Papers

Physical systems and distributed AI workflows can retain past information, receive delayed feedback and combine conflicting signals. These features pose a common modeling problem: determining what information about past interactions a state must retain to account for future responses. Active Neural Matter (ANM) addresses this problem through a finite-window response framework. Forward analysis uses each system's dynamics to predict state and output responses to perturbations, separating propagation from output sensitivity. Inverse analysis tests whether a candidate state retains enough information for these responses. With the system and future inputs or interaction rules fixed, distinguishable responses from histories assigned the same state identify distinctions the representation must preserve. Their distributional separation gives a lower bound on the worst-case error of approximating behavior from that state. Uncertainty bounds distinguish rejection, support within a fixed tolerance on tested comparisons, and unresolved cases. Delayed-consensus simulations check analytical predictions of response amplitude and phase over a narrow range of delays; flocking illustrates dependence on the chosen observable. An exact delayed-return construction establishes a sufficient state for a specified program. The AI studies identify distinctions in interaction history and evidence order that a representation must preserve to reproduce observed response distributions. Tests on new histories in a controlled feedback task provide bounded support for a state specified by the task instructions; the instructed rule predicts every observed action. ANM connects response prediction to measurable information requirements, separating behavioral fidelity from task correctness and initial-state adequacy from a closed evolution law for that state.

We extend Mark Kac's kinetic framework from single-species Euclidean spaces to a two-species, non-linear interaction space on the open cone $(0, \infty)^2$ governed by the Matsumoto–Yor rational collision involution. We identify a non-linear conserved energy $E(x,y) = x + y + 1/x$ and an invariant conic metric measure $d\mu = \frac{dx \, dy}{x(x+y)}$. By establishing a two-species $H$-theorem, we prove that entropy production vanishes if and only if post-collision velocities are statistically independent ($I(U;V)=0$). In contrast to the transform-domain proof of Letac and Wesołowski~\cite{letac2000}, which relies on joint Laplace transforms of general measures, our approach yields a direct physical derivation in state space via the transformation law for logarithmic gradients and separation of variables. This uncouples the functional-differential system to uniquely characterize the Generalized Inverse Gaussian (GIG) and Gamma distributions as the non-equilibrium thermodynamic equilibria.

We present a suite of 50 unpublished research-level theoretical physics problems with verifiable answers. Without AI assistance, a typical problem in our collection could be solved by a world expert in about a day or less but would take considerably longer for experts in adjacent specialties. We test the ability of off-the-shelf AI agents to tackle these problems autonomously, without human hints. When given access to tools available in a realistic setting, including the ability to search the literature and write scripts for numerical and symbolic work, the best-performing agent, GPT-6-Astra harnessed in Codex, obtains the final answers with a near-perfect accuracy of 98% at an average cost under USD 3 per problem. On average, agents show only small changes in accuracy when denied web search and access to background research papers, suggesting significant memorization of the literature. These results suggest that state-of-the-art agents can accurately perform concrete calculations required across theoretical physics research. We comment on the broader implications of these findings and discuss possible explanations for the current lack of agent-led breakthroughs in theoretical physics despite their outstanding problem-solving abilities.

In most theories of electrolyte and polyelectrolyte solutions, the solvent is treated implicitly as a dielectric continuum. While this approximation has enabled major advances, it neglects coupling between ionic and dipolar degrees of freedom. Using molecular dynamics simulations and the mean-spherical approximation, we show that this coupling qualitatively alters electrostatically driven phase behaviour. At finite ion concentrations, many-body ion-dipole correlations reverse the temperature dependence of effective electrostatic interactions, leading to upper-critical solution temperature (UCST) behaviour in direct contrast to the lower-critical solution temperature (LCST) behaviour predicted by implicit-solvent electrostatic theories. This phase reversal arises from ion-induced suppression of local dipolar screening, a collective effect inaccessible to pairwise implicit-solvent descriptions.

