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In this thesis, we investigate ergodicity breaking transitions in different models of closed quantum systems with quenched disorder. Based on the quantum chaos conjecture, a quantum system is considered ergodic if the statistical properties of its eigenvalues and eigenstates comply with predictions of the random matrix theory (RMT). An ergodicity breaking quantum phase transition can occur upon tuning some appropriate model parameter, for instance the degree of disorder in the system, which causes the departure of the said statistical properties from the RMT predictions. We first analyse one dimensional disordered interacting spin-1/2 chains, which are predicted to host a transition between an ergodic and many-body localized phase. For a range of different indicators, our results consistently display a linear drift of the critical transition parameter with the system size. This casts doubt onto the existence of the many-body localized phase and raises important questions about the stability of the transition in the thermodynamic limit. We validate our numerical methods by performing benchmarks on the noninteracting three dimensional Anderson model, for which the critical parameters of the Anderson localization transition are known to high accuracy. In two dimensional variant of the same model, localization should occur trivially at any nonzero disorder in the thermodynamic limit. Due to considerable finite-size effects, finite samples may appear delocalized, however. We show that our numerical analysis of finite systems can correctly reproduce absence of localization transition in the thermodynamic limit. Finally, we propose the zero dimensional quantum sun model as the toy model of an ergodicity breaking transition in an interacting system. We corroborate analytic predictions of the transition with numerical calculations. We observe several hallmarks of a transition already in small systems and hence propose the model as a benchmark for future studies of ergodicity breaking transitions in interacting systems. V doktorskem delu preučujemo prehode z zlomom ergodičnosti v izoliranih kvantnih sistemih s prisotnostjo nereda. V skladu z domnevo kvantnega kaosa se v ergodičnih sistemih statistične lastnosti njihovih energijskih spektrov in pripadajočih lastnih stanj ujemajo z napovedmi teorije naključnih matrik. Ob spreminjanju ustreznega modelskega parametra, denimo stopnje neurejenosti sistema, pričnejo omenjene statistične lastnosti odstopati od napovedi teorije naključnih matrik, kar lahko pripelje do kvantnega faznega prehoda z zlomom ergodičnosti. Uvodoma analiziramo enodimenzionalne verige spinov 1/2 ob prisotnosti nereda in meddelčnih interakcij. Za tovrstne sisteme je napovedan obstoj prehoda med ergodično in večdelčno lokalizirano neergodično fazo. V študiji analiziramo več različnih indikatorjev ergodičnosti, naši rezultati v končnih sistemih pa vseskozi skladno kažejo na linearno naraščanje kritičnega parametra prehoda skupaj z velikostjo sistema. S tem mečejo senco dvoma na obstoj večdelčno lokalizirane faze ter odpirajo nova vprašanja o obstoju prehoda v termodinamski limiti. V doktorskem delu uporabljene numerične metode preverimo na neinteragirajočem tridimenzionalnem Andersonovem modelu, v katerem so kritični parametri prehoda med lokalizirano in delokalizirano fazo znani z veliko natančnostjo. V dvodimenzionalni različici modela nastopi lokalizacija v termodinamski limiti trivialno pri vsakem neničelnem neredu. V končnih sistemih so prisotna znatna odstopanja od omenjene napovedi, zato so lahko končni vzorci pri majhnih neredih delokalizirani. V delu pokažemo, da lahko z našimi numeričnimi metodami s preučevanjem končnih sistemov pravilno napovemo odsotnost lokalizacijskega prehoda v termodinamski limiti. Nazadnje predlagamo ničdimenzionalni model kvantnega sonca kot vzorčni model za preučevanje prehodov z zlomom ergodičnosti v interagirajočih sistemih. Pri tem analitične napovedi potrdimo z rezultati numeričnih izračunov. Že v majhnih sistemih opazimo številne značilnosti prehoda in zato predlagamo model kot referenco za bodoče študije prehodov z zlomom ergodičnosti v sistemih z interakcijami.

We prove quantum ergodicity for the eigenfunctions of the pseudo-Laplacian on Riemannian surfaces with finitely many hyperbolic cusps and ergodic geodesic flow.

The Šeba billiard, a rectangular torus with a point scatterer, is a popular model to study the transition between integrability and chaos in quantum systems. Whereas such billiards are classically essentially integrable, they may display features such as quantum ergodicity [11], which are usually associated with quantum systems whose classical dynamics is chaotic. Šeba proposed that the eigenfunctions of toral point scatterers should also satisfy Berry’s random wave conjecture, which implies that the value distribution of the eigenfunctions ought to be Gaussian. However, Keating, Marklof, and Winn formulated a conjecture that suggested that Šeba billiards with irrational aspect ratio violate the random wave conjecture, and we show that this is indeed the case. More precisely, for tori having diophantine aspect ratio, we construct a subsequence of the set of new eigenfunctions having even/even symmetry, of essentially full density, and show that its 4th moment is not consistent with a Gaussian value distribution. In fact, given any set $\Lambda $ interlacing with the set of unperturbed eigenvalues, we show non-Gaussian value distribution of the Green’s functions $G_{\lambda }$, for $\lambda $ in an essentially full density subsequence of $\Lambda $.

