Multifractal Eigenfunctions for a Singular Quantum Billiard

The minima of quadratic forms and the behavior of Epstein and Dedekind zeta functions

Relaxation of few-body quantum systems can strongly depend on the initial state when the system's semiclassical phase space is mixed, i.e., regions of chaotic motion coexist with regular islands. In recent years, there has been much effort to understand the process of thermalization in strongly interacting quantum systems that often lack an obvious semiclassical limit. Time-dependent variational principle (TDVP) allows to systematically derive an effective classical (nonlinear) dynamical system by projecting unitary many-body dynamics onto a manifold of weakly-entangled variational states. We demonstrate that such dynamical systems generally possess mixed phase space. When TDVP errors are small, the mixed phase space leaves a footprint on the exact dynamics of the quantum model. For example, when the system is initialized in a state belonging to a stable periodic orbit or the surrounding regular region, it exhibits persistent many-body quantum revivals. As a proof of principle, we identify new types of "quantum many-body scars", i.e., initial states that lead to long-time oscillations in a model of interacting Rydberg atoms in one and two dimensions. Intriguingly, the initial states that give rise to most robust revivals are typically entangled states. On the other hand, even when TDVP errors are large, as in the thermalizing tilted-field Ising model, initializing the system in a regular region of phase space leads to slowdown of thermalization. Our work establishes TDVP as a method for identifying interacting quantum systems with anomalous dynamics in arbitrary dimensions. Moreover, the mixed-phase space classical variational equations allow to find slowly-thermalizing initial conditions in interacting models. Our results shed light on a link between classical and quantum chaos, pointing towards possible extensions of classical Kolmogorov-Arnold-Moser theorem to quantum systems.

We investigate the statistical distribution of the zeros of Dirichlet L-functions both analytically and numerically. Using the Hardy-Littlewood conjecture about the distribution of primes we show that the two-point correlation function of these zeros coincides with that for eigenvalues of the Gaussian unitary ensemble of random matrices, and that the distributions of zeros of different L-functions are statistically independent. Applications of these results to Epstein's zeta functions are briefly discussed.

On Epstein's Zeta Function

Can Bohmian trajectories account for quantum recurrences having classical periodicities?

Quantum chaos and interference

The richness and complexity of two-dimensional parameter space of the kicked Harper model is exploited to demonstrate quantum fingerprints of all classical transitions. The quantum system appears to follow the corresponding classical system in parameter but not in time in the localized, critical as well as in the ballistic regimes. Therefore, the relationship between quantum and classical systems appears to be universal when measured by their response to parameter changes. Additionally, a rather intriguing feature of quantum eigenstates is a set of critical points sprinkled in the regime where the classical dynamics is diffusive. These are the boundary points of the ballistic (localized) patches in the localized (ballistic) regime that survive in the semiclassical limit.

Classical and Quantum Chaos in Atomic Systems

Quantum manifestation of classical chaos has been one of the extensively studied subjects for more than a decade. Yet clear understanding of its nature still remains to be an open question partly due to the lack of a canonical definition of quantum chaos. The classical definition seems to be unsuitable in quantum mechanics partly because of the Heisenberg quantum uncertainty. In this regard, quantum chaos is somewhat misleading and needs to be clarified at the very fundamental level of physics. Since it is well known that quantum mechanics is more fundamental than classical mechanics, the quantum description of classically chaotic nature should be attainable in the limit of large quantum numbers. The focus of my research, therefore, lies on the correspondence principle for classically chaotic systems. The chaotic damped driven pendulum is mainly studied numerically using the split operator method that solves the time-dependent Schroedinger equation. For classically dissipative chaotic systems in which (multi)fractal strange attractors often emerge, several quantum dissipative mechanisms are also considered. For instance, Hoover`s and Kubo-Fox-Keizer`s approaches are studied with some computational analyses. But the notion of complex energy with non-Hermiticity is extensively applied. Moreover, the Wigner and Husimi distribution functions are examined with an equivalent classical distribution in phase-space, and dynamical properties of the wave packet in configuration and momentum spaces are also explored. The results indicate that quantum dynamics embraces classical dynamics although the classicalquantum correspondence fails to be observed in the classically chaotic regime. Even in the semi-classical limits, classically chaotic phenomena would eventually be suppressed by the quantum uncertainty.

