Research on the electrical transport properties of iron mononitrides (FeN), including ${\gamma }^{\prime \prime }\text{−}\mathrm{FeN}$ and ${\gamma }^{\prime \prime \prime }\text{−}\mathrm{FeN}$, has been hindered by difficulties in fabricating single-phase samples. In this study, nitrogen-deficient ${\gamma }^{\prime \prime }\text{−}\mathrm{FeN}$ and ${\gamma }^{\prime \prime \prime }\text{−}\mathrm{FeN}$ epitaxial films were successfully grown on ${\mathrm{Al}}_2{\mathrm{O}}_3\left(0001\right)$ substrates. Experimental characterization reveals that paramagnetic ${\gamma }^{\prime \prime }\text{−}\mathrm{FeN}$ exhibits metallic conduction, whereas antiferromagnetic ${\gamma }^{\prime \prime \prime }\text{−}\mathrm{FeN}$ behaves as a narrow-bandgap semiconductor. Remarkably, both polymorphs exhibit near-zero magnetoresistance ($\le 0.008\%$) at temperatures above 50 K, despite their distinct electronic natures. Hall effect measurements further indicate that the ordinary Hall coefficient of ${\gamma }^{\prime \prime }\text{−}\mathrm{FeN}$ undergoes a sequential sign reversal between 5 K and 400 K, whereas ${\gamma }^{\prime \prime \prime }\text{−}\mathrm{FeN}$ maintains hole-type carriers across the entire temperature range. First-principles calculations confirm that the contrasting metallic and semiconducting band structures of ${\gamma }^{\prime \prime }\text{−}\mathrm{FeN}$ and ${\gamma }^{\prime \prime \prime }\text{−}\mathrm{FeN}$, coupled with nitrogen vacancies acting as shallow electron traps, are responsible for these unique transport characteristics. This investigation provides a comparative analysis of the carrier transport behavior in FeN polymorphs, highlighting how nitrogen vacancies modulate carrier transport processes and demonstrating their intriguing immunity to magnetic-field perturbations.

Recently, a variational wave function based on the tensor representation of backflow corrections along with a Lanczos step has achieved state-of-the-art energy precision for Fermi-Hubbard-type models. However, solving the Fermi-Hubbard model remains challenging, and the validity of any method must be assessed through the ground state's physical properties. For simplicity, we refer this method as tensor-backflow. In this work, we apply the tensor-backflow method to investigate the Fermi-Hubbard model on two-dimensional lattices up to 256 sites, exploring various interaction strengths $U$, electron fillings $n$, next-nearest-neighbor hopping $t^{\prime }$, and boundary conditions. By considering backflow terms from nearest- or next-nearest-neighbor sites, we achieve competitive results without enforcing geometric symmetries on the variational wave function. The optimizations were stable from a prior unrestrictied Hartree-Fock state, followed by adding backflow corrections. However, changing interaction strengths in the prior unrestrictied Hartree-Fock state is helpful to bypass the local minima. When $t^{\prime }=0$, by considering nearest-neighbor backflow terms, linear stripe order emerges successfully for the case of $n=0.875$ and $U=8$ on a $16\times 16$ lattice with periodic boundary conditions. In a similar case with open boundary conditions, the energy obtained is only $4.5\times {10}^{-4}$ higher than the state-of-the-art method fPEPS with bond dimension $D=20$. Compared to state-of-the-art neural network methods, the energies obtained using the tensor-backflow approach are competitive, with relative errors below $5\times {10}^{-3}$. For $n=0.8$ and $n=0.9375$, direct optimizations yield results consistent with the phase diagram from AFQMC. When $t^{\prime }=-0.2$, considering next-nearest-neighbor backflow terms leads to energies that are either competitive with or even lower than those from state-of-the-art neural network approaches. For instance, for $n=0.875$ and $U=8$ on a $12\times 12$ lattice with periodic boundary conditions, the energy obtained is $8.1\times {10}^{-4}$ lower than that from the neural network result. Thus, the tensor-backflow method demonstrates strong representational capabilities for solving the Fermi-Hubbard model.