The low autocorrelation binary sequences (LABS) problem, often formulated as a spin-lattice energy-minimization problem, is a combinatorial optimization problem with a very rugged energy landscape. We demonstrate that iterative optimization of very small subsystems of n spins that are decoupled from the rest of the system achieves global optimization of LABS with a time-to-solution scaling that is competitive with heuristics that optimize the global system of N spins directly. Our isolated-subsystem heuristic with n = 4 achieves a time scaling of O(1.333^N ) on the interval N = [27, 60], while the current state-of-the-art classical and quantum scalings are respectively O(1.34^N ) and O(1.21^N ). Our results suggest that specialized optimization hardware with only small numbers of logical spins may be of practical relevance.

Self-assembled monolayers (SAMs), formed by the adsorption of a monomolecular layer of adsorbate onto a surface, have emerged as leading candidates for plat- forms for electrochemical biosensors. In order to characterize the working age of a SAM-based biosensor, it is necessary to understand the long-term degradation of the monolayer, with particular emphasis on the intermolecular forces present in the self-assembly process. In this work, discrete- and continuous-time proba- bilistic cellular automaton (PCA) models for the degradation of SAMs involving adsorbate-adsorbate interactions are proposed in which adsorbate molecules desorb at a rate dependent only on the number of adsorbed neighbors. Impor- tantly, we make no assumptions on the relationship between degradation rate and adsorbed neighbors. Using techniques from generating function theory and complex analysis, the expected degradation curve is solved for explicitly in one dimension and shown to be a sigmoid curve related to the Gompertz function.

Relevant IceCube recent data for astrophysical neutrinos are analyzed. Similarly to cosmic rays and proton-proton LHC collisions, the corresponding spectra are satisfactorily described, for the entire interval of measured energies, through q-Gaussians. This fact reinforces the already available evidence that nonadditive entropies (either of the $S_q$ class or of the $S_{delta}$ one, or even alternative ones) appear to play a crucial role in systems such as black holes and related cosmological phenomena (e.g., dark energy). The IceCube data yield sensibly high values for the momenta-distribution nonadditive index (q \simeq 1.6), in contrast with the cosmic rays values (q \simeq 1.2) and the LHC values (q \simeq 1.1). This fact neatly indicates very-long-ranged space-time correlations. The detailed associated physical mechanisms remain, at the present stage, as an instigating open issue.

Quantum mechanics seems to be an incredibly delicate theory. Slight alterations to nearly any part of its structure can lead to huge qualitative differences and wildly unphysical predictions. Here, we explore the many-body consequences of replacing the Born rule $p(m)\propto\|P_m|\psi \rangle|^2$ by a warped rule $p_b(m)\propto\|P_m|\psi\rangle\|^{2b}$. Specifically, in systems with conserved particle number, evolution punctuated by repeated local particle observations acquires a probabilistic bias over measurement histories with an effect controlled by the sign of $b-1$. For $b>1$, this bias generically drives the gas to collect into nearly frozen configurations. Similar to the action of Maxwell's demon, the simple act of observation seems to extract entropy from a closed system. For $b<1$, the bias is reversed, generically scouring out long-wavelength fluctuations to produce a hyperuniform gas. We study this effect in a one-dimensional quantum gas, where these two phases are derived analytically for a particular measurement scheme and corroborated numerically for more general procedures with discrete-time projective-measurement circuit models using matrix product state simulations. We additionally discuss how this bias manifests in a common technique that often employs this warped rule, the “replica trick,” implying qualitative differences between finite-replica and replica-limit calculations.

We construct a set of conserved charges which are deformations of quantum KdV charges. They are associated with the modular transformation of a GGE with the spin-$\sigma$ KdV charge inserted with chemical potential $\alpha_\sigma$. The charges are formal power series in the chemical potential, with coefficients given by local integrals of motion. We define local recursion relations and equivalent flow equations, prove commutativity to all orders, and establish uniqueness of the currents modulo total derivatives for a fixed generating series. We show that the eigenvalues of these deformed charges can be obtained from the thermodynamic Bethe Ansatz in two models. In the $c=1/2$ free fermion theory, the deformed charges reduce to explicit linear combinations of undeformed KdV charges, and their ground-state eigenvalues agree with the TBA expansion to all orders. In the Lee-Yang model, we find numerical agreement through cubic order in the spin-$5$ deformation for the first few charges evaluated on low-lying states.

