There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Topological insulators are new states of quantum matter which cannot be adiabatically connected to conventional insulators and semiconductors. They are characterized by a full insulating gap in the bulk and gapless edge or surface states which are protected by time-reversal symmetry. These topological materials have been theoretically predicted and experimentally observed in a variety of systems, including HgTe quantum wells, BiSb alloys, and Bi2Te3 and Bi2Se3 crystals. Theoretical models, materials properties, and experimental results on two-dimensional and three-dimensional topological insulators are reviewed, and both the topological band theory and the topological field theory are discussed. Topological superconductors have a full pairing gap in the bulk and gapless surface states consisting of Majorana fermions. The theory of topological superconductors is reviewed, in close analogy to the theory of topological insulators.

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Topological insulators and superconductors

We show that the quantum spin Hall (QSH) effect, a state of matter with topological properties distinct from those of conventional insulators, can be realized in mercury telluride–cadmium telluride semiconductor quantum wells. When the thickness of the quantum well is varied, the electronic state changes from a normal to an “inverted” type at a critical thickness d c . We show that this transition is a topological quantum phase transition between a conventional insulating phase and a phase exhibiting the QSH effect with a single pair of helical edge states. We also discuss methods for experimental detection of the QSH effect.

Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells

Recent theory predicted that the quantum spin Hall effect, a fundamentally new quantum state of matter that exists at zero external magnetic field, may be realized in HgTe/(Hg,Cd)Te quantum wells. We fabricated such sample structures with low density and high mobility in which we could tune, through an external gate voltage, the carrier conduction from n-type to p-type, passing through an insulating regime. For thin quantum wells with well width d 6.3 nanometers), the nominally insulating regime showed a plateau of residual conductance close to 2e(2)/h, where e is the electron charge and h is Planck's constant. The residual conductance was independent of the sample width, indicating that it is caused by edge states. Furthermore, the residual conductance was destroyed by a small external magnetic field. The quantum phase transition at the critical thickness, d = 6.3 nanometers, was also independently determined from the magnetic field-induced insulator-to-metal transition. These observations provide experimental evidence of the quantum spin Hall effect.

Quantum Spin Hall Insulator State in HgTe Quantum Wells

Weyl and Dirac semimetals are three-dimensional phases of matter with gapless electronic excitations that are protected by topology and symmetry. As three-dimensional analogs of graphene, they have generated much recent interest. Deep connections exist with particle physics models of relativistic chiral fermions, and, despite their gaplessness, to solid-state topological and Chern insulators. Their characteristic electronic properties lead to protected surface states and novel responses to applied electric and magnetic fields. The theoretical foundations of these phases, their proposed realizations in solid-state systems, and recent experiments on candidate materials as well as their relation to other states of matter are reviewed.

Weyl and Dirac semimetals in three-dimensional solids

Fermionic alkaline-earth atoms have unique properties that make them attractive candidates for the realization of atomic clocks and degenerate quantum gases. At the same time, they are attracting considerable theoretical attention in the context of quantum information processing. Here we demonstrate that when such atoms are loaded in optical lattices, they can be used as quantum simulators of unique many-body phenomena. In particular, we show that the decoupling of the nuclear spin from the electronic angular momentum can be used to implement many-body systems with an unprecedented degree of symmetry, characterizedbytheSU(N)groupwithNaslargeas10.Moreover,theinterplayofthenuclearspinwiththeelectronicdegreeof freedomprovidedbyastableopticallyexcitedstateshouldenablethestudyofphysicsgovernedbythespin‐orbitalinteraction. Such systems may provide valuable insights into the physics of strongly correlated transition-metal oxides, heavy-fermion materials and spin-liquid phases.

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Two-orbital SU(N) magnetism with ultracold alkaline-earth atoms

The surface states of a topological insulator are described by an emergent relativistic massless Dirac equation in 2 + 1 dimensions. In contrast with graphene, there is an odd number of Dirac points, and the electron spin is directly coupled to the momentum. We show that a magnetic impurity opens up a local gap and suppresses the local density of states. Furthermore, the Dirac electronic states mediate an RKKY interaction among the magnetic impurities which is always ferromagnetic, whenever the chemical potential lies near the Dirac point. Through this exchange mechanism, magnetic atoms uniformly deposited on the surface of a topological insulator could naturally form a ferromagnetically ordered film. These effects can be directly measured in STM experiments. We also study the case of quenched disorder through a renormalization group analysis.

Magnetic impurities on the surface of a topological insulator.

We study a minimal Hubbard model for electronically driven superconductivity in a correlated flat miniband resulting from the superlattice modulation of a twisted graphene multilayer. The valley degree of freedom drastically modifies the nature of the preferred pairing states, favoring spin triplet d+id order with a valley singlet structure. We identify two candidates in this class, which are both topological superconductors. These states support half-vortices carrying half the usual superconducting flux quantum hc/(4e), and have topologically protected gapless edge states.

Topological Superconductivity in Twisted Multilayer Graphene

The stability to interactions and disorder of the quantum spin Hall effect (QSHE) proposed for time-reversal-invariant two-dimensional systems is discussed. The QSHE requires an energy gap in the bulk and gapless edge modes that conduct spin-up and spin-down excitations in opposite directions. When the number of Kramers pairs of edge modes is odd, certain one-particle scattering processes are forbidden due to a topological ${\mathbb{Z}}_{2}$ index. We show that in a many-body description, there are other scattering processes that can localize the edge modes and destroy the QSHE: the region of stability for both classes of models (even or odd number of Kramers pairs) is obtained explicitly in the chiral boson theory. For a single Kramers pair, the QSHE is stable to weak interactions and disorder, while for two Kramers pairs it is not; however, the two-pair case can be stabilized by either finite attractive or repulsive interactions. For the simplest case of a single pair of edge modes, it is shown that changing the screening length in an edge with screened Coulomb interactions can be used to drive a phase transition between the QSHE state and the ordinary insulator.

Stability of the quantum spin Hall effect: Effects of interactions, disorder, and Z 2 topology

Motivated by recent neutron scattering experiments, we study the ordering of spins in the iron-based superconductors $\mathrm{LaFeAs}({\mathrm{O}}_{1\ensuremath{-}x}{\mathrm{F}}_{x})$, assuming them in proximity to a Mott insulator in the phase diagram. The ground state of the parent system with $x=0$ is a spin-density wave with ordering wave vector $\stackrel{P\vec}{Q}=(0,\ensuremath{\pi})$ or $(\ensuremath{\pi},0)$. Upon raising the temperature, we find that the system restores SU(2) symmetry, while an Ising symmetry remains broken, explaining the experimentally observed lattice distortion to a monoclinic crystal structure. Upon further temperature increase, the spins finally disorder at a second transition. The phase transition driven by doping with charge carriers similarly splits into an $\mathrm{O}(3)$ transition and an Ising transition with $z=3$ at larger doping.

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Cenke Xu

Papers

Two-orbital SU(N) magnetism with ultracold alkaline-earth atoms

Magnetic impurities on the surface of a topological insulator.

Topological Superconductivity in Twisted Multilayer Graphene

Stability of the quantum spin Hall effect: Effects of interactions, disorder, and Z 2 topology

Ising and Spin orders in Iron-Based Superconductors