We provide an extended account of the recent statistical mechanical theory of gauge invariance against operator shifting in quantum many-body systems [J. Müller and M. Schmidt, Phys. Rev. E 114, L022104 (2026)]. The gauge transformation is enacted by a shifting superoperator that displaces the fundamental position and momentum degrees of freedom. The shifting superoperator constitutes a map between Hilbert space operators, and it features Lie algebra commutator structure. Averages of general observables remain invariant under the shifting both in and out of thermal equilibrium, as well as in ground states. The gauge invariance induces exact sum rules that interconnect global observables and associated locally resolved correlation functions. In particular we describe the resulting one-body force, hyperforce, product, and two-body sum rules. We relate the shifting superoperator to a previously formulated quantum canonical transformation and present the generalization of quantum shifting to multicomponent systems. The gauge theory respects fundamental fermionic and bosonic particle properties, as we demonstrate by proving the compatibility of operator shifting and exchange symmetry. We formulate the quantum version of hyperdensity functional theory to provide formal access to hyperforces as well as to general averaged quantum observables via universal density functionals. For time-dependent situations, we describe quantum dynamical gauge invariance and prove exact dynamical sum rules for nonequilibrium situations, as generated by Hamiltonian time dependence. We argue for the fundamental status of statistical mechanical gauge invariance based on the compliance of the underlying geometry with canonical quantization according to Dirac's correspondence principle. Analogies and differences of the quantum mechanical sum rules with their classical counterparts remain indicative of the respective levels of description.

Let ${\left\{{\epsilon }_i\right\}}_{i=1}^{\infty }$ represent the set of unfolded energy eigenvalues of a quantized billiard system. Then, the set $\{{\delta }_n={\epsilon }_{n+1}-{\epsilon }_1-n\}$ may be considered as a finite time series of size $M=n_m-n_0$, in which $n$ plays the role of a discrete time and ${\delta }_n$ fluctuates around zero. These random fluctuations are known to exhibit the $1/f^{\alpha }$ noise characteristic, where $\alpha \in [1,2]$, the limit $\alpha =1\left(2\right)$ corresponding to a fully chaotic (regular) classical dynamics. Previously, the presence of $1/f^{\alpha }$ noise in the ${\delta }_n$ statistics has been demonstrated in a family of billiards for which the classical phase space is divided (Kolmogorov-Arnold-Moser systems). Here, the ${\delta }_n$ statistics is further explored, so as to include two new features, namely, (i) nonchaotic full ergodicity and (ii) symmetry. Four billiard families, classified as type $P$ and type $S$, are considered. The type-$P$ families are comprised of polygonal billiards, which are never chaotic, but may exhibit a mixing classical dynamics characterized quantitatively through the position autocorrelation function decay exponent $\sigma$. A type-$S$ billiard has a smooth boundary and, correspondingly, a divided phase space with normalized chaotic volume ${\rho }_{\text{c}}$. Possible correlations between the spectral noise parameter $\alpha$ and the classical quantities, $\sigma$ for type-$P$ and ${\rho }_{\text{c}}$ for type-$S$ domains, are numerically investigated. Spearman's rank correlation coefficients support that the quantum spectral exponent $\alpha$ scales with the classical quantities. In addition, one family in each class is $C_3$ symmetric. In this case, singlets and doublets have different spectral statistics in the chaotic limit, namely, the singlet (doublet) subspectrum follows the Gaussian orthogonal ensemble (Gaussian unitary ensemble) of random matrices. Let ${\alpha }_1\left({\alpha }_2\right)$ be the noise exponent in a singlet (doublet) time series. Numerical calculations of Pearson's correlation coefficients show that ${\alpha }_1$ and ${\alpha }_2$ are compatible with each other for both type-$S$ and type-$P$ within symmetrical families in the regime $1<\alpha <2$.

Analyzing the behavior of complex interdependent networks requires complete information about the network topology and the interdependent links across networks. For many applications, such as interdependent critical infrastructure (ICI) networks, understanding network interdependencies is crucial to anticipate cascading failures and mitigate the risk from disruptions. However, complete network data are often unavailable due to security concerns, and some important interdependent links are revealed only in the aftermath of a disruption. This study formulates and solves a network reconstruction problem to uncover uncertain network interdependencies during disruptions. We propose a scalable nonparametric Bayesian approach to reconstruct the topology of ICI networks from (observed) cascading failures. Metropolis-Hastings (M-H) algorithm coupled with the infrastructure-dependent proposal is employed to increase the efficiency of sampling possible graphs. Numerical results of reconstructing a synthetic system of ICI networks demonstrate that the proposed approach outperforms existing methods in both accuracy and computational time.