We find a robust scenario for the emergence of the Golden Ratio (GR) dynamical universality class, characterized by the dynamical exponent $z \equiv \varphi=\frac{1}{2}(1+\sqrt{5})$. Previously, its emergence required fine-tuning of system parameters. Here, the GR universality class appears generically in a wide class of Markov processes as a consequence of a global, circulant symmetry of the flux Jacobian. This symmetry leads to GR universality in systems where the total number of conservation laws is odd and larger than three. We introduce a multilane exclusion process with Bernoulli stationary measure as a minimal, parameter-free model, which features GR universality for all conserved modes, most of which are doubly degenerate. Our predictions, based on mode-coupling theory generalized to treat degenerate conserved modes, are corroborated by large-scale Monte Carlo simulations.

Performing measurements on gapless quantum ground states can give rise to new universal phenomena, whose origins remain poorly understood even for the paradigmatic Ising critical point. Replica methods offer a natural framework to study such post-measurement ensembles, with the replica limit $Q\to 1$ describing physical Born-averaged observables. Here we show that, for the monitored critical Ising chain, the replicated post-measurement ensemble of states itself undergoes a phase transition precisely at the physically relevant replica limit $Q \to 1$, preventing standard replica treatments. We use boundary conformal field theory (BCFT) methods to explain the underlying mechanism of this transition. Our methods yield a perturbative analytic calculation of the universal $O(1)$ constant in the Shannon entropy of the measurement record, previously known only numerically. We also apply our tools to study measurement-averaged spin correlators, providing a unified description of the universal properties of measurement outcomes and post-measurement states in the critical Ising chain.

Dissipative state preparation engineers quantum many-body states as steady states of an open-system dynamics. The spatial range of the required dissipation sets both the feasibility of analog implementations and the connectivity and control complexity of digital ones. Yet how this locality quantitatively constrains preparation times remains largely unknown. Here, we establish an exact locality-preparation time relation for quadratic fermionic systems and develop a construction that optimizes the locality of the dissipative dynamics while preserving the target state. Applied to the Kitaev chain, our framework reveals sharply different locality-preparation time tradeoffs in critical and gapped regimes. At criticality, suppressing the algebraic tails of the jump operators necessarily forces the preparation time to grow algebraically with system size. By contrast, both trivial and topological gapped phases admit exponentially localized dissipation with a finite preparation time in the thermodynamic limit. Our results identify when fast dissipative preparation is compatible with local control and when criticality imposes an unavoidable dynamical cost.

We establish that bond Hamiltonian densities, up to local difference terms whose sum vanishes under periodic boundary conditions, lying in the commutant of bond operators, $w_j\pm w_{j+1}$, is a necessary and sufficient condition for a one-dimensional nearest-neighbour Hamiltonian to conserve a uniform or staggered sum of the one-site operators $w_j$. Starting from these local commutants, we show that every such Hamiltonian and its conserved charge can be obtained from monodromy matrices constructed using regular and non-regular members of scalene Yang-Baxter triples. The results hold on periodic chains of length $L\geq3$, with even length required in the staggered case. The results do not require translation invariance, and hence the interaction on each bond can be chosen independently from the full corresponding commutant. Our representation-independent construction applies in arbitrary finite local dimension and exhausts the Hamiltonians with either prescribed additive symmetry, without requiring ordinary Yang-Baxter integrability. We illustrate the construction with spin-$\tfrac12$ and spin-1 systems and discuss its relation to the double-commutant theorem. Classifying the one-site generators $w$ and their associated bond commutants provides a route to classifying such Hamiltonians with additive conserved charges.

A novel physics-informed neural networks (PINNs) based approach for solving functional renormalization group flow equations is presented. We investigate the finite temperature phase transition in $O(N)$ models without relying on the large-$N$ approximation, which was used in earlier attempts with similar methods. We argue that these previous solutions cannot easily be generalized for more complex symmetries, and hence our approach can provide new insight. Our technique relies on a series of ansätze that combines the small- and large-field approximation schemes with a neural network, leading to a highly accurate approximation over the entire field range. We compare the PINNs approach with traditional grid-based Runge-Kutta results and show that the network can successfully learn the temperature, field, and scale dependencies of the effective potential in the local potential approximation.