In a classically chaotic system that is ergodic, any trajectory will be arbitrarily close to any point of the available phase space after a long time, filling it uniformly. Using Born's rules to connect quantum states with probabilities, one might then expect that all quantum states in the chaotic regime should be uniformly distributed in phase space. This simplified picture was shaken by the discovery of quantum scarring, where some eigenstates are concentrated along unstable periodic orbits. Despite of that, it is widely accepted that most eigenstates of chaotic models are indeed ergodic. Our results show instead that all eigenstates of the chaotic Dicke model are actually scarred. They also show that even the most random states of this interacting atom-photon system never occupy more than half of the available phase space. Quantum ergodicity is achievable only as an ensemble property, after temporal averages are performed., As published. 10 pages, 3 figures (main); 5 pages, 3 figures (supplementary information)

Let M τ 0 be the Grauert tube (of some fixed radius τ 0 ) of a compact, negatively curved, real analytic Riemannian manifold M without boundary. Let φ λ be a Laplacian eigenfunction on M of eigenvalue − λ 2 and let φ λ C be its holomorphic extension to M τ 0 . In this article, we prove that on M τ 0 ∖ M , there exists a dimensional constant α > 0 and a full density subsequence { λ j k } k = 1 ∞ of the spectrum for which the masses of the complexified eigenfunctions φ λ j k C are asymptotically equidistributed at length scale ( log ⁡ λ j k ) − α . Moreover, the complex zeros of φ λ j k C also become equidistributed on this logarithmic length scale.

Whereas much work in the mathematical literature on quantum chaos has focused on phenomena such as quantum ergodicity and scarring, relatively little is known at the rigorous level about the existence of eigenfunctions whose morphology is more complex. Quantum systems whose dynamics is intermediate between certain regimes - for example, at the transition between Anderson localized and delocalized eigenfunctions, or in systems whose classical dynamics is intermediate between integrability and chaos - have been conjectured in the physics literature to have eigenfunctions exhibiting multifractal, self-similar structure. To-date, no rigorous mathematical results have been obtained about systems of this kind in the context of quantum chaos. We give here the first rigorous proof of the existence of multifractal eigenfunctions for a widely studied class of intermediate quantum systems. Specifically, we derive an analytical formula for the Renyi entropy associated with the eigenfunctions of arithmetic Seba billiards, in the semiclassical limit, as the associated eigenvalues tend to infinity. We also prove multifractality of the ground state for more general, non-arithmetic billiards and show that the fractal exponent in this regime satisfies a symmetry relation, similar to the one predicted in the physics literature, by establishing a connection with the functional equation for Epstein's zeta function., 28 pages, to appear in Comm. Math. Phys

Quantum many-body scars have been put forward as counterexamples to the Eigenstate Thermalization Hypothesis. These atypical states are observed in a range of correlated models as long-lived oscillations of local observables in quench experiments starting from selected initial states. The long-time memory is a manifestation of quantum non-ergodicity generally linked to a sub-extensive generation of entanglement entropy, the latter of which is widely used as a diagnostic for identifying quantum many-body scars numerically as low entanglement outliers. Here we show that, by adding kinetic constraints to a fractionalized orthogonal metal, we can construct a minimal model with orthogonal quantum many-body scars leading to persistent oscillations with infinite lifetime coexisting with rapid volume-law entanglement generation. Our example provides new insights into the link between quantum ergodicity and many-body entanglement while opening new avenues for exotic non-equilibrium dynamics in strongly correlated multi-component quantum systems., 5 pages + 4 figures

We consider a sequence of finite quantum graphs with few loops, so that they converge, in the sense of Benjamini-Schramm, to a random infinite quantum tree. We assume these quantum trees are spectrally delocalized in some interval $I$, in the sense that their spectrum in $I$ is purely absolutely continuous and their Green's functions are well controlled near the real axis. We furthermore suppose that the underlying sequence of discrete graphs is expanding. We deduce a quantum ergodicity result, showing that the eigenfunctions with eigenvalues lying in $I$ are spatially delocalized., 64 pages

We discuss Shnirelman's Quantum Ergodicity Theorem, giving an outline of a proof and an overview of some of the recent developments in mathematical Quantum Chaos., Comment: 18 pages, 3 figures; minor revisions following the referee's comments. To appear special issue of Annales Math\'ematiques du Qu\'ebec in honor of Alexander Shnirelman's 75th birthday

We generalize Page's result on the entanglement entropy of random pure states to the many-body eigenstates of realistic disordered many-body systems subject to long range interactions. This extension leads to two principal conclusions: first, for increasing disorder the "shells" of constant energy supporting a system's eigenstates fill only a fraction of its full Fock space and are subject to intrinsic correlations absent in synthetic high-dimensional random lattice systems. Second, in all regimes preceding the many-body localization transition individual eigenstates are thermally distributed over these shells. These results, corroborated by comparison to exact diagonalization for an SYK model, are at variance with the concept of "non-ergodic extended states" in many-body systems discussed in the recent literature., Comment: 4+ pages, 2 figures, supplementary material

