We use exact diagonalization to explore the many-body localization transition in a random-field spin-1/2 chain. We examine the correlations within each many-body eigenstate, looking at all high-energy states and thus effectively working at infinite temperature. For weak random field the eigenstates are thermal, as expected in this nonlocalized, "ergodic" phase. For strong random field the eigenstates are localized, with only short-range entanglement. We roughly locate the localization transition and examine some of its finite-size scaling, finding that this quantum phase transition at nonzero temperature might be showing infinite-randomness scaling with a dynamic critical exponent z → ∞.

Closed quantum systems with quenched randomness exhibit many-body localized regimes wherein they do not equilibrate, even though prepared with macroscopic amounts of energy above their ground states. We show that such localized systems can order, in that individual many-body eigenstates can break symmetries or display topological order in the infinite volume limit. Indeed, isolated localized quantum systems can order even at energy densities where the corresponding thermally equilibrated system is disordered, i. e.: localization protects order. In addition, localized systems can move between ordered and disordered localized phases via non-thermodynamic transitions in the properties of the many-body eigenstates. We give evidence that such transitions may proceed via localized critical points. We note that localization provides protection against decoherence that may allow experimental manipulation of macroscopic quantum states. We also identify a 'spectral transition' involving a sharp change in the spectral statistics of the many-body Hamiltonian.

The quantum random energy model provides a mean-field description of the equilibrium spin glass transition. We show that it further exhibits a many-body localization -delocalization (MBLD) transition when viewed as a closed quantum system. The mean-field structure of the model allows an analytically tractable description of the MBLD transition using the forward-scattering approximation and replica techniques. The predictions are in good agreement with the numerics. The MBLD lies at energy density significantly above the equilibrium spin glass transition, indicating that the closed system dynamics freezes well outside of the traditional glass phase. We also observe that the structure of the eigenstates at the MBLD critical point changes continuously with the energy density, raising the possibility of a family of critical theories for the MBLD transition.a. Introduction-Equilibrium statistical mechanics applied to closed dynamical systems relies on the assumption of ergodicity. Until recently it was believed that even weak interaction between the elementary constituents of matter guarantees ergodicity. A notable counterexample was provided by the seminal work [1] where the authors showed that, for systems with quenched disorder, Anderson localization of noninteracting particles [2] can persist in the presence of (sufficiently weak) interactions leading precisely to the failure of ergodicity. The recent development of well-isolated experimental quantum many-body systems has spurred a great deal of numerical and theoretical work suggesting that many-body localization (MBL) is, indeed, a robust, universal phenomenon: it exists in any spatial dimension, for both bosons and fermions, and for generic short-range interactions [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. The MBL phase is characterized by the complete absence of transport (e.g. of particle, spin and energy) and by the permanence of the memory of the initial state in local observables for all time. In this respect, the MBL phase may be viewed as the quintessential quantum glass.By changing the control parameters such as energy, strength of interactions or disorder, a MBL system can transit into a delocalized phase, where transport is restored and the predictions of equilibrium statistical mechanics hold. In particular, the eigenstates satisfy the eigenstate thermalization hypothesis (ETH) [23][24][25], exhibiting thermal behavior for local observables. If the MBL-delocalization (MBLD) transition is obtained by changing just the energy of the state, all other parameters remaining the same, the critical energy density separating the two phases is also referred to, borrowing ter- * Correspondence: claumann@uw.edu † On leave from: Abdus Salam ICTP, Strada Costiera 11, 34151 Trieste, Italy minology from the theory of Anderson localization, as a many-body mobility edge. When one passes from the micro canonical to the canonical statistical description, the mobility edge defines a critical temperature T MBL .As MBL systems exhibit glas...

We propose a tensor network encoding the set of all eigenstates of a fully many-body localized system in one dimension. Our construction, conceptually based on the ansatz introduced in Phys. Rev. B 94, 041116(R) (2016), is built from two layers of unitary matrices which act on blocks of l contiguous sites. We argue that this yields an exponential reduction in computational time and memory requirement as compared to all previous approaches for finding a representation of the complete eigenspectrum of large many-body localized systems with a given accuracy. Concretely, we optimize the unitaries by minimizing the magnitude of the commutator of the approximate integrals of motion and the Hamiltonian, which can be done in a local fashion. This further reduces the computational complexity of the tensor networks arising in the minimization process compared to previous work. We test the accuracy of our method by comparing the approximate energy spectrum to exact diagonalization results for the random-field Heisenberg model on 16 sites. We find that the technique is highly accurate deep in the localized regime and maintains a surprising degree of accuracy in predicting certain local quantities even in the vicinity of the predicted dynamical phase transition. To demonstrate the power of our technique, we study a system of 72 sites, and we are able to see clear signatures of the phase transition. Our work opens a new avenue to study properties of the many-body localization transition in large systems.

We present a numerical study of the diffusion of energy at high temperature in strongly disordered chains of interacting classical spins evolving deterministically. We find that quenched randomness strongly suppresses transport, with the diffusion constant becoming reduced by several orders of magnitude upon the introduction of moderate disorder. We have also looked for but not found signs of a classical many-body localization transition at any nonzero strength of the spin-spin interactions.