Extensive systems have a simple thermodynamic signature: the logarithm of the partition function scales homogeneously with the size of the system. We show that the failure of this scaling, measured by the replica energy ${\cal E}$, provides a useful bridge between statistical mechanics and quantum field theory. The associated differential operator $(1-\frac1d L\partial_L)$ removes the leading bulk contribution to $W=\log Z$ and isolates the part that is sensitive to boundaries, topology, defects, long-range forces, or other sources of nonadditivity. In quantum field theory this thermodynamic idea has two closely related uses. For ordinary finite-volume or spherical partition functions, suitable higher-order versions of the same filter remove local counterterms and extract universal fixed-point data such as the central charge, the sphere free energy $F$, and the Euler anomaly coefficient $a$. For replica geometries with entangling defects, the same filtering principle gives the renormalized defect free energy. In $2+1$ dimensions, its $n\to1$ limit is precisely the entropic $F$-function. We use this perspective to distinguish ordinary finite-size corrections, topology-dependent constants in gapped phases, subextensive fracton degeneracies, and genuinely nonextensive systems with long-range interactions such as self-gravitating thermal matter. Replica energy therefore offers a common thermodynamic language for additivity, defect free energies, and renormalization-group irreversibility.

Renormalization group (RG) fixed points are commonly associated with scale invariance and a divergent correlation length. We show that this connection can fail in the tricritical $O(N)$ model in three dimensions in the limit $N\to\infty$. Revisiting the line of fixed points identified by Bardeen, Moshe, and Bander, we use the functional renormalization group to clarify the mechanism leading to mass generation at its singular endpoint (the BMB fixed point). We demonstrate that the generated mass is nonuniversal and originates from the nonanalytic structure of the effective potential. We show that the critical exponent $\nu$ which takes the value $\nu = 1/2$ along the regular part of the BMB line, that is, for $0 \leq \lambda < \lambda_{\rm BMB}$, jumps to $\nu = 1/3$ on the singular part of this line with the BMB FP, corresponding to $\lambda = \lambda_{\rm BMB}$, being the pivotal point between these two regimes. We also show how its singular potential emerges dynamically along the renormalization flow.

We discuss diffusion-model paths interpolating between a target distribution known only through p patterns and a reference distribution that can be sampled. These interpolating paths can be constructed symmetrically or else in forward direction (often referred to as a "noising") from the target patterns to the reference distribution or in backward direction (as a "denoising") from the reference distribution to the patterns. For backward paths with identical diffusion noise, we consider the path convergence in number of patterns p towards the path for infinitely many patterns. In a one-dimensional test case, we show that this convergence is on a scale 1/sqrt(p), but with infinite mean square deviation. We demonstrate that the path convergence allows for extrapolation towards the p=infinity path which samples the target distribution. We provide a proof-of-concept extrapolation algorithm and propose the convergence and extrapolation of paths as a possible strategy for density estimation and generalization. We illustrate all our algorithms through pseudo-codes and provide Python implementations.

Quantum integrability in a system is characterized by a large number of conserved charges. However, finding the conserved charges in a generic quantum system is highly challenging. The integrable matrix theory provides a unified framework to obtain the conserved charges in a certain class of systems. We demonstrate this framework in case of a single-particle system on a 1D finite-sized lattice with inhomogeneous nearest neighbor hopping to study the statistical properties of the system across its chaotic--integrable crossover from the perspective of the conserved charges. The eigenspectrum of our random matrix model is studied first. We then estimate the conserved charges and find their properties throughout the chaotic to integrable transition of the system. We calculate the number of conserved charges across this crossover and observe that it is nearly equal to the size of the system at its integrable limit. Our result suggests that the number of conserved charges, estimated using the integrable matrix theory, can be a measure of quantum integrability.

