This is a concise, pedagogical introduction to the dynamic field of open quantum systems governed by Markovian master equations. We focus on the mathematical and physical origins of the Lindblad equation, its unraveling in terms of pure-state trajectories, the structure of steady states with emphasis on the role of symmetry and conservation laws, and a sampling of the novel physical phenomena that arise from nonunitary dynamics (dissipation and measurements). This is far from a comprehensive summary of the field. Rather, the objective is to provide a conceptual foundation and physically illuminating examples that are useful to graduate students and researchers entering this subject. There are exercise problems and references for further reading throughout the notes.

We present a machine learning (ML) force-field framework for simulating the non-equilibrium dynamics of charge-density-wave (CDW) order driven by the Peierls instability. Since the Peierls distortion arises from the coupling between lattice displacements and itinerant electrons, evaluating the adiabatic forces during time evolution is computationally intensive, particularly for large systems. To overcome this bottleneck, we develop a generalized Behler-Parrinello neural-network architecture -- originally formulated for ab initio molecular dynamics -- to accurately and efficiently predict forces from local structural environments. Using the locality of electronic responses, the resulting ML force field achieves linear scaling efficiency while maintaining quantitative accuracy. Large-scale dynamical simulations using this framework uncover a two-stage coarsening behavior of CDW domains: an early-time regime characterized by a power-law growth $L \sim t^{\alpha}$ with an effective exponent $\alpha \approx 0.7$, followed by a crossover to the Allen-Cahn scaling $L \sim \sqrt{t}$ at late times. The enhanced early-time coarsening is attributed to anisotropic domain-wall motion arising from electron-mediated directional interactions. This work demonstrates the promise of ML-based force fields for multiscale dynamical modeling of condensed-matter lattice models.

The following model is studied analytically and numerically: point particles with masses $m,\mu,m, \dots$ ($m\geq\mu$) are distributed over the positive half-axis. Their dynamics is initiated by giving a positive velocity to the particle located at the origin; in its course the particles undergo elastic collisions. We show that, for certain values of $m/\mu$, starting from the initial state where the particles are equidistant the system evolves in a hydrodynamic way: (i) the rightmost particle (blast front) moves as $t^{\delta}$ with \delta < 1; (ii) recoiled particles behind the front enter the negative half-axis; (iii) the splatter -- the particles with locations $x\leq 0$ -- moves in the ballistic way and eventually takes over the whole energy of the system. These results agree with those obtained in S. Chakraborti et al, SciPost Phys. 2022, 13, 074, for $m/\mu=2$ and random initial particle positions. At the same time, we explicitly found the collection of positive numbers $\{\mathcal{M}_i, i \in \mathbf{N} \}$ such that, for $m/\mu = \mathcal{M}_i$, $i\leq 700$, the following holds: (a) the splatter is absent; (b) the number of simultaneously moving particles is at most three; (c) the blast front moves in the ballistic way. However, if, similarly as in S. Chakraborti et al, the particle positions are sampled from a uniformly distributed ensemble, for $m/\mu = \mathcal{M}_i$ the system evolves in a hydrodynamic way.

Drawing the quantum phase diagram of a many-body system in the parameter space of its Hamiltonian can be seen as a learning problem, which implies labelling the corresponding ground states according to some classification criterium that defines the phases. In this work we adopt unsupervised learning, where the algorithm has no access to any priorly labeled states, as a tool for determining quantum phase diagrams of many-body systems. The algorithm directly works with quantum states: given the ground-state configurations for different values of the Hamiltonian parameters, the process uncovers the most significant way of grouping them based on a similarity criterion that refers to the fidelity between quantum states, that can be easily estimated, even experimentally. We benchmark our method with two specific spin-$\frac{1}{2}$ chains, with states determined via tensor network techniques. We find that unsupervised learning algorithms based on spectral clustering, combined with ``silhouette'' and ``elbow'' methods for determining the optimal number of phases, can accurately reproduce the phase diagrams. Our results show how unsupervised learning can autonomously recognize and possibly unveil novel phases of quantum matter.

We investigate the adiabatic elimination of fast variables in relativistic stochastic mechanics, which is analyzed in the equation of motion and in the distribution function, with relativistic corrections explicitly derived. A new dimensionless parameter is introduced to characterize the timescale. The adiabatic elimination is also compared with the path integral coarse graining, which is more general yet computationally demanding.

Synchronization in quantum systems has been recently studied through persistent oscillations of local observables, which stem from undamped modes of the dissipative dynamics. However, the existence of such modes requires fine-tuning the system to satisfy specific symmetry constraints. We investigate the response of spin systems that possess such oscillating modes to generic, weak perturbations. We show that even when these perturbations break the symmetry and lead to a single steady state, the phase correlations in the resulting state exhibit signatures of synchronization. Our results therefore connect the persistent oscillation notion (dynamical) and the notion based on phase correlations (steady-state) of synchronization, which so far have been regarded as distinct phenomena. Furthermore, we demonstrate that steady-state synchronization in these systems can exhibit features that are absent in the dynamical synchronization. Our work suggests robustness of synchronization and points toward a potential unifying framework of quantum synchronization.

