Macroscopic Quantum Chaos from Topological Degradation We present numerical evidence that macroscopic quantum chaos can emerge without nonlinear dynamics in the evolution equation. Instead, it arises as a continuous phase transition driven solely by the topological degradation of the underlying spatial substrate. ◆ Key Features • 2D toroidal mesh: N = L × L sites (L = 20–50). • Tight-binding Hamiltonian with uncorrelated Anderson disorder (W as the primary control parameter). • Independent 10% bond-percolation channel confirming that the observed behavior is a generic consequence of connectivity loss. • Exact diagonalization on mobile ARM (Snapdragon 8 Gen 2) hardware without spectral unfolding: ~25–75 s per 50-realization ensemble, achieving ~25× speedup over conventional HPC workflows. ◆ Main Results • Perfect torus exhibits symmetry-induced spectral degeneracy, numerically close to Poisson statistics. • Moderate disorder (W ≈ 1–4) yields ⟨r⟩ = 0.530–0.534, matching GOE Wigner–Dyson statistics. • Strong disorder (W ≳ 8) restores genuine Poisson statistics (⟨r⟩ ≈ 0.39). • Critical regime (W ≈ 4–6): strongly multifractal eigenstates, monotonic decay of generalized dimensions Dq, and f(α) peaking near α ≈ 2.2–2.3. • Finite-size scaling confirms Anderson-transition-like criticality. ◆ Implications These results show that Wigner–Dyson level repulsion and algorithmic chaos are thermodynamic consequences of spatial frustration. The critical threshold (W ≈ 4–6) defines a quantitative failure boundary for distributed tensor-processing interconnects, while the unfolding-free methodology enables rapid, on-device spectral diagnostics for next-generation fault-tolerant AI accelerators.
Macroscopic Quantum Chaos from Topological Degradation We present numerical evidence that macroscopic quantum chaos can emerge without nonlinear dynamics in the evolution equation. Instead, it arises as a continuous phase transition driven solely by the topological degradation of the underlying spatial substrate. ◆ Key Features • 2D toroidal mesh: N = L × L sites (L = 20–50). • Tight-binding Hamiltonian with uncorrelated Anderson disorder (W as the primary control parameter). • Independent 10% bond-percolation channel confirming that the observed behavior is a generic consequence of connectivity loss. • Exact diagonalization on mobile ARM (Snapdragon 8 Gen 2) hardware without spectral unfolding: ~25–75 s per 50-realization ensemble, achieving ~25× speedup over conventional HPC workflows. ◆ Main Results • Perfect torus exhibits symmetry-induced spectral degeneracy, numerically close to Poisson statistics. • Moderate disorder (W ≈ 1–4) yields ⟨r⟩ = 0.530–0.534, matching GOE Wigner–Dyson statistics. • Strong disorder (W ≳ 8) restores genuine Poisson statistics (⟨r⟩ ≈ 0.39). • Critical regime (W ≈ 4–6): strongly multifractal eigenstates, monotonic decay of generalized dimensions Dq, and f(α) peaking near α ≈ 2.2–2.3. • Finite-size scaling confirms Anderson-transition-like criticality. ◆ Implications These results show that Wigner–Dyson level repulsion and algorithmic chaos are thermodynamic consequences of spatial frustration. The critical threshold (W ≈ 4–6) defines a quantitative failure boundary for distributed tensor-processing interconnects, while the unfolding-free methodology enables rapid, on-device spectral diagnostics for next-generation fault-tolerant AI accelerators.
