Papers

In this work, we theoretically investigate the non-equilibrium evolution of the correlation functions of a quantum scalar field following a mass quantum quench in a finite-size system. The self-consistent equation for the Keldysh (statistical) correlation function is solved numerically in the one-loop approximation. A specific spatio-temporal effect is observed: at a point in time approximately equal to half the system size, the effective mass experiences a strong disturbance. It is shown that this effect is driven by the interference of counter-propagating wave fronts of fluctuations that propagate within the light cone and is induced by periodic boundary conditions.

Exact statements about how a system far from equilibrium responds to changes in its rate constants are scarce beyond linear response, yet they are central to the control of biochemical networks. A notable exception is mutual linearity: in a jump process, controlling the rates of a single transition makes any two stationary probabilities obey an exact affine relation, arbitrarily far from equilibrium. With nonlinear mass-action kinetics, and conservation laws in place of a normalization, chemical reaction networks seem to defy mutual linearity since species concentrations obey no such relation. We show that it was there all along, in the complexes' activities (mass-action monomials) rather than in the concentrations of species. For zero-deficiency networks with a single linkage class, controlling one reaction makes any three stationary activities obey a linear relation whose coefficients are independent of the controlled rate constants, the conserved quantities (hence the initial state), and even of the stoichiometry of complexes. This predicts hidden activities and bounds the activity ratios the control can reach, while a violation signals nonzero deficiency or multiple linkage classes.

Is the relaxation of a modular network dictated by its modules or by the network that connects them? The answer fixes the time scales of diffusion, consensus, and synchronization, all set by the Fiedler eigenvalue. We show that, for bundled networks, the two levels act one after the other: a random walker must first escape from its module and then spread over the network that connects them. The equilibration time is the sum of these two times, the time needed to reach the base from inside a fiber and the relaxation time of the base, slowed down by the mass of the fibers. The consequences are unexpected. However large the modules are, a small-world core always imposes its own Fiedler dimension; the modules survive only as a logarithmic correction, which slows down equilibration and hides the true exponent up to sizes far beyond any real network. Effective dimensions measured on finite modular systems can therefore be systematically biased. We analytically work out all combinations of finite-dimensional and small-world bases and fibers, and compare them with numerically exact spectra of representatives of each category.

We study bond percolation on the one-dimensional lattice in which two sites at distance $r$ are connected with probability $p_r=C/r^{1+\sigma}$, where $0 < \sigma < 1$. An infinite cluster exists if $C$ exceeds a critical value $C_c(\sigma)$, which is not known exactly: the available results are the lower bound of Schulman, $C_c\ge1/[2\zeta(1+\sigma)]$, and numerical estimates. Here we prove explicit upper bounds. We apply a second-moment argument to a family of random monotone paths whose steps have a heavy-tailed (Sibuya) distribution. Two such paths meet only a finite number of times, and their overlap is computed exactly by renewal theory. The result is a closed-form bound, which gives $C_c\le\sigma+O(\sigma^2)$ as $\sigma\to0$ and is below $1$ for $\sigma < 0.667$. Since the paths are monotone, the bound holds also for oriented percolation, and it determines the oriented threshold asymptotically: $C_c^{\rightarrow}\zeta(1+\sigma)\to1$ as $\sigma\to0$. A coarse-grained version of the argument, with blocks of sites in place of sites, gives an explicit bound $C_c < 1$ for every $\sigma < 1$. We compare the bounds with the known numerical values of $C_c$.

We develop a family of control protocols for finite-time transitions between active nonequilibrium steady states using the noise-color (correlation rate) as the sole control parameter. By reverse-engineering the second moment dynamics of an active Ornstein-Uhlenbeck process, we determine the time-dependent correlation rate required to realize a prescribed evolution of the system's state. We experimentally implement these protocols with a micrometer-sized optically trapped particle coupled to an engineered active bath, demonstrating transitions substantially faster than the natural relaxation while keeping the confining potential, noise amplitude, and temperature fixed. Our approach extends engineered swift-equilibration methods to far-from-equilibrium steady states by directly controlling the temporal correlations of an active environment. We show how physical constraints impose a speed limit on finite-time transitions, while the freedom in choosing the prescribed system evolution can be exploited to eliminate control discontinuities, minimize the admissible transition time, or optimize a thermodynamic cost.

