Nonlinear response functions, formulated as multipoint correlation functions or Volterra kernels, encode the dynamical and spectroscopic properties of physical systems and underpin a wide range of nonlinear transport and optical phenomena. However, their evaluation rapidly becomes prohibitive at high orders because of combinatorial (often factorial) scaling or severe numerical errors. Here, we establish a systematic and efficient framework to compute nonlinear response functions directly from real-time dynamics, without explicitly constructing multipoint correlators or relying on numerically unstable finite-difference methods for order-resolved extraction. Our approach is based on the Gateaux derivative with respect to the external field in function space, which yields a closed hierarchy of tangent equations of motion (TEOM). Propagating the TEOM alongside the original dynamics isolates each perturbative order with high accuracy, providing a term-by-term decomposition of physical contributions. The computational cost scales exponentially with response order in the fully general setting and reduces to polynomial complexity when all perturbation directions are identical; both regimes avoid the factorial scaling of explicit multipoint-correlator evaluations. We demonstrate the power of TEOM by computing frequency-resolved fifth-order response functions for a solid-state electron model and by obtaining nonlinear response functions up to the 49th order with controlled accuracy in a classical Duffing oscillator. We further show that our time-evolution formulation allows optical conductivities to be evaluated directly while remaining numerically stable even near zero frequency. TEOM can be incorporated seamlessly into existing real-time evolution methods, yielding a general framework for computing nonlinear response functions in quantum and classical dynamical systems.

Quantum many-body scarring (QMBS) has emerged as an intriguing paradigm of weak ergodicity breaking in nonintegrable quantum many-body models, particularly lattice gauge theories (LGTs) in $1+1$ spacetime dimensions. However, an open question is whether QMBS has a true high-energy physics origin, and thus exists in higher-dimensional LGTs with dynamical matter. Given that nonergodic dynamics in $d=1$ spatial dimension tend to vanish in $d>1$, it is important to probe this question. Using matrix product state techniques for both finite and infinite systems, we show that QMBS occurs in the $2+1\mathrm{D} \mathrm{U}\left(1\right)$ quantum link model, as evidenced in persistent coherent oscillations in local observables, a marked slowdown in the growth of the bipartite entanglement entropy, and revivals in the fidelity. Interestingly, we see that QMBS is more robust when the matter degrees of freedom are bosonic rather than fermionic, which we argue is due to the strong nonlocality of the wave function in the latter case. Our results further shed light on the intimate connection between gauge symmetry and QMBS, and highlight the persistence of scarring in higher spatial dimensions. Our findings can be tested in near-term analog and digital quantum simulators, and we demonstrate their accessibility on a recently proposed cold-atom analog quantum simulator.

The interplay between disorder and topology has become a central theme in condensed matter physics. Disorder can not only destroy topological phases but also induce them, as exemplified by the topological Anderson insulator (TAI). Here, we show that, in close analogy, disorder can drive the clean-limit, time-reversal-broken (T-broken) quantum spin Hall state of ferromagnetic monolayer ${\mathrm{MnBi}}_4{\mathrm{Te}}_7$ into a quantum anomalous Hall phase, which was called topological Anderson Chern insulator (TACI). Using density functional theory and nonequilibrium Green's function calculations in the presence of disorder, we identify disorder-induced phases, including T-broken TAI, TACI, normal insulator, etc., and then construct a comprehensive phase diagram. To discriminate multiple phases in the strong-disorder regime, we further use the density of states computed within the self-consistent Born approximation, which, in particular, distinguishes gapped and ungapped topological phases. We find that the two effective band inversions of Hamiltonian are suppressed at distinct critical disorder strengths; the survival of a single inversion over a finite disorder window stabilizes the TACI. The region of quantized Hall resistance is not confined to the band gap but extends into a mobility gap, substantially enlarging the parameter space exhibiting quantization. Remarkably, at strong disorder, we further propose a zero Hall plateau insulating state characterized by an insulating bulk and edge channels subject to diffusive scattering that can coexist with the TACI. This behavior is distinct from a conventional band-gap Chern insulator and provides a clear experimental signature. Compared with the clean region, the expansion of the energy window supporting the Chern insulator of the disordered region enhances thermal robustness and enables observation of the quantum anomalous Hall effect at higher, technologically relevant temperatures, advancing both fundamental studies and potential applications.

