Papers

Renormalization group (RG) is a core concept in physics from statistical mechanics to quantum field theory, yet its schemes differ widely between field theory and many-body physics. We formulate the Wilsonian RG as a quantum channel, whose Kraus representation yields pure conditional trajectories for pure inputs and recovers mixed-state flows upon averaging. We then develop an algebraic extension applicable beyond the usual momentum-space, factorized setting: the low-energy algebra is constructed from the one-particle spectrum, and coarse graining is a conditional expectation onto it. We further construct a resource theory of RG whose free states are fixed points. For thermal states, the resulting relative-entropy monotone is proportional to $c-c_{\rm IR}$ to quadratic order along a single stable RG direction near a two-dimensional IR fixed point. We also introduce a complementary measure of the UV information discarded by coarse graining, establish its monotonicity for successive coarse-graining maps, and discuss its holographic realization.

We study the homology of percolation clusters on disk-shaped patches of the regular $\{3,q\}$ triangle tilings for $q\ge7$. Varying $q$ gives a one-parameter family of hyperbolic tilings, from mildly curved ($q=7$) to deeply hyperbolic ($q=20$), and this paper measures and derives loop content as a function of curvature. A single growth-rate parameter $\lambda(q)$, the asymptotic ratio of successive ring sizes, governs every result. Total persistent $H_1$ of the bond-percolation complex, per vertex, is measured across $q=7,\dots,20$ and found to be a nearly linear function of $1/\lambda(q)$. An exact identity reduces total loop persistence to the weight of the lattice's minimum spanning tree minus a boundary correction. The boundary correction follows from the ring recursion, and the spanning-tree weight is derived to first order in $1/\lambda(q)$ by attaching rings one at a time to a contracted interior, with a correction for loops that close through the ring outside. The resulting formula has no free parameters, gives the large-curvature intercept in closed form as $5/4-2\pi/(3\sqrt3)$, and agrees with the measurements to within the size of the next-order term.

We consider a fixed finite number of independent two-dimensional discrete Gaussian free fields on a box of side $n$, with zero boundary values, conditioned so that the vector-valued field stays outside a fixed ball on a macroscopic interior domain. We prove that the field, divided by $\log n$, converges in distribution in $L^2$ to twice the equilibrium potential of the domain multiplied by a uniform random unit vector. We also identify the exact capacity cost of the conditioning event and obtain alignment on a deterministic density-one set of sites, with no restriction on their mutual distance. The proof combines the norm repulsion theorem of Korzhenkova and Sepúlveda with Gaussian large deviations and the equality case in the capacity variational problem.

Weyl or Dirac fermions with the iconic linear energy-momentum relation and average density of states (ADOS) $\rho(E) \sim |E|^{d-1}$ at energy $E$ in $d$ spatial dimensions, constitute a unique setup to study the disorder-driven semimetal-to-metal quantum phase transition (QPT) between ballistic (realized for weak disorder) and diffusive (stabilized at stronger disorder) quasiparticles. Such a QPT takes place only for $d>2$ and falls beyond the realm of the Anderson metal-to-insulator transition. From numerically computed ADOS (using the kernel polynomial method) in dirty Weyl systems in $d=2$ to $6$, here we show that $d=2$ and $d=4$ are the lower ($d_\ell$) and upper ($d_u$) critical dimensions for such an unconventional QPT, respectively, as suggested from the solution of quasiparticle lifetime within the self-consistent Born approximation. Consequently, for $d \geq 4$ the associated correlation length exponent is found to be $\nu \approx 0.5$ (within numerical accuracy). However, the dynamic scaling exponent at the quantum critical point is pinned close to $z \approx d/2$ (numerically) for any $d \geq 3$, which is shown to be an exact result from a field-theoretic renormalization group calculation. Therefore, Weyl semimetal-to-metal QPTs can be studied field theoretically around both $d_\ell$ and $d_u$.

