Colloquium

Quantum error correcting codes and topological quantum order (TQO) are inter-connected fields that study non-local correlations in highly entangled many-body quantum states. In this talk I will argue that each of these fields offers valuable techniques for solving problems posed in the other one. First, we will discuss the zero-temperature stability of TQO and derive simple conditions that guarantee stability of the spectral gap and the ground state degeneracy under generic local perturbations. These conditions thus can be regarded as a rigorous definition of TQO. Our results apply to any quantum spin Hamiltonian that can be written as a sum of geometrically local commuting projectors on a D-dimensional lattice. This large class of Hamiltonians includes Levin-Wen string-net models and Kitaev's quantum double models. Secondly, we derive upper bounds on the parameters of quantum codes with local check operators and discuss the implications for feasibility of a quantum self-correcting memory.

The average quantum physicist on the street believes that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under combined matrix transposition and complex conjugation) in order to guarantee that the energy eigenvalues are real and that time evolution is unitary. However, the Hamiltonian $H=p^2+ix^3$, which is obviously not Dirac Hermitian, has a real positive discrete spectrum and generates unitary time evolution, and thus it defines a fully consistent and physical quantum theory. Evidently, the axiom of Dirac Hermiticity is too restrictive. While $H=p^2+ix^3$ is not Dirac Hermitian, it is PT symmetric; that is, invariant under combined space reflection P and time reversal T. The quantum mechanics defined by a PT-symmetric Hamiltonian is a complex generalization of ordinary quantum mechanics. When quantum mechanics is extended into the complex domain, new kinds of theories having strange and remarkable properties emerge. Some of these properties have recently been verified in laboratory experiments. If one generalizes classical mechanics into the complex domain, the resulting theories have equally remarkable properties.

This talk will describe the best current understanding of the interior structure of astronomically realistic black holes. A common misconception is that matter falling into a black hole simply falls to a central singularity, and that's that. Reality is much more interesting. Rotating black holes have not only outer horizons, but also inner horizons. Penrose (1968) first pointed out that an infaller falling through the inner horizon would see the outside Universe infinitely blueshifted, and he speculated that this would destabilize the inner horizon. The expectation was supported by linear perturbation theory, but it was not until 1990 that Poisson & Israel were able to clarify the nonlinear evolution of the instability at the inner horizon, which they called mass inflation. Inflation accelerates ingoing (positive energy) and outgoing (negative energy) streams to exponentially huge energies. The black hole thus behaves like a particle accelerator of extraordinary power, accelerating ingoing and outgoing particles to collide with each other at super-Planckian energies. The talk raises the fundamental question: What does Nature do with this remarkable accelerator?

The gravitational observatory LISA will detect radiation from massive black hole sources at cosmological distances, accurately measure their luminosity distance and help identify the electromagnetic counterparts that such sources may generate. I will describe various astrophysical scenarios for the generation of electromagnetic counterparts and discuss observational strategies aimed at identifying them. Successful identifications will enable novel studies of black hole astrophysics and cosmological physics.

A system of spins with complicated interactions between them can have many possible configurations. Many configurations will be local minima of the energy, and to get from one local minimum to another requires changing the state of very many spins. A system like this is called a spin glass, and at low temperatures tends to get caught for very long times at a local minimum of energy, rather than reaching its true ground state. Indeed, in many cases, finding the ground state energy of a spin glass is a computationally hard problem, too hard to be solved on a classical computer or even a quantum computer in any reasonable amount of time. Which types of interactions give us computationally hard problems and spin glasses? I will survey what is known as we close in on finding the simplest complex spin systems.

The AdS/CFT correspondence relates large-N, planar quantum gauge theories to string theory on the Anti-de-Sitter background. I will discuss exact results in field theories with AdS duals, which can be obtained with the help of diagram resummations, mapping to quantum spin chains and two-dimensional sigma-models.

"Conventional" superconductivity is one of the most dramatic phenomena in condensed matter physics, and yet by the 1970's it was fully understood - a solved problem much like quantum electrodynamics. The discovery of high temperature superconductivity changed all that and opened the door, not only to higher Tc's, but also to a wealth of even more exotic phenomena, including things like topologically ordered superconductors with factional vortices and non-Abelian statistics. I will describe some of the evolution of the field of exotic superconductivity, with a focus on recent theoretical and experimental work which sheds light on whether strontium ruthenate supports topological chiral superconductivity.

A Majorana fermion is a particle that is its own antiparticle. It has been studied in high energy physics for decades, but has not been definitely observed. In condensed matter physics, Majorana fermions appear as low energy fractionalized quasi-particles with non-Abelian statistics and inherent nonlocality. In this talk I will first discuss recent theoretical proposals of realizing Majorana fermions in solid-state systems, including topological insulators and nanowires. I will next propose experimental setups to detect the existence of Majorana fermions and their striking properties.

In this talk we will explore a "toy model" of quantum theory that is similar to actual quantum theory, but uses scalars drawn from a finite field. The set of possible states of a system is discrete and finite. Our theory does not have a quantitative notion of probability, but only makes the "modal" distinction between possible and impossible measurement results. Despite its very simple structure, our toy model nevertheless includes many of the key phenomena of actual quantum systems: interference, complementarity, entanglement, nonlocality, and the impossibility of cloning.

This talk will discuss some surprising links which have emerged in the last few years between two at first sight distinct areas of mathematical physics: the spectral properties of certain simple schroedinger-like equations, and the Bethe ansatz techniques which are used to compute the energies of states in integrable quantum field theories. No knowledge of either area will be assumed.

An entirely new kind of band insulator was discovered recently. These new electronic states - called "topological insulators" - are fundamentally different from standard band insulators. They are distinguished by the fact that their edges (in the 2D case) or surfaces (in the 3D case) support gapless transport which is extremely robust. In the two dimensional case, topological insulators can be thought of as time reversal invariant analogues of integer quantum Hall states. This analogy is intriguing since integer quantum Hall states are a special case of the far richer class of fractional quantum Hall states. It is natural to wonder: can topological insulators also be generalized? In this talk, I will investigate this question. I will show that, just as in quantum Hall systems, electron interactions allow for a whole new class of states - which we call "fractional topological insulators." These states have excitations with fractional charge and statistics, in addition to protected edge modes.