Recent discovery of persistent revivals in quantum simulators based on Rydberg atoms have pointed to the existence of a new type of dynamical behavior that challenged the conventional paradigms of integrability and thermalization. This novel collective effect has been named quantum many-body scars by analogy with weak ergodicity breaking of a single particle inside a stadium billiard. In this overview, we provide a pedagogical introduction to quantum many-body scars and highlight the newly emerged connections with the semiclassical quantization of many-body systems. We discuss the relation between scars and more general routes towards weak violations of ergodicity due to "embedded" algebras and non-thermal eigenstates, and highlight possible applications of scars in quantum technology., Comment: Review article, comments welcome

Intramolecular vibrational energy redistribution (IVR) impacts the dynamics of reactions in a profound way. Theoretical and experimental studies are increasingly indicating that accounting for the finite rate of energy flow is critical for uncovering the correct reaction mechanisms and calculating accurate rates. This requires an explicit understanding of the influence and interplay of the various anharmonic (Fermi) resonances that lead to the coupling of the vibrational modes. In this regard, the local random matrix theory (LRMT) and the related Bose-statistics triangle rule (BSTR) model have emerged as a powerful and predictive quantum theories for IVR. In this Perspective we highlight the close correspondence between LRMT and the classical phase space perspective on IVR, primarily using model Hamiltonians with three degrees of freedom. Our purpose for this is threefold. First, this clearly brings out the extent to which IVR pathways are essentially classical, and hence crucial towards attempts to control IVR. Second, given that LRMT and BSTR are designed to be applicable for large molecules, the exquisite correspondence observed even for small molecules allows for insights into the quantum ergodicity transition. Third, we showcase the power of modern nonlinear dynamics methods in analysing high dimensional phase spaces, thereby extending the deep insights into IVR that were earlier gained for systems with effectively two degrees of freedom. We begin with a brief overview of recent examples where IVR plays an important role and conclude by mentioning the outstanding problems and the potential connections to issues of interest in other fields., Comment: Accepted Phys. Chem. Chem. Phys Perspective article, 36 pages, 15 figures (reduced size)

Consider a sequence of finite regular graphs (GN) converging, in the sense of Benjamini-Schramm, to the infinite regular tree. We study the induced quantum graphs with equilateral edge lengths, Kirchhoff conditions (possibly with a non-zero coupling constant $\alpha$) and a symmetric potential U on the edges. We show that in the spectral regions where the infinite quantum tree has absolutely continuous spectrum, the eigenfunctions of the converging quantum graphs satisfy a quantum ergodicity theorem. In case $\alpha$ = 0 and U = 0, the limit measure is the uniform measure on the edges. In general, it has an explicit analytic density. We finally prove a stronger quantum ergodicity theorem involving integral operators, the purpose of which is to study eigenfunction correlations., Comment: To appear in J. Lond. Math. Soc

We study the Laplacian perturbed by two delta potentials on a two-dimensional flat torus. There are two types of eigenfunctions for this operator: old, or unperturbed eigenfunctions which are eigenfunctions of the standard Laplacian, and new, perturbed eigenfunctions which are affected by the scatterers. We prove that along a density one sequence, the new eigenfunctions are uniformly distributed in configuration space, provided that the difference of the scattering points is Diophantine., 17 pages. Revised according to referee's comments

The Laplace–Beltrami eigenfunctions on a compact Riemannian manifold M whose geodesic billiard flow has mixed character have been conjectured by Percival to split into two complementary families, with all semiclassical mass supported in the completely integrable and ergodic regions of phase space respectively. In this paper, we consider the Dirichlet Laplacian on a family of mushroom billiards M t parametrised by the length of their rectangular component. We prove that there exist eigenfunction subsequences of M t with full upper density that split as conjectured by Percival for almost all , providing the first example of a billiard known to satisfy this weak form of Percival's conjecture.

We show that a Wigner induced random orthonormal basis of spherical harmonics is almost surely quantum ergodic. Here, a random basis is identified with an element of the product probability space of unitary groups, each endowed with the measure induced by the generalized Wigner ensemble. This yields a semi-classical realization of the probabilistic quantum unique ergodicity result for Wigner eigenvectors of Bourgade-Yau. At the same time, this generalizes a similar result due to Zelditch, who uses Haar measure on the unitary groups in defining the notion of a random basis.

We give three different proofs of the main result of Anantharaman and Le Masson (Duke Math J 164(4):723–765, 2015), establishing quantum ergodicity—a form of delocalization—for eigenfunctions of the laplacian on large regular graphs of fixed degree. These three proofs are much shorter than the original one, quite different from one another, and we feel that each of the four proofs sheds a different light on the problem. The goal of this exploration is to find a proof that could be adapted for other models of interest in mathematical physics, such as the Anderson model on large regular graphs, regular graphs with weighted edges, or possibly certain models of non-regular graphs. A source of optimism in this direction is that we are able to extend the last proof to the case of anisotropic random walks on large regular graphs.