The physics of active matter, wherein constituent particles consume energy to generate autonomous motion, has revolutionized non-equilibrium statistical mechanics. While a large body of work has successfully elucidated the behavior of dilute active systems, the dense regime -- characterized by ``active glasses and active solids'' -- presents profound challenges that defy conventional theoretical frameworks. Recent observations reveal two striking features in these dense systems: an apparent enhancement of Mermin-Wagner-Hohenberg (MWH) fluctuations leading to anomalous long-wavelength density fluctuations, and a remarkable correspondence between activity-induced annealing and annealing via oscillatory shear. In this perspective article, we propose a novel approach toward a deeper understanding of dense active matter: by developing active Hamiltonian models as equilibrium reference frameworks, we map out pathways toward non-equilibrium active systems. This strategy allows us to elucidate both the correspondence between driven and active systems and the enhanced MWH fluctuations, which likely arise from a strong coupling between spatially random active forces and long-wavelength density (phonon) modes. We outline a comprehensive roadmap employing complementary approaches, including the active Hamiltonian formalism, comparative studies of oscillatory shear in active and passive solids, and investigations of chiral active matter. Establishing this activity-oscillatory shear correspondence across diverse systems is essential to demonstrate its universality, reveal the underlying large-scale emergent physics, and place our hypothesis on a firmer theoretical ground.

We characterize the quasi-stationary distribution (QSD) of the bond directed-percolation line of the Domany--Kinzel automaton using a matrix-product-state representation of the probability distribution, obtained by projecting out the absorbing state and iterating the transfer matrix. Unlike moment- or sampling-based methods, this yields the full conditional distribution and direct access to information-theoretic diagnostics. The spatial structure of the QSD changes sharply across the transition: the active phase is bulk-like with finite density, whereas in the inactive phase the surviving activity collapses into a single flock occupying a vanishing fraction of the chain, with an internal filling that ranges from a single cluster deep in the inactive phase to a loose, partially filled group near criticality. This picture carries a sharp information-theoretic signature: throughout the inactive phase the bipartite mutual information of the QSD equals the entropy of a single binary choice -- whether the flock lies to the left or right of the cut -- so the surviving clusters together encode just one bit of positional information, corresponding to a single effective cluster. The approach extends matrix-product-state techniques to the projected eigenvector defining a QSD, opening information-theoretic diagnostics for absorbing-state systems that bulk-observable methods cannot reach.

Trace norms are fundamental to quantum information theory, yet in many-body systems their evaluation remains a major computational bottleneck, as it generally requires diagonalizing exponentially large operators. Here, we overcome this bottleneck by introducing a controlled tensor-network algorithm for estimating the trace norm of matrix product operators without full diagonalization. The key idea is to combine Zolotarev's rational approximation to the sign function with a variational formulation solved using a density-matrix-renormalization-group-like algorithm. The resulting approximation is systematically improvable, with its accuracy controlled by the rational approximation parameters and the spectral weight near zero. Beyond the reach of exact diagonalization, we demonstrate controlled trace-norm calculations for entanglement negativity, quantum fidelity and quantum Fisher information, achieving substantially improved accuracy over polynomial-based Lanczos approaches. Our results establish trace-norm-based quantities as practical tensor-network observables, opening a route toward tensor-network studies of quantum information in mixed states.

The stress concentrations caused by inherent roughness of natural and manufactured surfaces often induce plastic deformation at contact interfaces, a challenge compounded by competing influences of the size effect of plastic deformation and self-affine rough surface topography. To address this, we developed a novel methodology based on stochastic theory using compounded Chapman-Kolmogorov equations, for the first time, to solve elastoplastic contact problems involving scale-dependent hardness. Our approach formulates three integral equations describing the evolution of probability density functions of elastic contact pressure, relative plastic contact area, and relative non-contact area across geometrical scales. We thoroughly investigate the effects of scale-dependent hardness on contact pressure distribution, relative elastic and plastic contact areas, and the area-to-load relationship. By adjusting various mechanical and material properties, our model predicts a smooth transition from linear elasticity to elastic-plastic behavior and finally to full plasticity. A key advancement is the derivation of a new topographic yield parameter incorporating a wider range of material and geometrical properties, aiding identification of contact status. Numerical solutions enable highly precise determination of elastic and plastic limits via curve-fitting, and we also provide a new diagram for rapid identification of contact status. This study pioneers a stochastic process framework for applying the compounded Chapman-Kolmogorov equation to rough surface contact analysis, and the integral equations characterizing how interfacial properties evolve with scale could offer valuable insights for other multidisciplinary fields where multiscale roughness is critical, such as earthquakes, electrical contact, and contact electrification.

