AI for Science

Antiferromagnetic skyrmions are swirling topological spin textures with rich dynamics and intriguing transport properties, yet their bidirectional dynamics remain largely unexplored. Here, we investigate the dynamics of antiferromagnetic skyrmions driven by current-induced spin-transfer and spin-orbit torques. We computationally demonstrate that antiferromagnetic skyrmions moving in one direction at low current densities can reverse their motion direction when the driving current is above a threshold. Based on the Thiele approach analysis, we show that this bidirectional motion originates from a change in the relative strengths of two effective forces arising from spin-transfer and spin-orbit torques. Furthermore, exploiting this bidirectional motion on a single racetrack, we design programmable logic gates. Our results not only uncover a hidden mechanism for bidirectional skyrmion motion but also facilitate the development of antiferromagnet-based logic devices.

Understanding strongly correlated quantum materials remains a central challenge in condensed matter physics and materials science. While angle-resolved photoemission spectroscopy (ARPES) has become an indispensable probe of single-quasiparticle excitations, it accesses electronic correlations only indirectly. Here we show that unlike one-photon in, two-electrons out coincidence ARPES ($γ\!\rightarrow\!2e$ 2eARPES), the two-photon in, two-electron out $2γ\!\rightarrow\!2e$ 2eARPES provides a direct and unambiguous probe of electronic pairing. We establish this on general theoretical grounds and substantiate it through large-scale numerical simulations of strongly correlated models with both paired and unpaired ground states. The key result is a model-independent separation in the $(ω_1,ω_2)$ plane of the two photoelectrons' energies, between signal from electrons emitted from the \emph{same} pair and signal from electrons emitted from \emph{different} pairs; this follows from energy conservation alone and is independent of any material-specific assumptions. Our findings demonstrate that $2γ\!\rightarrow\!2e$ 2eARPES can identify pairing and extract the pair binding energy as well as the energy of the 'glue' boson without any sophisticated data analysis or complementary measurements.

Terahertz radiation from $\text{Bi}_2\text{Sr}_2\text{CaCu}_2\text{O}_{8+δ}$ intrinsic Josephson junctions (IJJs) provides an ideal platform to study the synchronization of a macroscopic quantum system. Here, we present a spectral analysis of a frequency-modulated Josephson plasma emitter coupled with patch antennas. In the unmodulated intensity distribution as a function of radiation frequency $I_{\mathrm{UM}}(ω)$, we observe a double Gaussian peak structure. Crucially, double-peak spectra obtained at a constant bias voltage imply either a rapid temporal distribution of resonances or their simultaneous excitation, driven by the mutual electromagnetic coupling between the IJJ mesa and the antennas. At low modulation frequencies $f_m$, the spectra are well reproduced by the products of $I_{\mathrm{UM}}(ω)$ and frequency combs, yielding a synchronized relaxation time $τ_s \simeq 0.28\text{ ns}$. Incorporating $τ_s$ quantitatively reproduces a drastic spectral transformation observed around $f_m \sim 1\text{ GHz}$, unveiling the sub-nanosecond non-equilibrium dynamics of coupled Josephson plasma.

The complex Sachdev-Ye-Kitaev (SYK) model, featuring fermions with all-to-all interactions, serves as a dual paradigm for understanding non-Fermi liquid behavior and the holographic nature of charged black holes. Two defining characteristics of the standard SYK model are its maximal chaos (Lyapunov exponent $λ_{\mathrm{L}}=2πT$ at temperature $T$), and its finite zero-temperature residual entropy. While previous studies have largely focused on couplings drawn from a zero-mean Gaussian distribution, we investigate a generalized model with a finite mean-to-standard-deviation ratio, $g\equiv J_{0}/δJ$ of the coupling distribution in order to get deeper insight into the evolution of chaos. We find that increasing $g$ yields the following effects: (i) The system remains a fast scrambler with $λ_{\mathrm{L}}=A~T$, but with a suppressed coefficient $A<2π$. (ii) In the limit $g\to \infty$, out-of-time-ordered correlators (OTOCs) no longer exhibit exponential growth with $λ_{\mathrm{L}}\simeq 0$. (iii) The spectral correlations indicative of late-time chaos maintain Wigner-Dyson level spacing statistics for all values of $g$. (iv) The system preserves a finite residual entropy, albeit with reduced magnitude, for all $g$ values. We conclude that in this generalized SYK model, there is a chaotic to non-chaotic crossover. Moreover different measures of chaos decouple, demonstrating that the presence of finite residual entropy does not strictly imply maximal chaos.