We study a ferromagnetic XXZ Heisenberg model on a Lieb lattice. A set of exact eigenstates is constructed based on the restricted spectrum generating algebra (RSGA) when a resonant staggered magnetic field is applied. These states are identical to the eigenstates of a system of two coupled angular momenta. Furthermore, we find that the RSGA can be applied to other eigenstates of the Lieb lattice in an approximate manner. Numerical simulations reveal that there exist sets of eigenstates, which obey a quasi-RSGA. These states act as energy towers within the low-lying excited spectrum, indicating that they are quantum many-body scars. Our main results are the identification of both exact and approximate scar states and their dynamical signatures.

Neural quantum states (NQS) have emerged as a promising approach to solve second-quantized Hamiltonians, because of their scalability and flexibility. In this work, we design and benchmark an NQS impurity solver for the quantum embedding (QE) methods, focusing on the ghost Gutzwiller approximation (gGA) framework. We introduce a graph transformer-based NQS framework able to represent arbitrarily connected impurity orbitals of the embedding Hamiltonian and develop an error control mechanism to stabilize iterative updates throughout the QE loops. We validate the accuracy of our approach with benchmark gGA calculations of the Anderson lattice model, yielding results in excellent agreement with the exact diagonalization impurity solver. Finally, our analysis of the computational budget reveals the method's principal bottleneck to be the high-accuracy sampling of physical observables required by the embedding loop, rather than the NQS variational optimization, directly highlighting the critical need for more efficient inference techniques.

The weak interlayer bonding in van der Waals layered materials results in low energy barriers for sliding and twisting, facilitating access to diverse metastable stacking configurations. Since stacking order governs crystal symmetry and physical properties, it serves as an extra degree of freedom for engineering material functionalities. Recently, the van der Waals compound ${\mathrm{SnP}}_2{\mathrm{Se}}_6$, which comprises atomically thin layers with intrinsic chiral symmetry, was shown to exhibit layer-independent second-harmonic generation [Nat. Commun. 14, 2521 (2023)]. Here, using first-principles calculations, we investigate 16 distinct stacking configurations to elucidate the relationship between interlayer arrangement and nonlinear optical response. We reveal that while all considered structures lack inversion symmetry, the second-harmonic generation response of ${\mathrm{SnP}}_2{\mathrm{Se}}_6$ varies drastically with stacking sequence. Specifically, translational shifts relative to the ground state structure enhance the second-order susceptibility, whereas mirrored (${\mathrm{A}}^{\mathrm{m}}$) or rotated (${\mathrm{A}}^{\mathrm{r}}$) layer arrangements tend to suppress it. Moreover, we observe that significant interlayer coupling induces a strong stacking-dependence in both electronic structure and linear optical properties of ${\mathrm{SnP}}_2{\mathrm{Se}}_6$. Our results provide insights into chiral van der Waals materials and establish stacking engineering as an effective strategy for tailoring the optoelectronic performance of ${\mathrm{SnP}}_2{\mathrm{Se}}_6$.

Quantum thermodynamics defines the ideal quantum thermoelectric, with maximum possible efficiency at finite power output. However, such an ideal thermoelectric is challenging to implement experimentally. Instead, here we consider two types of thermoelectrics regularly implemented in experiments: (i) finite-height potential barriers or quantum point contacts, and (ii) double-barrier structures or single-level quantum dots. We model them with Landauer scattering theory as (i) step transmissions and (ii) Lorentzian transmissions, respectively. We optimize their thermodynamic efficiency for any given power output, when they are used as thermoelectric heat engines or refrigerators. The Lorentzian's efficiency is excellent at vanishing power, but we find that it is poor at the finite powers of practical interest. In contrast, the step transmission is remarkably close to ideal efficiency (typically within 15%) at all power outputs. The step transmission is also close to ideal in the presence of phonons and other heat leaks, for which the Lorentzian performs very poorly. Thus, a simple nanoscale thermoelectric—made with a potential barrier or quantum point contact—is almost as efficient as an ideal thermoelectric.

