layer. Bilayer graphene intercalated by Janus monolayers thus represents a promising platform for gate-tunable, layer-selective spintronic devices.

In this review we discuss semi-classical methods that are traditionally used to describe many-body systems in physics, but may also be used to describe partitions of integers in analytic number theory. Specifically, we explore the connection between the methods of statistical mechanics and number partitions. Though the two fields appear very different, their fundamental issues bear a close resemblance. In the former case it is the distribution of a given amount of energy among the particles in an ensemble at a given temperature with well defined properties, while in the latter case it is the way an integer is partitioned into other integers, with or without restrictions. We begin with a discussion of the single-particle quantum density of states, also called the level density, in which we illustrate the connection between the density of states and the classical periodic orbits through the semiclassical trace formula. This is then extended to many particle systems. We show that the asymptotic number partition is reproduced by the average (smooth) part of the level density at discrete integer values of the argument. In the especially interesting case of distinct square partitions, pronounced oscillations are well reproduced by the periodic orbit theory in terms of a few orbits characterised by Pythagorean number triples. We speculate on the connection to Fermat's theorem as to why such regular oscillations (though vanishing asymptotically) exist only in this special case. Finally, we discuss some new results for integer partitions of primes, both unrestricted and distinct.

Heterodyne Kelvin probe force microscopy (He-KPFM) enables high-sensitivity electrostatic measurements by converting a bias-modulated interaction into a resonant response at a higher cantilever eigenmode. While the "direct" heterodyne actuation of the second eigenmode is well established, the dynamical back-action of this heterodyne-driven motion on the fundamental eigenmode has remained largely unexplored, particularly in open-loop operation where the second mode is excited to a finite amplitude. Here, we demonstrate an inverse heterodyne effect: a force component generated by heterodyne frequency conversion acts back on the first eigenmode and produces measurable inter-mode energy exchange. The analysis combines a bimodal virial and power-balance framework with a non-truncated description of the tip-surface capacitance-gradient dynamics developed and validated in a companion manuscript submitted concurrently to the same journal. On this basis, we derive closed-form expressions linking inverse heterodyne coupling to the experimentally accessible observables of non-contact AFM open-loop amplitude-modulated He-KPFM. The theory predicts that inverse heterodyne coupling appears predominantly in the dissipation channel, with a sharply resonant dependence on the demodulation frequency near the second-eigenmode resonance, while its conservative contribution to the frequency shift remains comparatively weaker under typical conditions. Ultrahigh-vacuum experiments validate these predictions and isolate the inverse heterodyne signature through frequency- and voltage-dependent measurements. Beyond KPFM, this work connects heterodyne force microscopy to a broader class of driven multimode systems in which nonlinear coupling and frequency conversion produce inter-mode energy transfer, back-action, and dissipation-based observables.

We show that in a reciprocal Brownian motor the entropy production hidden behind a mechanically stalled coordinate can be reconstructed exactly from measurements of that coordinate alone. We introduce a minimal, analytically solvable Langevin motor in which an observed translational coordinate is coupled reciprocally to a hidden internal rotor: a single periodic potential $V(x-\ell\theta)$ generates both the force on the observed coordinate and the reaction torque on the hidden one, so that $\tau_{\rm int}=-\ell F_x$ holds identically. Force–torque reciprocity together with translational symmetry produces a local current identity that closes the hidden thermodynamic bookkeeping. From it we prove that the Harada–Sasa heat measured through the observed coordinate equals the positive current-square dissipation of that coordinate, with the information-flow correction vanishing identically. eciprocal class. ries in that it requires no separation of time scales.

We use the Ground-State Path Integral Monte Carlo method to study a Bose-Einstein condensate of strongly interacting NaCs polar molecules under the action of a fully anisotropic double microwave shielding potential characterized by a linear and an elliptical polarization field. In particular, we analyze the ground state of the system and its structure as a function of the ellipticity angle $\xi$. While for the circularly polarized case ($\xi=0$) a gas phase is realized, one or more self-bound droplets are observed for small $|\xi|$'s above a threshold value near $3^\circ$. With increasing $\xi$, the observed droplets rapidly become tightly bound and are estimated to form a superfluid array. Our results compare favorably to the experimental observations in [Zhang et al., Nature 651, 601 (2026)] for positive $\xi$, while moderate differences show up for $\xi<0$ where our simulations conform to the expected symmetries of the intermolecular potential.

