Condensed Matter

"Recently, exactly solvable 3D lattice models have been discovered for a new kind of phase, dubbed fracton topological order, in which the topological excitations are immobile or are bound to lines or surfaces. Unlike liquid topologically ordered phases (e.g. Z_2 gauge theory), which are only sensitive to topology (e.g. the ground state degeneracy only depends on the topology of spatial manifold), fracton orders are also sensitive to the geometry of the lattice. This geometry dependence allows for remarkably new physics which was forbidden in topologically invariant phases of matter.

In this talk, I will review the X-cube model [1] of fracton order. I will then explain how geometry dependence allows for braiding of point-like particles in this 3D phase. I'll summary how the X-cube model can be described by a quantum field theory, which is analogous to a topological quantum field theory (TQFT). [3] We will see that the gauge invariance of the field theory results in the mobility restrictions of the topological excitations by imposing a new kind of geometric charge conservation. I will conclude by briefly discussing current work on the remarkable geometry-dependent phenomenology of fracton order. For example, I will explain why even on a manifold with trivial topology, spatial curvature can induce a robust ground state degeneracy.

[1] Vijay, Haah, Fu 1603.04442

[2] Slagle, Kim 1704.03870"

Tensor network/spacetime correspondences explore the exciting idea that geometric information about a quantum state might be related to the actual geometry that the state describes in a quantum gravitational setting. I will give an overview of a new type of correspondence between global de Sitter spacetime and the MERA. This simple correspondence is already enough to see several features of de Sitter gravity emerge, such as cosmic no-hair and horizon complementarity. I will also comment on some more speculative topics like complexity = action and possible future directions.

I will describe the recently developed bimetric theory of fractional quantum Hall states. It is an effective theory that includes the Chern-Simons term that describes the topological properties of the fractional quantum Hall state, and a non-linear, a la bimetric massive gravity action that describes gapped Girvin-MacDonald-Platzman mode at long wavelengths. The theory reproduces the universal features of the chiral lowest Landau level (LLL) FQH states that lie beyond the reach of pure Chern-Simons theory, such as the projected static structure factor and the Girvin-MacDonald-Platzman algebra. The action of particle-hole transformation on the theory is particularly transparent. Familiar quantum Hall observables acquire a geometric interpretation in the bimetric language

BCS bogoliubon excitations, quantum Hall effect (QHE) and Landau levels (QHE handout)

Quantum critical points (QCP) beyond the Landau-Ginzburg paradigm are often called unconventional QCPs. There are in general two types of unconventional QCP: type I are QCPs between ordered phases that spontaneously break very different symmetries, and type II involve topological (or quasi-topological) phases on at least one side of the QCP. Recently significant progress has been made in understanding (2+1)-dimensional unconventional QCPs, using the recently developed (2+1)d dualities, i.e., seemingly different theories may actually be identical in the infrared limit. One group of dualities between unconventional QCPs have attracted particular interests in the field of condensed matter theory. This group of dualities include the so called deconfined QCP between the Neel and valence bond solid phases, and the topological transition between a bosonic topological insulator and a trivial Mott insulator. Each of the transitions mentioned above is also "self-dual". This group of dualities make extremely powerful predictions for numerical test. We will review the theoretical aspects and most recent numerical evidences for these new results.

Motivated by the close relations of the renormalization group with both the holography duality and the deep learning, we propose that the holographic geometry can emerge from deep learning the entanglement feature of a quantum many-body state. We develop a concrete algorithm, call the entanglement feature learning (EFL), based on the random tensor network (RTN) model for the tensor network holography. We show that each RTN can be mapped to a Boltzmann machine, trained by the entanglement entropies over all subregions of a given quantum many-body state. The goal is to construct the optimal RTN that best reproduce the entanglement feature. The RTN geometry can then be interpreted as the emergent holographic geometry. We demonstrate the EFL algorithm on 1D free fermion system and observe the emergence of the hyperbolic geometry (AdS_3 spatial geometry) as we tune the fermion system towards the gapless critical point (CFT_2 point).