We suggest that if a localized phase at nonzero temperature $T>0$ exists for strongly disordered and weakly interacting electrons, as recently argued, it will also occur when both disorder and interactions are strong and $T$ is very high. We show that in this high-$T$ regime the localization transition may be studied numerically through exact diagonalization of small systems. We obtain spectra for one-dimensional lattice models of interacting spinless fermions in a random potential. As expected, the spectral statistics of finite-size samples cross over from those of orthogonal random matrices in the diffusive regime at weak random potential to Poisson statistics in the localized regime at strong randomness. However, these data show deviations from simple one-parameter finite-size scaling: the apparent mobility edge ``drifts'' as the system's size is increased. Based on spectral statistics alone, we have thus been unable to make a strong numerical case for the presence of a many-body localized phase at nonzero $T$

We discuss fluctuating order in a quantum disordered phase proximate to a quantum critical point, with particular emphasis on fluctuating stripe order. Optimal strategies for extracting information concerning such local order from experiments are derived with emphasis on neutron scattering and scanning tunneling microscopy. These ideas are tested by application to two model systemsthe exactly solvable one dimensional electron gas with an impurity, and a weakly-interacting 2D electron gas. We extensively review experiments on the cuprate high-temperature superconductors which can be analyzed using these strategies. We adduce evidence that stripe correlations are widespread in the cuprates. Finally, we compare and contrast the advantages of two limiting perspectives on the high-temperature superconductor: weak coupling, in which correlation effects are treated as a perturbation on an underlying metallic (although renormalized) Fermi liquid state, and strong coupling, in which the magnetism is associated with well defined localized spins, and stripes are viewed as a form of micro-phase separation. We present quantitative indicators that the latter view better accounts for the observed stripe phenomena in the cuprates.

We consider isolated quantum systems with all of their many-body eigenstates localized. We define a sense in which such systems are integrable, and discuss a method for finding their localized conserved quantum numbers ("constants of motion"). These localized operators are interacting pseudospins and are subject to dephasing but not to dissipation, so any quantum states of these pseudospins can in principle be recovered via (spin) echo procedures. We also discuss the spreading of entanglement in many-body localized systems, which is another aspect of the dephasing due to interactions between these localized conserved operators. PACS numbers:Isolated quantum many-body systems with shortrange interactions and static randomness may be in a many-body localized phase where they do not thermally equilibrate under their own dynamics. While this possibility was pointed out long ago by Anderson [1], such localization of highly-excited states in systems with interactions did not receive a lot of attention until after Basko, et al.[2] forcefully brought the subject into focus. Isolated systems in the many-body localized phase have strictly zero thermal conductivity [2], so if some energy is added to the system locally, it only excites localized degrees of freedom and does not diffuse, even when the system's energy density corresponds to a nonzero (even infinite [3]) temperature.We expect that the many-body eigenstates of a system's Hamiltonian in the localized phase are product states of localized degrees of freedom, with some shortrange "area-law" entanglement between the "bare" local degrees of freedom. One goal of this paper is to explore how one can define suitably "dressed" localized pseudospin operators in terms of which the many-body eigenstates within the localized phase are indeed precisely product states with zero entanglement. When the Hamiltonian is then expressed in terms of these dressed localized pseudospins it has exponentially decaying long-range interactions, and it is these long-range interactions that cause decoherence and dephasing of local observables in the many-body insulator. These interactions also cause the spreading of entanglement for a nonentangled initial product state of the bare spins, as has been seen and explored in Refs. [4][5][6][7][8].To be concrete, assume we have a system of N spin-1/2's {σ i } on some lattice (say, in one, two or three dimensions). For an example, see, e.g., Ref. [9]. Our system has a specific random Hamiltonian H that contains only short-range interactions and strong enough static random fields on each spin so that, with probability one in the limit of large N , all 2 N many-body eigenstates of this H are localized. The construction we present below should be readily generalizable to local operators with more than two states. It should also be generalizable to systems where the dominant strong randomness is instead the spin-spin interactions rather than random fields. In those cases, the pseudospins we will construct may instead be localized domain wall operators [...

We develop a microscopic theory of the electronic nematic phase proximate to an isotropic Fermi liquid in both two and three dimensions. Explicit expressions are obtained for the small amplitude collective excitations in the ordered state; remarkably, the nematic Goldstone mode (the directorwave) is overdamped except along special directions dictated by symmetry. At the quantum critical point we find a dynamical exponent of z = 3, implying stability of the gaussian fixed point. The leading perturbative effect of the overdamped Goldstone modes leads to a breakdown of Fermi liquid theory in the nematic phase and to strongly angle dependent electronic self energies around the Fermi surface. Other metallic liquid crystal phases, e. g. a quantum hexatic, behave analogously.There is a growing body of both experimental and theoretical evidence for the relevance of inhomogeneous and/or anisotropic metallic phases in a wide array of highly correlated electronic systems. Quasi-one dimensional (stripe or "electronic smectic") phases have been observed in a large variety of transition metal oxides. 1More recently, 2,3 the dramatic discovery of a metallic phase with a strongly anisotropic resistivity tensor for a range of magnetic fields in ultra clean heterojunctions has provided clear evidence of the existence of a "quantum Hall nematic" phase. In parallel, theoretical work 4,5 has been carried out on electronic liquid crystal phases. These ground-state phases are classified, based on broken symmetries, by analogy with classical liquid crystals. So far, these studies have focused primarily on the smectic, which is a unidirectional density wave with broken translational symmetry in only one direction, but which supports liquid-like electron flow 6,7 , and to a lesser extent on the nematic, which is uniform but anisotropic (breaks rotational symmetry) 8,9 . A nematic state in the proximity to the smectic state can be visualized most naturally as a melted smectic, i. e. a smectic with dislocations. However, a theory of the nematic phase based on this picture has yet to be satisfactorily formalized.In this paper we approach the nematic metal via a complimentary route, from the isotropic and weakly correlated side. In this limit, the zero temperature isotropic to nematic transition is a Fermi surface instability. The director order parameter which characterizes the broken symmetry of the nematic state is a rank two symmetric traceless tensor which is even under time reversal. In two dimensions, but not in three, the order parameter is odd under 90• spatial rotation. For simplicity we first consider spinless fermions in two dimensions with full rotational symmetry, deferring until later any discussion of spin and the symmetry breaking effects of the crystal fields which are inevitable in actual solids. While many microscopic definitions of this order parameter are possible, we shall see that the natural one in the present context is the quadrupole densitŷ