Spin dynamics in ordered magnets with topological lattice defects is investigated. Using fracton--elasticity duality, we develop an effective field theory of magnons coupled to quantized lattice dislocations (dislons) in magnetic insulators. Within this framework, an elastic gauge field mediates a nonlocal interaction between dislocations and magnetization gradients. The resulting magnetoelastic coupling gives rise to coherent magnon-dislon hybridization whose properties are dictated by dislocation topology. Screw dislocations exhibit helicity-selective hybridization and symmetry-protected dark dislon sectors, while edge dislocations generate anisotropic hybrid excitations with finite spin-precession ellipticity through the glide constraint. Our results establish dislocations as dynamical topological defects with directly observable polarization fingerprints in magnon spectra, and reveal magnon-dislon hybridization as a new route to control spin dynamics.

We study the non-equilibrium dynamics of emergent dynamical axion quasiparticles (DAQ) coupled to a photon bath in equilibrium via a Chern-Simons term as a quantum open system. A quantum master equation (QME) is derived up to second order in this coupling implementing only a \emph{partial} Markov approximation, allowing time dependent rates in the Lindblad (QME). These are determined by the equilibrium correlation functions of the Chern-Simons density, and their time dependence allows us to explore transient dynamics in coherences and population: the formation of the quasiparticle on short time scales and its decay, and the build-up of population with an effective time dependent rate. Early time evolution features quantum \emph{anti} Zeno dynamics with enhanced quasiparticle decay and population growth. These phenomena describe transient violations of Fermi's Golden rule and of \emph{detailed balance}, and are distinct \emph{non-Markovian} effects directly related to the spectral density of the Chern-Simons correlators. We obtain the equation of motion of coherent (DAQ) condensates both with the (QME) and with quantum many body linear response establishing a direct bridge between both methods. As a corollary we obtain the expectation value of the Chern-Simons density \emph{induced} by a (DAQ) condensate in linear response, the topological susceptibility is shown to be proportional to the (DAQ) many body self-energy. We provide a Feynman diagram-based interpretation of approximations invoked in the (QME) and corrections from system-bath correlations in higher order.

In a magnetic field bilayer graphene hosts an octet of pseudo-zero-mode electron levels nearly degenerate in orbitals $n=(0,1)$, valleys and spins. They split in valleys by electrostatic gating. In gated bilayer graphene, in which the interlayer bias is set up to flip sign across a line, one has a line junction that traps a portion of pseudo-zero-mode electrons inside the insulating bulk band gap, giving rise to electron states localized along the junction, known as kink states. A close look is made into the spectra and electromagnetic response of such kink states. There are two species of valley current associated with them, a drift current driven by the bias gradient and a circulating current coming from cyclotron motion. It turns out that they both flow in essentially the same direction, with the circulating current exhibiting a magnetic character distinct from those of other higher levels. In equilibrium they spatially circulate within the kink states, creating a quasi-one-dimensional channel of orbital magnetization. The electric control of the orbital magnetization and valley currents via a network of gated junctions will find useful applications in valley electronics.

In this article, we develop a multi-dimensional two-layer thin film model extending the thin film model proposed in \cite{barthwal2025hyperbolic}. The model considered in \cite{barthwal2025hyperbolic} considered a very specific Marangoni scale by choosing Marangoni numbers in both layers to be $1$. We relax this condition here and prove that the obtained system possesses a full set of Riemann invariants. Based on these findings, we develop a Riemann Invariant-based Local Characteristic Decomposition WENO (RI-WENO) method for the two-layer thin film model in one and two dimensions. The method is built upon a specially designed variable transformation constructed from the derived Riemann invariants of the system. This transformation partially diagonalizes the governing equations and yields a sparse structure in the transformed eigenvector matrices. As a result, the proposed RI-WENO framework significantly reduces the computational cost of the standard Local Characteristic Decomposition WENO approach while retaining its strong capability to suppress spurious oscillations. Numerical experiments, including new benchmark test cases, demonstrate that the RI-WENO method achieves an effective balance between accuracy and computational efficiency, making it a promising and practical choice for solving the two-layer thin film model.