Superradiant phase transitions are fundamental phenomena in light-matter coupled systems. Here, we uncover a novel class of superradiant criticality that is topologically distinct from previously known superradiant transitions. Using cavity-mediated cluster-Ising chains as an illustrative example, we numerically demonstrate that the critical line separating the superradiant phase from the cavity topological phase hosts robust symmetry-protected edge degeneracies, a feature absent in conventional superradiant transitions. Remarkably, although its local bulk critical properties are indistinguishable from those of the previously identified topologically trivial superradiant transition, the two criticalities cannot be smoothly connected without breaking the protecting symmetries or passing through a multicritical point. We further show that such topological superradiant criticality is generic, also occurring in cavity-mediated $\mathbb{Z}_{n}^{}$ cluster-qudit chains, as supported by complementary lattice simulations.

Quantum annealing must be sufficiently slow to prevent defect formation. We study this problem in the random transverse-field Ising model via the adiabatic theorem to obtain the limiting rate at which defect formation can be avoided. Our approach based on the strong-disorder renormalization group (SDRG) technique views the annealing process as a gradual aggregation of strongly coupled clusters, giving access to the elementary events of error formation. We present a scaling theory of the statistics of these events in general dimensions and confront it to an extensive numerical analysis carried out in one, two, and three dimensions by an efficient implementation of the SDRG method. This approach is also validated by exact diagonalization in one dimension by means of the free-fermion technique. Our results consistently show a logarithmically slow decrease of the error density with the annealing time, characterized by the dimension-dependent critical exponents of the corresponding infinite-disorder fixed point of the model. Furthermore, we provide a map of individual events of potential defect formation in the excitation energy - control parameter plane, which may help in the design of optimal annealing paths.

Learned representations can lose latent degrees of freedom successively, suggesting a cascade of transitions whose underlying stability principle remains unclear. Here we formulate dimension-wise posterior collapse in variational autoencoder (VAE) as a fluctuation theory around partially collapsed states. Interpreting the negative evidence lower bound as an effective free energy, its quadratic expansion defines a Gaussian theory whose Hessian acts as a mass matrix for latent fluctuations. We show that the collapsed directions form an invariant fluctuation sector and derive its exact mass spectrum in terms of a conditional residual operator. A local reactivation direction lowers the free energy when the decoder variance falls below the residual spectral upper edge, with equality marking marginality. The criterion recovers principal component thresholds in the linear Gaussian VAE limit. Viewed in reverse along continuously connected branches, the reactivation boundary provides a local criterion for successive collapse. Numerical continuation experiments show successive loss of latent dimensions near these spectral marginalities. These results support a spectral cascade interpretation governed by residual information left unexplained by the surviving representation.

Conditional mutual information (CMI) probes spatial non-Markovianity and plays a central role in a recently developed framework for defining mixed-state phases based on local reversibility. This framework sharply distinguishes exponentially decaying from algebraically decaying CMI, since the latter obstructs equivalence to a state with finite Markov length. Here we show that states that flow to the same renormalization-group (RG) fixed point can nevertheless exhibit qualitatively different CMI. Starting from a critical Gibbs state of a finite-range local classical Hamiltonian, whose CMI vanishes beyond the interaction range, we show that an RG-irrelevant perturbation that breaks detailed balance generically generates algebraically decaying CMI. Crucially, these power-law tails are cutoff-suppressed: although they lead to an infinite Markov length at every fixed lattice cutoff, they vanish as a positive power of the cutoff in the continuum limit. We construct solvable models that exhibit this phenomenon and predict that the Ising-symmetric critical point of Toom's cellular automata generically exhibits such behavior. We also find analogous cutoff-suppressed CMI in the high-temperature paramagnetic phase of the long-range Ising paramagnet, in contrast with the genuine power-law CMI at the finite-temperature critical point in this same model that survives the continuum limit. We also derive a general result that in one spatial dimension, CMI decays faster than the inverse square of the buffer length if and only if the long-distance physics is Markovian.

