Understanding the inter-play of quenched disorder, in- teractions **and** quantum fluctuations has been a central theme of hard condensed matter for **many** years. Activ- ity on this topic boomed recently, in particular after that Basko, Aleiner **and** Altshuler (BAA) showed by using the self-consistent Born approximation that interacting **and** isolated quantum systems can fail to thermalize due to Anderson **localization** in Fock space [1]. This phe- nomenon, called **Many** **Body** **Localization** (MBL), repre- sents a new kind of ergodicity breaking transition, which is purely dynamical—indeed it can take place even at infi- nite temperature by increasing the amount of disorder— **and** which results from the interplay of disorder, interac- tions **and** quantum fluctuations [2, 3]. One of the most surprising results is that even the **delocalized** phase is unusual in a wide range of parameters already before the MBL transition. In fact both in numerical simulations [4– 8] **and** in experiments [9–11] it was found that transport appears to be sub-diffusive **and** that out-of-equilibrium relaxation toward thermal equilibrium is slow **and** power- law-like with exponents that gradually approach zero at the transition. Several works explained this behavior in terms of Griffiths regions, i.e., rare inclusions of the lo- calised phase which impede transport **and** relaxation [12– 15]. However, also quasi-periodic 1d **and** disordered 2d systems, in which Griffiths effects should be absent or milder [14, 15], do display analogous unusual transport **and** relaxation [10, 11, 16, 17]. It is therefore important to look for other explanations that might hold beyond the particular case of 1d disordered systems. Moreover, it is interesting to complement the real space Griffiths perspective to one directly based on quantum **dynamics** in Fock-space. These are the aims of our work.

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B. Multifractal exponents τ (q)
After its introduction in fluid **dynamics** in order to characterize the statistical properties of turbulence (see the book [61] **and** references therein), the notion of multifractality has turned out to be relevant in **many** areas of physics (see for instance [62–68]), in particular at critical points of disordered models like Anderson **localization** transitions [69, 70] or in random classical spin models [71–79]. More recently, the wavefunctions of manybody quantum systems have been analyzed via multifractality for ground states in pure quantum spin models [80–93], as well as for excited states in MBL models [10, 94–96].

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here, our scheme is compatible with renormalization group
methods **and** matrix product state representations [ 38 , 47 ],
which can potentially be applied to much larger systems **and** beyond one dimension. We showed the power of the constructed LIOMs by extracting the **localization** length of the LIOMs **and** the Hamiltonian interactions from their respective exponential decays. We also showed that in the MBL phase, the LIOMs **and** physical spin operators exhibit similar dephas- ing **dynamics**, even if it cannot be simply explained by the typical weights of LIOMs **and** typical interaction strengths.

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β P oisson = 0 (3)
In the field of quantum chaos, the Poisson statistics appears for systems whose classical **dynamics** is integrable [5], whereas the Wigner-Dyson statistics appears when the classical **dynamics** is chaotic [6].
In the field of **Many**-**Body**-**Localization** (MBL) (see the recent reviews [7, 8] **and** references therein), the Wigner- Dyson statistics appears in the **delocalized** phase where the Eigenstate Thermalization Hypothesis (E.T.H.) [9–13] holds, whereas the Poisson statistics appears in the **Many**-**Body** Localized phase, which is characterized by an extensive number of emergent localized conserved operators [14–22]. These conserved operators can be for instance constructed via the RSRG-X procedure [23–29] that generalize to the eigenstates the Fisher Strong Disorder Real Space RG for groundstates [30–32]. This RSRG-X approach breaks down as the MBL transition towards delocalization is approached as a consequence of resonances, **and** other types of RG have been proposed for the critical region [33, 34].

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So in the MBL phase, excited states are somewhat similar to ground states, with efficient representation via Density- Matrix-RG or Matrix Product States [9–12] **and** Tensor Networks [13]. The Fisher Strong Disorder Real Space RG to construct the ground states of random quantum spin models [14–16] has been extended into the Strong Disorder RG procedure for the unitary **dynamics** [17, 18], **and** into the RSRG-X procedure in order to construct the whole set of excited eigenstates [19–24]. This construction is actually possible only because the MBL phase is characterized by an extensive number of emergent localized conserved operators [25–32], but breaks down as the MBL transition towards delocalization is approached. As a consequence, the current RG descriptions of the MBL transition are based on different types of RG rules concerning either the entanglement [33] or the resonances [34].