It is remarkable that the quantum zeta function, defined as a sum over energy eigenvalues E: admits of exact evaluation in some situations for which not a single E be known. Herein we show how to evaluate instances of Z(s), and of an associated parity zeta function Y(s), for various quantum systems. For some systems both Z(n), Y(n) can be evaluated for infinitely many integers n. Such Z,Y values can be used, for example, to effect sharp numerical estimates of a system's ground energy. The difficult problem of evaluating the analytic continuation Z(s) for arbitrary complex s is discussed within the contexts of perturbation expansions, path integration, and quantum chaos.

We investigate the representation of the geometrical information of the Universe in terms of spectra, i.e., a set of eigenvalues of the Laplacian defined on the Universe. Here, we concentrate only on one specific problem along this line: to introduce a concept of distance between universes in terms of the difference in the spectra. We can find such a measure of closeness from a general discussion. First, we introduce a suitable functional ${\mathit{P}}_{\mathcal{G}}$[\ensuremath{\cdot}], where the geometrical information G (represented by the spectra) determines the detailed shape of the functional. Then, the overlapping functional integral between ${\mathit{P}}_{\mathcal{G}}$[\ensuremath{\cdot}] and ${\mathit{P}}_{\mathcal{G}\ensuremath{'}}$[\ensuremath{\cdot}] is taken, providing a measure of closeness between G and G\ensuremath{'}, d(G,G\ensuremath{'}). The basic properties of this distance (hereafter referred to as ``spectral distance,'' for brevity) are then investigated. First, it can be related to a reduced density-matrix element in quantum cosmology between G and G\ensuremath{'}. Thus, calculating the spectral distance d(G,G\ensuremath{'}) gives us insight into the quantum theoretical decoherence between two universes, corresponding to G and G\ensuremath{'}. Second, the spectral distance becomes divergent except for when G and G\ensuremath{'} have the same dimension and volume. This is very suggestive if the above-mentioned density-matrix interpretation is taken into account. Third, d(G,G\ensuremath{'}) does not satisfy the triangular inequality, which illustrates clearly that the spectral distance and the distance defined by the DeWitt metric on the superspace are not equivalent. We then pose a question: Do two universes with different topologies interfere with each other quantum mechanically? In particular, we concentrate on the difference in the orientabilities. To investigate this problem, several concrete models in two dimensions are set up, and the spectral distances between them are investigated: distances between tori and Klein's bottles, and those between spheres and real projective spaces. Quite surprisingly, we find many cases of spaces with different orientabilities in which the spectral distance turns out to be very short. This may suggest that, without any other special mechanism, two such universes interfere with each other quite strongly, contrary to our intuition. We discuss some curious features of the heat kernel for tori and Klein's bottles in terms of Epstein's theta and zeta functions. Differences and parallelisms between the spectral distance and the DeWitt distance are also discussed. \textcopyright{} 1996 The American Physical Society.

By giving the definition of the sum of a series indexed by a set on which a group operates, we prove that the sum of the series that defines the Riemann zeta function, the Epstein zeta function and a few other series indexed by $${\mathbb {Z}}^k$$ has an intrinsic meaning as a complex number, independent of the requirements of analytic continuation. The definition of the sum requires nothing more than algebra and the concept of absolute convergence. The analytical significance of the algebraically defined sum is then explained by an argument that relies on the Poisson formula for tempered distributions.

We consider an interacting collective spin model known as coupled top (CT), exhibiting a rich variety of phenomena related to quantum transitions, ergodicity, and formation of quantum scars, discussed in our previous work [Mondal, Sinha, and Sinha, Phys. Rev. E 102, 020101(R) (2020)2470-004510.1103/PhysRevE.102.020101]. In this work, we present a detailed analysis of the different type of transitions in the CT model, and find their connection with the underlying collective spin dynamics. Apart from the quantum scarring phenomena, we also identify another source of deviation from ergodicity due to the presence of nonergodic multifractal states. The degree of ergodicity of the eigenstates across the energy band is quantified from the relative entanglement entropy as well as multifractal dimensions, which can be probed from nonequilibrium dynamics. Finally, we discuss the detection of nonergodic behavior and different types of quantum scars using "out-of-time-order correlators," which has relevance in the recent experiments.

Classical and quantum dynamics in the classically chaotic stadium billiard are compared through studies of the evolving phase-space distributions and expectation values. Good agreement between the two mechanics is found for all distributions other than vertically directed wave packets which contain significant contributions from pseudoregular quantum eigenstates. Despite formal arguments as to the absence ``quantum chaos'' the quantum dynamics clearly shows a time-averaged relaxation to equilibrium, over the timescale studied, which is consistent with the classical K-system behavior.

Geometry and non-adiabatic response in quantum and classical systems