We introduce projected amorphous topological insulators (PATIs) that despite containing only a fraction ($x$) of otherwise randomly selected disconnected sites of a parent crystal, feature a quantized global topological invariant (strong PATIs) when its effective Hamiltonian is constructed by integrating out the residual sites of the original lattice. Within the topological regime of the parent model, additionally, there exists a critical $x$ below which such systems foster a fragile PATI that only supports quantized local topological marker on a small fraction of sites therein. Both strong and fragile PATIs accommodate in-gap modes near the boundary. The system also hosts a normal insulator in the entire trivial parameter regime of the lattice-based model for any $x$, devoid of gapless boundary modes, for which both global topological invariant and local topological marker vanish. We demonstrate these possibly generic outcomes by starting with a parent square lattice-based model for time-reversal symmetry breaking insulators. Then the strong-to-fragile PATI quantum phase transition, tunable via $x$, is characterized by the mean correlation length exponent $\nu \in (1.00,1.46)$.

We describe a probabilistic approach to gravitational clustering. Working purely in configuration space, we show that the two-point function can be formulated in terms of a transition probability describing the likely final position of a particle given its initial position. The transition probability is initially sharply peaked and then over time broadens out in a diffusion-like manner. The effect is more pronounced for widely-separated particles (e.g.\ at the baryon acoustic scale), leading to an effective suppression of the two-point function in what is normally referred to as infrared resummation.

Recent advances in quantum simulation have enabled the discovery of novel forms of universality in quantum many-body dynamics. In this Review, we discuss deep thermalization and Hilbert space ergodicity–two recently discovered phenomena characterized by the emergence of distributions of quantum states that are "maximally random" in a precise sense. They offer a new perspective on how irreversible statistical mechanics arises from reversible unitary quantum dynamics, going beyond conventional theories of quantum thermalization and equilibration. We review a unifying framework, rooted in quantum information theory and principles of maximum entropy, that explains the different forms of ergodicity which arise under various physical constraints. We discuss open questions and active research directions, including generalizations beyond ensembles of pure quantum states, phase transitions in deep thermalization tied to deeper forms of ergodicity-breaking, connections to broader topics in quantum thermalization and ergodicity, and applications in quantum information science, such as for benchmarking or tomography.

We propose a variational framework for the efficient simulation of stochastic quantum dynamics in interacting bosonic systems. Our method generalizes the time-dependent variational principle to accommodate quantum state diffusion processes and is tailored for describing quantum trajectories unraveled from a master equation. We adopt a wavefunction ansatz composed of a coherent superposition of Gaussian wavepackets, which allows the fully analytical assembly of the variational equations of motion while capturing both the quantum trajectory dynamics and the Lindblad evolution beyond semiclassical approximations. The method is carefully benchmarked on a Bose-Hubbard dimer, and applied to extended Bose-Hubbard lattices in both 1D and 2D. In particular, we show the emergence of a symmetry-breaking phase transition in 2D lattices, which is absent in 1D chains.

The stability of an ecological community is conventionally defined through species interactions, which quantify how species affect one another. Interaction strengths are notoriously difficult to measure in species-rich assemblages. Long-term monitoring of species-rich communities, however, provides species abundance time series, increasingly available across habitats and taxa but not yet connected to the stability properties of the communities they describe. Here we develop a statistical framework that infers stability regimes in large ecological communities directly from abundance time series. We study stochastic Generalized Lotka–Volterra dynamics with random interactions and environmental fluctuations using dynamical mean-field theory, reducing the multispecies system to an effective stochastic process for a representative species. The theory predicts three stability regimes (stable coexistence, intermittent dynamics close to extinction, and unbounded growth) separated by analytical boundaries, and shows that environmental stochasticity systematically destabilizes coexistence by promoting intermittent low-abundance dynamics. The resulting steady-state species abundance distribution is a Gamma law that ties the dynamical phases to the variability-based stability metrics used in empirical studies. Recasting the effective dynamics as a multivariate regression model, we infer interaction statistics, environmental variability and the characteristic timescale of the dynamics from community data, without reconstructing the interaction network. Applied to natural communities spanning a broad range of habitats and taxa, the method resolves contrasting stability regimes and yields quantitative estimates of species extinction risk.