We use molecular dynamics simulations to directly compute the effective diffusivity of hydrogen gas atoms in homogeneous distributions of monovacancies in tungsten and vanadium, and voids in tungsten. Rather than fitting the results to an Arrhenius law, we compare to an analytic approximation for the effective diffusivity recently derived for multi-occupancy traps [Kaur et al (2025), Phys. Rev. Mater. 9:125404]. We find good agreement between full atomistic simulation and our theory, validating the analytic model for diffusivity for materials containing nanoscale defects characteristic of radiation damage. There are no parameters fitted, only physically motivated quantities that can be computed with static density functional or atomistic potential calculations. In this study we prove rapid convergence of hydrogen trap occupation to the steady state using lattice kinetic Monte Carlo, the spontaneous emergence of voids in tungsten using atomistic simulation with empirical potentials, and molecular hydrogen formation in voids using molecular dynamics. We conclude with a prediction for diffusion and retention of hydrogen in voids in tungsten starting from first principles. This work shows that not only is the analytic form for diffusivity and retention in multi-occupancy traps a practical scheme for making predictive simulations of hydrogen isotope diffusion and retention in irradiated microstructures, derived and parameterized from first principles, it is superior to existing single-occupancy trap formalisms.

We investigate the completion dynamics of an overdamped dimer moving in a bistable potential under thermal fluctuations and a weak periodic force. Both monomers start in one of the two wells separated by a barrier. The transition is initiated when the monomer closer to the barrier makes a jump across it. The completion dynamics refers to the next part of the dynamics where the second monomer has to wait for some time before it can follow up. We use the Conditional Residence Time (CRT) to study the delay between the successive barrier crossing of the two monomers. The CRT distributions highlight qualitatively different regimes formed by the competition between the escape times of the lagging monomer and the time period of the external drive. The effect is strongest in the weak coupling regime where the delayed completion is spread across multiple forcing cycles. By partitioning this process into three windows, i.e. the immediate, first cycle and later cycles, we show that the probability that the lagging monomer will make a transition in the said cycle is redistributed among these pathways as we change the frequency of the drive. This leads to a non-monotonic dependence of the mean CRT on the frequency of the drive. Our results demonstrate that transition initiation and completion in a coupled system are two separate processes and establish CRT as a useful measure to quantify the sequential barrier crossing dynamics in coupled stochastic systems.

Altermagnets, characterized by spin-split electronic bands with compensated magnetic moments, have emerged as a new class of magnetic materials garnering attention in recent years. Here, using a minimal one-band Hubbard model, we show that the checkerboard lattice serves as a natural platform for altermagnetism for electrons. The instability towards altermagnetic order is denoted by diverging altermagnetic susceptibility at weak-coupling. Carrying out mean-field treatment of the Hubbard repulsion, we show phase transitions from the nonmagnetic to altermagnetic semimetal and then to altermagnetic insulating phase, allowing clear identification of spin-split states. We then examine magnetic excitations in the altermagnetic phases using a random-phase approximation treatment of the dynamical spin susceptibility. The altermagnetic order is found to be stable against spin-fluctuations with the excitation spectra showing well-defined magnon excitations, which decay into single-particle excitations with decreasing interaction strength. Remarkably, the magnetic excitations exhibit strong dependence on both chirality and direction, showing an alternating chirality splitting, similar to the alternating spin splitting of the electronic bands, which serves as a salient feature of altermagnetism.

When driven out of equilibrium, a Bose-Einstein condensate develops nonlinearly interacting density waves that trigger a turbulent cascade, transferring energy toward small scales. In this Letter, we investigate the nonstationary evolution of solutions to the two-dimensional Gross-Pitaevskii equation (GPE). Through numerical simulations of both the GPE and the corresponding Wave Kinetic Equation (WKE), we identify self-similar solutions relevant to turbulence in atomic and polariton Bose-Einstein Condensates. These solutions correspond to a new type of non-thermal fixed point and exhibit characteristics of both first and second kind self-similarity. In particular, we show that the dynamics of the propagating front is universal, governed by a dimensionless universal constant $\beta$, which we determine numerically.

The characteristics of graphene-metal interfaces play a decisive role in their electronic, optoelectronic, and mechanical applications. Properties such as charge transfer across the interface become particularly significant in the presence of topological defects. The stability of Stone Wales (SW) defects in graphene is governed by the balance between three energy descriptors, the activation energy, formation energy, and restoration energy. By comparing the energy parameters obtained from first-principles density functional theory calculations, we show that SW defect formation is energetically more favorable on metal-supported graphene. Our calculations for SW defects in graphene/Cu(111) and graphene/Al(111) systems indicate only a little dependence of energy profile on the type of metal. The presence of the metal substrate leads to a $\sim$ 12% increase in the formation energy and a $\sim$ 20% reduction in the activation energy, which together favor the formation of Stone Wales defects. Although the restoration energy decreases by about $\sim$ 35% in metal-supported graphene, it remains significantly higher to prevent self-healing. As a result, once formed, the Stone Wales defects are likely to remain stable, suggesting the possibility of terminal SW defect formation in metal-supported graphene.