Understanding the scrambling of quantum information is central to many areas of quantum physics, including quantum thermalization, entanglement growth, and quantum information processing. Insights from these studies have, in turn, inspired the development of novel quantum protocols and algorithms. Recently, a bidirectional teleportation protocol was proposed to implement a digital SWAP operation between qubits by leveraging chaotic Hamiltonian evolution combined with measurement and post-selection. In this work, we provide a comprehensive study of two central quantities that characterize the protocol, the post-selection probability and the fidelity, taking into account possible errors in time-reversed dynamics. We show that these quantities can be expressed in terms of standard diagnostics in quantum dynamics, including the Loschmidt echo and its subsystem variant. The results unveil (1) the initial-state dependence of the fidelity and (2) the stability of the post-selection probability in integrable models. Our findings offer practical guidance for the implementation of the protocol on realistic quantum devices.

In this work, we formulate self-similarity from the perspective of scale invariance, where a self-similar form is understood as the transformation of a function into a form invariant under scale transformations. By applying this formulation to physical parameters, which consist of numerical values and units, it is demonstrated that dimensional analysis works for physical problems because scale invariance is partially shared between units and physical parameters. This naturally leads to the distinction between similarity of the first kind and similarity of the second kind according to whether the scale functions induced by units and those associated with physical parameters are equivalent or not. Self-similar solutions of the second kind can be further classified according to whether the power exponents of the similarity parameters include functions of dimensionless numbers. This leads to the conclusion that there are three kinds of self-similar solutions. The present work provides a unified framework for understanding dimensional analysis and a universal classification of self-similarity in physical problems.

We study the moduli space of non-Hermitian Yang--Mills connections over a compact Kähler manifold. Using normalized harmonic metrics, we construct a natural Hermitian metric on the unobstructed locus and show that, near the Hermitian locus, the unobstructed locus carries an almost hypercomplex structure which is compatible with the associated Riemannian metric.

We give a dynamical proof of Benoist's non-arithmeticity theorem for Jordan spectra of Zariski dense subgroups of connected semisimple real algebraic groups. After passing to a Zariski dense Schottky subgroup, we use the coding of the limit set to realize Jordan projections as periods of a vector-valued Busemann return map for an expanding map on the Furstenberg boundary. The key step is to prove that a suitable two-branch asymptotic discrepancy is not locally constant on the limit set. We also show that the same criterion applies beyond Lie groups; in particular, it yields a direct density result for multiplier spectra of hyperbolic rational maps whose Julia set is not contained in a circle.

We prove, to all orders, maximal transcendentality of the strongly coupled large-N Principal Chiral Model in the double-scaling regime introduced in our earlier work. We also prove that, after a natural shift of the coupling constant, the coefficients of the vacuum-energy expansion are expressed purely as polynomials in odd zeta values with rational coefficients. The first 35 explicitly computed orders reveal further number-theoretic regularities, pointing to hidden structure beyond maximal transcendentality.

In this paper, we prove the local boundedness of solutions to systems of partial differential equations in divergence form. More specifically, we consider systems that include the first variations of functionals depending on the spatial variable and exhibiting nonstandard growth with respect to the gradient, such as $$\int_Ω \left( 1+ h(|Du|)\right) ^{α(x)} \, d x,$$ where the convex function $h=h(t)$ does not satisfy the so-called $Δ_2$ property and does not exhibit the conventional polynomial growth behavior.

This paper studies the problem of detecting a change in the distribution of a sequence of independent observations when the post-change distribution is unknown. We propose a novel change detection algorithm, termed Predictive-Mixture CuSum (PM-CuSum), which combines predictive distributions constructed from sliding windows of different lengths within a CuSum recursion. The predictive distributions are aggregated using adaptive weights based on their recent predictive performance. We show that PM-CuSum achieves first-order asymptotic optimality under mild conditions, and that its asymptotic delay bound has a smaller remainder order than what is achieved procedures using a single fixed (even oracle) window. Numerical simulations demonstrate that PM-CuSum performs well compared to existing methods. Moreover, it is demonstrated that forming likelihood ratios using full predictive distributions can substantially improve performance compared to plug-in likelihoods.