We present a high-resolution numerical study of the spin-1 Blume–Capel ferromagnet on the honeycomb lattice ($z=3$). Wang–Landau estimates of the joint density of states $\Omega(E_1,E_2)$, combined with field mixing, are used to map the first-order transition line and locate the tricritical point at $(\Delta_t,T_t)=[1.4819(2),\,0.4022(7)]$. Histogram reweighting of Metropolis simulations for $L=36$ and $48$ provides an independent consistency check, with a crossing temperature near $0.4039$, compatible with the Wang–Landau extrapolation. From the field-mixing analysis we obtain $r=-1.586(19)$, while the slope of the multicanonical coexistence line yields the consistent and more precise estimate $r=-1.587(5)$. Normalizing by the zero-temperature field $\Delta_0=zJ/2$, the available honeycomb, square, triangular, and simple-cubic estimates cluster near $r/\Delta_0\simeq-1.06$, corresponding to a magnitude about $6\%$ larger than the mean-field value $-1$; no systematic dependence on coordination number or dimensionality is resolved over the range examined. Finite-size scaling of $\sigma_Q$ gives $y_t=1.795(7)$, consistent with the tricritical Ising value $9/5$, while the fourth-order cumulant scaling is compatible with $y_g=4/5$. Deep in the first-order regime, the cumulant approaches the expected two-phase limit $U_Q\to2/3$. Multicanonical simulations up to $L=64$, using weights constructed directly from the Wang–Landau density of states for $L\le24$ and rescaled and refined for larger systems, yield an interface-tension exponent $\mu=1.265(26)$, consistent with the tricritical prediction $(d-1)/y_g=5/4$. Finally, we map the continuous branch of the phase boundary from the same density-of-states data, obtaining $T_c(\Delta=0)=1.157$; estimates sufficiently close to the tricritical point are affected by crossover from Ising to tricritical scaling.

We study general maximal hard-core models on periodic lattice-type graphs. While for standard models, phase coexistence may arise for high activity, for maximal models this behavior may occur also for sufficiently low activity values. Relying on the concept of volume allocation, we develop a unified set of assumptions that imply the Peierls condition, a convergent cluster expansion for the partition function, and hence the conclusions of Pirogov-Sinai theory for this class of models. We further check these assumptions for a number of examples including standard lattice-type periodic graphs, proving crystallization for both high and low activity. We also derive estimates for the values of activity that bound the phase coexistence regions in the phase diagram, rewriting for this purpose the proof of a technical result in the derivation of Pirogov-Sinai theory.

Dipolar quantum gases in elongated traps become supersolid through a transition that is continuous at intermediate density but first order at low and high density; at the continuous point, ramps of the scattering length follow Kibble–Zurek scaling. We study ramps across the first-order superfluid–supersolid transition of $^{164}$Dy in an infinite tube using the three-dimensional extended Gross–Pitaevskii equation with truncated-Wigner noise. With quantum noise only, the uniform superfluid survives the metastable window and transforms after its spinodal, where the roton softens. For onsets occurring during the linear ramp, the delay measured from this spinodal has a fitted exponent $\zeta = 0.36 \pm 0.02$, close to the continuous-transition result and the mean-field value $1/3$. Reverse ramps show a comparable delay beyond the fold of the crystal branch, producing dynamical hysteresis. Initial thermal noise with nominal temperatures of 2.5–10 nK produces crystalline regions inside the metastable window during holds, with faster nucleation closer to the spinodal and at higher initial temperature. In ramps at 5 nK, thermal fluctuations cross onset thresholds near $f_s = 1$ early, whereas the bulk of the crystal still forms after the zero-temperature spinodal with a similar scaling. Without a heat bath these effective rates depend on preparation: ramps crossing the window at the same rate nucleate early in 2 of 32 or 11 of 16 realizations depending on their starting point. At low density, slower ramps select larger spacings in a crystal of separated droplets, and droplets survive far above the transition on reverse ramps. A Maxwell construction gives a coexistence window $0.04 a_0$ wide around the first-order transition point, in which localized states exist; in the simulated tube they lie above the homogeneous states. At the continuous point there is no coexistence.