Type-II semi-Dirac fermions in two dimensions have been proposed to describe topologically nontrivial low-energy excitations in titanium/vanadium oxide heterostructures. These quasiparticles appear at the merger of three Dirac cones, resulting in a nonzero Berry phase. We find, by employing Hartree-Fock, renormalization group, and random phase approximation techniques, that the spectrum is very sensitive to long-range electron-electron interactions and can undergo a profound transformation. Our results indicate that at the topological phase boundary, long-range correlations stabilize a hybrid electronic phase displaying both Dirac and type-II semi-Dirac qualities, with physical characteristics exhibiting continuously varying critical exponents as a function of the Fermi energy; for example, Landau levels in a magnetic field vary with the energy scale: $|{\varepsilon }_n{\left(B\right)|\sim \left(nB\right)}^{1/2}\to {\left(nB\right)}^{3/4},n\in {\mathbb{N}}_0$. The quasiparticle spectrum evolves, driven by interactions, from anisotropic Dirac dispersion at the lowest energies toward the characteristic type-II semi-Dirac boomerang shape as the energy increases. The corresponding density of states concomitantly varies between linear and power one-third ($\rho \left(\varepsilon \right)\sim \left|\varepsilon \right|\to {\left|\varepsilon \right|}^{1/3}$). The crossover scale is controlled by the interaction strength $\alpha =e^2/\left(\hslash v\right)$ and the specifics of the effective interacting Hamiltonian.

Music is a structured and perceptually rich sequence of sounds in time, whose perception is shaped by the interplay of expectation and uncertainty about what comes next. Yet, the uncertainty we infer from music depends on how the musical piece is encoded as an event sequence. In this work, we use network representations, in which event types are nodes and observed transitions are directed edges, to compare how different feature encodings shape the transition structure we recover and how robust that structure is under modeled perceptual constraints. We systematically analyze eight encodings of piano music, from single-feature vocabularies to richer multifeature combinations. These representational choices reorganize the state space and fundamentally reshape network topology, shifting how uncertainty is distributed across transitions. To connect these descriptive differences to perception, we adopt a perceptual-constraint model that captures imperfect access to transition statistics. Overall, compressed single-feature representations yield dense transition structures with higher entropy rates, corresponding to higher average uncertainty per step, yet low model error, indicating that the constrained estimate stays close to the corpus transitions. In contrast, richer multifeature representations preserve finer distinctions but expand the state space, sharpen transition profiles, lower entropy rates, and increase model error. Finally, across representations, uncertainty concentrates in diffusion-central nodes while model error remains low there, suggesting an informational landscape in which predictable flow coexists with localized surprise. Overall, our results show that feature choice shapes not only the networks we reconstruct, but also which transition statistics are available and how vulnerable they are to distortion under perceptual constraints.

Most natural thermodynamic systems operate far from equilibrium, developing persistent currents and organizing into nonequilibrium stationary states (NESSs). Yet, the general principles governing how nonequilibrium driving reshapes probability distributions and breaks equilibrium symmetries remain poorly understood. Here, we establish a geometric framework connecting dissipation, current organization, and symmetry breaking in mesoscopic stochastic systems governed by Langevin dynamics. Using the inverse diffusion matrix as a metric, we derive the general geometric properties of stationary current velocities, employing a small-noise expansion to obtain further insights on their link to symmetry breaking. We then introduce a nonorthogonal decomposition of the velocity field into conservative and nonconservative components. The nonconservative contribution quantifies the violation of detailed balance induced by external driving, while the conservative one measures how the system reorganizes its internal state space relative to a reference equilibrium distribution. This decomposition naturally induces a splitting of entropy production and leads to an exact balance relation together with bounds defining the accessible thermodynamic region of a NESS. We show that the conservative contribution determines how nonequilibrium conditions amplify or generate symmetry-breaking phenomena, yielding a geometric classification of NESSs. We apply the framework to systems ranging from molecular machines to coupled oscillators and Brownian gyrators, and extend it to multiplicative-noise dynamics, where additional symmetry breaking emerges from curvature induced by space-dependent diffusion. Our results uncover a general connection between geometry, dissipation, and emergent organization in nonequilibrium thermodynamics.