Topological protection against many‑body quantum fluctuations remains an open fundamental question. Using exact diagonalization on a 6×6 periodic lattice, we investigate the robustness of topological vortex charge across the onset of many‑body quantum chaos for soft‑core Bose‑Hubbard and spinless fermion t‑V models. We track the central‑plaquette winding number over a wide interaction range U/t∈[0,15], covering the double‑N cusp catastrophe featured in level‑spacing ratio ⟨r⟩. Even deep inside chaotic parameter windows where ⟨r⟩ collapses far toward the Poisson limit, the ground‑state winding number stays numerically stable at 1.0000, showing decoupling between topological invariant and many‑body quantum chaos. Three distinct magnetic‑flux values Φ=Φ0,Φ0/4,Φ0/3 are compared. The observed number of chaos‑collapse valleys matches the number of resolvable Hofstadter subband gaps, furnishing exact‑diagonalization numerical evidence for the transition‑hierarchy conjecture. Further, we identify a finite‑size threshold for chaos activation controlled by Hilbert‑space subspace dimension. Our results demonstrate that topological vortex invariants can persist unperturbed through strong many‑body quantum‑chaos regimes, establishing that quantum chaos does not automatically destroy ground‑state topological quantization.
In the framework of Unified Time Scale, Matrix Universe THE\text{-MATRIX}, and Quantum Discrete Cellular Automaton Universe U_{qca}, we construct a systematic theory for Quantum Chaos and the Eigenstate Thermalization Hypothesis (ETH). The Unified Time Scale Mother Formula equation \kappa(\omega) =\varphi'(\omega)/\pi =\rho_{rel}(\omega) =(2\pi)^{-1}trQ(\omega) equation unifies the scattering semi-phase derivative, relative density of states, and Wigner--Smith group delay trace into a single time scale density, thereby providing a "Universe Time Mother Ruler" in the Matrix Universe U_{mat} independent of specific Hamiltonian choices. The Universe as a QCA object equation U_{qca}=(\Lambda,\mathcal H_{cell},\mathcal A_{qloc},\alpha,\omega_0) equation describes discrete time evolution via local unitary automorphisms \alpha on a countable lattice, and reconstructs relativistic quantum field theory and geometric structure in the continuum limit. Based on this, we propose an axiomatic system for "Postulated Chaos QCA" and prove the following main results: 1. On any finite region \Omega\subset\Lambda, the restricted finite-dimensional unitary operator U_\Omega satisfies Discrete Time ETH for its quasi-energy spectrum eigenstates with respect to any local operator O_X (support X\subset\Omega): for the vast majority of eigenstates \psi_n within an energy window, equation \langle\psi_n|O_X|\psi_n\rangle =\langle O_X\rangle_{micro}(\varepsilon_n) +\mathcal O(e^{-c|\Omega|}), equation and the squared average of off-diagonal matrix elements decays exponentially with volume, achieving eigenstate thermalization for local observables. 2. Under the axioms of "Postulated Chaos QCA" (finite propagation radius, translation symmetry, local gates generating high-order unitary designs, no extra extensive conserved quantities), the quasi-energy level statistics of the finite region U_\Omega converge after unfolding to the Wigner--Dyson distribution of the CUE random matrix class. Its spectral form factor exhibits a typical "ramp--plateau" structure, constituting a standard quantum chaos diagnosis at the QCA level. 3. Relating QCA--ETH to the Unified Time Scale: Under the unified time scale \tau, the "thermalization time scale" and the growth rate of local entropy density are jointly controlled by the energy shell average of the scale density \kappa(\omega) and the QCA light cone structure. We prove a "Unified Time--ETH--Entropy Growth Theorem": for any family of finite-density local initial states, the entropy density increases monotonically with \tau and approaches the microcanonical entropy density in the long-time limit. 4. On the global QCA universe object, embedding the above local results into the causal network and unified time scale mother structure shows that the "thermal time arrow" and "macroscopic irreversibility" on the cosmic scale can be understood as structural invariants in the unified Matrix--QCA Universe, rather than additionally introduced dynamical postulates. The appendix provides: rigorous definitions and equivalent forms of discrete time ETH; exact mapping between local random circuits and QCA; proof details and constant estimates for the QCA--ETH Theorem and CUE level statistics; and a 1D Postulated Chaos QCA model with specific predictions for numerical verification.