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Finally, it is interesting to notice that rare regions may also have deep destabilizing effects on quenched disorder spin chains. First, the possibility of hot mobile bubbles, as described above, was shown to imply the absence of **many**-**body** mobility edges [29], i.e. that the presence of thermal states at a given value of the thermodynamical parameters implies that states are thermal at all values of these parameters. Second, while for classical systems we concluded in Section 4 that the presence of chaotic spots due to fluctuations of the quenched disorder did not play a major role in the transport of the energy, their presence become relevant for quantum systems: They destabilize the MBL phase in d > 1 [28, 58] **and** are expected to drive the system to the thermal phase if the bare **localization** length exceeds some threshold value in d = 1 [58, 77]. Acknowledgements. This work of F.H. was partially supported by the grants ANR-15-CE40- 0020-01 LSD **and** ANR-14-CE25-0011 EDNHS of the French National Research Agency (ANR). W.D.R. acknowledges the support of the Flemish Research Fund FWO under grant G076216N.

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needed to solve the optimization problem up to a certain target infidelity. Intuitively transformations from a state with maximal diagonal entropy (i.e., completely **delocalized** in phase space) to one with low diagonal entropy (i.e., well localized) are expected to be more difficult than those between localized states or between adiabatically connected states. It turns out however that the complexity of an optimization protocol depends only weakly on the choice of initial **and** final states. Let us illustrate this considering two different state-to-state transformations: from a maximal entropy state to the ground state ( |M S → |g.s.) **and** from an eigenstate at

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CEREMADE, Universite Paris-Dauphine, France (Dated: October 31, 2014)
We consider disordered **many**-**body** systems with periodic time-dependent Hamiltonians in one spatial dimension. By studying the properties of the Floquet eigenstates, we identify two distinct phases: (i) a **many**-**body** localized (MBL) phase, in which almost all eigenstates have area-law entanglement entropy, **and** the eigenstate thermalization hypothesis (ETH) is violated, **and** (ii) a **delocalized** phase, in which eigenstates have volume-law entanglement **and** obey the ETH. MBL phase exhibits logarithmic in time growth of entanglement entropy for initial product states, which distinguishes it from the **delocalized** phase. We propose an effective model of the MBL phase in terms of an extensive number of emergent local integrals of motion (LIOM), which naturally explains the spectral **and** dynamical properties of this phase. Numerical data, obtained by exact diagonalization **and** time-evolving block decimation methods, suggests a direct transition between the two phases. Our results show that **many**-**body** **localization** is not destroyed by sufficiently weak periodic driving.

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One of the most important characterization of the **Many**-**Body** localized phase is that excited eigenstates display an area-law entanglement [3] instead of the volume-law entanglement of thermalized eigenstates. This property has been used numerically to identify the MBL phase [4, 5] **and** to show the consistency with other criteria of MBL [5]. The fact that excited eigenstates in the MBL phase are similar to the ground-states from the point of view of entanglement suggests that various approaches that have been developed for ground states in the past can be actually adapted to study the MBL phase. One first example is the efficient representation via Density-Matrix-RG or Matrix Product States [6–9] **and** Tensor Networks [10]. Another example is the Strong Disorder Real-Space RG approach (see [11, 12] for reviews) developed by Ma-Dasgupta-Hu [13] **and** Daniel Fisher [14, 15] to construct the ground states of random quantum spin chains, with its extension in higher dimensions d = 2, 3, 4 [16–26]. This approach has been extended into the Strong Disorder RG procedure for the unitary **dynamics** [32, 33], **and** into the RSRG-X procedure in order to construct the whole set of excited eigenstates [27–31]. It should be stressed that these two Strong Disorder RG procedures based on the spin variables are limited to the MBL phase, whereas the current RG descriptions of the MBL transition towards delocalization are based on RG rules for the entanglement [34] or for the resonances [35].

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PACS numbers: 05.30.Rt, 72.15.Rn, 72.20.Ee
I. INTRODUCTION
It is now almost mathematically proven that **many**- **body** **localization**, i.e., the absence of long-range trans- port in a thermodynamic **many**-**body** system, occurs in certain one-dimensional quantum lattice models at any energy density if sufficiently strong quenched disorder is present [1]. In this case, **many**-**body** **localization** (MBL) comes along with a complete set of conserved quasi- local quantities [2–4]. However, it remains less clear whether the originally predicted **localization** transition at finite temperature [5, 6] exists as a genuine dynam- ical phase transition defining a sharp **many**-**body** mo- bility edge in energy density. Even though several nu- merical investigations in small one-dimensional (1D) sys- tems have reported such mobility edges [7–9], studies in larger systems did not find similar evidence [10, 11] **and**, moreover, linked-cluster analysis [12] of the numer- ical data hint that the extent of the localized phase has been vastly overestimated. Furthermore, recent theoret- ical considerations [4, 13–16] have raised doubts about non-perturbative effects which might reduce the putative transition to a crossover. A related open issue concerns the **many**-**body** analog of Mott’s argument, which forbids the coexistence of localized **and** **delocalized** states at the same energy in single-particle problems.