Accurately simulating chaotic quantum many-body dynamics remains a major computational challenge for classical methods, due to the rapid buildup and spatial spreading of entanglement during the evolution. This raises the question of whether machine learning can provide an effective alternative for predicting quantum dynamics. We address this question by formulating quantum dynamics as a time-series forecasting problem, using a few-qubit PXP chain, realizable with Rydberg-atom arrays, as a benchmark. By varying the initial state, the system spans dynamical regimes ranging from ergodic behavior to quantum many-body scarring, providing a controlled setting for testing forecasting models across qualitatively different dynamics. We compare two contrasting architectures: an expressive nonlinear Transformer and DLinear, a simple linear forecasting model. The Transformer accurately predicts dynamics in the more ergodic regime, but its performance progressively deteriorates as the initial state approaches the scarred limit. In contrast, DLinear remains accurate across the entire family of initial states, with its main deviations consisting of small high-frequency oscillations that have little effect on the overall prediction error. Remarkably, these results show that observables generated by complex quantum many-body dynamics can be forecast with high accuracy through a simple linear mapping from past to future observations. This reveals that the complexity of the underlying quantum evolution need not translate into an equally complex forecasting problem.

Quantum systems are rarely isolated from their surroundings. These hidden environments can retain information and return it later, producing correlated errors in quantum computers and complicating the interpretation of experiments. On a quantum processor, for example, one job can therefore leave an environmental trace that affects the next, even after the qubits themselves have been reset. Learning how this hidden influence persists and affects later behavior is therefore important, but the environment is often impossible to measure directly, and fully reconstructing its effects through tomography rapidly becomes impractical. Here, we introduce a response matrix that uses controlled interventions and observations of the system alone to reveal the temporal structure of environmental memory. The matrix records how future observations change across different controlled histories. Its spectrum characterizes how memory is structured and how it persists under specified system resets, while suitable linear combinations of its entries can rule out specified classical models of that memory. Because its history-future queries can be evaluated independently, numerical response matrices can be assembled in parallel. We establish the framework theoretically, characterize response spectra in an interacting spin process, and certify quantum memory in a two-qubit benchmark. This approach turns system observations into a probe of inaccessible environments and could support the diagnosis of correlated noise, tests of open system models, and memory-aware control design.

Superconductivity and intrinsic topological order can intertwine in ways that go beyond simple coexistence. Here we develop a classification framework for (2+1)D gapped superconductors with condensate charge $2ne$. By focusing on the topological properties of superconducting vortices, we precisely characterize this interplay. We obtain a classification with increasing levels of intertwinement, from the well known invertible superconductors without topological order, to untwisted and twisted SC$^\ast$, and finally to SC$^{\sharp}$. Our main finding is the family SC$^{\sharp}$, in which the elementary vortex carrying $h/(2ne)$ flux cannot be factorized into an invertible superconductor vortex dressed by bulk anyons, and every elementary vortex is necessarily non-Abelian. For such phases, we also obtain no-go results: the elementary vortex sector has quantum dimension at least $\sqrt{6}$ and cannot satisfy Fibonacci fusion rules. We develop a systematic algebraic framework to obtain all these superconducting phases through anyon condensation, allowing us to construct several infinite families of examples and make connections to anyon superconductivity. Finally, we introduce primitive SC$^\sharp$ phases for which factorization of every nontrivial vortex sector is obstructed, and argue that they cannot arise by disordering the relative phases of lower-charge superconducting condensates. Our classification scheme based on vortex properties allows us to uncover new distinctions between topologically ordered superconductors and constrain how they can be realized.

