Talks

Topological phases of matter entail a prominent research theme, featuring
distinct characteristics that include protected metallic edge states and
the possibility of fractionalized excitations. With the advent of symmetry
protected topological (SPT) phases, many of these phenomena have
effectively become accessible in the form of readily available band
structures. Whereas the role of (anti-)unitary symmetries in such SPT
states has been thoroughly understood, the inclusion of lattice symmetries
provides for an active area of research.

In this talk, I will present a short overview of results on defects in SPT
states that directly motivate the existence of additional physics beyond
the characterization based on (anti-)unitary symmetries. More importantly,
I will then connect these ideas to recent work in which we were able to map
out all different gapped phases of free fermion systems in the presence of
solely lattice symmetries. This revolves around a very simple algorithm
that matches a rather involved mathematical perspective in terms of a
framework called K-theory. I will then discuss the implications of these
combinatorial arguments, such as their impact on the description of Weyl
phases, and sketch related ideas and future perspectives.

The entanglement properties of random quantum states or dynamics are important to the study of a broad spectrum of disciplines of physics, ranging from quantum information to high energy and many-body physics. This work investigates the interplay between the degrees of entanglement and randomness in pure states and unitary channels, by employing tools from random matrix theory, representation theory, combinatorics, Weingarten calculus etc. We reveal strong connections between designs (distributions of states or unitaries that match certain moments of the uniform Haar measure) and generalized entropies (entropic functions that depend on certain powers of the density operator), by showing that Renyi entanglement entropies averaged over designs of the same order are almost maximal. This strengthens the celebrated Page's theorem. Moreover, we find that designs of an order that is logarithmic in the dimension maximize all Renyi entanglement entropies, and so are completely random in terms of the entanglement spectrum. Our results relate the behaviors of Renyi entanglement entropies to the complexity beyond scrambling/thermalization/chaos in terms of the degree of randomness, and suggest a generalization of the fast scrambling conjecture.  (Refs: 1703.08104, 1709.04313)

Many strongly-correlated systems like high Tc cuprates and heavy fermions have interesting features going beyond quasi-particle description. While Sachdev-Ye-Kitaev(SYK) models are exactly solvable models that can provide a platform to study these physics. In this talk, I will discuss interesting features about the SYK models, including extensive zero temperature entropy and maximally chaos. I will also show some generalization of the SYK models and discuss physical insights from them.

The observables of the large-scale structure such as galaxy number density generally depends on the density environment (of a few hundred Mpc). The dependence can traditionally be studied by performing gigantic cosmological N-body simulations and measuring the observables in different density environments. Alternatively, we perform the so-called "separate universe simulations", in which the effect of the environment is absorbed into the change of the cosmological parameters. For example, an overdense region is equivalent to a universe with positive curvature, hence the structure formation changes accordingly compared to the region without overdensity. In this talk, I will introduce the "separate universe mapping", and present how the power spectrum
and halo mass function change in different density environments, which are equivalent to the squeezed bispectrum and the halo bias, respectively. I will then discuss the extension of this approach to inclusion of additional fluids such as massive neutrinos. This allows us to probe the novel scale-dependence of halo bias and squeezed bispectrum caused by different evolutions of the background overdensities of cold dark matter and the additional fluid. Finally, I will present one application of the separate universe simulations to predict the squeezed bispectrum formed by small-scale Lyman-alpha forest power spectrum and large-scale lensing convergence, and compare with the measurement from BOSS Lyman-alpha forest and Planck lensing map.

Finite-size spectra and entanglement both characterize nonlocal physics of quantum systems, and are universal properties of a CFT. I discuss the energy spectrum of the Wilson-Fisher CFT on the torus in the \epsilon and 1/N expansions. I also consider a class of deconfined quantum critical points where the torus spectrum contains signatures of proximate Z2 topological order. Finally, I compute the entanglement entropy of the Wilson-Fisher and Gross-Neveu CFTs in the large N limit, where an exact mapping to free field entanglement is obtained. Comparison is made with numerics.

We explain how a doubled version of the Beilinson-Bernstein localization functor can be understood using the geometry of the wonderful compactification of a group. Specifically, bimodules for the Lie algebra give rise to monodromic D-modules on the horocycle space, and to filtered D-modules on the group that respect a certain matrix coefficients filtration. These two categories of D-modules are related via an associated graded construction in a way compatible with localization, Verdier specialization, and additional structures. This is joint work with David Ben-Zvi and David Nadler.