We extensively study the phenomenology of one dimensional Nonreciprocal Cahn Hilliard model for varying nonreciprocity $(\alpha)$ and different boundary conditions. At small $\alpha$, a perturbed uniform state evolves to a defect laden configuration that lacks global polar order. Defects are the sources and sinks of travelling waves and nonreciprocity selects defects with a unique wave number that increases monotonically with $\alpha_c$. A critical threshold $\alpha_c$ marks the onset of a transition to states with finite global polar order. For periodic boundaries, above $\alpha_c$, the system shows travelling waves that are completely ordered. In contrast, travelling waves are incompatible with Dirichlet and Neumann boundaries. Instead, for $\alpha \gtrsim \alpha_c$, we find fluctuating domains that show intermittent polar order and at large $\alpha$, the system partitions into two domains with opposite polar order.

We study modular theory in hyperfinite von Neumann algebras, i.e. in those of type II or type III, from the viewpoint of a subregion charge sector decomposition. We address this symmetry resolution by considering infinite tensor products of finite-dimensional algebras with fixed subregion charge values. An important ingredient is the combination of these algebras using direct integrals. This allows us to obtain the symmetry-resolved modular operator, modular flow, and modular correlation functions for hyperfinite algebras. Our approach establishes a mathematical foundation for recent results on symmetry resolution and modular theory in conformal field theory. Our analysis applies both to charges defined on a continuous range, or on a discrete set. The latter is of interest for condensed matter theory. Moreover, within the AdS/CFT correspondence we expect our findings to be relevant as a new ingredient for bulk spacetime reconstruction, including information from different boundary charge sectors.

A key challenge in quantum complexity is how entanglement structure emerges from dynamics, highlighted by advances in simulators and information processing. The Lieb--Robinson bound sets a locality-based speed limit on information propagation, while the Small-Incremental-Entangling (SIE) theorem gives a universal constraint on entanglement growth. Yet, SIE bounds only total entanglement, leaving open the fine entanglement structure. In this work, we introduce Spectral-Entangling Strength, measuring the structural entangling power of an operator, and prove a Spectral SIE theorem: a universal limit for Rényi entanglement growth at $\alpha \ge 1/2$, revealing a robust $1/s^2$ tail in the entanglement spectrum. At $\alpha=1/2$ the bound is qualitatively and quantitatively optimal, identifying the universal threshold beyond which growth is unbounded. This exposes the detailed structure of Schmidt coefficients, enabling rigorous truncation-based error control and linking entanglement to computational complexity. Our framework further establishes a generalized entanglement area law under adiabatic paths, extending a central principle of many-body physics to general interactions. Practically, we show that 1D long-range interacting systems admit polynomial bond-dimension approximations for ground, time-evolved, and thermal states. This closes the quasi-polynomial gap and proves such systems are simulable with polynomial complexity comparable to short-range models. By controlling Rényi entanglement, we also derive the first rigorous precision-guarantee bound for the time-dependent density-matrix-renormalization-group algorithm. Overall, our results extend SIE and provide a unified framework that reveals the detailed structure of quantum complexity.

Synchronization in various complex networks is significantly influenced by higher-order interactions combined with non-Gaussian stochastic perturbations, yet their mechanisms remain mainly unclear. In this paper, we systematically investigate the synchronization and spike dynamics in a higher-order Kuramoto model subjected to non-Gaussian Lévy noise. Using the mean order parameter, mean first-passage time, and basin stability, we identify clear boundaries distingusihing synchronization and incoherent states under different stability indexes and scale parameters of Lévy noise. For fixed coupling parameters, synchronization weakens as the stability index decreases, and even completely disappears when the scale parameter exceeds a critical threshold. By varying coupling parameters, we find significant dynamical phenomena including bifurcations and hysteresis. Lévy noise smooths the synchronization transitions and requires stronger coupling compared to Gaussian white noise. Furthermore, we define spikes and systematically investigate their statistical and spectral characteristics. The maximum number of spikes is observed at small scale parameter. A generalized spectral analysis further reveals burst-like structure via an edit distance algorithm. Moreover, a power-law distribution is observed in the large inter-window intervals of spikes, showing great memory effects. These findings deepen the understanding of synchronization and extreme events in complex networks driven by non-Gaussian Lévy noise.

Performing thermodynamic tasks within finite time while minimizing thermodynamic costs is a central challenge in stochastic thermodynamics. Here, we develop a unified framework for optimizing the thermodynamic cost of performing various tasks in finite time for overdamped Langevin systems. Conventional optimization of thermodynamic cost based on optimal transport theory leaves room for varying the final distributions according to the intended task, enabling further optimization. Taking advantage of this freedom, we use Lagrange multipliers to derive the optimal final distribution that minimizes the thermodynamic cost. Our framework applies to a wide range of thermodynamic tasks, including particle transport, thermal squeezing, and information processing such as information erasure, measurement, and feedback. Our results are expected to provide design principles for information-processing devices and thermodynamic machines that operate at high speed with low energetic costs.

