Condensed Matter

Fracton order is a new kind of phase of matter which is similar to topological order, except its excitations have mobility constraints. The excitations are bound to various n-dimensional surfaces with exotic fusion rules that determine how excitations on intersecting surfaces can combine.

This talk will focus on one of the simplest fracton models: the X-cube model. I will explain how this fracton model can be thought of as starting from a 3D toric code and stacks of 2D toric code layers, and then condensing certain excitations on the 2D layers. This is the first TQFT picture of the X-cube model as a 3D order with non-invertible 2D "defects". When we zoom out so that the layers appear close together, I'll show that the model can be described by what I call a foliated field theory. I define a foliated field theory to be a field theory that couples to a foliation field, which describes the geometry of the foliating layers (in analogy to a metic which describes the geometry of spacetime). In this case, the foliated field theory takes the form of a 3+1D BF theory which is coupled to stacks of 2+1 BF theories, with stacking structure described by the foliation field.

This talk is based on arXiv:1812.01613 and another forthcoming work.

Using Monte Carlo methods we explore how well does the recent proposal for computing conformal dimensions, using a large charge expansion, work. We focus on the O(2) and the O(4) Wilson-Fisher fixed points as test cases. Since the traditional Monte Carlo approach suffers from a severe signal-to-noise ratio problem in the large charge sectors, we use worldline formulations that eliminate such problems. In particular we argue that the O(4) model can be simplified drastically by studying what we refer to as a "qubit" formulation. Such simpler  formulations of quantum field theories have become interesting recently from the perspective of quantum computing. Using our studies we confirm that the conformal dimensions of both conformal field theories with O(2) and O(4) symmetries obey a simple formula predicted by the large charge expansion. We also compute the two leading universal low energy constants in both cases , that play an important role in the large charge expansion.

We discover an infinite hierarchical web of the products of supersymmetric generators sustained by the superconformal Virasoro algebra. This hierarchy structure forms the mathematical foundation underpinning the explicit derivation of the character for the self-symmetric Ramond highest weight $c/24$. To consistently fit these exact results into the modular-invariant torus partition function, we advocate a necessary augmentation of the representation theory in the original Friedan--Qiu--Shenker construction via symmetrizing the ground-state manifold associated with the $c/24$ Verma module. Under the newly-proposed scheme, we invoke a quantum-interference mechanism between the two independent Ishibashi states to construct the boundary Cardy states for the whole family of the $\mathcal{N}=1$ superconformal minimal series, based on which the extra fusion channels are unveiled through the obtained Verlinde formula. Our work thus provides the first complete solution to this thirty-year-old question.