LaAlO 3 and SrTiO 3 are insulating, non-magnetic oxides, yet the interface between them exhibits a two-dimensional electron system with high electron mobility 1 , superconductivity at low temperatures 2-6 and electric-field-tuned metal-insulator and superconductor-insulator phase transitions 3,6-8. Bulk magnetization and magnetoresistance measurements also indicate some form of magnetism depending on preparation conditions 5,9-11 and a tendency towards nanoscale electronic phase separation 10. Here we use local imaging of the magnetization and magnetic susceptibility to directly observe a landscape of ferromagnetism, paramagnetism and superconductivity. We find submicrometre patches of ferromagnetism in a uniform background of paramagnetism, with a non-uniform, weak diamagnetic superconducting susceptibility at low temperature. These results demonstrate the existence of nanoscale phase separation as indicated by theoretical predictions based on nearly degenerate interface sub-bands associated with the Ti orbitals 12,13. The magnitude and temperature dependence of the paramagnetic response indicate that the vast majority of the electrons at the interface are localized 14 , and do not contribute to transport measurements 3,6,7. In addition to the implications for magnetism, the existence of a two-dimensional superconductor at an interface with highly broken inversion symmetry and a ferromagnetic landscape in the background indicates the potential for exotic superconducting phenomena. Coexistence of ferromagnetism and superconductivity in nature is rare 15-19. The LaAlO 3 /SrTiO 3 (LAO/STO) interface is a new system for studying this coexistence. LAO and STO are both perovskite band insulators with no magnetic order in their bulk form. For LAO grown on the TiO 2-terminated STO substrate, a high-mobility electron gas was observed at the interface 1. Electronic reconstruction, driven by the polar/nonpolar interface, is thought to move charge from the LAO layers across the interface into the STO, causing an effective electronic doping responsible for the observed conductivity 1. The interplay of this effect with oxygen vacancies and structural changes 20 , and the relative contribution of these three effects to the carrier concentration, remains a subject of debate. Significant variability in the physical properties in similar samples indicates that the ground state of this interface system is sensitive to small changes in growth conditions. Superconductivity 2-5 and features interpreted as interface magnetism 5,9,10 have been independently observed at the LAO/STO interface through transport and bulk magnetization measurements. One recent study inferred the existence of both LETTERS NATURE PHYSICS

Quantum mechanics predicts that the equilibrium state of a resistive electrical circuit contains a dissipationless current. This persistent current has been the focus of considerable theoretical and experimental work, but its basic properties remain a topic of controversy. The main experimental challenges in studying persistent currents have been the small signals they produce and their exceptional sensitivity to their environment. To address these issues we have developed a new technique for detecting persistent currents which offers greatly improved sensitivity and reduced measurement back action. This allows us to measure the persistent current in metal rings over a wider range of temperature, ring size, and magnetic field than has been possible previously. We find that measurements of both a single ring and arrays of rings agree well with calculations based on a model of non-interacting electrons. An electrical current induced in a resistive circuit will rapidly decay in the absence of an applied voltage. This decay reflects the tendency of the circuit's electrons to dissipate energy and relax to their ground state. However quantum mechanics predicts that the electrons' many-body ground state (and, at finite temperature, their thermal equilibrium state) may itself contain a "persistent" current which flows through the resistive circuit without dissipating energy or decaying. A dissipationless equilibrium current flowing through a resistive circuit is highly counterintuitive, but it has a familiar analog in atomic physics: some atomic species' electronic ground states possess non-zero orbital angular momentum, equivalent to a current circulating around the atom. Theoretical treatments of persistent currents (PC) in resistive metal rings have been developed over a number of decades (see [1,2] and references therein). Calculations which take 1

The quantum spin Hall (QSH) state is a state of matter characterized by a non-trivial topology of its band structure, and associated conducting edge channels. The QSH state was predicted and experimentally demonstrated to be realized in HgTe quantum wells. The existence of the edge channels has been inferred from local and non-local transport measurements in sufficiently small devices. Here we directly confirm the existence of the edge channels by imaging the magnetic fields produced by current flowing in large Hall bars made from HgTe quantum wells. These images distinguish between current that passes through each edge and the bulk. On tuning the bulk conductivity by gating or raising the temperature, we observe a regime in which the edge channels clearly coexist with the conducting bulk, providing input to the question of how ballistic transport may be limited in the edge channels. Our results represent a versatile method for characterization of new QSH materials systems.

These authors contributed equally to this work.The ability to control materials properties through interface engineering is demonstrated by the appearance of conductivity at the interface of certain insulators, most famously the {001} interface of the band insulators LaAlO 3 (LAO) and TiO 2 -terminated SrTiO 3 (STO) 1,2 . Transport and other measurements in this system display a plethora of diverse physical phenomena 3-14 . To better understand the interface conductivity, we used scanning superconducting quantum interference device (SQUID) microscopy to image the magnetic field locally generated by current in an interface. At low temperature, we found that the current flowed in highly conductive narrow paths oriented along the crystallographic axes, embedded in a less conductive background. The configuration of these paths changed upon thermal cycling above the STO cubic to tetragonal structural transition temperature, implying that local conductivity is strongly modified by STO tetragonal domain