We solve a minimal model for an ergodic phase in a spatially extended quantum many-body system. The model consists of a chain of sites with nearest-neighbour coupling under Floquet time evolution. Quantum states at each site span a q-dimensional Hilbert space and time evolution for a pair of sites is generated by a q 2 × q 2 random unitary matrix. The Floquet operator is specified by a quantum circuit of depth two, in which each site is coupled to its neighbour on one side during the first half of the evolution period, and to its neighbour on the other side during the second half of the period. We show how dynamical behaviour averaged over realisations of the random matrices can be evaluated using diagrammatic techniques, and how this approach leads to exact expressions in the large-q limit. We give results for the spectral form factor, relaxation of local observables, bipartite entanglement growth and operator spreading. CONTENTS arXiv:1712.06836v3 [cond-mat.stat-mech]

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 study spectral statistics in spatially extended chaotic quantum many-body systems, using simple lattice Floquet models without time-reversal symmetry. Computing the spectral form factor K(t) analytically and numerically, we show that it follows random matrix theory (RMT) at times longer than a many-body Thouless time, t_{Th}. We obtain a striking dependence of t_{Th} on the spatial dimension d and size of the system. For d>1, t_{Th} is finite in the thermodynamic limit and set by the intersite coupling strength. By contrast, in one dimension t_{Th} diverges with system size, and for large systems there is a wide window in which spectral correlations are not of RMT form. Lastly, our Floquet model exhibits a many-body localization transition, and we discuss the behavior of the spectral form factor in the localized phase.