We construct a complete set of quasi-local integrals of motion for the manybody localized phase of interacting fermions in a disordered potential. The integrals of motion can be chosen to have binary spectrum t0, 1u, thus constituting exact quasiparticle occupation number operators for the Fermi insulator. We map the problem onto a non-Hermitian hopping problem on a lattice in operator space. We show how the integrals of motion can be built, under certain approximations, as a convergent series in the interaction strength. An estimate of its radius of convergence is given, which also provides an estimate for the many-body localization-delocalization transition. Finally, we discuss how the properties of the operator expansion for the integrals of motion imply the presence or absence of a finite temperature transition.

We study high temperature spin transport in a disordered Heisenberg chain in the ergodic regime. By employing a density matrix renormalization group technique for the study of the stationary states of the boundary-driven Lindblad equation we are able to study extremely large systems (400 spins). We find both a diffusive and a subdiffusive phase depending on the strength of the disorder and on the anisotropy parameter of the Heisenberg chain. Studying finite-size effects we show numerically and theoretically that a very large crossover length exists that controls the passage of a clean-system dominated dynamics to one observed in the thermodynamic limit. Such a large length scale, being larger than the sizes studied before, explains previous conflicting results. We also predict spatial profiles of magnetization in steady states of generic nondiffusive systems.Introduction.-There are ever increasing technological capabilities in simulating isolated quantum systems through cold atomic gases [1] and, recently, through coupled, controlled superconducting qubits [2]. While there is a commensurately good theoretical handle on capturing ground state properties of such systems [3], understanding their dynamical properties, especially away from the ground state, is fraught with analytical and numerical challenges.Despite this, in the recent years we have witnessed a change in paradigm in the study of isolated quantum systems, in particular with regard to the role that disorder plays in such systems. The turning point came about from the study of Anderson localization [4] in interacting, many-body quantum systems [5]. The observation that disorder and quantum effects can hinder transport (of energy, charge or spin) even at an infinite temperature and in the presence of interactions [6] opened the door to a new phenomenology of a so-called manybody localized (MBL) phase exhibiting many unique and interesting properties. Slow growth of entanglement [7,8], emergent integrability [9], protection of symmetries [10], and change in the properties of eigenstates [11][12][13] are a few of the peculiar properties of this newly identified phase; see review [14] for a comprehensive list. The implications of the new MBL physics, being inherently robust, are far reaching, going from fundamental physics to the theory of quantum computation [15], some of which have already been experimentally probed [16].While the deep MBL region (in one dimensional systems) is well understood, much remains to be said about the conducting regime and the transition to it. Although both aspects are important, here we focus on characterizing the conducting phase, in particular its transport properties, in, what is by now, an archetypal model that harbors the MBL phase, i.e., the one dimensional anisotropic Heisenberg model.Generic arguments and numerical evidence on very small systems (about 20 spins) have been put forward for the existence of subdiffusive transport of spin [17][18][19][20] and energy [18,21,22]. A number of recent works have analyzed its sp...

Statistical analysis of the eigenfunctions of the Anderson tight-binding model with on-site disorder on regular random graphs strongly suggests that the extended states are multifractal at any finite disorder. The spectrum of fractal dimensions f (α) defined in Eq.(3), remains positive for α noticeably far from 1 even when the disorder is several times weaker than the one which leads to the Anderson localization, i.e. the ergodicity can be reached only in the absence of disorder. The oneparticle multifractality on the Bethe lattice signals on a possible inapplicability of the equipartition law to a generic many-body quantum system as long as it remains isolated.Introduction.-Anderson localization (AL) [1,2], in its broad sense, is one of the central paradigms of quantum theory. Diffusion, which is a generic asymptotic behavior of classical random walks [3], is inhibited in quantum case and under certain conditions it ceases to exist [2]. This concerns quantum transport of noninteracting particles subject to quenched disorder as well as transport and relaxation in many-body systems. In the latter case the many-body localization (MBL) [4] can be thought of as localization in the Fock space of Slater determinants, which play the role of lattice sites in a disordered tight-binding model. In contrast to a d-dimensional lattice, the structure of Fock space is hierarchical [5]: a twobody interaction couples a one-particle excitation with three one-particle excitations, which in turn are coupled with five-particle excitations, etc. This structure resembles a random regular graph (RRG) -a finite size Bethe lattice (BL) without boundary. Interest to the problem of single particle AL on the BL [6,7] has recently revived [8][9][10][11][12] largely in connection with MBL. It is a good approximation to consider hierarchical lattices as trees where any pair of sites is connected by only one path and loops are absent. Accordingly the sites in resonance with each other are much sparser than in ordinary d > 1-dimensional lattices. As a result even the extended wave functions can occupy zero fraction of the BL, i.e. be nonergodic. The nonergodic extended states on 3D lattices where loops are abundant are commonly believed [13][14][15][16] to exist but only at the critical point of the AL transition.In this paper we analyze the eigenstates of the Anderson model on RRG with connectivity K + 1 (K is commonly used to refer to the branching of the corresponding BL) and N sites:where ψ(i) (i = 1, ..., N ) can be characterized by the moments[13] (I 1 = 1 for the normalization). One can define the ergodicity as the convergence in the limit N → ∞ of the real space averaged |ψ(i)| 2q (equal to I q /N ) to its ensemble average value |ψ(i)| 2q = I q /N . This happens when the fluctuations of |ψ(i)| 2 are relatively weak andThe latter condition turns out to be both necessary and sufficient for the convergence of I q to I q (see Supplementary Materials for the discussion). Deviations of τ (q) from q − 1 are signatures of the nonergodic state. I...