We investigate the integrability-to-chaos transition and information scrambling in Ising spin networks via a graph-theoretic formulation. Modeling spins as vertices and interactions as edges encoded by adjacency matrices across path, Erdos-Renyi, and Watts-Strogatz topologies, we demonstrate that long-range couplings and heterogeneous degree distributions markedly accelerate quantum information propagation. The Hamiltonian comprises local and normalized non-local interactions; tuning the non-local coupling and field heterogeneity drives integrability breaking. To quantify scrambling, we employ bipartite mutual and tripartite information. Increasing non-local interactions drives tripartite information to large negative values, signaling deep information scrambling. The squared commutators constructed from out-of-time-order correlators (OTOCs) exhibit early-time exponential growth; equivalently, the OTOC four-point functions themselves decay exponentially, yielding quantum Lyapunov exponents that scale systematically with parameters governing the chaotic regime. Complementing this, Krylov complexity reveals rapid operator growth in the chaotic phase, synchronizing with OTOC and mutual information dynamics. Spectrally, the transition manifests as a shift from Poissonian to Wigner-Dyson level spacing statistics. The spectral form factor (SFF) exhibits the characteristic slope-dip-ramp-plateau structure, enabling the extraction of Thouless and Heisenberg times. Crucially, a reduced Thouless time strongly correlates with accelerated information and operator scrambling. Ultimately, this work establishes a unified framework bridging network topology with information-theoretic, operator, and spectral diagnostics, offering insights into thermalization and non-equilibrium dynamics in quantum many-body systems.
This work demonstrates that sparse long-range random bonds on a one-dimensional lattice alone can generate quantum-chaotic spectral correlations and also drive a localization transition in a noninteracting single-particle Hamiltonian. The model is a one-dimensional ring in which each pair of sites is connected independently with a probability pij=dij−(1+σ). Each bond carries identical unit hopping and on-site disorder is absent. Despite the absence of on-site disorder and interaction, the model displays quantum chaotic spectra with Gaussian orthogonal ensemble (GOE) level statistics at small σ and localized eigenstates with Poisson statistics at larger σ. The transition occurs in the range $ 0.80 \lesssim σ_c \lesssim 0.85,farabovethesummabilitythresholdofthemeanhoppingprofile(σ=0$). A Gaussian field theory retaining only the mean and variance of the Bernoulli bonds instead predicts a threshold at σ=1, suggesting that higher cumulants are infrared-relevant. Our findings hint towards a universality class that is distinct from both the power-law random banded matrix model and the standard Anderson transition.
In classical systems, chaos is clearly defined via the behavior of trajectories. In quantum systems with a classical analogue one finds that the transition from regular to chaotic dynamics is signified by a change in the spectral statistics. This has been found to remain true for quantum systems with no classical analogue, including many-body systems. Furthermore, quantum chaotic systems explore all the allowed configurations in the Hilbert space, i.e. they are ergodic, while integrable systems, and systems in the many-body localized phase, are restricted to a certain subspace of the available phase space, and hence strongly break ergodicity. In this dissertation, we study the intermediate behavior between ergodicity and localization, i.e. the weak breaking of ergodicity. The model examined is the PXP spin chain model, where spins are allowed to flip only under certain kinetic constraints. We start by reproducing some already established results. First, we explore the eigenstate thermalization hypothesis (ETH) for this model and demonstrate the existence of a small number of states, throughout the PXP spectrum, that violate the ETH. Then we study the level-spacing statistics of the model, a well-known quantum chaos diagnostic, which turns out to be close to semi-Poisson and approach Wigner--Dyson statistics for large system sizes. Moreover, we examine various aspects of the model that have not been studied before. For example, the eigenvector component statistics, another quantum chaos diagnostic, for the PXP model turn out to be non-Gaussian. Finally, we perform a quench, in order to study how the energy spreads throughout the system, and observe ballistic fronts.
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.