Janus-particle arrays provide a soft-matter platform in which particle orientations act as classical spins with anisotropic, bond-dependent interactions. Motivated by recent experiments on triangular arrays of metallodielectric Janus particles exhibiting sixfold orientational order and directly observable vortices and antivortices, we study effective models beyond the isotropic XY description. A classical $120^\circ$ compass model captures the minimal bond-directional interaction, while an effective Kern–Frenkel model incorporates the geometry of the Janus surface interaction more directly. Monte Carlo simulations show that both models exhibit a low-temperature sixfold ordered phase, a high-temperature disordered phase, and an intermediate quasi-long-range-ordered regime consistent with two Berezinskii–Kosterlitz–Thouless transitions. In both cases, the sixfold order arises through thermal order-by-disorder from a continuously degenerate ferromagnetic manifold. We further show that higher-order extensions of the $120^\circ$ interaction can alter the orientational selection: a quadratic term reverses the fluctuation-induced sixfold anisotropy, while a cubic term lifts the degeneracy energetically and selects the lattice-aligned directions that coincide with those observed experimentally. These results connect the microscopic anisotropy of Janus-particle interactions to emergent XY-like vortex physics and illustrate how distinct microscopic mechanisms can produce similar sixfold orientational order.

We study the relaxation dynamics of a one-dimensional system of hard rods at the level of one-body fields using superadiabatic dynamical density functional theory, and compare the results with Brownian dynamics simulations. The theory uses the exact equilibrium excess free energy functional and incorporates superadiabatic effects from first principles via an adiabatic closure relation at the three-particle level. The predicted superadiabatic response shows very good agreement with simulation data, provided that initial-state ensemble differences between theory and simulations remain small. Furthermore, we evaluate an approximate power functional theory based on a velocity-gradient expansion, which also yields an accurate superadiabatic response, although requiring a memory kernel extracted from simulation data.

Autocorrelation functions play a key role in the theoretical characterization of the dynamical properties of interacting many-body quantum systems. Recently, bringing together the eigenstate thermalization hypothesis and hydrodynamics, it was theoretically predicted that the relaxation of autocorrelators can be described by the relaxation-overlap inequality, which, when saturated, predicts a hierarchy of relaxation exponents for a set of operators that are easily identified and constructed. Here, we employ a trapped-ion quantum simulator to experimentally demonstrate it in the diffusive regime; to do so, we have extended the theory of the overlap-relaxation inequality to systems with multiple conservation laws. Our study thus opens the path to a thorough characterization of the relaxation to equilibrium and appearance of hydrodynamic behavior in quantum matter.

Entangled polymer melts exhibit the robust two-branch Baumgärtel-Schausberger-Winter (BSW) relaxation spectrum, while related spectra occur in nonpolymeric monodisperse disordered systems. In spite of the successful application of BSW to many different materials, a molecular derivation of these spectra is lacking. We construct a molecular theory in which a chain segment moves relative to a screened, dynamically responding multichain environment. Gaussian-chain preaveraging gives $M_{\rm seg}(\Delta m)\sim(\Delta m)^{-1/2}$, hence $\lambda_p\sim p^{3/2}$ and, after stress projection, $H(\tau)\sim\tau^{-2/3}$. Independently, longitudinal primitive-path diffusion gives contour-length fluctuations with $H(\tau)\sim\tau^{1/4}$. A molecular-weight-constrained implementation is tested simultaneously against experimental $G'(\omega)$ and $G''(\omega)$ data for four monodisperse polybutadiene (PBD) melts, without fitting spectral exponents or individual modal weights. The resulting BSW spectrum exhibits a continuous transfer from the fast cooperative to the slow constraint-renewal cascade before a finite-chain terminal edge. A common two-sector caged dynamics then connects polymers to particle rafts without assuming identical microscopic mechanisms.

Diffusivity of passive tracers in complex mixtures is widely relevant for industrial applications and for probing biological systems. Interactions with the surrounding medium typically generate a drag force that suppresses tracer diffusion, although self-propulsion can accelerate tracers via active fluctuations. Similar effects are not understood in mixtures with particle conversion and exchange, although these are particularly relevant in biological contexts, where actively driven reactions prevail. By studying a thermodynamically consistent model of chemical reactions in mixtures, we show that reactions provide an additional relaxation pathway that suppresses interaction-induced memory, reducing the drag on tracers and restoring their diffusivity toward the value expected in the absence of solutes. Moreover, active reactions generate nonequilibrium fluctuations that can push tracer diffusivity beyond this limit, an effect we confirm with particle-based simulations. Our results identify chemical activity as a distinct route to controlling mass transport and offer a framework for interpreting microrheology experiments in chemically active mixtures.