If (M; g) is a compact Riemannian manifold of dimension n ≥ 2 we give necessary and sufficient conditions for improved Lp(M)-norms of eigenfunctions for all 2 < p ≠ pc = 2(n+1)/n-1, the critical exponent. Since improved Lpc (M) bounds imply improvement all other exponents, these conditions are necessary for improved bounds for the critical space. We also show that improved Lpc (M) bounds are valid if these conditions are met and if the half-wave operators, U(t), have no caustics when t ≠ 0. The problem of finding a necessary and sufficient condition for Lpc (M) improvement remains an interesting open problem., "Harmonic Analysis and Nonlinear Partial Differential Equations". July 4～6, 2016. edited by Hideo Kubo and Hideo Takaoka. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

Let M be a closed Riemannian manifold carrying an effective and isometric action of a compact connected Lie group G. We derive a refined remainder estimate in the stationary phase approximation of certain oscillatory integrals on T ∗ ⁢ M × G T^{\ast}M\times G with singular critical sets that were examined in [7] in order to determine the asymptotic distribution of eigenvalues of an invariant elliptic operator on M. As an immediate consequence, we deduce from this an asymptotic multiplicity formula for families of irreducible representations in L 2 ⁢ ( M ) \mathrm{L}^{2}(M) . The improved remainder is used in [4] to prove an equivariant semiclassical Weyl law and a corresponding equivariant quantum ergodicity theorem.

We show that in a generic, ergodic quantum many-body system the interactions induce a non-trivial topology for an arbitrarily small non-hermitean component of the Hamiltonian. This is due to an exponential-in-system-size proliferation of exceptional points which have the hermitian limit as an accumulation (hyper-)surface. The nearest-neighbour level repulsion characterizing hermitian ergodic many-body sytems is thus shown to be a projection of a richer phenomenology where actually all the exponentially many pairs of eigenvalues interact. The proliferation and accumulation of exceptional points also implies an exponential difficulty in isolating a local ergodic quantum many-body system from a bath, as a robust topological signature remains in the form of exceptional points arbitrarily close to the hermitian limit.

The Floquet-Magnus expansion is a useful tool to calculate an effective Hamiltonian for periodically driven systems. In this study, we investigate the convergence of the expansion for a one-body nonlinear system in a continuous space, a driven anharmonic oscillator. In this model, all eigenstates of the time evolution operator are found to be localized in energy space, and the expectation value of the energy is bounded from above. We first propose a general procedure to estimate the radius of convergence of the Floquet-Magnus expansion for periodically driven systems with an unbounded energy spectrum. By applying it to the driven anharmonic oscillator, we numerically show that the expansion diverges for all driving frequencies even if the anharmonicity is arbitrarily small. This conclusion contradicts the widely accepted belief that the divergence of the Floquet-Magnus expansion is a direct consequence of quantum ergodicity, which implies that each eigenstate of the time evolution operator is a linear combination of all available eigenstates of the unperturbed Hamiltonian and the system heats up to infinite temperature after long intervals., 13 pages, 8 figures

We establish the first moment bound (phi)Sigma L(phi circle times phi Psi, 1/2) << (epsilon) p(5/4+epsilon) for triple product L-functions, where Psi is a fixed Hecke-Maass form on SL2 (Z) and phi runs over the Hecke-Maass newforms on Gamma(0) (p) of bounded eigenvalue. The proof is via the theta correspondence and analysis of periods of half-integral weight modular forms. This estimate is not expected to be optimal, but the exponent 5/4 is the strongest obtained to date for a moment problem of this shape. We show that the expected upper bound follows if one assumes the Ramanujan conjecture in both the integral and half-integral weight cases. Under the triple product formula, our result may be understood as a strong level aspect form of quantum ergodicity: for a large prime p, all but very few Hecke-Maass newforms on Gamma(0)(p)\H of bounded eigenvalue have very uniformly distributed mass after pushforward to SL2(Z)\H. Our main result turns out to be closely related to estimates such as (vertical bar n vertical bar p)Sigma L(Psi circle times chi(np), 1/2) << p where the sum is over those n for which np is a fundamental discriminant and chi(np) denotes the corresponding quadratic character. Such estimates improve upon bounds of Duke-Iwaniec., Forum of Mathematics, Sigma, 8, ISSN:2050-5094

We show that for every prime d and α ∈ (0, 1/6), there is an infinite sequence of (d + 1)-regular graphs G = (V, E) with high girth Ω(α logd(∣V∣), second adjacency matrix eigenvalue bounded by $$(3/\sqrt 2)\sqrt d $$ , and many eigenvectors fully localized on small sets of size O(mα). This strengthens the results of [GS18], who constructed high girth (but not expanding) graphs with similar properties, and may be viewed as a discrete analogue of the “scarring” phenomenon observed in the study of quantum ergodicity on manifolds. Key ingredients in the proof are a technique of Kahale [Kah92] for bounding the growth rate of eigenfunctions of graphs, discovered in the context of vertex expansion and a method of Erdős and Sachs for constructing high girth regular graphs.