Complex disordered conductors can display highly structured and parameter-sensitive transport even in the absence of a thermodynamically stable glass phase. Overlap distributions originally developed for replica symmetry breaking (RSB) provide a useful diagnostic of this organization, provided that their interpretation is distinguished from the equilibrium Parisi construction. In this study, we report RSB-like overlap phenomenology and parametric mesoscopic glassiness in a graphene-based electronic Lévy glass, a device that mimics optical phenomena via electronic processes. By tuning the Fermi energy, we can control the number of propagating wave modes and observe reentrant changes from narrowly centered to broad or bimodal overlap distributions. The phenomenology is observed in both graphene-based and conventional two-dimensional electron-gas electronic Lévy glasses, indicating its generality. Sample-resolved overlap distributions show that the effect is intrinsic to each disorder landscape, whereas the normalized conductance variance shows that the growth of the absolute variance is mainly caused by the increasing mean conductance. We therefore interpret the overlap as a measure of correlation among hierarchically organized transport pathways and the reentrant behavior as evidence of pinning, parameter-induced reorganizations, and mesoscopic chaos in coherent transport.

The Kelvin-Helmholtz (KH) instability is a fundamental signature of the natural world, manifesting in phenomena ranging from atmospheric clouds to the iconic turbulence of Jupiter's Great Red Spot. While the linear dynamics of this instability is well-understood, the ultimate fate of the fluid — the final stationary state to which it relaxes — remains a formidable challenge. In this paper, we present a general non-equilibrium statistical mechanics framework that allows us to predict the asymptotic vorticity distribution of a sheared 2D Euler fluid. By constructing a self-consistent ansatz for the solution of the Vlasov equation, we determine the stationary stream function without explicitly solving the time-dependent Euler equations. We demonstrate that the relaxation leads to a population-inverted core, where high-energy states are fully occupied, surrounded by a tenuous halo of vortices in the lower-energy regions of configuration space. This result establishes the core-halo structure as a robust statistical consequence of the system's evolution, offering a universal framework for understanding the saturation of shear-induced turbulence in inviscid 2D fluids.

We present, to our knowledge, the first theoretical realization of an independent charge-4e phase in a two-dimensional classical array of conventional superconductors coupled through altermagnetic junctions. Tuning the altermagnetic splitting suppresses the leading single-pair tunneling harmonic while retaining strong double-pair coupling, producing a finite charge-4e region between staggered and uniform charge-2e states. Using the complete temperature-dependent microscopic Josephson spectrum, we find short-range ordinary pair correlations alongside algebraic coherence of pairs of Cooper pairs. Berezinskii–Kosterlitz–Thouless scaling and a half-flux response further identify the charge-4e phase. Our results establish altermagnetism as a microscopic means to select between ordinary and composite superconducting coherence.

Semi-Markov networks provide a natural framework for describing stochastic transitions between coarse-grained states with non-exponential dwell times, with applications ranging from biophysics and neuroscience to epidemiology and finance. Here, it is shown that finite semi-Markov networks admit explicit combinatorial representations of their Laplace-transformed occupation probabilities and hitting-time distributions as sums over weighted spanning trees and forests of an augmented graph. These representations yield asymptotic results, including a generalization of the Markov Chain Tree Theorem to semi-Markov networks, in which the rate constants replaced by pseudo-rates given by the inverse mean sojourn time weighted by the exit probability through this channel, as well as the long-time scaling of hitting-time survival probabilities in the presence of infinite-mean sojourn times. They also extend recently derived combinatorial formulas for hitting-time moments to the semi-Markov setting. In particular, the mean hitting-time formula coincides with that of a Markov network with the corresponding pseudo-rates, which also yields a short proof of the constancy of Kemeny's constant for finite-mean semi-Markov networks. Finally, we use the same framework to show that weak ergodicity breaking occurs generically for time-additive state observables that are nonconstant on the support of the limiting distribution in heavy-tailed semi-Markov networks.

