Papers

Collective non-equilibrium phenomena in many-body systems are strongly influenced by fluctuations, particularly in the vicinity of phase transitions, where the interplay between classical and quantum effects can alter cooperative behavior. Here, we investigate this classical-quantum competition in a kinetically constrained spin chain, the dissipative far-East model. In absence of fluctuations, the system exhibits emergent collective dynamics, resulting in phase coexistence. We find that this phenomenon is robust to quantum fluctuations, which further stabilize it, whereas classical fluctuations suppress phase coexistence at long times. We map out the steady-state phase diagram using a cluster mean-field approach and characterize the dynamics with stochastic tensor-network methods. Within the bistable phase we identify signatures of dynamical heterogeneity in both the local magnetization and the entanglement dynamics. This highlights how competing quantum fluctuations and classical kinetic constraints shape the relaxation dynamics and the phase structure of many-body systems.

Energy transport across conformal interfaces is universal in two spacetime dimensions, but its higher-dimensional counterpart has remained largely unexplored. In higher dimensions, scattering can depend on additional kinematic data, such as the angle of incidence, and conformal symmetry is far less restrictive. We use gravitational holography to study this problem. For a broad class of AdS$_d$-sliced domain-wall geometries, we derive a closed-form expression for the energy transmission coefficient in terms of the bulk warp factor. Our formula reproduces known two-dimensional holographic results and extends them to arbitrary dimension. Within this class, we establish different aspects of higher-dimensional universality. In particular, we show that the transmission is independent of the incidence angle and of the profile of the incident perturbation. We also derive bounds on transmission controlled by central-charge-like quantities characterizing the number of degrees of freedom of the theories on the two sides of the interface. This provides the first explicit result for reflection and transmission across conformal interfaces in higher dimensions, where no CFT-based results are currently available.

Matchgate circuits describe classically tractable dynamics in a physically relevant setting while nonetheless exhibiting features characteristic of complex quantum systems, such as high amounts of magic. This invites the broader question: to what extent are typical complexity measures saturated under trivial dynamics? Here, we address this by studying the behaviour of different Heisenberg-picture complexity measures in random matchgate ensembles. Our main technical contribution is a tractable and generalisible free-fermionic unitary Weingarten calculus formulated entirely in the Majorana basis. This method is tailored to computing moments of operators, and simplifies to a combinatorics problem which depends on the boundary conditions prescribed by a given operator functional. Using this framework, we derive closed-form expressions for the typical local-operator entanglement, operator stabiliser entropy, and higher-order out-of-time-ordered correlators. For extensive initial Majorana strings, both operator entanglement and stabiliser entropy obey volume laws, even though the underlying circuits remain classically simulable. Out-of-time-ordered correlators can likewise become exponentially small, showing that their late-time saturation value alone does not witness computational hardness and constituting a classically tractable example of emergent free independence. Finally, we relate these separations to operator resource theories, showing how operator entanglement beyond the Gaussian baseline lower-bounds the number of non-Gaussian gates in doped matchgate circuits. Our results present a practical toolbox for computing exact averages over random matchgates, while clarifying the distinction between quantum resources versus simulation complexity, and their dependence on the representation being exploited.

Recent progress in network renormalization has identified various ways to transform network representations consistently across resolution levels. In particular, the renormalization flow describing how random graph models transform under node aggregation has identified a fixed point corresponding to an aggregation-invariant model that preserves the form of the connection probability across arbitrary scales. While this probabilistic multiscale approach has successfully been used to model real-world networks, construct latent-space node embeddings, and reconstruct finer-grained versions of aggregate network data in a principled resolution-invariant way, it has so far been restricted to binary graphs. Here we reformulate multiscale network renormalization for weighted random graphs. Via a generating-function approach, we identify a weighted multiscale model whose full probability law, encoding both link creation and weight assignment, has an invariant compound-Poisson form under arbitrary node aggregation. The renormalization rules that transform parameters across aggregation levels allow the ensemble to be calibrated at one level and applied at another, enabling consistent modeling and reconstruction of weighted networks across arbitrary scales.