In this paper, the semigroup is defined by the continuous transformation on the compact metric space. We use the chain recurrence to study the Lipschitz ergodicity and generalized ergodicity. Some criterions for a system to be Lipschitz ergodic or generalized ergodic are given. At last, we illustrate the relationship between ergodicity, generalized ergodicity and Lipschitz ergodicity.

We investigate eigenfunctions of the Laplacian perturbed by a delta potential on the standard tori $${\mathbb{R}^d/2\pi\mathbb{Z}^d}$$ in dimensions $${d = 2,3}$$ . Despite quantum ergodicity holding for the set of “new” eigenfunctions we show that superscars occur—there is phase space localization along families of closed orbits, in the sense that some semiclassical measures contain a finite number of Lagrangian components of the form $${c_{i} \cdot dx\delta({\xi}-{\xi}_{i})}$$ , for $${c_{i} > 0}$$ uniformly bounded from below. In particular, for both $${d = 2}$$ and $${d = 3}$$ , eigenfunctions fail to equidistribute in phase space along an infinite subsequence of new eigenvalues. For $${d = 2}$$ , we also show that some semiclassical measures have both strongly localized momentum marginals and non-uniform quantum limits (i.e., the position marginals are non-uniform). For $${d = 3}$$ , superscarred eigenstates are quite rare, but for $${d = 2}$$ we show that the phenomenon is quite common—with $${N_{2}(x) \sim x/\sqrt{\log x}}$$ denoting the counting function for the new eigenvalues below x, there are $${\gg N_{2}(x)/{\rm log}^A x}$$ eigenvalues $${\lambda}$$ with the property that any semiclassical limit along these eigenvalues exhibits superscarring.

The hyperbolic lattice point problem asks to estimate the size of the orbit \(\Gamma z\) inside a hyperbolic disk of radius \(\cosh ^{-1}(X/2)\) for \(\Gamma \) a discrete subgroup of \({\hbox {PSL}_2( {{\mathbb {R}}})} \). Selberg proved the estimate \(O(X^{2/3})\) for the error term for cofinite or cocompact groups. This has not been improved for any group and any center. In this paper local averaging over the center is investigated for \({\hbox {PSL}_2( {{\mathbb {Z}}})} \). The result is that the error term can be improved to \(O(X^{7/12+{\varepsilon }})\). The proof uses surprisingly strong input e.g. results on the quantum ergodicity of Maas cusp forms and estimates on spectral exponential sums. We also prove omega results for this averaging, consistent with the conjectural best error bound \(O(X^{1/2+{\varepsilon }})\). In the appendix the relevant exponential sum over the spectral parameters is investigated.

We review the fundamental concepts of quantum chaos in Hamiltonian systems. The quantum evolution of bound systems does not possess the sensitive dependence on initial conditions, and thus no chaotic behaviour occurs, whereas the study of the stationary solutions of the Schrodinger equation in the quantum phase space (Wigner functions) reveals precise analogy of the structure of the classical phase portrait. We analyze the regular eigenstates associated with invariant tori in the classical phase space, and the chaotic eigenstates associated with the classically chaotic regions, and the corresponding energy spectra. The effects of quantum localization of the chaotic eigenstates are treated phenomenologically, resulting in Brody-like level statistics, which can be found also at very high-lying levels, while the coupling between the regular and the irregular eigenstates due to tunneling, and of the corresponding levels, manifests itself only in low-lying levels.

We introduce the concept of ergodicity space of a measure-preserving transformation and will present some of its properties as an algebraic weight for measuring the size of the ergodicity of a measure-preserving transformation. We will also prove the invariance of the ergodicity space under conjugacy of dynamical systems.

We prove a quantitative statement of the quantum ergodicity for Hecke–Maass cusp forms on the modular surface. As an application of our result, along a density 1 subsequence of even Hecke–Maass cusp forms, we obtain a sharp lower bound for the L 2-norm of the restriction to a fixed compact geodesic segment of $${\eta=\{iy : y > 0\} \subset {\mathbb{H}}}$$ . We also obtain an upper bound of $${O_\epsilon\left(t_\phi^{3/8+\epsilon} \right)}$$ for the $${L^\infty}$$ norm along a density 1 subsequence of Hecke–Maass cusp forms; for such forms, this is an improvement over the upper bound of $${O_\epsilon\left(t_\phi^{5/12+\epsilon} \right)}$$ given by Iwaniec and Sarnak. In a recent work of Ghosh, Reznikov, and Sarnak, the authors proved for all even Hecke–Maass forms that the number of nodal domains, which intersect a geodesic segment of $${\eta}$$ , grows faster than $${t_\phi^{1/12-\epsilon}}$$ for any $${\epsilon > 0}$$ , under the assumption that the Lindelof Hypothesis is true and that the geodesic segment is long enough. Upon removing a density zero subset of even Hecke–Maass forms, we prove without making any assumptions that the number of nodal domains grows faster than $${t_\phi^{1/8-\epsilon}}$$ for any $${\epsilon > 0}$$ .