We study magnets composed of molecules with an arbitrary number $m$ of magnetically active atoms in a chain configuration interacting with nearest-neighbor couplings. We derive the thermodynamics of a chain of spin-$j$ $SU(2)$ magnetic atoms using a mean-field approach and determine the equilibrium states and stability conditions. The system exhibits a rich phase structure with a cascade of ferromagnetic phase transitions, their number ranging from $m$, for maximal ferromagnets, to zero, for paramagnets, depending on the relative strengths of the couplings. In the presence of impurities (modified couplings) at the ends of the chain, boundary states emerge that coexist with the paramagnetic phase or the fully ferromagnetic phase. For long chains, a continuum approximation can be taken that clarifies the structure of the phases. In the continuum description, magnetized states at temperatures lower than the critical temperature at which they appear correspond to kink solitons.

We have presented an analytical and numerical study of efficiency fluctuations in quantum Carnot and Otto engines with pistons operating under non-quasi-static adiabatic expansion and compression processes at constant speed, along with quasi-static isothermal (for the Carnot engine) or isochoric processes (for the Otto engine). We have considered the two-point measurement scheme during the adiabatic strokes in this regard. By considering each of the pistons as a single quantum particle in a 1-D box with a non-quasi-statically movable boundary, we capture the stochastic nature of the work output resulting from the unitary evolution of the quantum mechanical state of the system during non-equilibrium processes. Furthermore, we have explicitly shown how the piston speed during quantum non-equilibrium strokes introduces additional quantum fluctuations that modify the average efficiency of both the engines.

A rigorous macroscopic analysis of classical thermodynamics reveals an overlooked contradiction at the foundations of ideal gas theory: the traditional volume dependent formula for entropy violates the fundamental axiom of thermal equilibrium during reversible isothermal processes. This paper resolves this inconsistency by establishing a strict operational distinction between strictly isothermal and quasi isothermal processes. We show that in order for the perfect gas to be strictly consistent with the laws of classical thermodynamics, the thermodynamic entropy of a perfect gas must be a function of temperature only. This independence of volume is also confirmed by a new analysis of the adiabatic free expansion of an ideal gas, which is found to be intrinsically reversible. To reconcile these results with standard thermodynamic practice, we introduce the concept of a pseudo isothermal process (a finite sequence of small adiabatic and isochoric steps) and show that the classical equations for work and heat apply rigorously to these real processes, as opposed to strictly isothermal ones. We conclude by discussing the profound implications of these findings for statistical mechanics, showing that while the corrected thermodynamic entropy is in perfect agreement with Maxwell's kinetic theory, it is essentially different from the Planck Boltzmann formula. Thus, thermodynamic entropy and statistical microstate counting are two different physical quantities, and the present analysis provides a simple way to free classical thermodynamics from the paradoxes of microscopic disorder.

Achieving global coordination without a central controller or dense global communication is a defining challenge for both biological collectives and engineered swarms. We introduce and analyze a minimal model in which self-propelled agents in a bounded domain periodically reorient their motions toward the centroid of a small, randomly chosen subset of their peers, with no direct sensing of any individual neighbor's position or heading. We show that this sparse, non-local sampling rule reliably drives an initially disordered population to a globally aligned, nematic state, and that shrinking the sampled subset – down to the minimum of two agents – accelerates ordering rather than impeding it: the resulting estimation noise, filtered through a geometric turning rule, is itself the engine of symmetry breaking. We derive an analytical criterion, in quantitative agreement with simulation, that predicts when this ordering succeeds as a function of the sampling size and sampling frequency. We confirm the mechanism experimentally in a swarm of up to fifteen differential-drive robots. These results identify sparse random sampling as an information-efficient route to collective coordination, with implications for understanding animal collectives and for designing communication-limited robotic swarms.