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u pP i,j x q “ ´upxq, if D 1 ď l ď s, s.t. i, j P I l . (7)
In fact, in **many**-**body** quantum mechanics, fruitful results derive from the anti- symmetry. In the past three decades, the stability of Coulomb systems has been studied extensively (see [ 13 ] for a textbook presentation). For all normalized, anti-symmetric wave function ψ with s spin state,

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Introduction - The solution of the Schr¨odinger equation in interacting **many**-**body** systems is a formidable problem in condensed matter **and** nuclear physics, as well as in quantum chemistry. Starting with the Hartree-Fock (HF) method as one of the first ap- proaches to treat the quantum **many**-**body** problem, dif- ferent theoretical formalisms have been developed over time [1]: approaches relying on the **many**-**body** wave function, like quantum Monte Carlo or quantum chem- istry methods, or on the electronic density, like density- functional theory (DFT) in its static or time-dependent (TDDFT) form or, finally, quantum field theoretical, Green function based methods. While exact in princi- ple, all of these methods rely in practical calculations on approximations **and** recipes whose validity are difficult to judge.

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I. INTRODUCTION
Recent advances in laser cooling techniques have re- sulted in experimental realizations of well-isolated, highly tunable quantum **many**-**body** systems of cold atoms 1 . A rich experimental toolbox of available quantum optics, combined with the systems’ slow intrinsic time scales, allow for a preparation of non-equilibrium **many**-**body** states **and** also a precise characterization of their quan- tum evolution. This has made the study of different dy- namical regimes in **many**-**body** systems one of the fore- front directions in modern condensed matter physics (for a review, see Ref. 2 ).

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During my thesis I worked **and** met a lot of people. Some of them were of particular importance to the realization of this thesis. I particularly enjoyed the presence of Dr. Arturo Argüelles in my oﬃce **and** the long conversations about Density Matrix Renormalization Group **and** Matrix Product States. I also would like to thank Thomas Engl **and** Juan-Diego Urbina from the University of Regensburg for the stimulating collabora- tion we had for the last four years. I would also like to thank Boris Nowak for suggesting to consider the truncated Wigner method for studying the transport of Bose–Einstein condensates. The implementation of smooth exterior complex scaling instead of transparent boundary conditons was suggested by Prof. Alejandro Saenz. I’m deeply grateful for his input **and** also for arranging a one-month visit in Berlin to learn about complex scal- ing.

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The analysis of the data in Table 2 clearly indicates that our results (beDeft-P14) are within 20 meV of the reference TM-noRI HOMO values. For the present systems, the error associated with the (P14) analytic continuation is smaller than 2 meV (BeO) for the HOMO quasiparticle energies. As a result most of the discrepancy with TM-noRI data originates from the resolution-of-the-identity approximation. In comparison, the AIMS-P16 approach could not identify all possible solutions, yielding further errors of the order of 120 meV, 560 meV **and** 320 meV for BN, BeO **and** MgO, respectively. Increasing the number of Pad´e parameters up to 128 (AIMS-P128), requiring thus the calculation of the self-energy matrix elements at 128 imaginary frequencies, could restore excellent results. However, the AIMS-P128 calculations does not identify the correct solutions for the O 3 **and** MgO systems,

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DOI: 10.1103/PhysRevLett.117.225301
First predicted in 1970 [1] , the Efimov effect describes the behavior of three strongly interacting bosons when any two of them cannot bind. At unitarity, when the scattering length diverges, the three-**body** bound states are scale invariant **and** they form a sequence up to vanishing binding energy **and** infinite spatial extension. Efimov trimers had been intensely discussed in nuclear physics, but it was in an ultracold gas of caesium atoms that they were finally discovered [2] . To observe Efimov trimers, experiments in atomic physics rely on Feshbach resonances [3] , which allow one to instantly switch a gas between weak inter- actions **and** the unitary limit. Such a control of interactions is lacking in nuclear physics or condensed matter experi- ments, **and** singular interactions can be probed there only in the presence of accidental fine tuning [4] . Beyond the original system [2] , Efimov trimers have now been observed for several multicomponent systems, including bosonic, fermionic, **and** Bose-Fermi mixtures [5–7] . These experimental findings are interpreted in terms of the theory of few-**body** strongly interacting quantum systems. For three identical bosons in three dimensions, a complete universal theory is available, on **and** off unitarity [4] . Further theoretical work is aimed at understanding bound states for more than three bosons, mixtures, **and** the effects of dimensionality.