The dynamics of generative models exhibit two apparently distinct temporal windows: a speciation window, in which a sample commits to a semantic class, and a nonlocality window, in which local context windows become insufficient for generation. Motivated by evidence of their near-concurrence in a variety of frontier models, we investigate their relationship through the spatial distribution of semantic information. Under a "common cause" hypothesis, we prove that the nonlocality window must lie in the speciation window. This hypothesis postulates that semantic labels explain a fraction of the correlations between distant tokens, a condition that is natural for many real datasets. We further give conditions under which both windows shrink to a single limiting time as system size grows, defining a "phase transition", and verify this behavior analytically in Gaussian mixtures. Together, these results identify conditions under which semantic information explains the concurrence of speciation and nonlocality, connecting two complementary perspectives on the emergence of semantic structure in generative modeling.

Topological phases of matter support edge states protected from random perturbations, and the identification of the discrete non-spatial symmetries and dimensions that support such phases was systematically developed for quantum systems and analogous platforms. While topological phases were also discovered in stochastic systems and posited as mechanisms for key biochemical processes, a classification consistent with their Markovian constraints remains lacking, impeding generalization to new scenarios. To address this, we develop new methods including a new homotopy approach, real-space invariants and a redefinition of the point gap to classify both transient and steady states. We find robust transient states in two symmetry classes, no symmetry and pseudo-Hermiticity respectively, while topological steady states are expected only in systems without symmetry. We identify the dimensions with non-trivial phases and their group structure, creating a rigorous framework for predicting robust behavior in active and living matter.

It is shown that Shannon entropy, Fisher information, and quantum-mechanical kinetic energy may all be viewed as measures of correlation. We begin with a fundamental correlation-destroying map: a joint probability density $\rho(x,y)$ is replaced by $\rho_x(x)\rho_y(y)$, the product of its marginal distributions. In the discrete case, Shannon's entropy is nondecreasing under this map. For continuous distributions, this fundamental map leads to a well-behaved, coordinate-invariant correlation measure, analogous to the discrete Shannon entropy: \[ I[\rho] = h[\rho_x] + h[\rho_y] - h[\rho]. \] In the continuous case, however, another measure appears: Fisher information. The relative Fisher information $J(\rho\|\rho_x\rho_y)$ behaves similarly under the fundamental map. For a normalized real quantum wavefunction, this decrease is exactly proportional to the decrease in mean kinetic energy, \[ \langle T\rangle_\psi - \langle T\rangle_\Phi = \frac{\hbar^2}{8m} J(\rho\|\rho_x\rho_y), \qquad \Phi = \sqrt{\rho_x \rho_y}. \] In Jaynes's language, the result supports a principle of minimum justified correlation: given physical constraints and a set of possible distributions or related amplitudes satisfying those constraints, select from that set those with the least correlation. This is Part I of a two-part paper. Here we develop the entropy, Fisher-information, and kinetic-energy identities above, and state the principle they support. Part II addresses questions which Part I raises but leaves unanswered: multiple solutions, time dependence, the role of spin, a route to the Schrödinger equation itself, and a proposed experimental test. AI has been used.

An emergent gauge field can control boundary critical behavior without changing the bulk quantum spin liquid. We investigate the boundary criticality in $(2+1)$-dimensional $U(1)$ Dirac spin liquid, whose low-energy physics is described by massless quantum electrodynamics. Using a perturbative renormalization-group analysis in a half-space, we show that Neumann and Dirichlet boundary conditions for the emergent gauge field lead to distinct boundary universality classes. We determine the boundary scaling dimensions of the fundamental fields and gauge-invariant operators that provide observable signatures. We further propose a fermion-gauge lattice model with tunable boundary interactions as a microscopic setting for realizing and probing these boundary universality classes.

