Condensed Matter Theory

Abstract:

We construct models hosting classical fractal spin liquids on two realistic three-dimensional (3D) lattices of corner-sharing triangles: trillium and hyperhyperkagome (HHK). Both models involve the same form of three-spin Ising interactions on triangular plaquettes as the Newman-Moore (NM) model on the 2D triangular lattice. However, in contrast to the NM model and its 3D generalizations, their degenerate ground states and low-lying excitations cannot be described in terms of scalar cellular automata (CA), because the corresponding fractal structures lack a simplifying algebraic property, often termed the "freshman's dream."By identifying a link to matrix CAs - that makes essential use of the crystallographic structure - we show that both models exhibit fractal symmetries of a distinct class to the NM-type models. We devise a procedure to explicitly construct low-energy excitations consisting of finite sets of immobile defects or "fractons,"by flipping arbitrarily large self-similar subsets of spins, whose fractal dimensions we compute analytically. We show that these excitations are associated with energetic barriers which increase logarithmically with system size, leading to "fragile"glassy dynamics, whose existence we confirm via classical Monte Carlo simulations. We also discuss consequences for spontaneous fractal symmetry breaking when quantum fluctuations are introduced by a transverse magnetic field, and propose multispin correlation function diagnostics for such transitions. Our findings suggest that matrix CAs may provide a fruitful route to identifying fractal symmetries and fractonlike behavior in lattice models, with possible implications for the study of fracton topological order.

Abstract:

These are notes based on lectures given at the 2021 summer school on Fundamental Problems in Statistical Physics XV. Their purpose is to give a very brief introduction to Generalized Hydrodynamics, which provides a description of the large scale structure of the dynamics in quantum integrable models. The notes are not meant to be comprehensive or provide an overview of all relevant literature, but rather give an exposition of the key ideas for non-experts, using a simple fermionic tight-binding model as the main example.