Participation entropy (PE) quantifies the spread of a many-body wavefunction across configuration space. While PE relaxes rapidly in generic chaotic systems, we show that $\mathrm{U}(1)$ conservation laws slow it down by imprinting with the slow hydrodynamic modes. Using a cluster expansion around equilibrium, we show that, after local density inhomogeneities decay, the leading PE deficit is dominated by squared connected density correlations. The long time relaxation is therefore controlled by diffusive correlation spreading, giving $\Delta S(t)\sim t^{-1/2}$ in the hydrodynamic regime and crossing over to $\sim \exp[-O(t/L^2)]$ when $t\geq L^2$. We confirm this entropy correlation relation using exact computation and infinite system tensor network simulations in various quantum $\mathrm{U}(1)$ conserving circuits. Our results establish PE as a sensitive probe of hydrodynamic memory and suggest that slow relaxation is a generic consequence of conservation laws.

Unitary time evolution of initially simple quantum many-body states rapidly generates entanglement and complex correlations, which limits direct numerical simulations. The late-time dynamics of physical observables, however, typically exhibits an effective simplicity in the form of hydrodynamics or kinetic theory. This leads to the question whether microscopic equations of motion can remain accurate and tractable up to long time scales by discarding irrelevant information in a controlled manner. Here, we introduce compressed minimum-purity time evolution (CoMPuTE) as an approach to keep track of a consistent set of reduced local density matrices, closing the hierarchical equations of motion using a minimum-purity principle. In benchmark applications we demonstrate (i) accurate description of energy diffusion in the one-dimensional mixed-field Ising model, (ii) the applicability to genuinely out-of-equilibrium Floquet dynamics starting from a pure state, and (iii) the limitations of the local reduced density matrix approximation when describing transport in the XXZ chain at $\Delta=1$ that is governed by increasingly non-local integrals of motion. The CoMPuTE method enhances computational efficiency in comparison to the closely related local-information time evolution algorithm, opening a possible route towards an extension to systems in higher spatial dimensions.

We study stochastic interacting particle systems whose interaction structure is described by dense weighted directed graphs converging to a graphon. In the thermodynamic limit, we prove a law of large numbers for the empirical measure process and derive a deterministic nonlinear master equation describing the macroscopic evolution. The limiting equation retains the heterogeneous interaction structure of the microscopic system through the limiting graphon, allowing for spatially non-homogeneous behaviors such as localized or community-type interactions.

Ideal Bose-Einstein condensation (BEC) remains a paradigmatic example of a continuous phase transition and a cornerstone for understanding quantum degenerate bosonic matter. We demonstrate that critical behavior of the ideal Bose gas near the BEC phase transition falls into three distinct classes, determined exclusively by the low-energy scaling of the density of states. Depending on its scaling exponent, which is controlled by dimensionality and confinement, the transition displays either the usual algebraic divergences of thermodynamic susceptibilities, divergent behavior with marginal logarithmic corrections, or a more subtle form of criticality, where only the correlation length diverges. Our work provides a unified framework for criticality in noninteracting bosonic systems. This classification applies broadly to atomic, photonic, polaritonic, and magnonic condensates, where dimensionality, confinement, and spectral engineering can strongly reshape the density of states.

Exact results using dual-unitarity largely rely on nearest-neighbor structures, while finite-range interactions typically lead to complications. Going beyond the usual nearest-neighbor setting, we introduce an analytically tractable family of finite-range kicked Ising models that admit exact closed-form entanglement dynamics. The construction is based on a staggered structure in which dual-unitarity is present on sublattices that are then coupled to each other. The central observation is that these inter-sublattice couplings do not obstruct the dual-unitarity of the resulting model. For the minimal interaction range of $r= 2$, we derive exact expressions for all the $n-$Rényi entanglement entropies at all times and show that the result is the sum of the two coupled sublattice contributions. Our framework extends naturally to larger finite interaction ranges and to systems with heterogeneous local Hilbert spaces, without additional assumptions. It thus provides a controlled setting for studying exact entanglement growth beyond strictly nearest-neighbor dual-unitary models.