We introduce a mean-reverting stochastic differential equation with a three-component stochastic term and show that it generates a hierarchy of steady-state (stationary) distributions. At the top level, the hierarchy is described by a modified-Beta distribution, while one- and two-parameter reductions produce compact-support, power-law-tailed, and exponential-type limiting families within a single stochastic framework. We then construct two generalized extensions of this hierarchy. In the first, the power transformation is applied directly at the level of the stochastic differential equation; in the second, the same transformation is applied only after the stationary modified-Beta hierarchy has been obtained. While these two procedures agree on important lower branches and limiting cases they generally differ at the top level. The generalized hierarchy is therefore not unique: nonlinear transformation and stationary-state reduction do not commute. For both routes, we derive the probability density and cumulative distribution functions, express their parameters in terms of the underlying stochastic dynamics, clarify the relations among their limiting cases, and compare the resulting families with the traditional Generalized Beta framework.

We study Bernoulli percolation and its universality classes on hierarchical lattices generated by edge iterated graph systems (EIGS). By examining eight critical exponents, we prove that no finite family of substitution-multiplicative dimensions completely determines universality, providing a rigorous proof of the failure of dimension-based universality discussed by Hauser and Saxena [Phys. Rev. B 34 (1986), 8193-8195]. Although the six exponentials theorem provides number-theoretic criteria for scale commensurability, we further disprove the conjecture that critical-exponent universality requires commensurate discrete renormalisation scales.

We study the two-dimensional anisotropic three-state Potts ferromagnet on the square lattice using Wolff single-cluster Monte Carlo simulations and finite-size scaling. For coupling ratios $\lambda=J_y/J_x=0.5$, $0.75$, and $1$, finite-size scaling of the correlation ratio yields critical behavior consistent with the two-dimensional three-state Potts universality class. The scaling of the leading Fisher zeros gives a correlation-length exponent consistent with $\nu=5/6$, while their cumulative density is consistent with the expected specific-heat exponent $\alpha=1/3$. We further characterize the anisotropy at criticality using directional correlation ratios $R_x$ and $R_y$ together with directional FK wrapping probabilities. For $\lambda=0.5$ on a physically square lattice, $R_x$ and $R_y$ approach distinct critical values while yielding a common correlation-length exponent. From the wrapping probabilities, we independently determine an effective aspect ratio $\rho_e^\square=0.6413(5)$, in close agreement with the theoretical value $\rho_e^{\square,\mathrm{th}}\simeq0.64150030$ obtained from the isoradial representation. Using the theoretical value to set the physical aspect ratio restores directional equivalence, $R_x\simeq R_y$, and brings the overall correlation ratio toward the isotropic-square reference. The results show that spatial anisotropy changes the effective critical geometry without altering the bulk three-state Potts universality class.

We present results from a detailed finite-size scaling analysis of the $3$-$d$, $Z(2)$ and $O(2)$ spin models in an external field $H$. Using high statistics Monte Carlo data, we obtain the leading finite-size scaling correction to the infinite volume scaling functions. We show that these corrections are proportional to $\widetilde{z}_L^2=1/(H L^{3/(1+1/\delta))})^{2}$. This provides the parametric form for finite-size corrections to bulk thermodynamic observables as well as the pseudo-critical temperatures determined at non-vanishing $H$. In particular, it allows to eliminate systematic errors in the analysis of the chiral phase transition temperature in (2+1)-flavor QCD, that arose from the previously not well-controlled ansatz for infinite volume extrapolations. We also establish the validity range of this leading order correction and point out that there are significant differences between the $3$-$d$, $O(2)$ and $Z(2)$ universality classes.