We review mathematical results about the spectrum of spin-$s$ chains with a nearest neighbor interaction given by the projection onto the singlet state of two spin-$s$ degrees of freedom. Up to normalization, these interaction terms generate a Temperley-Lieb algebra at loop parameter $d=2s+1$. We also present several new results. First, we prove a ferromagnetic ordering of energy levels across symmetry sectors of the model, thus generalizing a previous result of Nachtergaele, Spitzer and Starr (2004) for the spin-$\tfrac12$ $XXZ$ chain. This enables us to determine a uniform lower bound on the finite volume spectral gap for these spin chains. By combining a variational estimate with Knabe's finite size criterion, we show that the exact spectral gap of the GNS Hamiltonian of the fully polarized state is $1-2/d$. Furthermore, this value serves as a lower bound for the gap of the maximally mixed ground state. Finally, we show that the ground state projectors on open chains satisfy the Jones-Wenzl recursion, thereby recovering a known result that the degeneracies are the quantum integers $[L+1]_q$ for $q+q^{-1}=d$. We discuss how the high degeneracy of the ground states allows the model to break a continuous symmetry while remaining gapped, and renders that gap unstable.

Strong-to-weak spontaneous symmetry breaking (SWSSB) is a novel phase transition in mixed states associated with the loss of global charge information. In this work we study fermions with $U(1)$ symmetry under infinite density dephasing and derive the SWSSB properties of the resulting mixed states. Our central observation is that at infinite dephasing every Renyi correlator is exactly the spin-spin correlator of a compact XY model, whose Boltzmann weight is the squared full-counting-statistics generating function of a classical ensemble built from the parent state. For the Renyi-2 correlator it is the full counting statistics of the parent state itself. We complement the infinite dephasing mapping with a novel diagrammatic expansion for the Renyi-1 correlator, applicable to any diagonal mixed state, which computes the couplings of the Renyi-1 XY model. We also study a replica field theory for even Renyi index quantities, applicable at finite dephasing. Our analysis indicates that fully dephased metals display long-range SWSSB in $d\geq 2$, as confirmed numerically and experimentally in $d=2$, and quasi-long-range SWSSB in $d=1$. For insulating parent states both an SWSSB and a trivial phase are possible at infinite dephasing depending on UV details. Long-range SWSSB in $d\geq 3$, or quasi-long-range SWSSB in $d=2$, survives above a nonuniversal, Renyi-index-dependent threshold and is lost when the parent state is deep in the insulating phase, with the exception of quantum Hall insulators which generically have quasi-long-range SWSSB. For even Renyi indices we also determine the stability of the SWSSB phase away from infinite dephasing and derive the universality class of the SWSSB transition.

Network reconstruction from partial information is usually performed at the same resolution level at which constraints are observable. This becomes problematic when only coarse-grained information is available, while the relevant process occurs at a finer scale. Here we employ the multiscale model of weighted networks introduced in a companion paper and turn it into a probabilistic framework for reconstructing weighted networks across arbitrary aggregation levels. The model is built to preserve its functional form under coarse-graining, so that global parameters calibrated on an observable aggregate layer can be transferred to finer layers without refitting. We test the method on two empirical systems with different aggregation mechanisms. In the International Trade Network, countries are aggregated into geographic macro-regions and the observed coarse-grained layer is used to infer the underlying country-level network. In the Dutch production network, sectoral flows are reconstructed across the hierarchical industrial classification, using coarser sectoral layers to infer finer ones. In both geographical and sectoral settings, we benchmark our genuinely multiscale reconstruction method against a state-of-the-art weighted reconstruction model calibrated directly at the target resolution, and therefore using additional information available at the same (finer) scale at which performance is evaluated. Remarkably, despite this informational disadvantage, our method recovers the fine-scale binary structure with high accuracy, improving precision, specificity, accuracy and maximum degree-error diagnostics, while remaining nearly equivalent in sensitivity.

Pairs of spheres sharing an acoustic trap sometimes spin intermittently despite having no self-propulsion, no obvious power source, and no symmetry-breaking mechanism, switching between rocking and spinning within a single trajectory. Nonreciprocal wave scattering powers this emergent activity, and drag shapes their dynamics by coupling rotations to translations. Mismatched pairs can pass through a supercritical Hopf bifurcation into a quasiperiodic torus on which the pair's orbit reverses regularly and, farther from equilibrium, into deterministic chaos. Experiments motivate this mechanism and share its qualitative signatures; simulations reveal a full noise-free route from passivity to chaos, and suggest how analogous dynamics can emerge in other active-matter systems.