A Devil's staircase is a sequence of transitions between topological sectors as a control parameter is varied. Such staircases are predicted in frustrated magnets and lattice gauge theories, but observing them is hard: the winding number that labels each sector is topologically protected, so local quantum dynamics alone cannot move the system from one step to the next. Here we propose and analyze an experimentally feasible way to realize a thermal quasi-Devil's staircase in a finite Rydberg-atom array. We study the anisotropic triangular-lattice Ising antiferromagnet using a dimer mapping, a directed-string description, exact thermodynamics, and Monte Carlo simulations. In the defect-free string manifold, the commensurate–incommensurate onset follows from a simple energy–entropy balance, and the exact Ising critical relation reduces to the same condition when triangle-rule defects are rare. On a finite lattice, this onset appears as a quasi-Devil's staircase of winding-sector crossovers, which we resolve through winding distributions, structure factors, and directional correlations. The key point is that thermal fluctuations overcome the topological barriers that block sector changes under purely quantum dynamics, and the staircase exists precisely at the finite sizes and tunable effective temperatures of current Rydberg platforms. This makes the effect directly observable in existing experiments. Our comparison also shows that the finite-size steps are not phase transitions: in the thermodynamic limit only a single phase transition survives, and the system crosses over to a defect-dominated paramagnet.

On the $\{5,4\}$ hyperbolic lattice, the outermost generation holds 73% of the sites at all system sizes, making macroscopic averages strictly boundary-dependent. We study the ferromagnetic Ising model on this geometry with free boundaries by Monte Carlo simulation, demonstrating that conventional observables cease to identify a single critical scale. Instead, the finite lattices are organized by two distinct temperature scales. The susceptibility maximum identifies the lower transition scale at $T_{c2}=1.4782(15)J/k_{B}$ without extrapolation. However, the order-parameter distribution lacks a conventional fixed point at this scale. A distinct pseudocritical scale emerges instead at $T^{*}\simeq1.67(3)$ inside the boundary-sensitive intermediate phase, where Binder cumulant curves cross and effective exponents are compatible with mean-field criticality. Between $T_{c2}$ and $T^{*}$, the interior amplifies boundary fluctuations into induced order. Furthermore, because the volume grows exponentially with depth, finite-size scaling must be formulated in the generation index rather than the total number of sites. The lack of asymptotic power-law convergence in the scaling stretch confirms through an independent observable that $T^{*}$ is a finite-size crossover rather than a thermodynamic fixed point. Finally, by applying a coherent field to the outermost generation alone, we recover the upper thermodynamic transition at $T_{pt}=2.81(1)J/k_{B}$, demonstrating that it remains accessible under an appropriate boundary perturbation.

Random matrix spectral statistics are widely used to diagnose quantum chaos in open systems, but whether chaotic eigenvalue correlations are accompanied by Haar-typical eigenoperators remains unclear. We study the operator-entanglement statistics of Liouvillian eigenoperators with near-maximal entanglement, analogous to states near the middle of the spectrum in Hamiltonian systems. Using the Kullback-Leibler divergence, we show that Haar-random statistics accurately describe both the Ginibre and class-AI$^\dagger$ Gaussian ensembles. For purely dissipative random Lindbladians, we find that locality of the jump operators drastically alters the eigenoperator-entanglement statistics. Nonlocal Lindbladians remain relatively close to Haar-random behavior, whereas local Lindbladians deviate increasingly from random-matrix statistics with system size. Moreover, unlike in Hamiltonian systems, the most highly entangled eigenoperators of local Lindbladians do not generally lie in the region of largest density of states, owing to the clustered structure of the complex eigenspectrum. We introduce an iterative $\sigma$-clipping scheme to extract the high-entanglement distribution without preselecting a spectral window, and find that it is Gaussian for nonlocal Lindbladians but log-normal for local ones. Remarkably, the same log-normal distribution, with no additional fitting, also describes dissipative mixed-field Ising chains belonging to distinct non-Hermitian symmetry classes. Our results therefore point to a universal eigenoperator-entanglement distribution for local Lindbladians that is not captured by generic non-Hermitian random-matrix ensembles.