A definition of classical and quantum chaos on the basis of the Liouville–Arnold theorem is proposed. According to this definition, a chaotic quantum system that has N degrees of freedom should have M < N independent first integrals of motion (good quantum numbers) that are determined by the symmetry of the Hamiltonian for the system being considered. Quantitative measures of quantum chaos are established. In the classical limit, they go over to the Lyapunov exponent or the classical stability parameter. The use of quantum-chaos parameters in nuclear physics is demonstrated.

We use a straightforward variation on a recent argument of Hezari and Riviere [8] to obtain localized L p -estimates for all exponents larger than or equal to the critical exponent p c = 2 ( n + 1 ) n − 1 . We are able to do this directly by just using the L p -bounds for spectral projection operators from our much earlier work [12] . The localized bounds we obtain here imply, for instance, that, for a density one sequence of eigenvalues on a manifold whose geodesic flow is ergodic, all of the L p , 2 p ≤ ∞ , bounds of the corresponding eigenfunctions are relatively small compared to the general ones in [12] , which are saturated on round spheres. The connection with quantum ergodicity was established for exponents 2 p p c in the recent results of the author [13] and Blair and the author [3] ; however, the article of Hezari and Riviere [8] was the first one to make this connection (in the case of negatively curved manifolds) for the critical exponent, p c . As is well known, and we indicate here, bounds for the critical exponent, p c , imply ones for all the other exponents 2 p ≤ ∞ . The localized estimates involve L 2 -norms over small geodesic balls B r of radius r , and we shall go over what happens for these in certain model cases on the sphere and on manifolds of nonpositive curvature. We shall also state a problem of determining when one can improve on the trivial O ( r 1 2 ) estimates for these L 2 ( B r ) bounds. If r = λ − 1 , one can improve on the trivial estimates if one has improved L p c ( M ) bounds just by using Holder's inequality; however, obtaining improved bounds for r ≫ λ − 1 seems to be subtle.

We prove that the number of nodal domains of a density one subsequence of eigenfunctions grows at least logarithmically with the eigenvalue on negatively curved `real Riemann surfaces'. The geometric model is the same as in prior joint work with Junehyuk Jung (arXiv:1310.2919, to appear in J. Diff. Geom), where the number of nodal domains was shown to tend to infinity, but without a specified rate. The proof of the logarithmic rate uses the new logarithmic scale quantum ergodicity results of Hezari-Riviere (arXiv:1411.4078) and X. Han (arXiv:1410.3911).

We study the ergodic properties of a two-dimensional self-gravitating system using molecular dynamics simulations. We apply three different tests for ergodicity: a direct method comparing the time average of a particle momentum and position to the respective ensemble average, sojourn times statistics and the dynamical functional method. For comparison purposes they are also applied to a short-range interacting system and to the Hamiltonian mean-field model. Our results show that a two-dimensional self-gravitating system takes a very long time to establish ergodicity. If a Kac factor is used in the potential energy, such that the total energy is extensive, then this time is independent of particle number, and diverges with N without a Kac factor.

We compare the entire classical and quantum evolutions of the Dicke model in its regular and chaotic domains. This is a paradigmatic interacting spin-boson model of great experimental interest. By studying the classical and quantum survival probabilities of initial coherent states, we identify features of the long-time dynamics that are purely quantum and discuss their impact on the equilibration times. We show that the ratio between the quantum and classical asymptotic values of the survival probability serves as a metric to determine the proximity to a separatrix in the regular regime and to distinguish between two manifestations of quantum chaos: scarring and ergodicity. In the case of maximal quantum ergodicity, our results are analytical and show that quantum equilibration takes longer than classical equilibration., Comment: 25 pages, 10 figures, 4 supplementary animations can be requested to d.v.pcf.cu@gmail.com

The goal of this article is to draw new applications of small scale quantum ergodicity in nodal sets of eigenfunctions. We show that if quantum ergodicity holds on balls of shrinking radius $r(\lambda) \to 0$, then one can achieve improvements on the recent upper bounds of Logunov and Logunov-Malinnikova on the size of nodal sets, according to a certain power of $r(\lambda)$. We also show that the order of vanishing results of Donnelly-Fefferman and Dong can be improved. Since by the results of Han and Hezari-Rivi\`ere small scale QE holds on negatively curved manifolds at logarithmically shrinking rates, we get logarithmic improvements on such manifolds for the above measurements of eigenfunctions. We also get $o(1)$ improvements for manifolds with ergodic geodesic flows. Our results work for a full density subsequence of any given orthonormal basis of eigenfunctions., Comment: 19 pages. Comments are greatly appreciated