Thermodynamic uncertainty relations bound the fluctuations of time-antisymmetric currents. Stochastic efficiency is not one: a ratio of two odd quantities, it is even under path reversal, and for the conventional ratio $-W/Q_h$ no moment exists. We work with the exergetic ratio $\eta=W/(W+T_0S)$, whose moments exist, and ask how much of its variance a finite perturbation of the dynamics can certify when linear response cannot. The question is specific to nonlinear observables: for every current, and every observable linear in transition counts and residence times, the multiparameter Cramér-Rao bound over rate and site perturbations already equals the variance. For $\eta$ it does not. The Cramér-Rao bound vanishes on a hyperplane of perturbation directions, while the Barankin multi-point bound, applied to tilted path measures, stays positive there and recovers 75% of the variance of $\eta$ in a three-state motor model where every linear-response bound is zero. The Gram matrix of tilted Markov-jump path measures is a Feynman-Kac matrix exponential whose potential is the Hellinger integrand of the jump intensities, and the cost of a perturbation is its $\chi^2$ divergence, which no dissipation bound controls. The bound captures 1.25-1.57 times the best linear-response bound, and the gain persists on networks of up to sixteen states. For an Ornstein-Uhlenbeck process the construction is exactly solvable: the optimal tilt scales as $T^{-1/2}$ and three test points nearly reach the supremum. Two limits are established. The gain is a short-window effect that disappears as the counts become Gaussian, and the bound cannot be turned into inference: coarse measurements bound $\chi^2$ only from below. The construction is a computational instrument for a model in hand, not an uncertainty relation.

We consider a gradient, speed-change, symmetric exclusion process on the discrete torus $T^d_n$, $d\le 3$, whose empirical measure evolves, on the diffusive scale, according to a non-linear parabolic equation. We prove that, starting from a sequence of initial states whose relative entropy with respect to the inhomogeneous product measure associated with the initial profile is $o(n^{d/2})$, and whose density fluctuation field converges, the density fluctuation field converges, in $L^1(0,T; H_{- r})$, to the solution of a linear stochastic partial differential equation with time-dependent coefficients driven by a conservative space-time white noise.

Curvature and topology jointly organize defects in two-dimensional crystals, but their combined role remains unresolved when competing lattice symmetries coexist with spatially varying curvature. We use simulated-annealing Langevin dynamics to study Hertzian particles forming coexisting hexagonal (Hex) and square (Sq) lattices on prolate and oblate ellipsoids. Mapping reduced density and aspect ratio reveals a broad sequence of scar and domain-based morphologies in both Hex-dominant and Sq-dominant backgrounds. Latitude-resolved comparisons show that Gaussian curvature biases defects toward its maxima under weak deformation. Strong prolateness, however, confines high curvature to small polar caps that cannot independently accommodate all defect motifs. Defects then spread toward lower-curvature latitudes to relieve defect crowding and elastic repulsion. In the Hex-dominant regime, this competition drives vertex-contacted domains with neutralized corner contacts, and compensating positive defects locate away from the poles. The Sq-dominant regime features Hex-rich triangular domains, bridged states, and linear or open scars similarly reorganized by curvature anisotropy. On oblate ellipsoids, the extended equatorial high-curvature belt allows defects to separate azimuthally while remaining curvature-localized. This work elucidates that nonuniform curvature can engineer rich defect patterns by selecting the spatial distribution of topological charge and the connectivity of finite defect motifs.

We propose a quantum heat engine based on a two-component Fermi gas with s-wave contact interaction, operating across the BCS-BEC crossover. The work is extracted from the statistical properties of the gas, which are controlled by the interaction, rather than relying solely on compression and expansion stages. Using the functional renormalization group formalism, we obtain the equation of state in a non-perturbative form along the crossover, encompassing the superfluid-normal phase transition. The cycle combines isentropic density strokes, isochoric thermalization, and isothermal interaction sweeps. This construction makes it possible to integrate features of both Otto and Carnot cycles, in which the system simultaneously saturates both efficiency limits without the net work vanishing. In the absence of density variations, the cycle reduces to a statistical Stirling-like engine, in which the work generated arises exclusively from the interaction, achieving efficiencies up to $38\%$. A pronounced asymmetry emerges through the crossover, giving rise to distinct operating regimes depending on the trajectory followed in the phase diagram. Consequently, the same architecture can be tuned to function as an engine, refrigerator, accelerator, or heater. These findings highlight pairing correlations as a versatile thermodynamic resource for quantum heat machines.