We investigate the nonequilibrium dynamics of a one-dimensional interacting system across a transition from a charge-density-wave (CDW) phase to a symmetry-protected topological (SPT) phase. Starting from a CDW initial state, we study both sudden quenches and slow ramps into the SPT regime. While the CDW order melts under both protocols, the fate of topological order is sharply different. Following a sudden quench, long-range SPT order does not emerge because the post-quench state contains a finite density of excitations above the topological ground state. In contrast, slow ramps allow the system to follow the instantaneous ground state away from the critical region, enabling the buildup of SPT order with deviations governed by Kibble-Zurek defect production. The dynamics is solvable via a unitary mapping to a quadratic fermionic Hamiltonian, allowing us to compute the Loschmidt echo, correlation functions, and string correlator. The Loschmidt rate function exhibits cusps signaling dynamical quantum phase transitions, while the correlation dynamics reveal the contrasting mechanisms governing quenches and ramps across the transition. These results demonstrate that entering the topological regime is not sufficient for the emergence of topological order; the decisive factor is the suppression of excitation production during the evolution.

For groups without strict hierarchy, collective decisions often emerge through compromise. We develop a second-order network model of collective decision-making using a dissipative Hamiltonian formulation, in which informed agents introduce preferred directions while uninformed participants contribute only direction-free dissipation. We show that under low conflict, the model admits a locally unique, exponentially stable compromise state. Using a structured modular network we further show that as conflict increases the local compromise branch terminates through a saddle-node fold rather than through a smooth mean-field symmetry-breaking transition. Modular polarized states persist on branches that are locally separated from the compromise branch. Direction-free dissipation does not shift the static structural threshold, but it delays escape from the saddle-node ghost and pushes the observable onset of polarization to larger conflicts. Our work identifies a dissipation-mediated mechanism, complementary to connectivity-based accounts, through which uninformed participants stabilize collective behavior in biological and engineered swarms.

The Schramm-Loewner Evolution (SLE) describes a family of fractal curves that arise in the study of the scaling limits of many planar Statistical Physics models. These curves are modeled using the Loewner Differential Equation for the conformal maps $g_t(z)$ with a Brownian motion driver. Using Euler's Method, in the current work we performed numerical experiments to study at a fixed time the quantities $|g_t(z) - \overline{g_t(z)}|$ and $Re(g_t(z)) - Re(\overline{g_t(z)})$, where $Re$ denotes the real part and $\overline{g_t(z)}$ refers to the sample average. These random variables measure the 'spread' of the dynamics from the average behavior at fixed time. One of the scopes of this work is to give numerical predictions for future theoretical investigations on these quantities. When investigating these quantities in the SLE case our experiments predict that the distribution is bimodal when the dynamics started close to the origin, and it can become bell-shaped if the dynamics is started further from the origin. In the second part, we performed experiments for a Multiple SLE model whose driver is Dyson Brownian Motion. Due to singularity in the dynamics of the drivers and the many data points needed, this part is challenging from a computational perspective. In the multiple SLE case, our experiments predict that the distribution is bell-shaped in all cases. In addition, we check the changes in the distributions as we vary the parameter $\kappa$ in the SLE case and $\beta$ in the Multiple SLE case.

We establish a universal scaling law for collective charging of quantum batteries, independent of microscopic details. We prove that the ergotropy-to-energy ratio approaches unity at least as fast as $\sim N^{-1}$ with the number of batteries $N$, implying generic asymptotic freedom. We further show how the universal $1/N$ scaling can be overcome: when the battery state becomes asymptotically pure, the convergence can be substantially faster, including $\sim N^{-b}$ with $b>1$ and even exponential scaling in $N^2$. Rigorous finite-$N$ upper and lower bounds on the ergotropy-to-energy ratio are further derived, providing nonasymptotic guarantees for the universal $1/N$ scaling.