We study how a neural network reorganizes its learned function as training proceeds and as previously unseen parity data accumulate. The trained model is itself a Boolean function on binary strings of finite length. We then evaluate it on all data, inside and outside the training set, and read off its complete sign table. Its $\pm1$ entries are the amplitudes of a normalized state, dubbed the function-state, whose inverse subsystem purity $K_f=1/\textrm{Tr}(\rho_A^2)$ is an effective Schmidt rank. We find that small data additions broaden the Schmidt spectrum, whereas larger additions reconcentrate it in the target mode. The former is called differentiation because the model distinguishes inputs more sharply. The latter is called target-selective recoherence, where the model settles on the single rule shared by the added data. As the dataset grows, the connectivity of the constraint graph defined by the training set increases monotonically, whereas the learned inverse purity changes nonmonotonically. On a fixed connected dataset, as learning proceeds, the Schmidt spectrum begins to reorganize, followed by gains in target alignment and later by improvement in predictive accuracy. This ordering distinguishes three aspects of learning: constraint identifiability, function-state organization, and predictive accuracy. At fixed data size, a function that has already been learned can re-differentiate, an event we call retreat, without changing graph connectivity. Recovery reconcentrates the Schmidt spectrum in a similar target-selective pattern, without exact recurrence of the Boolean state. Thus the percolation of data constraints fixes the identifiable support, while the reduced state produced by training continues to reorganize even after that support stops changing.

The real-time dynamics of quantum phase transitions in open systems coupled to finite-temperature environments sits at the heart of contemporary condensed-matter and quantum-information science. Here, invoking a Schwinger-Keldysh formalism, we uncover that, within the coherence-dominated regime of critical real-time evolution, a dissipative coupling deemed naively irrelevant determines, counterintuitively, the longest timescale that holds sway. Possessing a well-defined temporal scaling dimension, it engenders a novel dissipation-regression scaling that systematically reproduces the unusual Kibble-Zurek scaling and relaxation observed in recent numerical studies. By contrast, dissipation-dominated dynamics behaves qualitatively differently. There, furthermore, we show that the discretized and diagonalized formulations of the same open quantum Ising chain realize distinct universality classes.

The Markov length, the length scale governing the exponential decay of conditional mutual information (CMI) in a quantum many-body state, has proven useful for characterizing mixed-state phases of matter. Nevertheless, we show that a state's Markov length is not generally stable under weak local channels. The failure of stability occurs already at a pure product state. We demonstrate this explicitly in a one-dimensional fermionic chain, where a fine-tuned depth-two circuit of nearest-neighbor Gaussian channels maps the vacuum to a state whose CMI decays algebraically and whose Markov length is therefore infinite, even when the channels are taken arbitrarily close to the identity. The algebraic decay is rooted in the nonanalyticity of the von Neumann entropy at rank-deficient density operators. We exhibit an analogous instability on an ordinary qubit chain. More generally, we show that the Markov length is unstable in any translation-invariant state with even a single rank-deficient reduced density operator. These results pose a challenge to recent definitions of mixed-state phase equivalence based on the Markov length.

The staggered order parameter of the gapped XYZ spin-$1/2$ chain follows from the eight-vertex solution, but in elliptic variables that fix the Hamiltonian couplings only implicitly. We derive a closed form for the coefficient governing the leading quadratic response of the spontaneous staggered magnetization to the XY anisotropy at fixed longitudinal anisotropy in the gapped antiferromagnetic regime. Starting from the Baxter–Kelland–Jimbo–Miwa–Nakayashiki polarization formula, we show that at fixed longitudinal anisotropy both the modular nome and the crossing-parameter variable of the formula move with the XY anisotropy, and that the motion of the crossing-parameter variable reverses the sign of the expansion at fixed elliptic variables. The coefficient is a rational term plus a Lambert series. It is strictly negative and bounded above by a rational function of the longitudinal anisotropy, to which it is asymptotic in the Ising limit. At the isotropic point it diverges as the inverse five-halves power of the distance to that point, and since the Lambert sum has an asymptotic expansion with exponentially small remainder, the subleading terms of the divergence also follow in closed form. When the XY anisotropy vanishes as the five-fourths power of the distance to the isotropic point, the ratio of the perturbed to the unperturbed order parameter tends to a Gaussian in the scaling variable, and the quadratic truncation is controlled when that variable is small. A fourth-order perturbative calculation in the spin chain, with no input from the eight-vertex solution, reproduces the leading strong-coupling behaviour, and infinite-DMRG calculations at four anisotropies in the gapped regime are consistent with the closed form.