Radical pairs (also known as spin qubit pairs, electron-hole pairs) are transient reaction intermediates that are found and utilised in all areas of science. Radical pair spin dynamics simulations including all nuclear spins have been a computational barrier due to exponential scaling memory requirements. We address this issue with a tensor network method for accurately simulating the full open quantum dynamics of radical pair systems, explicitly accounting for hyperfine interactions with up to 30 nuclear spins with additional benchmarking including 60 nuclei. By employing the matrix product state (MPS) and matrix product density operator (MPDO) representations, we mitigate the exponential scaling of Hilbert and Liouville spaces typically encountered in full quantum non-Markovian treatments. We demonstrate the power of these methods with biologically relevant flavin-tryptophan radical pair systems, where we investigate electron hopping processes between multiple radical pairs using Lindblad jump operators. These simulations precisely capture anisotropic spin dynamics, clearly identifying orientational dependence of the magnetic field, which enhances or diminishes the spin-selective product yield. These directional sensitivities highlight the critical dependence of the nuclear environment and underscore the necessity of fully quantum treatments in spin biophysics, offering critical insights into avian magnetoreception mechanisms. This work provides a robust computational framework applicable to a broad range of scientific realms, which include spin chemistry, quantum biology, and spintronics.

We introduce families of classical stochastic dynamics in two and higher dimensions which stabilize order in the absence of any symmetry. Our dynamics are qualitatively distinct from Toom's rule, and have the unusual feature of being fluctuation-stabilized: their order becomes increasingly fragile in larger dimensions. One of our models maintains an ordered phase only in two dimensions. The phase transitions that occur as the order is lost realize new dynamical universality classes which are fundamentally non-equilibrium in character.

Understanding the timescales associated with relaxation to equilibrium in closed quantum many-body systems is one of the central focuses in the study of their non-equilibrium dynamics. At late times, these relaxation processes exhibit universal behavior, emerging from the inherent randomness of chaotic Hamiltonians. In this work, we investigate a disordered spin-$1/2$ XX ladder - an experimentally realizable model known for its diffusive dynamics - to explore the connection between transport properties and spectral measures derived solely from the system's energy levels via these relaxation timescales. We begin by analyzing the spectral form factor, which yields the time when the system begins to follow the random matrix theory (RMT) statistics, known as the RMT time. We then determine the Thouless times - the average times for a local excitation to diffuse across the entire finite system - through the linear-response theory for both spin and energy transport. Our numerical results confirm that the RMT time scales quadratically with system size and upper bounds the Thouless times. Interestingly, we also find that, unlike other non-integrable models, spin diffusion proceeds faster than energy diffusion.

The two-star random graph is the simplest exponential random graph model with nontrivial interactions between the graph edges. We propose a set of auxiliary variables that control the thermodynamic limit where the number of vertices N tends to infinity. Such 'master variables' are usually highly desirable in treatments of 'large N' statistical field theory problems. For the dense regime when a finite fraction of all possible edges are filled, this construction recovers the mean field solution of Park and Newman, but with an explicit control over the 1/N corrections. We use this advantage to compute the first subleading correction to the Park-Newman result, which encodes the finite, nonextensive contribution to the free energy. For the sparse regime with a finite mean degree, we obtain a very compact derivation of the Annibale-Courtney solution, originally developed with the use of functional integrals, which is comfortably bypassed in our treatment.

In this work, we investigate the impact of conserved charges on the dynamics of spread complexity of quantum states. Building on the notion of symmetry-resolved Krylov complexity [1], we extend the framework to general quantum states and analyze the relation between the total spread complexity and its decomposition into fixed-charge sectors. After exploring a range of analytical examples and using orthogonal polynomial approach, we identify conditions under which spread complexity exhibits equipartition across sectors. Finally, we discuss quantum speed limits that constrain the growth of complexity in the presence of conserved charges.

Lossy data transformations by definition lose information. Yet, in modern machine learning, methods like data pruning and lossy data augmentation can help improve generalization performance. We study this paradox using a solvable model of high-dimensional, ridge-regularized linear regression under 'data coarse graining.' Inspired by the renormalization group in statistical physics, we analyze coarse-graining schemes that systematically discard features based on their relevance to the learning task. Our results reveal a nonmonotonic dependence of the prediction risk on the degree of coarse graining. A 'high-pass' scheme--which filters out less relevant, lower-signal features--can help models generalize better. By contrast, a 'low-pass' scheme that integrates out more relevant, higher-signal features is purely detrimental. Crucially, using optimal regularization, we demonstrate that this nonmonotonicity is a distinct effect of data coarse graining and not an artifact of double descent. Our framework offers a clear, analytical explanation for why careful data augmentation works: it strips away less relevant degrees of freedom and isolates more predictive signals. Our results highlight a complex, nonmonotonic risk landscape shaped by the structure of the data, and illustrate how ideas from statistical physics provide a principled lens for understanding modern machine learning phenomena.