We consider the $SO(d)$ and $O(d)$ invariant sectors of the bosonic $d$-matrix harmonic oscillator with $U(N)$ gauge symmetry. The micro-canonical degeneracy $\mathcal{Z}( N, d, k )$ for fixed energy $k$ is expressed as a pairing between an $N$-dependent vector and a $d$-dependent vector in the space of partitions of the integer $k$. This pairing formula is derived by counting invariant words in multi-matrix variables $X^i_{j,a}$, using properties of Clebsch-Gordan multiplicities (Kronecker coefficients) for the symmetric group $S_k$, Schur-Weyl duality and harmonic analysis on the homogeneous space $U(d)/SO(d)$. Analytic formulae for large $N$ and $k$ with $ k \le N $ are obtained using group integrals over $U(N)$ and $SO(d)$ (or $ O(d)$). The micro-canonical heat capacity in this regime is negative and turns positive, at a critical value $k_{\rm crit}$, due to finite $N$ modifications to the counting, thus forming what we denote as a characteristic caloric fold in the $ E $ versus $T$ curve. Data from the pairing formula is well fitted by $k_{\rm crit} \sim { N^2 \over 4 }$ for small values of $d$. A derivation of this large $N$ formula is given using a matrix model approximation and semi-classical analysis of the eigenvalue density. The large $N,d$ limit of the degeneracies reveals a key role for ribbon graph combinatorics. The caloric fold is also notably a property of black hole thermodynamics in anti-de-Sitter spaces. We propose the spherically symmetric \(SO(d)\) and \(O(d)\) invariant sectors of \(d\)-matrix quantum mechanics as tractable matrix systems for capturing key features of dual descriptions of black-hole thermodynamics.

We develop a systematic theory of spectral decimation for quantum many-body Hamiltonians and show that it provides a quantitative probe of emergent symmetries in statistically mixed spectra. Building on an analytical description of statistical mixtures, we derive an explicit expression for the size of a characteristic symmetry sector (CSS), defined as the largest subsequence of levels exhibiting non-Poissonian correlations. The CSS dimension is shown to be the size-biased average of the underlying symmetry sectors, establishing a direct link between spectral statistics and Hilbert-space structure. We apply this framework to two paradigmatic settings: Hilbert-space fragmentation and disorder-induced many-body localization (MBL). In fragmented systems, the CSS reproduces the mixture prediction and isolates correlated subsectors even when the full spectrum appears nearly Poissonian. In the disordered Heisenberg chain, spectral decimation reveals the gradual emergence of integrability through a shrinking CSS, whose statistics exhibit signatures consistent with local integrals of motion. We introduce a characteristic symmetry entropy (CSE) as a finite-size scaling observable and extract, within accessible system sizes, the crossover exponents. Our results establish spectral decimation as a controlled, unbiased and computationally inexpensive diagnostic of hidden structure in many-body spectra, capable of distinguishing between chaotic dynamics, statistical mixtures, and emergent integrability.

While energy scale competition dictates equilibrium phase behavior, kinetic barriers often drive soft-matter mixtures into dynamically arrested states, rendering conventional phase diagrams incomplete. We extend the classification of binary systems into regions of thermodynamic instability to explore the interplay between phase separation and dynamical arrest. Strong cross-attractions kinetically suppress demixing, whereas competitive energy scales yield either condensation- or demixing-driven amorphous states. We show that this morphological crossover is parameterized by a structural order parameter, $\chi$, providing a unified non-equilibrium framework that reconciles theoretical predictions with experimentally observed arrested mixtures.

We introduce a non-equilibrium version of the Caldeira-Leggett model in which a quantum particle is strongly coupled to a set of engineered reservoirs. The reservoirs are composed by collections of squeezed and displaced thermal modes, in contrast to the standard case in which the modes are assumed to be at equilibrium. The model proves to be very versatile. Strongly displaced/squeezed reservoirs can be used to generate an effective time dependence in the system Hamiltonian and can be identified as sources of pure work. In the case of squeezing, the time dependence is stochastic and breaks the fluctuation-dissipation relation, this can be reconciled with the second law of thermodynamics by correctly accounting for the energy used to generate the initial non-equilibrium conditions. To go beyond the average description and compute the full heat statistics, we treat squeezing and displacement as generalized Hamiltonians on a modified Keldysh contour. As an application of this technique, we show the quantum-classical correspondence between the heat statistics in the non-equilibrium Caldeira-Leggett model and the statistics of a classical Langevin particle under the action of squeezed and displaced colored noises. Finally, we discuss thermodynamic symmetries of the heat generating function, proving a fluctuation theorem for the energy balance and showing that the conservation of energy at the trajectory level emerges in the classical limit.