We develop an optimization algorithm for spherical lattice discretizations based on the spectrum of the discretized Laplace operator. Starting from refined icosahedral lattices, we minimize the splitting of the $(2\ell+1)$, $\ell=0,1,2,\dots$, continuum multiplets expected from $\mathrm{SO}(3)$ symmetry via gradient descent. This approach simultaneously produces lattices based on triangles with nearly uniform circumradii and perimeters. Although the optimization uses only the free-particle spectrum, we show that the resulting lattices carry over to interacting models. On the optimized lattices, rotational symmetry is restored for the critical Ising and Potts models and primary scaling dimensions are accurately reproduced for the three-dimensional Ising conformal field theory, when appropriate geometry-dependent interaction strengths are used. Higher-angular-momentum descendant states reveal weak residual symmetry breaking beyond the sectors protected by the underlying icosahedral symmetry.

A tropical quantum field theory is an exactly solvable deformation limit of an ordinary QFT, governed by a non-linear partial differential equation. We use the integrable two-dimensional sine-Gordon theory as a testbench to compare the exact features of an undeformed model with its tropical counterpart. We classify scale-invariant tropical QFTs in $D=2$ and leverage functional renormalization group methods to demonstrate that all scale-invariant theories with periodic effective actions are infrared fixed points of flows originating from sine-Gordon Lagrangians. For the tropical sine-Gordon model, we compute the $\beta$-functions exactly, solve the renormalized theory both via a convergent expansion and numerically for all admissible couplings, and illustrate the complete renormalization group flow from the Gaussian to the IR fixed point. Finally, we study the large-order behaviour of the loop expansion and relate it to the Hamilton-Jacobi equation, whose solutions we interpret as tropical instantons.

We investigate the full statistics of entropy production in coherent mesoscopic conductors beyond the weak-coupling regime. Using the inside-outside perspective and the Landauer-Büttiker scattering framework, we relate entropy production to the full counting statistics of particle transport and show that the corresponding cumulant-generating function satisfies the fluctuation symmetry independently of the coupling strength. We obtain closed analytical expressions for the cumulant-generating function in the weak- and strong-coupling limits, enabling a detailed characterization of entropy production distributions and higher-order cumulants. These cumulants reveal distinct signatures of resonant transport and coherent fermionic partitioning. While coupling-induced broadening substantially modifies the transport statistics, the fluctuation symmetry imposed by microscopic reversibility remains preserved in the long-time limit.

We present a minimal classical lattice model of superfluidity at arbitrarily high temperatures, using a modified three-dimensional XY model with “soft" rotors. The global U(1) symmetry is spontaneously broken above a positive critical temperature due to entropic order. We study hydrodynamics in this high-temperature regime and verify consistency with the fluctuation-dissipation theorem, explaining how the large noise variance at very high temperatures is not sufficient to destroy superfluidity or the long lifetime of Goldstone modes.

We study symmetries and anomalies of dissipative Majorana chains described by Lindblad dynamics. Using the operator-state mapping, we represent the Lindbladian as a non-Hermitian operator on a doubled Hilbert space. In this formulation, the ket and bra contours admit strong fermion parities and strong translations, while their diagonal combinations define weak symmetries. For even and odd chain lengths, we diagnose anomalies by comparing symmetry algebras with and without weak fermion-parity defects. These algebras retain the lattice precursor of the modulo-$8$ anomaly of the closed Majorana chain and exhibit projective relations between strong fermion parity and weak translation, as well as between weak fermion parity and strong translation, which are intrinsic to open quantum systems. We establish the IR correspondence between the lattice symmetry algebras and those of the doubled Majorana CFT in four spin sectors, identifying chiral fermion parities emanated from the lattice translations. Finally, we develop the lattice bosonization of the dissipative chain. For even length with collective energy dephasing, combining the defect and no-defect sectors and imposing GSO projections gives rise to an Ising Lindbladian with non-invertible Kramers–Wannier symmetries descending from the anomalous strong translation and fermion parity of the Majorana chain. For odd length, constructing mutually commuting ket- and bra-sector spin operators instead yields an Ising Lindbladian with a Kramers–Wannier duality defect. Our results provide a controlled framework for extending lattice anomalies and non-invertible symmetries to open quantum systems.