The extension of moire physics to complex oxides offers new ways to manipulate electronic states, but the large oxide moire supercells make systematic first-principles calculations demanding. Here, we combine density functional theory with the E(3)-equivariant deep-learning Hamiltonian framework DeepH-E3 to investigate the twist-angle-dependent electronic structure and optical responses of twisted bilayer SrTiO3. The model is trained on untwisted bilayers with different interlayer-sliding configurations and then applied to commensurate twisted bilayers with twist angles from 8.80 degrees to 53.13 degrees. Compared with the untwisted bilayer, decreasing twist angle systematically flattens the valence bands and leads to nearly dispersionless bands at the smallest angles studied. Based on the predicted Hamiltonians, we evaluate the dielectric response, second-harmonic generation (SHG), shift current, and spin Hall conductivity. The dielectric response and spin Hall conductivity remain close to those of the untwisted bilayer, whereas the nonlinear optical responses are more strongly affected by twisting. SHG is strongly enhanced relative to the weak untwisted response, and the shift current shows a clear twist-angle dependence within the response-calculation range (53.13 degrees-22.62 degrees). These results show that twist engineering can control electronic and optoelectronic responses in oxide moire systems.

The study of active matter has reshaped our understanding of collective states of matter far from equilibrium by proving that energy pumped into the microscopic scale leads to order on the macroscopic scale, collective motion, and anomalous mechanical responses. More recently, the discovery of odd elasticity and nonreciprocal mechanical couplings has extended these ideas to solid-like active systems, revealing materials with nonconservative elastic response. Simultaneously, innovative developments in swarm robotics , programmable metamaterials , and learning algorithms have led to the emergence of a new frontier in which collective behavior and mechanical response are no longer fixed by design, but adapted, optimized, and learned toward functional goals. This Perspective proposes a unifying trajectory, from active to odd to smart matter, organized along two intertwined axes: the traditional gas–liquid–solid progression of condensed matter, and the more recentparadigm shift from spontaneous collective dynamics to task-driven functionality. We try to highlight emerging principles, conceptual shifts, and open challenges that come along this trajectory, and argue that learning may play the role of a specific form of emergence, which could advantageously replace the more traditional view of control, at least in the realm of physics.

We study genuine multi-entropy as a diagnostic of multipartite entanglement in the toric code, which provides a controlled setting for probing multipartite structures in topologically ordered states. Our main question is whether genuine multi-entropy captures information that is not reducible to conventional lower-party entropic data, such as topological entanglement entropy. We first analyze toric-code ground states that admit a stabilizer-state description, where the relevant quantities can be evaluated exactly. We conjecture that, for stabilizer states and $\mathtt{q}\ge4$, the $\mathtt{q}$-partite genuine multi-entropy at replica index $n<\mathtt{q}$ collapses to a linear combination of multi-entropies involving at most $\mathtt{q}-2$ parties. We establish this pattern explicitly for $\mathtt{q}=4$ in the toric code stabilizer sector: for $n=2,3$, the genuine multi-entropy is proportional to the tripartite information $I_3$ and, for the Kitaev-Preskill partition, contains no independent genuine four-partite information beyond that captured by the topological entanglement entropy. At $n=4$, however, this reduction breaks down: the genuine multi-entropy is no longer proportional to $I_3$, but remains a topological invariant of the toric-code stabilizer ground states under Kitaev-Preskill-preserving deformations that do not change whether the numbers of vertex- and plaquette-type junctions coincide. For generic non-stabilizer superpositions within the ground-state manifold and for coherent superpositions of local excitations, the low-$n$ reduction also fails. These results show that genuine multi-entropy probes multipartite entanglement structure beyond the tripartite information, and hence beyond the topological entanglement entropy in the Kitaev-Preskill partition, whereas for stabilizer states at low replica index it reduces to lower-partite entropic data.

We develop a general method, based on the construction of a kinetic potential serving as a Lyapunov function, to establish when diffusion necessarily equilibrates in non-ideal reaction-diffusion systems under arbitrary driving by autonomous homogeneous chemostats. This equilibration of diffusion implies vanishing diffusion currents and homogeneous chemical potentials, while reactions can remain far from equilibrium. Using this method, we generalize the results of J. Chem. Phys. 161, 174108 (2024) by relaxing some of the underlying assumptions. Specifically, we show that diffusion equilibrates in reaction-diffusion systems whose chemical reaction network is either pseudo-detailed balanced, with reaction fluxes controlled by the stoichiometry of reactants and products, or complex balanced, with reaction fluxes controlled only by the stoichiometry of the reactants. These different constraints on the reaction fluxes are shown to originate from the distinct stoichiometric properties of the two classes of networks.