We study spin-current fluctuations of the folded XXZ chain, which describes the strong-anisotropy limit of the XXZ chain. We obtain an exact expression for the moment-generating function of the time-integrated spin current at infinite temperature. A long-time asymptotic analysis establishes that the rescaled time-integrated current converges to a non-Gaussian distribution described by the M-Wright function. The derivation relies on a nonlocal map to the one-dimensional t-0 model and its exact spin-charge separation. By extending this approach, we also show that the dynamical correlation function of the spins summed over two adjacent sites exhibits a diffusive Gaussian profile at long times.

We study the three-dimensional active Potts model (APM) on a cubic lattice with 2, 4, and 6 orientational states. The steady-state phase diagrams exhibit low-density gaseous, high-density polar liquid, and liquid–gas coexistence phases, with the phase behavior controlled by density, temperature, and propulsion speed and retaining the main features of the previously studied two-dimensional APM. At weak propulsion, the dense phase forms a slab perpendicular to its direction of motion. With increasing propulsion speed, this morphology persists for the 2-state model, whereas the dense phases for 4 and 6 states reorient and propagate along their longitudinal direction. For the 6-state model, the reoriented dense phase further develops a cylindrical morphology. We also examine the stability of polar liquids against artificially introduced droplets using microscopic simulations and coarse-grained hydrodynamic equations. At high propulsion speeds, counter-propagating droplets can partially modify the initial polar state, while transverse droplets can completely alter it, demonstrating the metastable nature of polar liquids in the three-dimensional active Potts model.

This dissertation investigates the transport of Brownian particles driven by active fluctuations in periodic potential landscapes. Unlike thermal noise, active fluctuations break detailed balance and violate the fluctuation-dissipation theorem. They model self-propulsion mechanisms and active-bath interactions in biological and synthetic micro-systems. By focusing on white Poisson shot noise with non-Gaussian amplitude statistics, this work systematically uncovers the mechanisms governing three major transport anomalies: giant transport enhancement, absolute negative mobility, and the ratchet effect. Combining Monte Carlo simulations with analytical modeling, this work demonstrates that active fluctuations adhering to a bidirectional skew-normal amplitude distribution enhance directed velocity compared to the free-particle by several orders of magnitude. This giant transport enhancement is fully quantified via a phenomenological jump-relaxation model that decouples particle displacement into instantaneous stochastic jumps and deterministic relaxation towards potential minimum. Extending this model to inertial dynamics reveals that finite particle mass m acts as a dynamic control parameter capable of strengthening, weakening, constructively inducing, or completely suppressing transport enhancement depending on noise variance. Furthermore, an ultra-minimal model for absolute negative mobility is established in a one-dimensional, purely overdamped system, showing that discontinuous shot noise drives transport opposite to an applied bias without requiring particle inertia, nonlinearity of spatially periodic potential, or explicit time-dependent periodic driving. Conversely, under an alternative scaling of the average stochastic force, an asymmetric distribution of fluctuation amplitudes generates net directed transport through a tilting ratchet effect in the zero-bias limit.

We propose that, at criticality, equilibrium dynamics effectively evolves on a fractal subspace rather than throughout the full Euclidean space. Starting from Fisher's formulation of the order-parameter correlation function, we interpret the anomalous critical exponent $\eta$ geometrically through a correlation fractal dimension associated with this effective subspace. Using fractional-calculus tools, we derive a form of the correlation function that recovers critical behavior below the upper critical dimension and obtain an explicit relation between $\eta$ and a fractal dimension $d_R$ linked to the Riesz fractional derivative. We also examine the Rushbrooke scaling relation and investigate critical behavior in non-integer-dimensional and disordered systems. In the disordered case, controlled by a parameter $\sigma$, we test the range over which the proposed geometric interpretation remains valid. The results connect scaling laws, critical exponents, correlations, and fractal geometry, suggesting that the anomalous behavior observed at criticality can be understood as a consequence of dynamics constrained to an effective fractal subspace.