Conventional centrality measures provide compact descriptions of node importance, but they emphasize specific structural relations and do not directly resolve the temporal and spectral organization of dynamical perturbation responses. Building on the Green function of a linearized networked system, we develop a multidimensional framework for dynamical node characterization. From the same response function, we extract three complementary indicators—Influence, Efficiency, and Distortion—that quantify cumulative response strength, temporal rate of response, and spectral concentration, respectively, together with a contribution matrix that resolves these quantities into source–target pathways. As a numerical demonstration, we apply the framework to weighted Kuramoto–Sakaguchi dynamics on heterogeneous Barabási–Albert networks. The three indicators distinguish implanted dynamical node classes through complementary dimensions of the perturbation response, while remaining strongly coupled to conventional topological centralities. These results demonstrate how a common dynamical response can be decomposed into distinct, physically interpretable characteristics rather than represented by a single aggregate descriptor. We finally discuss the scope and limitations of the linear-response assumption and extensions to time-dependent and nonlinear regimes.

We study the roles of broken spectra and exceptional points in finite-time driven non-Hermitian fermionic dynamics. We compute the Nambu biorthogonal correlation matrix for this purpose. The imbalanced-pairing Kitaev chain serves as our concrete realization. Transient passage through a broken-spectrum region amplifies preparation memory. The effect survives when the final Hamiltonian returns to a real-spectrum regime. Negative imbalance forces spectral nonpositivity in the static endpoint state. We isolate the genuine drive history by subtracting out this baseline, leaving an excess that is strictly controlled by the accumulated imaginary-energy action and persists over the entire post-ramp time window. A connected longitudinal correlation mirrors this physics. Its slow-ramp growth tracks the corresponding doubled action. Unstable sectors instead continue amplifying post-ramp if the drive halts inside the broken-spectrum region. Exceptional points yield distinct physics. The finite-time propagator and subsystem correlation matrix remain entirely regular near an exceptional endpoint, even as the quasiparticle gap exhibits its characteristic square-root closing. This finite-time regularity reflects the analyticity of the matrix evolution in the endpoint parameter; a diagonalizable endpoint is strictly not required. A diverging long-time crossover eventually reveals the exceptional scale. We halt the drive exactly at the exceptional point to find that the correlation projector and the connected longitudinal correlation share an identical ballisti front. The subsystem saturation length establishes a distinct but comparable spatial scale. Memory and exceptional-endpoint scalings show no divergence across the tested negative-imbalance range. Memory scaling is fixed by the drive and remains insensitive to the specific choice of real-spectrum final endpoint.

Jarzynski's equality recovers an equilibrium free-energy difference from nonequilibrium work trajectories, but its exponential average can converge very slowly because rare trajectories carry large weight. We show that coarse thermodynamic information measured at the nonequilibrium endpoint can reduce this sampling burden while keeping the same microscopic trajectories and the same free-energy target. The endpoint mean energy and a coarse spatial record define a maximum-entropy state and select one member of an exact family of fluctuation relations. That physically selected member provides an exact statistical control for the ordinary Jarzynski estimator, even when the final energy and retained record do not commute. In a finite Bose–Hubbard system, this control substantially lowers the finite-confidence sampling cost, while a small controlled departure from exact endpoint matching produces a square-root information–sampling tradeoff.

Self-similar dynamics are often described in terms of a single asymptotic scaling regime. Here we experimentally investigate the post-breakup recovery of a hydrodynamic interface over substantially wider temporal and spatial ranges than previously accessible. Beyond the previously identified scaling behavior, we find four successive self-similar regimes characterized by the $R_{\beta}$-scaling with $\beta=1/6$, 1/2, 1, and 2, connected by three distinct crossovers. All four regimes exhibit robust collapse near the interface tip, while geometrical dependence becomes detectable farther from the singular region. This systematic variation across space and scaling regimes is consistent with scale separation and enhanced universality closer to the breakup singularity. The characteristic scales associated with the different $R_{\beta}$-scalings become comparable at the common geometric scale $z_m \simeq R$. These findings demonstrate that self-similar dynamics can undergo multiple successive crossovers among distinct scaling regimes rather than directly approaching a single asymptotic self-similar state.

