We study the emergence of generic quantum chaos from an arithmetic billiard in the hyperbolic plane. Starting from an arithmetic triangle, we introduce small area-preserving geometric perturbations and compute long consecutive sequences of eigenvalues and eigenstates. The spectral statistics show a perturbation-dependent crossover from Poisson-like arithmetic behaviour to the Gaussian orthogonal ensemble (GOE), with the crossover scale moving to higher wavenumbers as the perturbation decreases. To characterize the nearest-neighbour spacing distribution, we compare the data with Brody and generalized-gamma fits. The generalized-gamma form is used only as a two-parameter diagnostic, allowing the small- and intermediate-spacing structure to vary independently from the large-spacing tail, rather than as a universal crossover law. For the weakest perturbation, we also compare the data with an effective block-GOE model, which diagnoses residual block-like structure deep into the computed spectral range. We complement the spectral analysis with Poincaré-Husimi representations of the eigenstates and quantify phase-space localization using an entropy-based measure. Its distribution is well described by Beta distributions whose width decays as a power law in the wavenumber. Before the spectral crossover, the localization statistics retain memory of the near-arithmetic regime. After the GOE regime is reached, the localization-fluctuation widths collapse onto a common decay law, sigma(Beta) ~ k^{-eta}, with eta approximately 0.37--0.38. These results provide a combined spectral and eigenfunction-level picture of the arithmetic-to-GOE crossover in hyperbolic quantum billiards.
We study the emergence of generic quantum chaos from an arithmetic billiard in the hyperbolic plane. Starting from an arithmetic triangle, we introduce small area-preserving geometric perturbations and compute long consecutive sequences of eigenvalues and eigenstates. The spectral statistics show a perturbation-dependent crossover from Poisson-like arithmetic behaviour to the Gaussian orthogonal ensemble (GOE), with the crossover scale moving to higher wavenumbers as the perturbation decreases. To characterize the nearest-neighbour spacing distribution, we compare the data with Brody and generalized-gamma fits. The generalized-gamma form is used only as a two-parameter diagnostic, allowing the small- and intermediate-spacing structure to vary independently from the large-spacing tail, rather than as a universal crossover law. For the weakest perturbation, we also compare the data with an effective block-GOE model, which diagnoses residual block-like structure deep into the computed spectral range. We complement the spectral analysis with Poincaré-Husimi representations of the eigenstates and quantify phase-space localization using an entropy-based measure. Its distribution is well described by Beta distributions whose width decays as a power law in the wavenumber. Before the spectral crossover, the localization statistics retain memory of the near-arithmetic regime. After the GOE regime is reached, the localization-fluctuation widths collapse onto a common decay law, sigma(Beta) ~ k^{-eta}, with eta approximately 0.37--0.38. These results provide a combined spectral and eigenfunction-level picture of the arithmetic-to-GOE crossover in hyperbolic quantum billiards.
In this thesis, we have explored the commonalities and connections between different classes of quantum systems that do not thermalize. Specifically, we have (1) shown that localized systems possess conservation laws like integrable systems, which can be constructed in a systematic way and used to detect localization-delocalization transitions
, (2) studied the phenomenon of many-body localization in a model with a single
particle mobility edge, (3) shown that interesting finite-size scaling emerges, with universal exponents, when athermal quantum systems are forced to thermalize through the
application of perturbations and (4) shown that these scaling laws also arise when a perturbation causes a crossover between quantum systems described by different random
matrix ensembles. We conclude with a brief summary of each chapter.
In Chapter 2, we have investigated the effects of finite size on the crossover between quantum integrable systems and non-integrable systems. Using exact diagonalization of finite-sized systems, we have studied this crossover by obtaining the energy level statistics and Drude weight associated with transport. Our results reinforce the idea that for system size L → ∞, non-integrability sets in for an arbitrarily small integrabilitybreaking
perturbation. The crossover value of the perturbation scales as a power law
∼ L−3 when the integrable system is gapless and the scaling appears to be robust to
microscopic details and the precise form of the perturbation.
In Chapter 3, we have studied the crossover among different random matrix ensembles
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[Poissonian, Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE)
and Gaussian Symplectic Ensemble (GSE)] realized in different microscopic models. We
have found that the perturbation causing the crossover among the different ensembles
scales to zero with system size as a power law with an exponent that depends on the
ensembles between which the crossover takes place. This exponent is independent of
microscopic details of the perturbation. We have also found that the crossover from the
Poissonian ensemble to the other three is dominated by the Poissonian to GOE crossover
which introduces level repulsion while the crossover from GOE to GUE or GOE to GSE
associated with symmetry breaking introduces a subdominant contribution. Finally,we
have conjectured that the exponent is dependent on whether the system contains interactions among the elementary degrees of freedom or not and is independent of the
dimensionality of the system.