We experimentally investigate the dynamics of quasi-two-dimensional vibrated granular systems composed of geometric solids with identical mass. We analyze both the motion of individual particles and the collective behavior at different packing fractions. For single particles, we find that the number of vertices determines the type of motion. Solids with few vertices exhibit Brownian-like motion, while shapes with many or no vertices display more superdiffusivelike behavior. At the collective level, particle geometry and packing fraction jointly determine whether the system exhibits diffusive, subdiffusive, or superdiffusive dynamics, as revealed by the mean squared displacement, diffusion coefficients, the non-Gaussian parameter, and the probability distribution functions. We conclude that while the packing fraction controls the overall progression across dynamical regimes, the particle geometry dictates the specific pathways and extent of these transitions. These findings establish particle geometry as a design parameter for controlling the dynamics of driven granular media.

We address gauge invariance in the statistical mechanics of quantum many-body systems. The gauge transformation acts on the position and momentum degrees of freedom, and it is represented by a quantum shifting superoperator that maps quantum observables onto each other. The shifting superoperator is anti-self-adjoint, and it has noncommutative Lie algebra structure. These properties induce exact equilibrium sum rules that connect locally resolved force and hyperforce densities for any given observable. We argue that the framework is amenable to tight integration into quantum hyperdensity functional theory and that it generalizes naturally to nonequilibrium.

We study quantum spin chains solvable via hidden free fermionic structures. We review the algebras behind such models, establishing connections to the mathematical literature of the so-called “graph-Clifford” or “quasi-Clifford” algebras. We also introduce the “defining representation” for such algebras, and show that this representation actually coincides with the terms of the Hamiltonian in two relevant models: the XY model and the “free fermions in disguise” model of Fendley. Afterwards we study a particular antisymmetric combination of commuting Hamiltonians; this is performed in a model-independent way. We show that for this combination there exists a reference state, and few-body eigenstates can be created by the fermionic operators. Concrete application is presented in the case of the “free fermions in disguise” model.

Quantum and classical systems can consistently be coupled via nonunitary time-irreversible mechanisms. In this paper, we characterize which kind of corresponding dynamics converge in the stationary regime to a thermal hybrid state, that is, a density matrix that maximizes the hybrid arrangement entropy under the constraints of a canonical ensemble. Introducing a detailed balance condition, it is found that a specific subclass of hybrid Lindblad equations fulfill the demanded requirement. The main theoretical results are exemplified through a set of specific dynamics that in addition enlighten how the thermal state of each subsystem in isolation is affected by their mutual coupling. In particular, a Gaussian thermal state could become a bimodal distribution when increasing the interaction strength of a classical subsystem with a quantum two-level subsystem.

Conformal symmetry is expected to be realized in many equilibrium statistical mechanical systems at criticality. Although this is certainly true in two-dimensional systems, the three-dimensional case is subtler, and only a few proofs exist, only so in very specific cases. In this work, we give two proofs for the large-$N$ limit of the $\mathrm{O}\left(N\right)$ universality class within the nonperturbative renormalization group framework: one functional, and one vertex-by-vertex in Fourier space. While doing so, we unveil how the theory is structured in order for conformal symmetry to be realized. As a consequence, we shed light on what to expect, on rather general grounds, for a theory to be conformally invariant.

We investigate thermalization in a tight-binding chain with an on-site defect subject to local dephasing noise implemented as random phase kicks. For a single linear defect of strength $\epsilon$, we obtain an exact analytical description of the system spectrum and formulate the dephasing-induced dynamics in the eigenstate basis. We derive an approximate kinetic equation for mode populations that describes a continuous-time random walk in action space. The walk transition rates are defined by the overlap matrix encoding the spatial structure of eigenstates that can be computed exactly. Analyzing the spectral properties of the equation, we show that defect-induced localized modes act as bottlenecks that strongly slow down relaxation, with rates scaling as ${\epsilon }^{-2}$ for strong defects. Using large-deviation theory, we characterize rare dynamical trajectories and identify distinct relaxation pathways associated with low- and high-activity regimes in action space. We provide numerical evidence that the large-deviation function exhibits a dynamical phase transition in the limit $\epsilon \to \infty$. We then extend our analysis to the nonlinear case, considering a single nonlinear defect embedded in either a linear or a fully nonlinear discrete Schrödinger equation. Numerical simulations reveal a qualitatively faster approach to equilibrium than in the corresponding linear-defect model, driven by the amplitude-dependent weakening of the defect. Our results provide a unified framework for understanding thermalization, rare fluctuations, and relaxation pathways in stochastic tight-binding systems.