${\mathrm{SnP}}_2{\mathrm{Se}}_6$, a recently synthesized two-dimensional (2D) polar semiconductor, exhibits significant discrepancies in reported carrier mobilities between experimental and theoretical studies. Here, based on first-principles calculations, we first investigate the electron-phonon (el-ph) scattering-limited intrinsic carrier mobility of ${\mathrm{SnP}}_2{\mathrm{Se}}_6$ monolayer. Our calculated electron and hole mobilities are both much lower than previous deformation potential (DP) model predictions, which indicates that the DP model is unsuitable for describing the carrier transport properties in such a 2D polar material. Then, we develop a theoretical scheme to quantitatively estimate the intrinsic carrier mobility of ${\mathrm{SnP}}_2{\mathrm{Se}}_6$ multilayers, as direct first-principles calculations on such a multilayer system are infeasible. The electron mobility for a 15-layer film predicted by our theoretical scheme is in excellent agreement with the experimental value measured on a sample of the same thickness. In addition, the scheme reveals that the intrinsic electron mobility of 2D ${\mathrm{SnP}}_2{\mathrm{Se}}_6$ initially decreases as the layer number increases, and then turns to increase. This predicted layer dependence successfully explains the experimental result that thin multilayers have a notably smaller electron mobility than thick multilayers. Our scheme provides an efficient and simple theoretical framework for quantitatively studying the intrinsic carrier mobility of 2D layered polar materials, particularly for relatively thick films for which first-principles calculations are impractical.

We propose a superconducting phase-difference-free approach to realizing second-order topological superconductivity in bilayers of coupled two-dimensional electron gases. In this setup, one layer hosts Rashba and the other hosts Dresselhaus spin-orbit coupling. With proximity-induced conventional $s$-wave pairing of the identical phase and a perpendicular Zeeman field, the two layers effectively emulate chiral $p$-wave superconductors of opposite chirality. The misalignment between the Rashba and Dresselhaus spin textures, combined with interlayer tunneling, gives rise to mass domain walls at the sample corners, which bind robust Majorana zero modes protected by a mirror-symmetry-graded ${\mathbb{Z}}_2$ invariant. Unlike previous proposals, our scheme requires no superconducting phase difference between the layers, thereby eliminating a major experimental challenge. Furthermore, the spatial positions of the Majorana corner modes can be electrically controlled by engineering orthogonal Rashba channels via ferroelectric polarization. This electrical tunability naturally enables the movement and exchange of Majorana modes in multiterminal geometries, positioning Rashba-Dresselhaus bilayers as a practical and versatile platform for higher-order topological superconductivity and Majorana-based quantum computation.

Density functional theory (DFT) in chemistry and materials science aims for “chemical accuracy,” but this goal is challenged by the need to approximate the exact exchange-correlation (XC) energy functional. The restored-regularized strongly constrained and appropriately normed (${\mathrm{r}}^2\mathrm{SCAN}$), meta-generalized gradient approximation to the XC functional fulfills 17 exact constraints of the XC energy, and has significantly boosted prediction accuracy for molecules and materials. However, ${\mathrm{r}}^2\mathrm{SCAN}$ remains inadequate at predicting the properties of open $d$ and $f$ transition-metal strongly correlated compounds, such as band gaps, magnetic moments, and oxidation energies. Prediction inaccuracies of ${\mathrm{r}}^2\mathrm{SCAN}$ energies arise from functional- and density-driven errors, mainly resulting from the DFT self-interaction error. Here, we propose a method termed ${\mathrm{r}}^2\mathrm{SCANY}$@${\mathrm{r}}^2\mathrm{SCANX}$ to mitigate the self-interaction error of XC functionals for the accurate simulations of electronic, magnetic, and thermochemical properties of transition-metal oxides. ${\mathrm{r}}^2\mathrm{SCANY}$@${\mathrm{r}}^2\mathrm{SCANX}$ uses different fractions of exact Hartree-Fock exchange, X for the electronic density and Y for the density functional approximation of the total energy, thereby simultaneously addressing functional-driven and density-driven inaccuracies. Building on just one (or maximum two) parameters that apply unchanged to s-p-bonded systems, we demonstrate that ${\mathrm{r}}^2\mathrm{SCANY}$@${\mathrm{r}}^2\mathrm{SCANX}$ improves upon the ${\mathrm{r}}^2\mathrm{SCAN}$ predictions for 20 highly correlated oxides and even outperforms the highly parametrized DFT(${\mathrm{r}}^2\mathrm{SCAN}$)$+U$ method—the state-of-the-art approach to predict strongly correlated materials. Prediction uncertainties for oxidation energies and magnetic moments of transition-metal oxides are significantly reduced by ${\mathrm{r}}^2\mathrm{SCAN}10@{\mathrm{r}}^2\mathrm{SCAN}50$ and band gaps with ${\mathrm{r}}^2\mathrm{SCAN}10@{\mathrm{r}}^2\mathrm{SCAN}$. ${\mathrm{r}}^2\mathrm{SCAN}10@{\mathrm{r}}^2\mathrm{SCAN}50$ diminishes the density-driven error of the energy in ${\mathrm{r}}^2\mathrm{SCAN}$ and ${\mathrm{r}}^2\mathrm{SCAN}10$. We demonstrate that the computationally efficient ${\mathrm{r}}^2\mathrm{SCAN}10@{\mathrm{r}}^2\mathrm{SCAN}$ is nearly as accurate as the global hybrid ${\mathrm{r}}^2\mathrm{SCAN}10$ for oxidation energies. This indicates that accurate energy differences can be obtained through rate-limiting self-consistent iterations and geometry optimizations with the efficient ${\mathrm{r}}^2\mathrm{SCAN}$. Subsequently, a more expensive nonlocal functional, such as a hybrid or self-interaction correction, can be applied in a fast, single post-self-consistent calculation, as in ${\mathrm{r}}^2\mathrm{SCAN}10$@${\mathrm{r}}^2\mathrm{SCAN}$.

