Colloquium

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.

Theoretical insights originated from the study of black holes combined with developments in string theory indicate that space time and gravity are emergent. A central role in these developments is played by the holographic principle. I will present a heuristic argument that indicates that at a microscopic level gravity is an entropic force caused by changes in the available phase space due to the displacement of material bodies. Refinement of the argument makes clear that this entropic view on gravity is consistent with quantum mechanics and supported by various results in string theory. I end with some thoughts on the possible emergence of the other forces of Nature.

Frustrated magnets are materials in which localized magnetic moments, or spins, interact through competing exchange interactions that cannot be simultaneously satisfied, giving rise to a large degeneracy of the system ground state. Under certain conditions, this can lead to the formation of fluid-like states of matter, so-called spin liquids, in which the constituent spins are highly correlated but still fluctuate strongly down to a temperature of absolute zero. The fluctuations of the spins in a spin liquid can be classical or quantum and show remarkable collective phenomena such as emergent gauge fields and fractional particle excitations. This exotic behaviour is now being uncovered in the laboratory, providing insight into the properties of spin liquids and challenges to the theoretical description of these materials.

The primordial density fluctuations that seeded large-scale structure are known to be nearly Gaussian, as predicted by most early universe models like slow-roll inflation. Many of these models predict a small (but nonzero!) amount of primordial non-gaussianity, which can subtly affect the statistics of CMB anisotropies. Surprisingly, even a small primordial non-gaussianity can produce enormous changes in the large-scale clustering of galaxies and quasars at late times. I will describe the origin of this effect, and review recent constraints on non-gaussianity using measurements of the clustering of galaxies and quasars in SDSS.

We study a novel state of matter: algebraic Bose liquid (ABL). An ABL is a quantum bosonic system on a 2d or 3d lattice that does not break any symmetry in its ground state, but still able to stabilize a gapless spectrum. At high energy these boson systems only have the simplest U(1) global symmetry associated with the conservation of boson number, but at low energy the system is described by self-dual gauge fields. In this talk we will present two new ABL phases emerged from a quantum Boson model on the cubic lattice. At low energy these two ABL phases are described by the linearized z=2 and z=3 Lifshitz gravity respectively. We will show that the self-duality of the field theory is crucial to guarantee the stability of the ABL. References: Cenke Xu, arXiv:cond-mat/0602443, Phys. Rev. B. 74, 224433 Cenke Xu and Petr Horava, arXiv:1003.0009

Recently, a new class of topological states has been theoretically predicted and experimentally realized. The topological insulators have an insulating gap in the bulk, but have topologically protected edge or surface states due to the time reversal symmetry. In two dimensions the edge states give rise to the quantum spin Hall (QSH) effect, in the absence of any external magnetic field. I shall review the theoretical prediction[1] of the QSH state in HgTe/CdTe semiconductor quantum wells, and its recent experimental observation[2]. The edge states of the QSH state supports fractionally charged excitations[3]. The QSH effect can be generalized to three dimensions as the topological magneto-electric effect (TME) of the topological insulators[4]. Bi2Te3, Bi2Se3 and Sb2Te3 are theoretically predicted to be topological insulators with a single Dirac cone on the surface[5]. I shall present a realistic experimental proposals to observe the magnetic monopoles on the surface of topological insulators[6]. Topological superconductors and superfluid have been theoretically proposed recently [7], in both two and three dimensions. They have a full pairing gap in the bulk, and their mean field Hamiltonian look identical to that of the topological insulators. However, the gapless surface states consists of a single Majorana cone, containing only half the degree of freedom compared to the single Dirac cone on the surface of a topological insulators. I shall discuss their physics properties and the search for these novel states in real materials. [1] A. Bernevig, T. Hughes and S. C. Zhang, Science, 314, 1757, (2006) [2] M. Koenig et al, Science 318, 766, (2007) [3] J. Maciejko, Chaoxing Liu, Yuval Oreg, Xiao-Liang Qi, Congjun Wu, and Shou-Cheng Zhang, , Phys. Rev. Lett. {\bf 102}, 256803 (2009). [4] Xiao-Liang Qi, Taylor Hughes and Shou-Cheng Zhang, Phys. Rev B. 78, 195424 (2008) [5] Haijun Zhang, Chao-Xing Liu, Xiao-Liang Qi, Xi Dai, Zhong Fang, and Shou-Cheng Zhang, Nature Physics 5, 438 (2009). [6] Xiao-Liang Qi, Run-Dong Li, Jiadong Zang and Shou-Cheng Zhang, Science 323, 1184 (2009). [7] Xiao-Liang Qi, Taylor L. Hughes, Srinivas Raghu and Shou-Cheng Zhang, Phys. Rev. Lett. 102, 187001 (2009)

The hot, gaseous atmospheres of galaxies and clusters of galaxies are repositories for the energy output from accreting, supermassive black holes located in the nuclei of galaxies. X-ray observations show that star formation fueled by gas condensing out of hot atmospheres is strongly suppressed by feedback from active galactic nuclei (AGN). This mechanism may solve several outstanding problems in astrophysics, including the numbers of luminous galaxies and their colors, and the excess number of hot baryons in the Universe. The most energetic AGN outbursts may be powered by rapidly-spinning, ultra-massive black holes.

I will present recent numerical results obtained in collaboration with Frans Pretorius that describe head-on collisions of two solitons coupled to the general relativistic gravitational field and boosted to ultra relativistic energies. The calculations show, for the first time, that at sufficiently high energies such a collision leads to black hole formation, consistent with hoop conjecture arguments. This implies that the non-linear gravitational interaction between the kinetic energy of the solitons results in gravitational collapse, and the arguments for black hole formation in super-Planck scale particle collisions are robust. I will also speculate on the nature of the threshold of black hole formation in the model.

Quantum Bayesianism is a point of view on quantum foundations that says that there is no such thing as a “measurement problem” because there is no such THING as a quantum state: Quantum states are not things---instead information. But the view doesn’t stop there; it starts there! Taking the idea seriously over the last 15 years has been the direct motivation for a number of theorems and objects in quantum information theory: from the no-broadcasting theorem, to the quantum de Finetti theorem, and even some quantum cryptographic alphabets. I will review some of this, and then move on to the holy grail of present efforts: Finding an efficient representation of quantum states in terms of a singular probability function. Doing so leads to the hard technical problem of demonstrating the existence of a certain very symmetric sets of quantum states, and holds out the hope of understanding the amount of “quantum stuff” in a physical system in terms of a single parameter. (I.e., there is the THING that the quantum state is not).

Self-assembly refers to any thermodynamic process in which a bunch of particles (molecules, biomolecules, polymers, colloids) come together in solution to form an ordered structure. In living things it is a widely used and robust manufacturing tool: DNA, RNA and proteins spontaneously form three dimensional structures, and supramolecular structures emerge from protein aggregates with staggering degrees of ordering and specificity. By contrast, most synthetic systems in soft condensed matter do not assemble robustly. In this talk I will discuss experiments on simple systems that allow us to probe the physics and thermodynamics of self-assembly. We use systems consisting of small numbers (N ⇐ 12) of confined spherical colloidal particles to understand what physical parameters (interactions) determine how a system will assemble. We find that the probability of self-assembling a particular configuration can be understood in terms of the geometry of sphere packings. The geometrical model gives some insights into how phase transitions emerge as N approaches the bulk limit. At the same time, it yields some general insights into the design principles for robust self-assembly.