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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.