We study the breaking of ergodicity measured in terms of return probability in the evolution of a quantum state of a spin chain. In the non ergodic phase a quantum state evolves in a much smaller fraction of the Hilbert space than would be allowed by the conservation of extensive observables. By the anomalous scaling of the participation ratios with system size we are led to consider the distribution of the wave function coefficients, a standard observable in modern studies of Anderson localization. We finally present a criterion for the identification of the ergodicity breaking (many-body localization) transition based on these distributions which is quite robust and well suited for numerical investigations of a broad class of problems.

We find the exact Casimir force between a plate and a cylinder, a geometry intermediate between parallel plates, where the force is known exactly, and the plate-sphere, where it is known at large separations. The force has an unexpectedly weak decay ∼ L/(H 3 ln(H/R)) at large plate-cylinder separations H (L and R are the cylinder length and radius), due to transverse magnetic modes. Path integral quantization with a partial wave expansion additionally gives a qualitative difference for the density of states of electric and magnetic modes, and corrections at finite temperatures.PACS numbers: 42.25. Fx, 03.70.+k, With recent advances in the fabrication of electronic and mechanical systems on the nanometer scale quantum effects like Casimir forces have become increasingly important [1,2]. These systems can probe mechanical oscillation modes of quasi one-dimensional structures such as nano wires or carbon nanotubes with high precision [3]. However, thorough theoretical investigations of Casimir forces are to date limited to "closed" geometries such as parallel plates [4] or, recently, a rectilinear "piston" [5], where the zero point fluctuations are not diffracted into regions which are inaccessible to classical rays. A notable exception is the original work by Casimir and Polder on the interaction between a plate and an atom (sphere) at asymptotically large separation [6].

We review the current (as of Fall 2016) status of the studies on the emergent integrability in many-body localized models. We start by explaining how the phenomenology of fully many-body localized systems can be recovered if one assumes the existence of a complete set of (quasi)local operators which commute with the Hamiltonian (local integrals of motions, or LIOMs). We describe the evolution of this idea from the initial conjecture, to the perturbative constructions, to the mathematical proof given for a disordered spin chain. We discuss the proposed numerical algorithms for the construction of LIOMs and the status of the debate on the existence and nature of such operators in systems with a many-body mobility edge, and in dimensions larger than one.

In two remarkable recent papers the planar perturbative expansion was proposed for the universal function of the coupling appearing in the dimensions of high-spin operators of the N=4 super Yang-Mills theory. We study numerically the integral equation derived by Beisert, Eden, and Staudacher, which resumes the perturbative series. In a confirmation of the anti-de Sitter-space/conformal-field-theory (AdS/CFT) correspondence, we find a smooth function whose two leading terms at strong coupling match the results obtained for the semiclassical folded string spinning in AdS5. We also make a numerical prediction for the third term in the strong coupling series.

Abstract.It is well known that one can map certain properties of random matrices, fermionic gases, and zeros of the Riemann zeta function to a unique point process on the real line R. Here we analytically provide exact generalizations of such a point process in d-dimensional Euclidean space R d for any d, which are special cases of determinantal processes. In particular, we obtain the n-particle correlation functions for any n, which completely specify the point processes in R d . We also demonstrate that spinpolarized fermionic systems in R d have these same n-particle correlation functions in each dimension. The point processes for any d are shown to be hyperuniform, i.e., infinite wavelength density fluctuations vanish, and the structure factor (or power spectrum) S(k) has a nonanalytic behavior at the origin given by S(k) ∼ |k| (k → 0). The latter result implies that the pair correlation function g 2 (r) tends to unity for large pair distances with a decay rate that is controlled by the power law 1/r d+1 , which is a well-known property of bosonic ground states and more recently has been shown to characterize maximally random jammed sphere packings. We graphically display one-and two-dimensional realizations of the point processes in order to vividly reveal their "repulsive" nature. Indeed, we show that the point processes can be characterized by an effective "hard-core" diameter that grows like the square root of d. The nearest-neighbor distribution functions for these point processes are also evaluated and rigorously bounded. Among other results, this analysis reveals that the probability of finding a large spherical cavity of radius r in dimension d behaves like a Poisson point process but in dimension d + 1, i.e., this probability is given by expPoint processes in arbitrary dimension 2 for large r and finite d, where κ(d) is a positive d-dependent constant. We also show that as d increases, the point process behaves effectively like a sphere packing with a coverage fraction of space that is no denser than 1/2 d . This coverage fraction has a special significance in the study of sphere packings in high-dimensional Euclidean spaces.