Many natural systems exhibit dynamics characterized by alternating phases or recurring sets of states. Describing the fluctuations of such systems over stochastic trajectories is necessary across diverse fields, from biological motors to quantum thermal machines. In an accompanying letter, we introduced the notion of stochastic excursions—a framework to analyze out of equilibrium processes via subtrajectories. Through counting observables, this framework captures finite-time fluctuations and trajectory-level behavior, which provides insights into thermodynamical tradeoffs between thermodynamic quantities of interest, such as entropy production and dynamical activity. In this work, we enhance this formalism by providing a suite of technical results on how to efficiently compute excursion-related quantities. Our analytical results provide explicit formulas for general moments of counting variables and excursion duration, as well as their covariance and conditional moments. We show that excursion counting statistics recover full counting statistics results, and uncover a relation between fluctuations of counting observables at the single-excursion level and the steady-state diffusion coefficient (noise). We also discuss a fluctuation theorem for individual excursions and show that it implies a modified thermodynamic uncertainty relation at the excursion level. In addition, we explore how analyzing excursions and using the results developed here can yield insights into three problems of interest: the three-qubit absorption refrigerator, cellular sensing, and birth-and-death processes.

We investigate quantum percolation in a honeycomb lattice with site dilution and random spin-orbit coupling. Using exact diagonalization combined with finite-size scaling analysis, we study the metal-insulator transition, extracting the quantum percolation threshold $p_q$, and the correlation-length exponent $\nu$. In the absence of spin-orbit coupling, we find that $p_q$ remains finite and demonstrate that the quantum threshold is significantly higher than the classical site-percolation threshold $p_c$ of the honeycomb lattice. When spin-orbit coupling is present, the spectral statistics exhibit a crossover from the Gaussian orthogonal ensemble to the Gaussian symplectic ensemble, reflecting the change in symmetry class. Simultaneously, the quantum percolation threshold shifts systematically to lower occupation probabilities, indicating that the spin-orbit coupling favors delocalization. For sufficiently strong spin-orbit coupling, $p_q$ tends to saturate, while the critical exponent approaches the expected one of the two-dimensional symplectic universality class.

We report the experimental realization of minimal chimera states in a system of three uncoupled oscillators driven solely by frequency-modulated forcing. Unlike conventional scenarios where chimera states emerge due to interactions among oscillators, here the coexistence of coherent and incoherent dynamics arises entirely from a common external modulation of a system parameter. By tuning the modulation amplitude and frequency, the system exhibits transitions between global synchronization, global incoherence, and minimal chimera states. The stability of these regimes is quantified using the maximal Lyapunov exponent, while a synchronization order parameter is employed to characterize the degree of coherence. A systematic exploration of the parameter space reveals well-defined regions associated with distinct dynamical behaviors. To provide analytical understanding, we employ a phase-reduction approach and derive the corresponding phase dynamics, which elucidate the mechanisms underlying phase locking and desynchronization. The robustness of the proposed mechanism is further demonstrated in a time-delayed chaotic system. Finally, experimental results obtained from an electronic circuit realization confirm the emergence of minimal chimera states under frequency-modulated driving. These findings establish external modulation as a viable route to chimera formation without coupling, offering a unique perspective on collective dynamics in driven nonlinear systems.

We study the emergence of a unique dynamical state, termed the asymmetric coherent cluster (ACC), in ring networks of identical Stuart–Landau oscillators arising from the interplay of symmetric and asymmetric couplings. The ACC state constitutes an intermediate regime between the splay (SP) and complete synchronization (CS) states, exhibiting macroscopic coherence despite persistent microscopic phase drift of the coupled oscillators. Remarkably, in the ACC state, oscillators self-organize into a stationary, symmetry-broken macroscopic structure while remaining dynamically nonstationary at the microscopic level. The transition from the SP state to the ACC state occurs via an abrupt frequency-unlocking process and is mediated by the breakdown of a stable limit cycle into an invariant torus. In contrast, the ACC–CS transition proceeds through gradual frequency locking accompanied by a continuous increase in global coherence. In this regime, the invariant torus collapses smoothly onto a fully synchronized limit cycle via a reverse supercritical Neimark–Sacker bifurcation. Numerical simulations support the theoretical analysis and demonstrate the robustness of the ACC state. These results demonstrate how competing symmetric and asymmetric interactions give rise to nontrivial collective dynamics, offering a route to coherent macroscopic states beyond conventional phase-locked synchronization. Furthermore, the robustness of ACCs and their existence over a broad range of coupling parameters strongly suggest their experimental realizability.