Fluctuation–response relations (FRRs) provide a fundamental relation between spontaneous fluctuations and the linear response to external perturbations, yet their validity in non-equilibrium systems remains an open problem. In a general class of noisy, finite-size, mean-field models of coupled phase oscillators, we reveal two types of FRRs in the incoherent phase below the synchronization transition. The first, corresponding to the infinite-size limit (I-FRR), involves the convolution between the external force and the correlation function in the time domain. The second, the finite-size counterpart (F-FRR), is governed by finite-size fluctuations and carries a correction factor: a spectrum function whose roots correspond to eigenvalues, or Landau poles, of the linear operator. The finite-size fluctuations are suppressed in the weak-coupling or strong-noise limits, and F-FRR reduces to I-FRR. Our analytical findings are verified by extensive numerical simulations.

Particle accelerators have carried fundamental physics to remarkable precision, but the Planck scale remains fifteen orders of magnitude beyond any foreseeable collider. We argue that this impasse is not a temporary shortfall but an insurmountable one. The way out runs through condensed matter physics: laboratory tests of generalized uncertainty relations, analog systems, and — most provocatively — frameworks in which spacetime itself is a condensate. We review the string-loop picture, in which quantization turns the world sheet into a three-dimensional fractal, the corpuscular description of black holes as self-sustained graviton condensates, and the phase-transition structure connecting the Riemannian phase to a pre-geometric one. The conclusion is deliberately provocative: the question of quantum gravity is no longer a question of energy, but of crossing the bridge.

A new simulation approach (J. Chem. Phys. 163, 164113), based on the seamless combination of the electron transfer at the electrode/electrolyte interface with the charge and mass transport in electrolyte solutions, holds the promise to advance the comprehensive modeling of basic electrochemical devices such as electrolytic cells and batteries. The underlying model also includes a fully consistent treatment of structural, thermodynamic and statistical properties of the electrical double-layer at non-ideal electron-conducting interfaces. As a result, subtle aspects such as Tafel and Butler-Volmer equations, the overpotential concept, as well as (stationary) non-equilibrium features such as the entropy production, arise naturally from the model, with a minimum of ad-hoc assumptions and elaborations of the simulation results. We illustrate these aspects for a simple model of electrochemical interface, and discuss further improvements of the method meant to enhance its ability to model systems and phenomena of interest for electrochemistry. One such improvement, concerning the effect of fluctuating electric fields on the electron transfer rate, is explicitly developed.

We introduce a generalized classical long-range stochastic cellular automaton that exactly captures the entanglement dynamics and transitions of a measurement-only monitored quantum system. The corresponding quantum model consists of a one-dimensional qubit chain subject to competing single-site and long-range Bell measurements, with separations drawn from a power-law distribution. At every step of each trajectory, the state remains a tensor product of Bell pairs and unpaired qubits, with bipartite entanglement entropy fully encoded in a classical matching of the sites. This exact representation allows us to solve the dynamics analytically in several limits and to determine the steady-state scaling of the entanglement entropy. We find volume-law, fractal, and area-law regimes. Numerical results show that these regimes persist beyond the analytically solvable limit. Because Bell measurement outcomes change only the Bell-state labels and leave the matching unchanged, all subsystem entropies are outcome independent, avoiding the exponential cost of trajectory postselection often encountered in experimental studies of measurement-induced entanglement dynamics.

We generalize the canonical Hopfield model of associative memory, which stores classical states of individual spins, to store entangled quantum states. The construction is based on truncated projector-sum Hamiltonian, and we explicitly consider a special case that stores dimer singlet coverings of a spin-1/2 system. We analyze the stability of the stored memories and find that the capacity – the number of stable patterns – scales faster with the system size than one would expect from a naive analogy with the standard Hopfield model, suggesting a possible quantum advantage.

We study the Bayesian monitoring of a single particle diffusing along a line, while an observer tracks it continuously using noisy measurements at each spatial point. The average Shannon entropy $\overline{S(t)}$ associated with the posterior distribution of the particle's position quantifies the uncertainty that remains after conditioning on the sequence of measurements. We map the problem to the calculation of the moments of the partition functions of directed polymers in $1 + 1$ dimensions. For a constant monitoring rate, the entropy remains bounded for any finite measurement intensity and can be quantified using exact results from the Kardar-Parisi-Zhang equation, providing both the saturation value and the asymptotic behavior for long times. For a monitoring intensity that decreases according to a power law $\sim t^{-\alpha}$, two distinct asymptotic regimes emerge: $\overline{S(t)} \simeq \tfrac12 \ln t$ for $\alpha > 1/2$ and $\overline{S(t)} \simeq \alpha \ln t$ for $0 < \alpha < 1/2$. Both results are obtained using the replica method, which maps the problem onto an attractive Lieb-Liniger Hamiltonian with a time-dependent coupling: the two regimes can be understood from a perturbative expansion around free diffusion and around the attractive Lieb-Liniger ground state, respectively. We discuss the limiting case $\alpha = 1/2$ and compare the predictions with simulations of discrete Gaussian and log-gamma polymer models.