We prove quantum ergodicity for certain orthonormal bases of $L^2(\mathbb{S}^2)$, consisting of joint eigenfunctions of the Laplacian on $\mathbb{S}^2$ and the discrete averaging operator over a finite set of rotations, generating a free group. If in addition the rotations are algebraic we give a quantified version of this result. The methods used also give a new, simplified proof of quantum ergodicity for large regular graphs., 27 pages

We give an estimate of the quantum variance for $d$-regular graphs quantised with boundary scattering matrices that prohibit back-scattering. For families of graphs that are expanders, with few short cycles, our estimate leads to quantum ergodicity for these families of graphs. Our proof is based on a uniform control of an associated random walk on the bonds of the graph. We show that recent constructions of Ramanujan graphs, and asymptotically almost surely, random $d$-regular graphs, satisfy the necessary conditions to conclude that quantum ergodicity holds., Comment: 28 pages, 5 figures

We review a theory for coupled many-nonlinear oscillator systems that describes quantum ergodicity and energy flow in molecules. The theory exploits the isomorphism between quantum energy flow in Fock space, that is, vibrational state space, and single-particle quantum transport in disordered solid-state systems. The quantum ergodicity transition in molecules is thereby analogous to the Anderson transition in disordered solids. The theory reviewed here, local random matrix theory (LRMT), describes the nature of the quantum ergodicity transition, statistical properties of vibrational eigenstates, and quantum energy flow through the vibrational states of molecules. Predictions of LRMT have been observed in computational studies of coupled nonlinear oscillator systems, which are summarized here. We also review applications of LRMT to molecular spectroscopy and chemical reaction rate theory, including adoption of LRMT in theories that predict rates of conformational change of molecules taking place at energie...

In the present paper, we define an ergodicity coefficient of a positive mapping defined on ordered Banach space with a base , and study its properties. The defined coefficient is a generalization of the well-known the Dobrushin’s ergodicity coefficient. By means of the ergodicity coefficient we provide uniform asymptotical stability conditions for nonhomogeneous discrete Markov chains (NDMC). These results are even new in case of von Neumann algebras. Moreover, we find necessary and sufficient conditions for the weak ergodicity of NDMC. Certain relations between uniform asymptotical stability and weak ergodicity are considered.

In this paper we study Spectral Decomposition Theorem [1] and translate it to quantum language by means of the Wigner transform. We obtain a quantum version of Spectral Decomposition Theorem (QSDT) which enables us to achieve three distinct goals: First, to rank Quantum Ergodic Hierarchy levels [2,3]. Second, to analyze the classical limit in quantum ergodic systems and quantum mixing systems. And third, and maybe most important feature, to find a relevant and simple connection between the first three levels of quantum ergodic hierarchy (ergodic, exact and mixing) and quantum spectrum. Finally, we illustrate the physical relevance of QSDT applying it to two examples: Microwave billiards [4,5] and a phenomenological Gamow model type [6,7]., Comment: Chaos, Solitons & Fractals (2014)

We present a quantum ergodicity theorem for fixed spectral window and sequences of compact hyperbolic surfaces converging to the hyperbolic plane in the sense of Benjamini and Schramm. This addresses a question posed by Colin de Verdi\`{e}re. Our theorem is inspired by results for eigenfunctions on large regular graphs by Anantharaman and the first-named author. It applies in particular to eigenfunctions on compact arithmetic surfaces in the level aspect, which connects it to a question of Nelson on Maass forms. The proof is based on a wave propagation approach recently considered by Brooks, Lindenstrauss and the first-named author on discrete graphs. It does not use any microlocal analysis, making it quite different from the usual proof of quantum ergodicity in the large eigenvalue limit. Moreover, we replace the wave propagator with renormalised averaging operators over discs, which simplifies the analysis and allows us to make use of a general ergodic theorem of Nevo. As a consequence of this approach, we require little regularity on the observables., Comment: v3: 25 pages, 3 figures, the proof in Section 9 has been corrected, to appear in Duke Math. J

We outline some recent proofs of quantum ergodicity on large graphs and give new applications in the context of irregular graphs. We also discuss some remaining questions., To appear in "Annales de la facult\'e des Sciences de Toulouse"

Ce mémoire d'habilitation traite de problèmes à l'interface de la théorie des systèmes dynamiques, de la théorie spectrale et de l'analyse microlocale avec en toile de fond des questions de physique mathématique. Dans le premier chapitre, on décrit quelques résultats de théorie du contrôle en lien avec la question de l'unique ergodicité quantique. Les chapitres 2 et 3 sont eux consacrés à la dynamique en temps longs de l'équation de Schrödinger semi-classique pour des systèmes chaotiques et intégrables. Le chapitre 4 contient des variations du théorème d'ergodicité quantique ainsi que des applications à l'analyse harmonique sur les variétés. Le chapitre 5 concerne le problème des ondes aléatoires du point de vue des ensembles nodaux. Enfin, le chapitre 6 traite de la dynamique fine des flots de gradient Morse-Smale à travers les résonances de Pollicott-Ruelle et on y décrit quelques applications à la topologie différentielle.