In a recently introduced bond-percolation model, Sousa et al. [Phys. Rev. Res. 7, L022067 (2025)] showed that multiple spanning clusters can coexist in three dimensions, with their number remaining unbounded in the thermodynamical limit. Here, we investigate further this multigel percolation model by showing that the fractal dimension of the shortest path is the same as standard three-dimensional percolation, while the fractal dimension of the contact between percolating clusters at the critical point is consistent with that expected for the intersection of two fractal objects. When the initial density is treated as a control parameter, the behavior of both the largest cluster and the second moment follows the same scaling relations as in standard percolation. For systems with multiple species, the minimum density required for all clusters to percolate scales with system size in agreement with the fractal dimension. In the case of two species with different initial densities, we find a transition from a regime where only one species percolates to one where both percolate. These results strengthen the connection between multigel percolation and classical percolation theory and suggest possible routes for designing materials with multiple interpenetrating percolating structures, such as trigels and tetragels.

Collective oscillations in neuronal systems often arise from interactions between excitatory and inhibitory populations rather than from recurrent coupling within a single ensemble. Motivated by the coexistence of strongly and partially synchronized regimes in such systems, we study the Kuramoto-Sakaguchi model on a bipartite network. Despite its minimal structure, the model exhibits rich collective dynamics, including both continuous and discontinuous transitions from full synchrony to partial synchrony (PS). In the PS regime, global oscillations fail to entrain one of the two populations, whose oscillators display quasiperiodic dynamics with an average frequency that can significantly deviate from that of the global field, as observed in neuronal networks. We show that this PS state constitutes an example of self-organized quasiperiodicity, arising here in the bimodal Kuramoto-Sakaguchi model despite its purely linear global coupling.

We developed an open-source scalar wave transport model to estimate the generalized scattering matrix ($S$-matrix) of a disordered medium in the diffusion regime. The term generalized refers to the incorporation of evanescent wave field modes alongside propagating modes in the estimation of the $S$-matrix. To achieve this, we employed the scalar Kirchhoff-Helmholtz boundary integral formulation together with the Green’s function perturbation method, thereby extending the conventional Fisher-Lee relations to include evanescent modes. The estimated $S$-matrix, which satisfies the generalized unitarity and reciprocity relations, is modeled for a two-dimensional disordered waveguide. The generalized transmission matrix contained within the $S$-matrix is utilized to estimate the optimal phase-conjugate wavefront for focusing onto an evanescent mode. The phenomenon of a universal transmission value of 2/3 for such an optimal phase conjugate wavefront is demonstrated in the context of evanescent wave mode focusing through a diffusive disorder. The presented code framework may be of interest to wavefront shaping researchers for visualizing and estimating wave transport properties in general.

The forces that mixtures of motorized and passive crosslinking proteins collectively generate between cytoskeletal filaments within our cells are the key drivers of active cellular mechanics. Despite their importance, a unified theory connecting measurable crosslinker properties to force generation between filament pairs has so far been missing. In this paper, we derive a theory that predicts the forces generated collectively by crosslinking proteins linking two biopolymer filaments from experimentally accessible filament and crosslinker properties, using out-of-equilibrium thermodynamics. Our framework allows us to decompose the forces generated by crosslinkers into three separate components: entropic, active, and frictional. In doing so, it offers a clear physical interpretation of the fundamental mechanisms by which crosslinking proteins self-organize and collectively generate forces. We demonstrate the robustness and utility of this framework by applying it to four different experiments that probe the combined roles of passive and motorized crosslinkers. For each experiment, our theoretical approach allows us to disentangle the relative contributions of entropic, active, and frictional forces, clarifying how different physical processes underpin collective force production. In turn, this makes it possible to quantitatively compare and predict how various crosslinker combinations influence force generation between filaments, pattern formation along filaments, and the dynamics of filament pairs.

Quantum state control is a fundamental tool for quantum technologies. In this work, we propose and analyze the use of quantum optimal control to exploit the dipolar interaction of ultracold atoms on a lattice ring, focusing on the generation of selected states with entangled circulation. This scheme requires time-dependent control over the orientation of the magnetic field, a technique that is feasible in ultracold atom laboratories. The system's evolution is driven by just two independent control functions. We describe the symmetry constraints of this approach and numerically test them using the extended Bose-Hubbard model. We find that the proposed control can engineer entangled current states with perfect fidelity across a wide range of systems, and that in the remaining cases, the theoretical upper bounds for fidelity are reached.