Modeling how networks change under structural perturbations can yield foundational insights into network robustness, which is critical in many real-world applications. The largest connected component is a popular measure of network performance. Percolation theory provides a theoretical framework to establish statistical properties of the largest connected component of large random graphs. However, this theoretical framework is typically only exact in the large-$N$ limit, failing to capture the statistical properties of largest connected components in small networks, which many real-world networks are. We derive expected values for the largest connected component of small $G(N,p)$ random graphs from which nodes are either removed uniformly at random or targeted by highest degree and compare these values with existing theory. We also visualize the performance of our expected values compared to existing theory for predicting the largest connected component of various real-world, small graphs.

The stationary equilibria of $N$ point vortices in a rectangular domain with doubly periodic boundary conditions offer a foundational framework for understanding the stationary lattices of coherent vortex structures. In this paper, we obtain new continuous families of these lattices. These solutions are constructed using methods from complex analysis including conformal mapping and special function theory. First, the doubly periodic rectangular domain is mapped to an annulus by a conformal mapping. Next, a loxodromic function is used to construct a solution consisting of stationary point vortices embedded in a smooth background vorticity field which is exponentially related to the stream function (Liouville-type background vorticity). This solution contains three continuous parameters, and by considering the limits of one of these parameters, a stationary point vortex solution without a background vorticity field is constructed. Until now, stationary point vortex solutions under doubly periodic boundary conditions have been mainly symmetric. Varying the other two parameters noted above, we generate families of asymmetric stationary point vortex lattices.

Wave-like spatial statistics in walking-droplet systems are often associated with wave-mediated interactions and wave-memory effects. Here we explore how similar statistical structure can arise from the low-dimensional nonlinear dynamics of an inertial active particle with internal degrees of freedom. In this framework, steady propulsion corresponds to internal-state fixed points whose spiral or transiently chaotic relaxation organizes oscillatory ensemble densities. Local perturbations then generate wave-like statistics in both open and closed geometries, suggesting that wave-like ensemble behavior may emerge more generally from internal-state attractor dynamics in inertial active matter.

The critical behavior of the equilibrium Ising model, famously governed by the Wilson-Fisher universality class (UC), is a cornerstone of statistical physics. A natural question is whether its properties are modified when the interacting spins are allowed to move. This can be explored within the framework of active motility by, e.g., prescribing that the spins self-propel along the direction of their orientation. In such a system, the spin number density necessarily constitutes a soft hydrodynamic mode and can therefore modify the scaling behavior. In this letter, we show that this is indeed the case in a critical active Ising model where density impedes the system's collective motion. Using a perturbative dynamic renormalization group analysis at a one-loop level, We identify six UCs, including three previously unidentified ones, one of which supersedes the Wilson–Fisher UC as the generic UC governing the critical behavior.

Fingerprint labyrinthine patterns exhibit a level of structural complexity beyond simple stripe phases, combining local stripe order with a dense network of pointlike defects. Unlike symmetry-breaking phases, where coarsening proceeds via diffusive defect annihilation, or conventional stripe phases, where curvature-driven motion of extended grain boundaries dominates, the coarsening of fingerprint labyrinths is governed primarily by localized junction and terminal defects. Using the Turing-Swift-Hohenberg equation, we study the nonequilibrium relaxation of fingerprint labyrinthine patterns following a quench. To go beyond conventional Fourier-based diagnostics, we employ a template-matching convolutional neural network to identify and track junctions and terminals directly in real space, enabling a quantitative characterization of defect statistics and spatial correlations. We show that, although these pointlike defects drive coarsening, their motion is strongly constrained by the surrounding stripe geometry, leading to slow, nondiffusive dynamics that are qualitatively distinct from both conventional phase ordering and stripe coarsening. Together, these results establish defect-mediated dynamics as the central organizing principle of fingerprint labyrinthine coarsening and demonstrate the effectiveness of machine learning–assisted approaches for complex pattern-forming systems.

Changing geometry creates its own overdamped thermodynamic anomaly. This is the geometric analog of the underdamped-overdamped anomaly known for Langevin systems in nonuniform or time-dependent temperature fields. We show this for Langevin dynamics in ${\mathbb{R}}^3$ under a time-dependent potential and a moving holonomic constraint. The moving constraint imposes a normal control velocity on the overdamped motion. Starting from a one-parameter family of underdamped parent models indexed by $\chi$, with exact hard constraints, we show that the small-mass overdamped limit leaves a finite geometric contribution proportional to the surface divergence of the control velocity. This term is absent from the naive overdamped heat-work split. It is therefore a genuine geometric anomaly. Once restored, it reorganizes the first law into a modified heat-work pair. On the heat side, the detailed fluctuation theorem recovers its Clausius form. On the work side, the Jarzynski equality is closed on effective work. The same identification also clarifies the relative-entropy balance. Our results identify changing geometry as a source of thermodynamic structure in the overdamped Langevin dynamics.