Quantum error correction was invented to allow for fault-tolerant quantum computation. Systems with topological order turned out to give a natural physical realization of quantum error correcting codes (QECC) in their groundspaces. More recently, in the context of the AdS/CFT correspondence, it has been argued that eigenstates of CFTs with a holographic dual should also form QECCs. These two examples raise the question of how generally eigenstates of many-body models form quantum codes. In this work we establish new connections between quantum chaos and translation-invariance in many-body spin systems, on one hand, and approximate quantum error correcting codes (AQECC), on the other hand. We first observe that quantum chaotic systems exhibiting the Eigenstate Thermalization Hypothesis (ETH) have eigenstates forming approximate quantum error-correcting codes. Then we show that AQECC can be obtained probabilistically from translation-invariant energy eigenstates of every translation-invariant spin chain, including integrable models. Applying this result to 1D classical systems, we describe a method for using local symmetries to construct parent Hamiltonians that embed these codes into the low-energy subspace of gapless 1D quantum spin chains. As explicit examples we obtain local AQECC in the ground space of the 1D ferromagnetic Heisenberg model and the Motzkin spin chain model with periodic boundary conditions, thereby yielding non-stabilizer codes in the ground space and low energy subspace of physically plausible 1D gapless models., 13 pages, no figures. v2: added references, corrected typos

We prove quantitative equidistribution properties for orthonormal bases of eigenfunctions of the Laplacian on the rational $d$-torus. We show that the rate of equidistribution of such eigenfunctions is of polynomial decay. We also prove that equidistribution of eigenfunctions holds for symbols supported in balls with a radius shrinking at a polynomial rate., This article is based on the appendix of our previous preprint: arXiv:1411.4078. We have included improvements and have simplified the proofs (no semiclassical/microlocal techniques are necessary)

The work distribution is a fundamental quantity in nonequilibrium thermodynamics mainly due to its connection with fluctuations theorems. Here we develop a semiclassical approximation to the work distribution for a quench process in chaotic systems. The approach is based on the dephasing representation of the quantum Loschmidt echo and on the quantum ergodic conjecture, which states that the Wigner function of a typical eigenstate of a classically chaotic Hamiltonian is equidistributed on the energy shell. We show that our semiclassical approximation is accurate in describing the quantum distribution as we increase the temperature. Moreover, we also show that this semiclassical approximation provides a link between the quantum and classical work distributions., Comment: 4+ pages, 4 figures. 2+ pages Sup. Mat

We present a replica path integral approach describing the quantum chaotic dynamics of the SYK model at large time scales. The theory leads to the identification of non-ergodic collective modes which relax and eventually give way to an ergodic long time regime (describable by random matrix theory). These modes, which play a role conceptually similar to the diffusion modes of dirty metals, carry quantum numbers which we identify as the generators of the Clifford algebra: each of the $2^N$ different products that can be formed from $N$ Majorana operators defines one effective mode. The competition between a decay rate quickly growing in the order of the product and a density of modes exponentially growing in the same parameter explains the characteristics of the system's approach to the ergodic long time regime. We probe this dynamics through various spectral correlation functions and obtain favorable agreement with existing numerical data., Comment: 23 pages, 5 figures

The diagonal ensemble is the infinite time average of a quantum state following unitary dynamics. In analogy to the time average of a classical phase space dynamics, it is intimately related to the ergodic properties of the quantum system giving information on the spreading of the initial state in the eigenstates of the Hamiltonian. In this work we apply a concept from quantum information, known as total correlations, to the diagonal ensemble. Forming an upper-bound on the multipartite entanglement, it quantifies the combination of both classical and quantum correlations in a mixed state. We generalize the total correlations of the diagonal ensemble to more general $\alpha$-Renyi entropies and focus on the the cases $\alpha=1$ and $\alpha=2$ with further numerical extensions in mind. Here we show that the total correlations of the diagonal ensemble is a generic indicator of ergodicity breaking, displaying a sub-extensive behaviour when the system is ergodic. We demonstrate this by investigating its scaling in a range of spin chain models focusing not only on the cases of integrability breaking but also emphasize its role in understanding the transition from an ergodic to a many body localized phase in systems with disorder or quasi-periodicity., Comment: v3: several minor improvements

We generalize the definition of a “quantum ergodic sequence” of sections of ample line bundles $$L \rightarrow M$$ from the case of positively curved Hermitian metrics h on L to general smooth metrics. A choice of smooth Hermitian metric h on L and a Bernstein–Markov measure $$\nu $$ on M induces an inner product on $$H^0(M, L^N)$$ . When $$||s_N||_{L^2} =1$$ , quantum ergodicity is the condition that $$|s_N(z)|^2 d\nu \rightarrow d\mu _{\varphi _{eq}} $$ weakly, where $$d\mu _{\varphi _{eq}} $$ is the equilibrium measure associated with $$(h, \nu )$$ . The main results are that normalized logarithms $$\frac{1}{N} \log |s_N|^2$$ of quantum ergodic sections tend to the equilibrium potential, and that random orthonormal bases of $$H^0(M, L^N)$$ are quantum ergodic.