Coincidence imaging, also known as ghost imaging, is a technique that exploits correlations between two particles to reconstruct information about a specimen. The particle that relays the spatial information about the object remains completely noninteracting, while the particle used to probe the object is not spatially resolved. While ghost imaging has been primarily implemented on photonic platforms, applying it to particles with fundamentally different properties opens up scientific directions. Mixing massive, charged electrons with massless, neutral photons introduces a hybrid architecture that unites two fundamental microscopic platforms, each serving as a cornerstone of highly advanced imaging systems. In this work, we investigate coincidence imaging using electron–cathodoluminescence photon pairs generated within a transmission electron microscope. Utilizing a custom-built free-space cathodoluminescence setup, we demonstrate two-dimensional ghost imaging of complex patterns. We are able to obtain a spatial resolution down to 2 $\mu \mathrm{m}$, paving the way for adaptation of quantum-enhanced imaging techniques from photonic quantum optics to electron microscopy.

Qubit-based quantum simulators naturally target two-level systems, whereas many quantum many-body problems are intrinsically $d$ level. Encodings from qudits to qubits then enlarge the Hilbert space and can introduce unphysical states that interfere with variational optimization. We formulate a variational framework for encoded $d$-level models that suppresses these illegitimate states with penalty terms and benchmark it for spin-1 and spin-3/2 bilinear-biquadratic Heisenberg chains. We compare binary encoding, which minimizes the qubit overhead, with symmetry encoding, which preserves the relevant spin symmetries and enables symmetry-conserving ansätze. Although binary encoding is more qubit efficient, its hardware-efficient ansatz is harder to train and less effective at exploiting conserved quantities. In contrast, symmetry encoding requires more qubits but reaches substantially higher fidelities, converges faster, and exhibits better trainability than the binary hardware-efficient ansatz. These results identify symmetry-preserving encodings as a practical route to simulating higher-spin models on existing qubit platforms.

We analyze how simple local constraints in two dimensions lead a defect to exhibit robust, nontransient, and tunable subdiffusion. We uncover a rich dynamical phenomenology realized in ice- and dimer-type models. On the microscopic scale, a single defect exhibits anomalously long retractions. Such retractions result in a form of dynamical caging and can be captured through an effective fractional diffusion equation. Mapping to a height field yields an effective random walk subject to an emergent (entropic) logarithmic potential, whose strength is tunable, related to the exponent of algebraic ground-state correlations. The defect's path, viewed as nonequilibrium growth process, yields a frontier of fractal dimension of $5/4$, the value for a loop-erased random walk, rather than $4/3$ for simple and self-avoiding random walks. Such frustration/constraint-induced subdiffusion is expected to be relevant to platforms, such as artificial spin ice and quantum simulators aiming to realize discrete link models and emergent gauge theories.

Making use of the simple fact that all separable complex Hilbert spaces of given dimension are isomorphic, we show that there are just six basic ways to define generalized coordinate operators in quantum mechanics. In each case, a canonically conjugate generalized momentum operator can be defined, but it may not be self-adjoint. Even in those cases, we show there is always either a self-adjoint extension of the operator or a Neumark extension of the Hilbert space that produces a self-adjoint momentum operator. In one of the six cases, both extensions work, thus leading to seven basic pairs of coordinate and momentum operators. We also show why there are more ways of defining basic coordinate and momentum measurements. A special role is reserved for measurements that simultaneously but imperfectly measure both.

We develop a unified field-theoretical description of the Bose-Hubbard model that captures low-energy dynamics across the insulator-to-superfluid transition. We show that commonly used cumulant expansions about the Mott ground state fail in the superfluid phase not only quantitatively but also fundamentally, because they impose an incorrect canonical structure. Expanding instead about the appropriate mean-field saddle point, we derive an effective action in which the dynamics of fluctuations are governed by a Berry-phase term, giving rise to an emergent magnetic field in the associated oscillator description. This field encodes particle-hole symmetry breaking in both the Mott-insulator phase and the superfluid, and vanishes at the point where a pure Goldstone mode emerges. The resulting effective hydrodynamic theory admits a speed of sound that approaches the Bogoliubov limit in the weakly interacting regime, demonstrating consistency across the entire phase diagram, from the Mott-insulator to the deep superfluid regime. While the total density remains continuous within mean-field theory, we find that quantum fluctuations generate a discontinuous contribution at the quantum phase transition, which vanishes at the tip of the Mott lobe.