A detector that cannot take two snapshots closer than a time $\tau_0$ still identifies a fully observed finite Markov generator exactly, provided its gaps are randomized by two clocks. Identification is not detection. For a normal nonequilibrium network, the number of snapshots needed to reject every hidden equilibrium explanation grows as $e^{2\gamma_{\mathrm{irr}}\tau_0}$, where $\gamma_{\mathrm{irr}}$ is the slowest damping of rotating modes. No adaptive schedule beats this rate, and a test on one dependent trajectory attains it.

We demonstrate how the population annealing Monte Carlo simulation method can be used to compute the ground-state entropy to high precision. This approach is tested for the classical antiferromagnetic Ising model on the Shastry-Sutherland lattice with diagonal couplings chosen twice as strong as the nearest-neighbor interactions. In the absence of an external magnetic field, the model is known to be disordered at all temperatures, with a macroscopically degenerate ground state. We find the ground-state entropy to be $0.458\,777\,78(10)$ per site, which is in good agreement with the estimate of a recent study using the corner transfer matrix renormalization group method. The sampled ground-state configurations are analyzed by considering the bond configurations, the local-energy configurations, and the spin-spin correlations. While all observations are consistent with the disordered nature of the ground state, the numerical data suggest a power-law decay of the size distribution of clusters of sites with local energies different from the ground-state level.

Stochastic resetting can generate strong correlations in many-body systems through a shared fluctuating environment. Here, we investigate how such dynamically emergent correlations (DEC) coexist with intrinsic (direct) interactions in a weakly repulsive Bose-Einstein condensate described within the Gross-Pitaevskii framework. The condensate undergoes free expansion from an initial Thomas-Fermi state and is stochastically reset to this initial state at a constant rate. We show that the resetting protocol drives the system into a unique nonequilibrium steady state and obtain its density profile, edge statistics, and full counting statistics analytically. The steady-state density retains an inverted-parabolic form within the core, while developing exponentially decaying tails outside it. We further show that resetting induces nontrivial fluctuations of the condensate edge and particle number, yielding exact scaling forms for the edge distribution and the full counting statistics. Our results provide a tractable setting for exploring the interplay between intrinsic repulsive interactions and attractive DEC generated by a common stochastic environment in quantum many-body systems.

This paper extends the formalism of stochastic path-space thermodynamics by systematically expanding the space of thermodynamic variables with a spectrum of boundary and relative functionals of random processes. Generalizing the approaches introduced in preprints arXiv:2607.24078 and arXiv:2609.08477, we investigate the fluctuation statistics of a one-dimensional Ornstein-Uhlenbeck process with an asymmetric linear drift and a step penalty potential at the origin, which models the waiting phase of a molecular Brownian motor. Utilizing the Feynman-Kac formalism and the parabolic cylinder theory, a numerical scheme based on matching logarithmic derivatives via the secant method is constructed. A high-precision eigenvalues of the effective Hamiltonian is obtained, which define the cumulant generating function. We formulate a modified path-space fluctuation theorem of the Gallavotti-Cohen type for the conjugate drift current at a fixed integral occupation time of the system in the dissipative half-plane. The physical and thermodynamic significance of the new variables is elucidated. A combination of "residence time" and "integral current" functionals is applied to describe an ion channel sensitive to mechanical or electrical stimuli.

We study the site-diluted Ising model on the kagome lattice using the Wolff single-cluster algorithm, focusing on both equilibrium and dynamic critical properties. The equilibrium critical exponents $\nu = 1$ and $\gamma/\nu = 7/4$ retain their exact pure Ising values for all spin concentrations $p$ considered, consistent with the marginal irrelevance of disorder at $\alpha = 0$, with the specific heat crossing over from logarithmic to double-logarithmic growth upon dilution. These static results are shared between the kagome and square lattices. On the dynamic side, $z$ decreases upon dilution on both lattice, opposite to what is observed under local dynamics, with evidence of saturation to a $p$-independent value at strong dilution, suggesting a diluted dynamic fixed point. While $z$ is consistent between the two lattices in the pure case, it differs in the diluted regime, an effect we attribute to the two lattices sitting at different positions along the renormalization-group flow between the Ising and percolation fixed points, rather than to a true asymptotic breakdown of dynamic universality.