We construct locally compact non-unital spectral triples for a noncommutative planar system determined by a fixed nondegenerate irreducible unitary sector of the kinematical symmetry group \(G_{\mathrm{NC}}\). The sector is labelled by central parameters \((\hbar_0,\vartheta_0,B_0)\), with \(\hbar_0,\vartheta_0,B_0\neq0\) and \(\hbar_0-\vartheta_0B_0\neq0\). For this sector, the triples \((\mathcal S_{\hbar_0,\vartheta_0,B_0},\mathcal H, D^{r,s}_{\hbar_0,\vartheta_0,B_0})\) form an even two-parameter family indexed by \((r,s)\), and different choices of \((r,s)\) give unitarily equivalent realizations. The unperturbed Dirac operators have Landau-type spectral levels of infinite multiplicity; hence local compactness, rather than compact resolvent, is the relevant analytic condition. We then identify the represented algebra \(π(\mathcal S_{\hbar_0,\vartheta_0,B_0})\) with the effective Moyal Fréchet \(\ast\)-algebra with deformation parameter \(\vartheta_{\mathrm{eff}} =\frac{\vartheta_0}{1-\vartheta_0B_0/\hbar_0}.\) For each star-product realization parameter \(\varrho\), this yields spectral triples over the involutive Moyal algebra \(\mathcal A_{\vartheta_{\mathrm{eff}},\varrho}\). External \(U(1)_{\star_{\vartheta_{\mathrm{eff}},\varrho}}\)-gauge potentials are incorporated by localizing the affine gauge potentials with smooth cutoffs. The resulting bounded self-adjoint perturbations \(B_R^{(\varrho)}\) define Dirac operators \(D_R^{\varrho,r,s}=D^{\prime\,r,s}+B_R^{(\varrho)}.\) Finally, as \(R\to\infty\), these operators converge in the strong resolvent sense to a self-adjoint limiting operator \(D_\infty\), the closure of the formal minimally coupled operator. Thus the finite-cutoff triples rigorously approximate the limiting minimally coupled Dirac operator associated with the fixed nondegenerate \(G_{\mathrm{NC}}\)-sector.

Quantum fluctuations are often suppressed in ferromagnetic materials because they admit a simple unfrustrated ground state, greatly limiting the scope of the phenomena that can be observed. In this work, we show how naturally occurring magnetization-non-conserving couplings fundamentally alter this paradigm by demonstrating the existence of a chiral ferromagnet that is stabilized by quantum fluctuations. More specifically, we show how these spin-orbit interactions modify the classical phase diagram; whereas a classical analysis predicts only achiral collinear states, we observe fluctuation-stabilized phases, including a ferromagnet with large orbital chirality and a chiral stripe. We elucidate how such couplings generate a scalar orbital chirality spontaneously, in contrast to conventional mechanisms which rely upon a field-induced canting of vector chiral order. The resultant chiral states exhibit distinct transport signatures, namely an enhanced thermal Hall effect, and are of direct relevance to moiré heterostructures, Rydberg-atom arrays, and solid-state materials featuring non-Kramers spins

We present a self-consistent two-fluid theory demonstrating that pressure gradients simultaneously generate plasma flows and magnetic fields. We show that compatibility between ion momentum balance and mass conservation imposes a previously unrecognized constraint on plasma evolution: the total pressure must satisfy the Laplace equation, $\nabla^2 p = 0$. This condition yields a class of exact analytical solutions in which pressure-driven flows and Biermann-type magnetic fields emerge together. Application of the model to a galactic gas clump reveals that, under thermal pressure, electrons and ions move almost together, giving rise to weak currents and consequently very small seed magnetic fields. Ion dynamics are also important for determining the seed magnetic-field generation time $τ_B$ and for estimating the ion flow velocity. The model is further applied to laser-produced plasma to describe its short-time evolution. The present theory provides a unified, self-consistent description of pressure-driven flow generation and magnetogenesis in both astrophysical and laboratory plasmas.

We derive a priori error estimates for Nitsche's method applied to elliptic problems on approximate domains. Such approximations arise, for example, in unfitted finite element methods, data-driven simulations, and evolving domain problems, where the computational domain does not coincide exactly with the physical one. We quantify geometric errors in terms of boundary location and normal perturbations and carry out the analysis in an abstract CutFEM framework under standard stability assumptions. In the energy norm, we obtain an estimate exhibiting an $h^{-1/2}$ amplification of the boundary location error. We then prove a refined $H^1$-seminorm estimate that removes this amplification, yielding a sharper bound with additive contributions from boundary location and normal errors. Finally, we establish an optimal order $L^2$-error estimate based on a refined duality argument, where the geometry contribution appears as a separate additive term, decoupled from the mesh size $h$. The results reveal a fundamental distinction between the norms: the energy norm amplifies boundary location errors while remaining insensitive to normal perturbations, the $H^1$-seminorm separates location and normal errors, and the $L^2$-norm is insensitive to normal perturbations. This provides a clear characterization of how geometric approximation affects convergence in Nitsche-based finite element methods, with particular relevance for unfitted discretizations.