Deep thermalization indicates the emergence of universal ensembles of pure states in a subsystem after measuring its environment. We investigate this phenomenon in a free-fermion chain using local measurements that preserve Gaussianity and break particle-number conservation. Although the measurement outcomes provide partial information about local conserved charges within a finite subsystem, we argue that this information becomes negligible in the thermodynamic limit. This effective non-revelation underlies the emergence of a fermionic Gaussian Scrooge ensemble, determined by the subsystem's generalized Gibbs ensemble. We construct this universal ensemble directly at the subsystem level and provide numerical evidence that it describes the late-time projected ensemble at finite temperature. Our results extend the description of thermalization in non-interacting integrable systems beyond the reduced density matrix to universal statistics of conditional pure states.

The stability of the Berezinskii–Kosterlitz–Thouless phase against long-range interactions remains an open problem. We address this question in the two-dimensional XY model with couplings decaying as $r^{-2-\sigma}$, by developing a real-space renormalization-group treatment near the Gaussian manifold of quasi-long-range ordered (QLRO) states. We retain the full many-body vortex interaction and show that, at leading order in the fugacity, screening by small vortex–antivortex pairs preserves its Gaussian-average representation without an expansion in the long-range coupling. Including spin waves perturbatively then yields a local stability criterion: a stable QLRO interval exists only for $\sigma>7/4$. We also investigate a Wilson–Fisher-like fixed point at finite coupling for $\sigma<7/4$, for which the retained flow predicts $\nu^{-1}\sim\sqrt{7/4-\sigma}$ as it merges with the Gaussian line.

Understanding the non-equilibrium dynamics of quantum many-body systems plays a central role in modern many-body physics. Intriguing insights emerge from investigating how quantum resources, which are essential for achieving unambiguous quantumness, are generated and spread under chaotic quantum dynamics. In fermionic systems, non-Gaussianity constitutes an important resource beyond entanglement, characterizing the deviation of generic quantum states from free-fermion states. In this Letter, we propose using the replica twist entropy, defined through the expectation value of a replica rotation operator with a tunable angle $\alpha$, as a natural probe of non-Gaussianity in pure fermionic states. After deriving its behavior for typical many-body states, we establish a general framework for analyzing the replica twist entropy in solvable Sachdev-Ye-Kitaev models in the large-$N$ limit. Applying this framework to the dynamics of the thermofield double state, we uncover a novel dynamical phase diagram in the parameter space of $\alpha$ and evolution time $t$, exhibiting spontaneous $Z_2$ symmetry breaking at $\alpha=\pi/4$ and universal cusps as a function of $\alpha$ in the long-time regime. Our results pave the way for studying non-Gaussianity in strongly correlated fermionic systems and reveal a new class of dynamical transitions probed by non-Gaussianity.

We study the emergence of a mutual giant component in a multiplex network comprised of two one-dimensional long-range networks on a lattice. The probability that two nodes on one lattice, with a distance $r$ between them, are connected with an edge falls algebraically with distance as $Cr^{-(1+\sigma)}$, where $0 < \sigma < 1$ and $0 < C \leq 1$. A mutual giant component exists if $C > C_c^{\mathrm{mul}}$, where $C_c^{\mathrm{mul}}(\sigma)$ is a critical value of $C$ that is dependent on $\sigma$. We find rigorous lower bounds on $C_c^{\mathrm{mul}}$, and a tree approximation whose estimate of $C_c^{\mathrm{mul}}$ agrees with simulations within 0.6% for $\sigma \leq 0.5$. The multiplex network is significantly less stable than the single network case, as is evident by the larger value of $C_c^{\mathrm{mul}}$, which is up to 2.4 times the single-layer value. Our simulations indicate that the order of the transition depends on $\sigma$. For $\sigma \leq 0.35$ the mutual giant component collapses abruptly, in a first-order phase transition, while for $\sigma \geq 0.4$ the transition is continuous. When the dependency links connect random pairs of nodes, the transition is first order for all $\sigma$.