We present a comprehensive theoretical analysis of the spin-dependent thermoelectric properties of a double quantum dot system coupled to a topological superconducting nanowire and ferromagnetic leads. The study focuses on the behavior of the Seebeck coefficient and its spin-resolved counterparts, with calculations performed by means of the numerical renormalization group method. We investigate the low-temperature transport regime, where a complex interplay between the two-stage Kondo effect, the ferromagnet-induced exchange field, and the Majorana coupling occurs. We demonstrate that thermoelectric measurements can reveal unique signatures of the Majorana interaction that are challenging to isolate in conductance measurements alone. It is shown that the exchange field fundamentally alters the thermoelectric response, leading to a rich, nonmonotonic temperature evolution of the thermopower, which is driven by a temperature-dependent competition between the spin channels. Furthermore, we have identified qualitatively different regimes of spin thermopower generation, controlled by the interplay between the Majorana-induced asymmetry and the spin polarization of the leads. Finally, by connecting the system's thermoelectric response to the underlying transport asymmetries quantified by the conductance spin polarization, we provide a consistent and unified physical picture, proposing thermoelectric transport as a sensitive probe for Majorana signatures.

Classical U(1) fracton spin liquids feature an extensive ground state degeneracy and follow an effective description in terms of a tensor Gauss's law where charges, so-called fractons, have restricted mobility. Here we introduce a simple spin model that realizes such a state by straightforward discretization of the higher-rank gauge theory on a square lattice. The simplicity of this construction offers direct insights into the system's fundamental fractonic properties, such as real-space fracton configurations, height-field representation of the classical ground state manifold as well as properties of local and nonlocal fluctuations within the fracton-free subspace. By sampling classical Ising states from the extensive ground state manifold, we show that the effective tensor Gauss's law remains intact when explicitly enforcing the spin-1/2 length constraint, demonstrating the existence of a classical Ising fracton spin liquid. However, we observe that perturbative quantum effects are insufficient to efficiently tunnel between classical ground states, leading to severe Hilbert space fragmentation which obscures fractonic quantum behavior. Specifically, by simulating the spin-1/2 quantum model with Green's function Monte Carlo as a function of the Rokhsar-Kivelson potential, we find that the system supports either magnetic long-range order or a classical spin liquid. Our findings highlight the crucial role of Hilbert-space fragmentation in fractonic spin systems but also indicate ways to mitigate such effects via increasing the spin magnitude to $S=1$, investigated in a companion paper.

Using matrix product state techniques we study the nonequilibrium dynamical response of the half-filled Hubbard ladder when subject to an optical pump. Optical pumping offers a way of producing and manipulating new strongly correlated phenomena by suppressing existing magnetic correlations. The ladder allows the effects of pump directionality to be investigated, and compared to a single chain it has strong spin-charge coupling and a fully gapped excitation spectrum, promising different nonequilibrium physics. We compute time-dependent correlations, including the nonequilibrium dynamical structure factors for spin and charge. By deriving a combined spin-charge sum rule that applies both in and out-of-equilibrium, we show that spectral weight is pumped directly from the antiferromagnetic spin response into a low energy $\omega \sim 0$ charge response below the Mott gap. The transfer of weight is pump direction dependent: pumping directed along the legs disrupts magnetic correlations more than pumping in the rung-direction, even if the post-pump energy density is similar. The charge correlation length is dramatically enhanced by the pump, whilst the spin correlations are most strongly suppressed at nearest- and next-nearest-neighbor spacings. After the pump the system is in a nonthermal correlated metallic state, with gapless charge excitations and approximately equal spin and charge correlation lengths, emphasizing the importance of treating these degrees of freedom on an equal footing in nonequilibrium systems.