We present a discrete Helmholtz--Hodge decomposition for H(div)-conforming Brezzi--Douglas--Marini (BDM) finite elements on triangulated surfaces of arbitrary topology. The divergence-free BDM subspace is split L2-orthogonally into rotated gradients of a continuous streamfunction space and a finite-dimensional space of discrete harmonic fields whose dimension equals the first Betti number of the surface. Consequently, any incompressible flow discretized on this subspace can be reformulated with a scalar streamfunction and finitely many harmonic coefficients as the only unknowns. This eliminates the pressure and the saddle-point structure while ensuring exact tangentiality, pointwise divergence-freeness, and pressure-robustness. We present a randomized algorithm for constructing the harmonic basis and discuss implementation aspects including hybridization, efficient treatment of the harmonic unknowns, and pressure reconstruction. Numerical experiments for unsteady surface Navier--Stokes equations on a trefoil knot and a multiply-connected sculpture surface demonstrate the method and illustrate the physical role of the harmonic velocity component.

We propose a ground state entanglement probe for gapped, two-dimensional quantum many-body systems that involves taking powers of reduced density matrices in a particular geometric configuration. This quantity, which we denote by $ω_{α,β}$, is parameterized by two positive real numbers $α, β$, and can be seen as a ``Rényi-like" generalization of the modular commutator -- another entanglement probe proposed as a way to compute the chiral central charge from a bulk wave function. We obtain analytic expressions for $ω_{α,β}$ for gapped ground states of non-interacting fermion Hamiltonians as well as ground states of string-net models. In both cases, we find that $ω_{α,β}$ takes a universal value related to the chiral central charge. For integer values of $α$ and $β$, our quantity $ω_{α,β}$ can be expressed as an expectation value of permutation operators acting on an appropriate replica system, providing a natural route to measuring $ω_{α,β}$ in numerical simulations and potentially, experiments.

Sampling a quantum system's underlying probability distributions is an important computational task, e.g., for quantum advantage experiments and quantum Monte Carlo algorithms. Tensor networks are an invaluable tool for efficiently representing states of large quantum systems with limited entanglement. Algorithms for sampling one-dimensional (1D) tensor networks are well-established and utilized in several 1D tensor network methods. In this paper we introduce two novel sampling algorithms for two-dimensional (2D) isometric tensor network states (isoTNS) that generalize existing 1D tensor networks sampling algorithms. Our first proposed algorithm performs independent sampling and yields a single configuration together with its associated probability. The second algorithm employs a greedy search strategy to identify $K$ high-probability configurations and their corresponding probabilities. Numerical results demonstrate the effectiveness of these algorithms across quantum states with varying entanglement and system size.

We propose analogs of the generalized MICZ-Kepler system on the three-dimensional sphere and the two-sheet hyperboloid. We construct their energy spectra and normalized wave functions and find that they depend on two quantum numbers, which suggests that the systems are minimally superintegrable.

Power-free integers and related lattice subsets give rise to interesting dynamical systems. They are revisited from a spectral perspective, in the setting of the Halmos--von Neumann theorem. With respect to the natural patch frequency measure, also known as the Mirsky measure, many of these systems have pure-point dynamical spectrum, but trivial topological point spectrum. We calculate the spectra explicitly, in additive notation, and derive their group structure, both for a large class of $\cB$-free lattice systems in $\RR^d$ and for power-free integers in quadratic number fields. Further, in all cases, the eigenfunctions can be given in closed form, via the Fourier--Bohr coefficients of generic elements and their translates, which form a subset of full Mirsky measure. Based on a simple argument via Kolmogorov's strong law of large numbers, we show how the Fourier--Bohr coefficients also provide the eigenfunctions for the unique measure of maximal entropy, and that we get phase consistency for both measures.

Deep neural networks are known to suffer from exploding or vanishing gradients as depth increases, a phenomenon closely tied to the spectral behavior of the input-output Jacobian. Prior work has identified critical initialization schemes that ensure Jacobian stability, but these analyses are typically restricted to fully connected networks with i.i.d. weights. In this work, we go significantly beyond these limitations: we establish a general stability theorem for deep neural networks that accommodates sparsity (such as that introduced by pruning) and non-i.i.d., weakly correlated weights (e.g. induced by training). Our results rely on recent advances in random matrix theory, and provide rigorous guarantees for spectral stability in a much broader class of network models. This extends the theoretical foundation for initialization schemes in modern neural networks with structured and dependent randomness.