Energy landscapes of non-convex optimization problems are high-dimensional surfaces that are difficult to reveal, visualize, or analyze. Message-passing algorithms, or equivalently cavity approaches in statistical physics, may fail to identify local minima as messages do not converge owing to the ruggedness of the energy landscapes. Here we aim to reveal the characteristics of these complex energy landscapes by limiting the amount of information passed in local messages, improving the convergence of messages for local minima. By studying a routing optimization problem with its convexity governed by a single parameter, we introduce a cavity message-passing algorithm with a trust region, and sample ensembles of converged local minima of rugged energy landscapes across many independent realizations of the same problem instances. We observe three regimes with different characteristics of the energy landscapes: (1) a maximally rugged regime with a peak in the diversity of low-energy minima; (2) an intermediate regime with a hierarchical multi-cluster organization of low-energy solutions; and (3) a smooth regime dominated by a single minimum. Additional tests show that the converged energies are largely insensitive to the size of the trust region if the size is small or moderate, demonstrating the robustness of the proposed approach in identifying characteristics of rugged energy landscapes.

The square-root impact law (SRIL), $I = Y\sigma\sqrt{Q/V}$, bundles two facts that a single mechanism must explain at once: a shape (impact proportional to square-root of traded volume $Q$) and an amplitude ($Y=O(1)$, independent of the participation rate $\varphi$). Bonart has recently proposed an elegant solution: if realized and counterfactual prices are both diffusive, information-neutral impact must have white increments and the SRIL follows without the wart. We reformulate and simplify his argument, and foreground the ingredient he himself regards as essential — that the market whitens the stationary stream of each participant's metaorders, not any isolated one. The Lillo–Mike–Farmer model satisfies his premises, yet gives an isolated metaorder the super-square-root impact $\varphi^{(1-\gamma)/2}Q^{(1+\gamma)/2}$, $\gamma\in(0,1)$, because single metaorders (i.e. not part of a stream) are statistically invisible and thus priced mechanically. We then construct an explicit multi-agent propagator that realizes Bonart's whitening, and show it reproduces the clean SRIL precisely when the market can attribute trades to their issuer. But correct re-attribution of anonymous trades is a formidable task, which requires markets to behave as implausibly efficient signal processing machines. The stream hypothesis makes a sharp, falsifiable prediction — isolated metaorders, well separated in time, should not obey the SRIL — that does not seem agree with empirical data known to us. Independently, a volume-conservation argument singles out the square-root law and lays bare the step the diffusivity route must assume: diffusivity fixes the variance of impact — an additive, stationary, age-blind quantity — whereas the SRIL is a law for the mean impact, which marginally decays as $1/\sqrt{\text{age}}$ and therefore demands an origin of time.

We investigate anomalous heat transport in the one-dimensional $\beta$-Fermi–Pasta–Ulam–Tsingou chain in the weakly anharmonic regime using large-scale equilibrium molecular dynamics. By resolving the heat current into phonon-mode contributions, we show that its long-time autocorrelation is controlled by diagonal mode correlations, whose decay is quantitatively determined by the corresponding phonon damping rates. Extrapolation to the thermodynamic limit gives $\Gamma_k \propto (\beta T)^{4/3} k^{5/3}$, with a subleading $k^2$ correction. The latter produces an extended pre-asymptotic regime, $C(t) \sim t^{-4/5}$, before the asymptotic decay $C(t) \sim t^{-3/5}$ is reached, leading to $\kappa(N) \sim N^{2/5} + O(N^{1/5})$. At weak anharmonicity, off-diagonal mode correlations persist up to a characteristic time $\tau_{\mathrm{od}} \sim (\beta T)^{-2}$, while increasingly large systems are required to resolve the long-wavelength phonon contribution. Weak anharmonicity therefore does not introduce a distinct asymptotic transport mechanism, but shifts the onset of the phonon-dominated regime to progressively longer times and larger length scales.