In Chapter 4, we have outlined a procedure to construct conservation laws for Anderson
localized systems. These conservation laws are found as power series in the hopping
parameters. We have also obtained the conservation laws for the disorder free Aubry-Andre model, where the states are either localized or extended depending on the strength of a coupling constant. We have formulated a specific procedure for averaging over disorder, in order to examine the convergence of the power series. Using…
We study signatures of quantum chaos in (1+1)D quantum field theory (QFT) models. Our analysis is based on the method of Hamiltonian truncation, a numerical approach for the construction of low-energy spectra and eigenstates of QFTs that can be considered as perturbations of exactly solvable models. We focus on the double sine-Gordon, also studying the massive sine-Gordon and ϕ^{4} model, all of which are nonintegrable and can be studied by this method with sufficiently high precision from small to intermediate perturbation strength. We analyze the statistics of level spacings and of eigenvector components, which are expected to follow random matrix theory predictions. While level spacing statistics are close to the Gaussian orthogonal ensemble (GOE) as expected, on the contrary, the eigenvector components follow a distribution markedly different from the expected Gaussian. Unlike in the typical quantum chaos scenario, the transition of level spacing statistics to chaotic behavior takes place already in the perturbative regime. Moreover, the distribution of eigenvector components does not appear to change or approach Gaussian behavior, even for relatively large perturbations. Our results suggest that these features are independent of the choice of model and basis.
Spectral statistics and localization properties of a C3-symmetric billiard
We revisit the spectral statistics of the C3--symmetric billiard introduced by Dembowski [Phys. Rev. E, R4516 (2000)], which exhibits both GOE and GUE statistics depending on the symmetry block. Using high--precision Beyn's contour--integral method for the nonlinear Fredholm eigenvalue problem with built-in separation of irreducible subspaces, we compute 2.8x105 eigenvalues in each symmetry subspace, enabling statistically meaningful comparisons with random matrix theory. The improved spectra reveal clear GOE--GUE correspondence and resolve previously observed deviations in long--range spectral correlations. Furthermore, we analyze phase--space eigenstate localization through the distribution of entropy localization measures, which, for chaotic states follow a Beta distribution whose standard deviation decays as a power--law with energy, consistent with the onset of quantum ergodicity as described by Schnirelman's theorem.
In the field of quantum chaos, spectral statistics is one of the most extensively investigated characteristics. Despite a large body of existing literature, the effects of many-body interactions on the spectral statistics of relativistic quantum systems remain poorly understood. Treating electron-electron interactions through the one-orbital mean-field Hubbard model, we address this fundamental issue using graphene billiards with the geometric shape of a circular sector as prototypical systems. Our approach is to consider the two characteristically different cases where the statistics are Poisson and Gaussian orthogonal ensemble (GOE) so the corresponding classical dynamics are typically integrable and chaotic, respectively, and to systematically investigate how the statistics change as the Hubbard interaction strength increases from zero. We find that, for energies near the Dirac point, the Hubbard interactions have a significant effect on the spectral statistics. Regardless of the type of spectral statistics to begin with, increasing the Hubbard interaction strength up to a critical value causes the statistics to approach GOE, rendering more applicable the random matrix theory. As the interaction strength increases beyond the critical value, the statistics evolve toward Poisson, due to the emergence of an energy gap rendering the quasiparticles massive. We also find that the energy levels above and below the Dirac point can exhibit different statistics, and the many-body interactions have little effect on the statistics for levels far from the Dirac point. These results reveal the intriguing interplay between many-body interactions and the spectral statistics, which we develop a physical picture to understand.
Directionally inter-contained causal networks carry Gaussian-Orthogonal- Ensemble (GOE) nearest-neighbour spacing statistics as a parameter- insensitive attractor, with the orthogonal class protected by the causal time-layering itself [2]. Here we quantize the growth network by its symmetrized adjacency, H = A, and test the eigenstate- thermalization battery and long-range spectral probes. Three results. (i) Eigenstates are window-selective: extended but non-GOE in the upper window (Porter–Thomas variance 3.74 vs. 2), recovering GOE variance in an interior window (2.13–2.25) though not the full component law (KS p ≤ 0.001): a three-layer structure, window-dependent throughout. (ii) A localized excitation relaxes to a return probability consistent with its own eigenstate ensemble to within a factor of two, with the residual excess tracking the quasi-degenerate spectrum centre created by the bipartite pairing — dynamics senses the very sector excluded in the static study. (iii) The spectral form factor is window- dependent: a partial ramp in the upper window (fitted slope 0.47) and no ramp at any scale in the interior window (slopes −0.09 to +0.20, statistically consistent with zero): rigidity is absent where the thermodynamic limit would have to build it. We argue the interior- window flatness is the expected spectral fingerprint of a system without a thermodynamic limit, corroborating the nocontinuum-limit statement of the metric study, and we register the discriminator experiment (hidden quasi-conservation vs. finite size) as open. All protocols were pre- registered and all data are deposited.
The Eigenstate Thermalization Hypothesis explains thermalization in isolated quantum systems through the statistical properties of observables in the energy eigenbasis. We investigate the crossover from integrability to chaos in the spin-1/2 XXZ chain, establishing a direct correspondence between the spectral correlations of the Hamiltonian and local observables expressed in the energy eigenbasis as a signature of ergodicity breaking. By introducing a local perturbation that drives the system from integrability to chaos, we track the standard ETH indicators and the eigenstate entanglement entropy. We introduce a submatrix-based framework for analyzing local observables in the energy eigenbasis. By extracting real-symmetric blocks along the diagonal of the local observables represented in eigenbasis, we show that these submatrices exhibit both the short-range and long-range spectral features of the Hamiltonian. Remarkably, this correspondence persists even in a partially ergodic regime, indicating that the emergence of chaos is already encoded locally within the observables' matrix structure and that small blocks are sufficient to capture the underlying spectral correlations.