As a powerful theoretical construct, the entanglement Hamiltonian (EH) encapsulates the essential entanglement properties of a quantum many-body system. From the EH, one can extract a variety of entanglement quantities, such as entanglement entropies, negativity, and the entanglement spectrum. However, its general analytical form remains largely unknown. While the Bisognano-Wichmann theorem gives an exact EH form for Lorentz-invariant field theories, its validity on lattice systems is limited, especially when Lorentz invariance is absent. In this work, we propose a general scheme based on the lattice-Bisognano-Wichmann (LBW) ansatz and multi-replica-trick quantum Monte Carlo methods to numerically reconstruct the entanglement Hamiltonian in two-dimensional systems and systematically explore its applicability to systems without translational invariance, going beyond the original scope of the primordial Bisognano-Wichmann theorem. Various quantum phases--including gapped and gapless phases, critical points, and phases with either discrete or continuous symmetry breaking--are investigated, demonstrating the versatility of our method in reconstructing entanglement Hamiltonians. Furthermore, we find that when the entanglement boundary of a system is ordinary (i.e., free from surface anomalies), the LBW ansatz provides an accurate approximation well beyond Lorentz-invariant cases. Our work thus establishes a general framework for investigating the analytical structure of entanglement in the complex quantum many-body systems.

A binary fluid mixture in contact with lateral particle reservoirs is considered. By imposing different particle concentrations in these reservoirs, the system can be maintained under controlled non-equilibrium conditions. Previous stochastic approaches have revealed an unexpected property of the system's state trajectory, namely that it remains time-reversible even when the system is driven out of equilibrium. In the absence of relevant experimental evidence, we employ microscopic molecular dynamics simulations to assess the validity of this surprising result. Remarkably, the simulation results unambiguously confirm the prediction of the stochastic analysis.

One-dimensional integrable fermions can be classified into massless and massive regimes, and the $R$-operator for the latter can be constructed from that of the former. Here, I define integrable massless fermions by the simultaneous satisfaction of the Yang-Baxter equation (YBE) and Shastry's decorated YBE (DYBE) by the $R$-matrix. This notion is strictly more general than Maassarani's `free-fermion algebra', yet more restrictive than the notion of free fermions in exactly solvable quantum models or in integrable two-dimensional classical vertex models dual to quantum spin chains. Within this framework, there emerge two archetypal mechanisms for opening a spectral gap and generating massive fermions: (i) breaking time-reversal symmetry by coupling to external field, and (ii) introducing time-reversal symmetric interactions. These paradigms are realized, respectively, in the XY chain in a longitudinal field and in the Hubbard model, both of which possess non-relativistic, bivariate $R$-matrices. Integrability conditions on local Hamiltonians for both massless and massive fermions are identified, and schematic procedures for uniquely determining their $R$-matrices are proposed.

A quantum mechanical setting consisting of a frozen qubit composed with a fully thermalized chaotic system of N states is proposed, with potential relevance to quantum control. Observing the states of the composed system selectively retaining the states leads to the observation of novel localization in the subsystem. At a tuning parameter of 1.0, implying no selection, the system exhibits Wigner-Dyson level spacing statistics, indicative of quantum chaos. As the tuning parameter is reduced and selection occurs at a cutoff, the nearest-neighbor level spacing distribution develops heavier tails, a signature of suppressed spectral mixing and the emergence of non-thermal dynamics. In these regimes, the eigendensity develops a pronounced "cat-ears" structure, reflecting the formation of spatially localized bimodal eigenstates. These topological features persist without transitioning to Poisson statistics, indicating a transition from quantum chaos to a non-thermal, novel many-body localized (MBL) regime-referred to as Wigner Cat Phases. The proposed mixed random matrix ensemble offers a practical probe for sustaining this novel quantum localization setting. Results from our rigorous spectral statistics analysis show how "cat-ears" form in spectral densities based on the degree of selection or disorder and indicate that gap ratio statistics must be used with caution in detecting the full integrable limit due to the possibility of heavy-tailed Wigner-Dyson distributions.