With the intriguing coexistence of local magnetic orders and nonlocal valence bond patterns, unconventional magnetic plateau phases in geometrically frustrated quantum systems have attracted significant attention. This work investigates the spin-1/2 anisotropic kagome antiferromagnet, uncovering an exotic transition between two distinct valence bond patterns within the 1/9 magnetization plateau, where the fidelity between them is nearly equal to zero. By tuning the exchange anisotropy, we observe a spatial inversion of the valence bond pattern, accompanied by singularities at the transition boundary that is indicated by peaks in local magnetizations, nearest-neighbor correlations, entanglement entropy, and correlation length. Particularly in an external field $0.33<h<0.4$ along the spin-$z$ direction, tuning the exchange anisotropy induces a reentrant phenomenon, where the system enters from the type-$\mathcal{A}$ 1/9 plateau to a nonplateau region and back to the type-$\mathcal{B}$ 1/9 plateau with an inverted valence bond pattern. As the external field decreases to $0.29<h<0.33$, the nonplateau region narrows, reducing to a direct phase transition between these two types of 1/9 plateaus. Our findings highlight the intricate interplay of quantum fluctuations arising from nonlocal geometrical frustration and local exchange anisotropy, shedding light on exotic quantum phenomena in frustrated spin systems.

We present far-infrared magnetospectroscopy measurements of a HgTe quantum well in the inverted band structure regime over the temperature range of 2 to 60 K. The particularly low electron concentration enables us to probe the temperature evolution of all four possible optical transitions originating from zero-mode Landau levels, which are split off from the edges of the electronlike and hole-like bands. By analyzing their resonance energies, we reveal an unambiguous breakdown of the single-particle picture, indicating that the explanation of the anticrossing of zero-mode Landau levels in terms of bulk and interface inversion asymmetries is insufficient. Instead, the observed behavior of the optical transitions is well explained by their hybridization driven by electron-electron interaction. We emphasize that our proposed many-particle mechanism is intrinsic to HgTe quantum wells of any crystallographic orientation, including (110) and (111) wells, where bulk and interface inversion asymmetries do not induce the anticrossing of zero-mode Landau levels.

Open quantum systems governed by non-Hermitian effective Hamiltonians exhibit unique phenomena, such as the non-Hermitian skin effect, where eigenstates localize at system boundaries. We investigate this effect in a Rashba nanowire coupled to a ferromagnetic lead and demonstrate that it can be detected via nonlocal transport spectroscopy: while local conductance remains symmetric, the nonlocal conductance becomes nonreciprocal. We account for this behavior using both conventional transport arguments and the framework of non-Hermitian physics. Furthermore, we explain that exceptional points shift in parameter space when transitioning from periodic to open boundary conditions, a phenomenon observed in other non-Hermitian systems but so far not explained. Our results establish transport spectroscopy as a tool to probe non-Hermitian effects in open electronic systems.