When a system of many interacting constituents is extended across space, every region carries the same disordered landscape, and it is unclear whether distant regions settle into the same state. We study copies of the Random Energy Model (REM), sharing one disorder realization, coupled along a chain. A first-order transition separates a phase with short-range correlations, from a locked phase in which the entire chain occupies the lowest-energy state with probability one, even at positive temperature. No intermediate ordering lengths occur. In the Generalized REM the chain locks either to the lowest-energy state or to the lowest free-energy valley.

It is widely believed that there are no exact global symmetries in quantum gravity. We review recent works showing that global symmetries can nevertheless emerge after ensemble averages of theories, as encountered in holography. As a precision case study, we discuss the ensemble average of two-dimensional Narain-type conformal field theories associated with an even quadratic form of general signature, which is computed by the Siegel-Weil formula and interpreted as a sum over geometries in a three-dimensional Abelian Chern-Simons theory. Global symmetries of the bulk anyons can emerge after the average, as T-dualities relating different theories in the ensemble are "folded" into symmetries of a single theory (duality origami). We discuss the relation to the Swampland program, including the emergence of global symmetries at infinite distance. Finally, we recast strong-to-weak spontaneous symmetry breaking (SWSSB) of mixed states as an emergence of symmetries in ensemble averages; in holography, wormholes connecting the two sides of a thermofield-double construction provide its bulk realization. This motivates a new Swampland conjecture.

We derive the finite-temperature characteristic function of work (CFW) for a driven $O(N)$ Gaussian field theory and obtain closed-form expressions for all zero-temperature excess-work cumulants. For gapped protocols, we use adiabatic perturbation theory (APT) to derive the $\tau_Q^{-2}$ scaling for protocols with a nonzero first derivative at either boundary, where $\tau_Q$ is the protocol duration; smoother boundaries lead to faster decay. For power-law protocols approaching the critical point with exponent $p$, we determine the competition between critical excitations and the regular contribution. The $n$th cumulant's critical contribution follows Kibble–Zurek (KZ) scaling $\tau_Q^{-p(d+n)/(p+2)}$ for $d+n<2p+4$, acquires a logarithmic correction $\tau_Q^{-2p}\log\tau_Q$ at $d+n=2p+4$, and scales as $\tau_Q^{-2p}$ above this condition, where $d$ is the spatial dimension. This analytical study of the APT–KZ crossover in a solvable model provides a basis for studying their competition in interacting field theories.

There are infinitely many functions that satisfy the Black-Scholes partial differential equation and the terminal condition corresponding to the European call option. This means that the Black-Scholes formula, which led to the award of the 1997 Nobel Prize in Economic Sciences, is not the unique solution, as was once assumed. Consequently, it violates the law of one price, one of the fundamental laws of economics and finance. In this article, we present a rigorous derivation of these solutions to the Black-Scholes boundary value problem.

We propose a $q$-deformation in the single-particle kinetic energy and investigate how it modifies the localization in the one-dimensional Aubry-André (AA) model. We construct a Hermitian $q$-deformed kinetic operator as a nonlinear function of the lattice translation operator, preserving the uniform lattice and recovering the conventional AA Hamiltonian continuously in the undeformed limit $q\to1$. The deformation generates an infinite set of correlated odd-range kinetic processes, controlled by a single parameter q, rather than phenomenologically involving long-range hopping. Under the dual transformation, this long-range hopping appears as higher harmonics of the dual quasiperiodic potential, providing a controlled route for breaking the exact self-duality of the AA model, with $q$ as the control parameter, consequently modifying the localization structure for $q\neq1$. In contrast to the conventional AA model, where all eigenstates localize simultaneously at $\lambda_c=2$, the deformed model exhibits a fraction of delocalized states even beyond $\lambda_c =2$. An intermediate regime also emerges in the $q-\lambda$ plane where localized and extended eigenstates coexist across the spectrum. We also propose a possible experimental realization of the hierarchy produced in the $q$-deformed kinetic setting in a periodically driven AA model.