Quantum Matter

The standard theory of topological insulators and superfluids (or superconductors) assumes that the fermionic elementary excitations in these systems – electrons in the insulator and Bogoliubov quasiparticles in the superfluid – do not interact with one another. In this talk I will discuss extensions of this theory to include the effects of interparticle interactions on the topological surface states of 3D topological insulators and superfluids. First, I will argue that bulk quantum fluctuations of the superfluid order parameter in the only 3D topological superfluid known to date, 3He-B, can generate an effective two-body interaction between the surface Majorana fermions.

Possible consequences of this interaction will be discussed. Second, I will propose an extension of the phenomenological Landau theory of Fermi liquids to the surface states of 3D topological insulators, leading to the concept of helical Fermi liquid.

Recent progress on the construction of holographic lattices and its applications to AdS/CMT correspondence will be briefly reviewed. Our special interests will focus on the building of bulk geometry of gravity whose holographic duals exhibit metal-insulator transitions (MIT).  In particular, the Peierls phase transition induced by charge density waves is implemented in a holographic manner. The holographic entanglement entropy close to quantum critical points will be discussed as well.

Symmetry-protected topological (SPT) phases can be thought of as generalizations of topological insulators. Just as topological insulators have robust boundary modes protected by time reversal and charge conservation symmetry, SPT phases have boundary modes protected by more general symmetries. In this talk, I will describe a method for analyzing 2D and 3D SPT phases using braiding statistics. More specifically, I will show that 2D and 3D SPT phases can be characterized by gauging their symmetries and studying the braiding statistics of their gauge flux excitations. The 3D case is of particular interest as it involves a generalization of quasiparticle braiding statistics to three dimensions.

I will show how hydrodynamics is modified if the underlying fluid constituents are massless Weyl fermions, which are anomalous at the quantum level. Because of the nondissipative nature of the modification I will construct a partition function which compactly describes the transport properties of the system and I will explain how the anomalous properties can be understood in terms of kinetic theory and heat kernels. Finally I will comment on possible applications to heavy-ion collisions, condensed matter systems such as Weyl-semi metals and potential future questions such as anomalous corrections to entanglement entropy.

We examine the interplay of symmetry and topological order in 2+1D topological phases of matter. We define the topological symmetry group, characterizing symmetry of the emergent topological quantum numbers, and describe its relation with the microscopic symmetry of the physical system.

We then derive a general classification of symmetry fractionalization in topological phases, including phases that are non-Abelian and symmetries that permute the quasiparticle types and/or are anti-unitary. We develop a general algebraic theory of extrinsic defects (fluxes) associated with elements of the symmetry group, which provides a general classification of symmetry-enriched topological phases derived from a topological phase of matter with symmetry. We also examine the promotion of the global symmetry to a local gauge invariance, wherein the extrinsic defects are turned into deconfined quasiparticle excitations, which results in a different topological phase.