Quantum error-correcting codes on graph geometries beyond regular euclidean lattices have emerged as a promising route toward robust quantum information processing. A major challenge in quantum error correction is the trade-off between favorable properties of codes constructed in high dimensionality and the need for locality in practical implementations on near-term hardware. Sparse intermediate geometries may combine the advantages of both, but how their structure governs quantum information dynamics and code performance remains unclear. Here we show that the spectral dimension $d_s$, defined from the low-energy spectrum of the graph Laplacian, is the universal parameter connecting these properties. Using Floquet random Clifford circuits on two distinct sparse graph architectures with tunable spectral dimension, we find that $d_s$ governs both quantum information spreading and the scaling of the resulting quantum error-correcting code distance. We identify a geometry-induced entanglement phase transition from localized area-law to ergodic volume-law, whose universality is determined by $d_s$ and exhibits long-range percolation criticality. In the ergodic phase, the circuit acts as a CSS code encoder whose code distance is set by the competition between two logical sectors — the cost of severing the graph and the length of its shortest winding cycle — which connectivity pushes in opposite directions. The spectral dimension organizes this trade-off, highlighting a maximum at intermediate $d_s$ retaining sparse connectivity and physical realizability.

Entanglement and non-stabilizerness are complementary resources underlying the complexity of quantum states, yet extracting either quantity from large quantum circuits generally requires exponentially scaling resources. In diagrammatic approaches to quantum computation, such as the ZX-calculus, both quantities likewise typically require expensive tensor contractions that are not native to graphical rewriting. Here we show that both entanglement and magic can be efficiently estimated directly from ZX-diagrams. Flow enables efficient extraction of a normal form separating a potentially extensively entangled graph-state backbone from non-Clifford Pauli gadgets. This structure yields additive upper and lower bounds on bipartite entanglement entropy, which we further tighten through a preprocessing procedure that removes or merges redundant non-Clifford contributions. The same normal form gives an upper bound on logarithmic stabilizer extent. We benchmark these methods in random unitary and monitored circuits, and a circuit combining Trotterized Hamiltonian evolution with Clifford layers, finding that the resulting bounds remain informative even at large qubit numbers, circuit depths, and internal spider counts. Our results provide scalable probes of entanglement and non-stabilizerness of ZX-diagrams and establish a bridge between ZX-calculus and quantum many-body resource characterization, with applications across quantum computing, quantum information, and condensed-matter physics.

Replica symmetry breaking (RSB) is the organizing principle of mean-field spin-glass theory, yet whether a continuous Parisi hierarchy can arise in a genuinely short-range system in finite spatial dimension remains unresolved. Here we introduce a disordered model of $N$-component vector spins on a $D$-dimensional hypercubic lattice, with random matrix couplings on nearest-neighbor links and random tensor couplings on elementary plaquettes. In the large-$N$ limit, the model supports thermodynamically stable one-step, full, and one-full RSB phases, alongside replica-symmetric paramagnetic and ferromagnetic phases. We find that spatial locality constrains these ordered phases through two distinct infrared mechanisms. ($i$) Quenched random anisotropy destroys uniform ferromagnetism for $D\leq4$ through an Imry-Ma mechanism. ($ii$) A continuous Parisi hierarchy gives rise to replica Goldstone modes whose gaplessness is protected by a Ward identity. At leading order in $1/N$, these modes behave as a band of free spin waves, producing infrared-divergent fluctuations in $D\leq2$ and a Mermin-Wagner instability of the continuously broken phase. Remarkably, the Parisi hierarchy also acquires a direct real-space interpretation. For two pure states with mutual overlap $q$, a characteristic length $\xi(q)$ describes how far the differences between their frozen magnetization patterns remain correlated. Throughout a continuous sector of the Parisi hierarchy, $\xi(q)$ is independent of temperature and decreases monotonically with $q$. The ultrametric organization of pure states is therefore accompanied by an ultrametric hierarchy of real-space correlation lengths: states that separate near the root of the Parisi tree differ through long-wavelength frozen patterns, whereas states that separate farther from the root differ only on progressively shorter scales.