We propose a semismooth Newton-based augmented Lagrangian framework for reconstructing sparse sources in inverse acoustic scattering problems. Rather than working in the unknown source space, our semismooth Newton updates operate in the measurement (adjoint) space, which is especially efficient when the number of measurements is much smaller than the discretized source dimension. The source is then recovered via Fenchel-Rockafellar duality. Our approach substantially accelerates computation and reduces costs. Numerical experiments in two and three dimensions demonstrate the high efficiency of the proposed method.

We propose a new perspective on the asymptotic regimes of fast and slow extinction in the contact process on locally converging sequences of sparse finite graphs. We characterise the phase boundary by the existence of a metastable density, which makes the study of the phase transition particularly amenable to local-convergence techniques. We use this approach to derive general conditions for the coincidence of the critical threshold with the survival/extinction threshold in the local limit. We further argue that the correct time scale to separate fast extinction from slow extinction in sparse graphs is, in general, the exponential scale, by showing that fast extinction may occur on stretched exponential time scales in sparse scale-free spatial networks. Together with {the results of} Nam, Nguyen and Sly (Trans.\ Am.\ Math.\ Soc.\ 375, 2022), our methods can be applied to deduce that the fast/slow threshold in sparse configuration models coincides with the survival/extinction threshold on the limiting Galton-Watson tree.

We analyse the metastable behaviour of the disordered Curie--Weiss--Potts (DCWP) model subject to a Glauber dynamics. The model is a randomly disordered version of the mean-field $q$-spin Potts model (CWP), where the interaction coefficients between spins are general independent random variables. These random variables are chosen to have fixed mean (for simplicity taken to be $1$) and well defined cumulant generating function, with a fixed distribution not depending on the number of particles. The system evolves as a discrete-time Markov chain with single spin flip Metropolis dynamics at finite inverse temperature $β$. We provide a comparison of the metastable behaviour of the CWP and DCWP models, when $N \to \infty$. First, we establish the metastability of the CWP model and, using this result, prove metastability for the DCWP model (with high probability). We then determine the ratio between the metastable transition time for the DCWP model and the corresponding time for the CWP model. Specifically, we derive the asymptotic tail behavior and moments of this ratio. Our proof combines the potential-theoretic approach to metastability with concentration of measure techniques, the latter adapted to our specific context.

It is known that Green's functions can be expressed as continued fractions; the content at the $n$-th level of the fraction is encoded in a coefficient $b_n$, which can be recursively obtained using the Lanczos algorithm. We present a theory concerning errors in approximating Green's functions using continued fractions when only the first $N$ coefficients are known exactly. Our focus lies on the stitching approximation (also known as the recursion method), wherein truncated continued fractions are completed with a sequence of coefficients for which exact solutions are available. We assume a now standard conjecture about the growth of the Lanczos coefficients in chaotic many-body systems, and that the stitching approximation converges to the correct answer. Given these assumptions, we show that the rate of convergence of the stitching approximation to a Green's function depends strongly on the decay of staggered subleading terms in the Lanczos cofficients. Typically, the decay of the error term ranges from $1/\mathrm{poly}(N)$ in the best case to $1/\mathrm{poly}(\log N)$ in the worst case, depending on the differentiability of the spectral function at the origin. We present different variants of this error estimate for different asymptotic behaviours of the $b_n$, and we also conjecture a relationship between the asymptotic behavior of the $b_n$'s and the smoothness of the Green's function. Lastly, with the above assumptions, we prove a formula linking the spectral function's value at the origin to a product of continued fraction coefficients, which we then apply to estimate the diffusion constant in the mixed field Ising model.

We construct a weighted Riemannian manifold $(\mathbb R^2,g,μ)$ satisfying $\mathrm{CD}(1/2,\infty)$, the curvature-dimension condition, with the following property: if $γ$ denotes a centered Gaussian measure on $\mathbb R^2$, then there is no Lipschitz map $T:(\mathbb R^2,\|\cdot\|) \to (\mathbb R^2,g)$ satisfying $T_\#γ=μ$. Building on this, we prove a Weyl-type asymptotic law for the eigenvalues of the weighted Laplacian $-Δ_{g,μ}$ and show that they are asymptotically negligible when compared to the eigenvalues of $-Δ_γ$. These results give strong counterexamples to two questions of E. Milman and complement the recent counterexample of Aryan.