Feldspar is the most abundant mineral group in the Earth's continental crust and almost invariably contains ${\mathrm{Fe}}^{3+}$ impurities in natural samples. These impurities dominate the characteristic deep-red luminescence of feldspar, while the host lattice itself also exhibits intrinsic luminescence associated with self-trapped excitons (STEs). Despite the central role of feldspar in luminescence dosimetry and geochronology, the microscopic mechanisms underlying both intrinsic and ${\mathrm{Fe}}^{3+}$-related luminescence remain incompletely understood. Here, we present a comprehensive first-principles investigation of disorder, intrinsic defects, and impurity-related charge trapping in K-feldspar. Using a cluster expansion trained on density functional theory calculations, we determine the ground-state Al–Si ordering and quantify the formation of Al–O–Al linkages at high temperatures. Based on this structure, we identify energetically favorable intrinsic defects and demonstrate that hole polarons localize preferentially on oxygen atoms bridging ${\mathrm{AlO}}_4$ and ${\mathrm{SiO}}_4$ tetrahedra, while Si–O–Si bridges do not stabilize holes. Rare Al–O–Al bridges formed at melting temperatures are shown to be particularly efficient hole traps. We compute STE configurations and vertical emission energies using a hybrid functional approach, obtaining emission energies in excellent agreement with experimental measurements. Furthermore, we show that ${\mathrm{Fe}}^{3+}$ impurities strongly modify local charge trapping: ${\mathrm{Fe}}^{3+}$ preferentially captures electrons to form ${\mathrm{Fe}}^{2+}$, while also stabilizing hole polarons on neighboring oxygen sites, forming metal-oxyl complexes. Based on these results, we propose a microscopic mechanism for ${\mathrm{Fe}}^{3+}$-related luminescence in which electron capture by ${\mathrm{Fe}}^{3+}$ is followed by spin-selective recombination that populates the ${\mathrm{Fe}}^{3+}$ excited state ${}^{4}T_1$. Our results provide a unified atomistic picture of intrinsic and impurity-driven luminescence in feldspar and establish a quantitative foundation for modeling charge trapping and recombination processes relevant to luminescence dating.

Critical points with emergent symmetry exhibit intriguing scaling properties induced by two divergent length scales, attracting extensive investigations recently. We study the driven critical dynamics in three-dimensional $q$-state clock models, in which the ordered phase breaks the $Z_q$ discrete symmetry, while an emergent $\text{U}\left(1\right)$ symmetry appears at the critical point. The ${\mathrm{Z}}_5$-clock model with both isotropic and anisotropic couplings is taken as an example. By increasing the temperature at a finite velocity $v$ to traverse the critical point from the ordered phase, we find that in the driven process, there exist two finite-time scaling (FTS) regions, characterized by two driving-induced time scales ${\zeta }_d\propto v^{-z/r}$ and ${\zeta }_d^{\prime }\propto v^{-z/r^{\prime }}$, respectively. Here $z$ is the dynamic exponent, $r$ is the usual critical exponent of $v$, and $r^{\prime }$ represents an additional critical exponent of $v$ associated with the dangerously irrelevant scaling variable. While the square of the order parameter $M^2$ obeys the usual FTS form, the angular order parameter ${\varphi }_q$ shows remarkably distinct scaling behaviors controlled by both FTS regions. For small $v, {\varphi }_q$ is dominated by the time scale ${\zeta }_d$, whereas for large $v, {\varphi }_q$ is governed by the second time scale ${\zeta }_d^{\prime }$. These results confirm the universality of the dynamic scaling form proposed in Nat. Commun. 16, 3402 (2025), wherein only the isotropic ${\mathrm{Z}}_6$-clock model is considered. Besides, we find that the value of $r^{\prime }$ changes from the value of the isotropic case to another value as the strength of anisotropy increases. The crossover effect between weak and strong anisotropy is also explored. Our theoretical insights provide a promising foundation for further experimental investigations in the hexagonal $R{\mathrm{MnO}}_3$ ($R$=rare earth) materials.

The quantum Pontus-Mpemba effect (QPME) is a counterintuitive phenomenon wherein a quantum system relaxes more rapidly through a two-step evolution protocol than through direct evolution under a symmetric Hamiltonian alone. In this protocol, the system first evolves under a symmetry-breaking Hamiltonian and then switches to a symmetric one. We demonstrate that QPME occurs under both real-time and imaginary-time dynamics with respect to $U\left(1\right)$ symmetry. Using tilted ferromagnetic initial states, we demonstrate that a transient asymmetric evolution significantly accelerates thermalization or convergence to the ground state for both real-time and imaginary-time evolutions, respectively. The effect is pronounced for small tilt angles, while larger tilts or antiferromagnetic initial states suppress it. Further numerical evidence confirms the consistency of QPME across different system sizes studied. Finally, by implementing a variational optimization procedure, we identify an optimized dynamical path, demonstrating that the relaxation rate can be further enhanced through tailored symmetry-breaking protocols. This work extends the framework of nonequilibrium quantum phenomena to incorporate active state preparation, with direct implications for the implementation of quantum simulation.