Active adaptive matter has attracted considerable interest due to its rich, largely unexplained dynamics and its relevance to a wide range of synthetic and biological materials. An important subclass of such systems consists of active particles that can remodel the network in which they move. Here, we introduce a minimal yet versatile model of active particles moving on an adjustable network. In this model, particles undergo discrete run-and-tumble motion along the links of a triangular lattice and leave behind a trail of temporarily blocked links. These closed links cannot be traversed by other particles and reopen only after a characteristic healing time. The resulting trail-mediated blocking mechanism is fundamentally distinct from more familiar interactions such as excluded-volume effects. In the high-persistence limit, we find a qualitative contrast between the two mechanisms: while steric blocking leads to reduced diffusivity with increasing persistence, trail-induced blocking causes diffusivity to increase monotonically. We characterize this fundamental difference and the associated, unexpected transport properties, and discuss potential applications of our findings.

In this paper we introduce an electron ptychography reconstruction framework, CAVIAR -- Correlated Atomic Vibration Imaging with sub-Angstrom Resolution -- that reveals an entirely new channel of information: spatial correlations in atomic displacements at the atomic scale. We show reconstructions of a symmetric $Σ$9 grain boundary in silicon from realistically simulated data and experimental data for hexagonal boron nitride. By reconstructing the object as an ensemble of multiple states we are able to observe correlations between movements of atoms in the range of 10-20 pm at room temperature in agreement with the expectation. Moreover, using only the masses of the atomic species and the temperature as input, we obtain average frequencies of $10.8\pm0.1$, $13.6\pm0.6$, $18.0\pm0.2$, $25.5\pm1.5$ THz for the longitudinal and transversal acoustic and optic phonons, respectively, in agreement with inelastic neutron scattering, albeit from just a few nm$^3$ volume. This ability to spatially resolve correlated atomic motion makes CAVIAR a unique tool to explore atom dynamics at the finest scale with the potential to be instrumental in the development of phononic devices, in studying phonon-based decoherence in quantum systems, or other emerging phonon-based applications.

Since the advent of quantum mechanics, classical probability interpretations have faced significant challenges. A notable issue arises with the emergence of negative probabilities when attempting to define the joint probability of non-commutative observables. In this work, we propose a resolution to this dilemma for quantum spin chains, by introducing an exact representation of their dynamics in terms of classical continuous-time Markov chains (CTMCs). These CTMCs effectively model the creation, annihilation, and propagation of pairs of classical particles and antiparticles. The quantum dynamics then emerges by averaging over various realizations of this classical process.

We investigate 1D and 2D cluster states under local decoherence to assess the robustness of their mixed-state subsystem symmetry-protected topological (SSPT) order. By exactly computing fidelity correlators via dimensional reduction of effective statistical mechanics models, we pinpoint the critical error rate for strong-to-weak spontaneous breaking of strong subsystem symmetry. Without resorting to the replica trick, we demonstrate that mixed-state SSPT order remains remarkably robust up to the maximal decoherence rate when noise respects strong subsystem symmetry. Furthermore, we propose that the mixed-state SSPT order can be detected by a constant correction to the area-law scaling of entanglement negativity, termed spurious topological entanglement negativity. This also highlights that topological entanglement negativity, a widely used diagnostic for mixed-state topological order, is generally not invariant under finite-depth quantum channels.

Mixed-state phases of matter under local decoherence have recently garnered significant attention due to the ubiquitous presence of noise in current quantum processors. One of the key issues is understanding how topological quantum memory is affected by realistic coherent noise, such as random rotation noise and amplitude-damping noise. In this work, we investigate the intrinsic error threshold of the two-dimensional toric code (TC), a paradigmatic topological quantum memory, under these types of coherent noise by employing both analytical and numerical methods based on the doubled-Hilbert-space formalism. A connection between the mixed-state phase of the decohered TC and a non-Hermitian Ashkin-Teller-type statistical-mechanics model is established, and the mixed-state phase diagrams under the coherent noise are obtained. We find remarkable stability of mixed-state topological order under random rotation noise with axes near the $Y$-axis of qubits. We also identify intriguing extended critical regions at the phase boundaries, highlighting a connection with non-Hermitian physics. We argue that these phase boundaries provide upper bounds for the intrinsic error threshold, beyond which quantum error correction becomes impossible. We complement these findings by estimating the error thresholds for random rotation noise under standard quantum error correction, thereby providing lower bounds on the intrinsic error threshold.