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of Jerison **and** Kenig [12] covers the case p = d/2 in dimension d. It was later improved by Koch **and** Tataru in [15].
Unfortunately, these results are not well adapted to the situation of Schr¨ odinger operators describing N particles, which are defined on R dN . In order to apply the existing results, one would need assumptions on the potentials depending on N . To the best of our knowledge, two works, due to Georgescu [9] **and** Schechter-Simon [28], provide a unique continuation property for **many**-particle Hamiltonians with an assumption on the poten- tials independent of N . However, they require the wavefunction to vanish on an open set (weak UCP), **and** for the Hohenberg-Kohn theorem strong UCP is needed.

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nonlinear core-corrections. I have found similar parameters as the V ‘semico’ pseudopotential in the one adopted in Ref. [ 334 ].
iii. Considering for vanadium only the 3d **and** 4s states in valence could be problematic. In fact, the all-electron wavefunctions of the V atom for the valence states 3d **and** 4s **and** the semicore states 3s **and** 3p have a large spatial overlap (see Fig. 6.7 ). For this reason a better approxima- tion would be to treat the semicore states explicitly as valence states, excluding them from the atomic frozen core. This turns out to be mandatory in particular for GW calculations, where in the exchange term it is rightly the spatial overlap between wavefunctions that really matters. The same situation has been found also for other transition metals, such as copper [ 327 ] or in the case of cadmium sulphide [ 150 ]. Pseudopotentials of the Troullier-Martins type have been chosen [ 328 ] in the fully separable Kleinman-Bylander form [ 329 ]. Using the fhi98pp code [ 330 ][ 331 ] to tune the cutoff radii, the transferability of these pseudopoten- tials has been optimized in order to find a compromise with the energy cutoff required for the convergence (the “softness” of the pseudopotential). After a proper choice of the reference component of the pseudopotential, the absence of spurious ghost states [ 332 ] has been carefully checked.

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interacting case. It is an interesting open problem to extend our result to more physical interactions **and** to k > 2.
Open problems **and** the link with QFT. Nonlinear Gibbs measures have also played an important role in constructive quantum field theory (QFT) [24, 47, 64]. By an argument similar to the Feynman-Kac formula, one can write the (formal) grand-canonical partition function of a quantum field in space dimension d, by means of a (classical) nonlinear Gibbs measure in dimension d + 1, where the additional variable plays the role of time [48, 49]. The rigorous construction of quantum fields then sometimes boils down to the proper definition of the corresponding nonlinear measure. This so-called Euclidean approach to QFT was very successful for some particular models **and** the literature on the subject is very vast (see, e.g., [48, 1, 55, 28] for a few famous examples **and** [61] for a recent review). Like here, the problem becomes more **and** more difficult when the dimension grows. The main difficulties are to define the measures in the whole space **and** to renormalize the (divergent) physical interactions. We have not yet tried to renormalize physical interactions in our context or to take the thermodynamic limit at the same time as T → ∞. These questions are however important **and** some tools from constructive QFT could then be useful.

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The exploration of energy landscapes of complex systems is, for these reasons, a very broad research ﬁeld, **and** several exploration techniques have been proposed in the past. The main distinction can be made between zero **and** ﬁnite temperature methods, implying the exploration of the potential energy surface the ﬁrst, **and** of the free energy surface the second. Obviously, the latter is more indicated to a proper study of thermally activated events. The ﬁrst category contains for instance several eigenvector-following methods: in these techniques, saddle points are searched following the unstable direction of the potential energy surface indicated by its negative curvature. In other words, this requires the determination of the matrix of second derivatives of the potential energy - the hessian matrix - that presents a spectra with (some) negative eigenvalues corresponding to unstable directions on saddle points. Thus, a preliminary step prior to calculating ﬁnite-temperature reaction paths using these sampling techniques may consist in locating the saddle points of the energy **and** the corresponding energy minima using one of the eigenvector following methods described in the literature (e.g. the activation-relaxation technique, [ 9 ] Optim [ 11 ] or the dimer method [ 12 ]). A limitation of these methods is that energy saddles only correspond directly to the actual barriers for the **dynamics** if the temperature is very low, otherwise the entropic contribution to the **dynamics** becomes relevant.

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