Universal statistical laws are a hallmark of quantum chaos, usually associated with the loss of spatial structure. Here we show, theoretically and experimentally, that universality emerges already at shallow depths in chaotic quantum circuits. We derive universal distributions of output-probability fluctuations and test them on a programmable superconducting processor. Two-dimensional lattices, open one-dimensional chains, and periodic rings exhibit distinct universality classes, retaining signatures of geometry and boundary topology before full random-matrix behavior emerges. Within each class, the distributions depend on only two parameters: an effective many-body Thouless length and the accumulated number of incoherent errors. Extracting these directly from measured bitstring frequencies provides access to scrambling, entanglement growth of the target state, and global fidelity, without reconstructing or classically simulating the implemented circuit. We also identify a noise-driven transition between output statistics retaining system-wide quantum correlations and those reproducible by independent small patches. Our results establish finite-depth universality as a practical tool for probing deep properties of quantum processors.

Active matter, i.e., nonequilibrium systems that transform non-thermal energy from the environment into self-propulsion or other functional mechanisms, has attracted the attention of the statistical physics community in recent decades. Flocking, as shown by the aerial displays of starling flocks, is perhaps one of the most intriguing collective behaviors exhibited by active matter. While the bulk behavior of free collective motion is now fairly well understood, at least when the surrounding fluid can be neglected (the so-called dry approximation), much less is known when collective motion explicitly breaks a continuous rotational symmetry, either globally or locally. This thesis explores the effects of such explicit symmetry breaking on the dynamics of collective motion. Global symmetry breaking may arise from an anisotropic environment, where a favored direction sets the mean flocking direction. A key question addressed here is how to detect small anisotropies without prior knowledge of the underlying environmental asymmetry. The thesis then examines boundary-induced symmetry breaking. In confined flocking systems, local anisotropies arise at the boundaries and significantly affect both bulk and boundary behavior, especially in finite-sized setups. We focus in particular on a polar active fluid confined between two parallel repelling walls, showing that the influence of the boundaries extends far into the bulk. Finally, we consider a more subtle confinement inspired by the phototactic behavior of certain cyanobacteria, showing that key features of active matter, such as accumulation at boundaries, can arise even without mechanical confinement. The results shed light on how symmetry-breaking perturbations, whether imposed globally or locally, alter the dynamics of active matter systems and offer new insights into the control of collective motion.

We study the nonequilibrium dynamics of a minimal recurrent transformer with $N$ normalized tokens, $Q=K=I$, and a negative value map $V=-I$. Similarity-based attention selects nearby representations, while the negative value map drives tokens away from the selected field. This feedback can continually reorganize both the representation geometry and the attention network. For $d=2$, the tokens lie on a circle, where the regular polygon is an exact fixed point. As the attention feedback strength $\gamma$ is increased, the polygon loses stability through a flip bifurcation, giving rise to period-two motion, chaos, and cluster-exchange or cluster-flip states. Despite this temporal complexity, attention remains diffuse as $N\to\infty$ at finite fixed softmax sharpness $\beta$. Attention condensation instead emerges in the scaling regime $\beta\sim N^2$. In the hard-routing limit, repulsive updates amplify local perturbations and routing-partner switches transmit them ballistically, producing an emergent butterfly cone in representation space. High-dimensional geometry provides a distinct route to localization. For $d=N\to\infty$, simulations from Gaussian initial conditions provide evidence for a condensation transition at $\beta=O(1)$, driven by dynamically generated finite overlap gaps. Depending on $\gamma$, the resulting phases include diffuse simplex-like states, consensus flips, condensed active routing with signatures of chaos, and fragmented cluster flips. These results establish temporal activity, attention condensation, and geometric clustering as distinct collective phenomena, and show that sparse attention can sustain persistent dynamics rather than freeze it.