Han Purple, a pigment first obtained in ancient China, is one of the earliest known synthetic pigments. Also naturally occurring, as a mineral, it is known as colinowensite (Cwn). Its chemical formula is BaCuSi$_{2}$O$_{6}$, and its structure is a layered cyclosilicate in which magnetic Cu$^{2+}$ ions (S = 1/2) form dimers arranged on a square lattice, making it also the first known synthetic metal dimer compound. Most interesting magnetic properties arise from a strong intradimer spin coupling, accompanied by weaker interdimer interactions within and between Cu-dimer layers. In zero or small magnetic fields, BaCuSi$_{2}$O$_{6}$ remains a quantum paramagnet. However, under high magnetic fields between 23 and 49 Tesla – about a million times stronger than Earth's magnetic field – it undergoes magnetic ordering at subliquid Helium temperatures into an almost ideal easy-plane (XY) antiferromagnetic state regarded as a realization of a Bose-Einstein condensate of magnons. Within this experimentally accessible field range, BaCuSi$_{2}$O$_{6}$ serves as an extraordinary playground for testing predictions of quantum many-body physics.

Scalable plasmonic technologies face a critical trade-off: few-body architectures offer high enhancement but are sensitive to fabrication flaws, while scalable methods like solid-state dewetting yield large, low-enhancement gaps. We introduce a paradigm shift using a many-body plasmonic architecture inspired by statistical mechanics. By moving toward the continuum limit (N>>1), local geometric variations are statistically averaged out, effectively decoupling optical performance from microscopic disorder. We implement this concept via a lithography- and etching-free, multi-step dewetting strategy, creating wafer-scale nanoclusters. This process strategically forms a robust many-body system by introducing numerous small satellite nanoparticles between larger particles. Crucially, this design achieves a high collective enhancement that surpasses even optimized few-body systems, despite having larger individual gaps. Under optimized conditions, these substrates exhibit a surface-enhanced Raman scattering enhancement factor approaching 4 x 108 with unprecedented reproducibility (RSD of ~10%). This scalable, low-cost concept establishes a practical route toward reproducible wafer-scale nanophotonic platforms for sensing, spectroscopy, and quantum technologies.

Magnetic van der Waals materials have mainly been investigated in their bulk form or as few-layers flakes. Due to the challenges in producing atomically thin films, only a few have been isolated as monolayers, which typically exhibit long-range magnetic order below 150 K. In this work, we use molecular beam epitaxy to synthesize Co-doped Fe5GeTe2, achieving precise control over both thickness and composition. We demonstrate ferromagnetism well above room temperature in multilayer samples and present clear evidence of ferromagnetic ordering in monolayers up to $\sim$200 K. The changes in Curie temperature and magnetic anisotropy with composition exhibit similar trends in both monolayers and thicker films, indicating that the magnetic properties are primarily governed by intralayer magnetic interactions. Through element-specific X-ray magnetic circular dichroism and density functional theory, we identify the substitution site of Co dopants and reveal the mechanism behind the Curie temperature enhancement induced by Co doping. Our findings suggest that, despite their weak magnetic moment, Co dopants strengthen the magnetic moments on neighboring Fe atoms and enhance the intralayer ferromagnetic exchange interactions.

The Ramsey community number $r_k$ is the smallest size at which a network is better described by communities than by none, under a Bayesian detection rule. On the diamond hierarchical lattice we show that $r_k$ is an exact renormalization-group crossing: the block-model sufficient statistics obey a linear map with eigenvalues $\{bs,b\}$, the degree-corrected evidence density flows to $\ln K$ at a community fixed point, and $r_k$ is the generation at which the running evidence clears the detection threshold. Degree correction advances detection by two generations. We derive $r_k(b,s;q)$ in closed form for the whole family. Finally, placing on the lattice the Reichardt–Bornholdt community Hamiltonian – whose ground state is the partition itself – we find an exact community-ordered phase: below the ferromagnetic critical temperature the two hubs lock into opposite communities for any resolution $\gamma>0$, a staggered order that persists as $n\to\infty$. Allowing each nested sub-community its own label, the optimal partition is a hierarchy of $q_{\rm opt}\sim\sqrt{n}$ communities, so the number of Potts states that best describes the network grows with the network. This hierarchy orders thermally level by level, through a cascade of first-order transitions whose temperatures fall as $1/\ln q$, so every stable level persists as $n\to\infty$: the emergent partition is detectable, optimal, and thermodynamically ordered.