Hyperbolic fracton model, subsystem symmetry, and holography. II. The dual eight-vertex model

Abstract

We propose that the fracton topological order is a class of toy models for holography. The discovery of AdS/CFT correspondence as a concrete construction of holography, and the subsequent developments including the Ryu-Takanayagi formula of entanglement entropy have revolutionized our understanding of quantum gravity, and provided a powerful tool set for solving various strongly-coupled quantum field theory problems. To resolve many of the mysteries of holography, toy models can be very helpful. One example is the holographic tensor-network constructions which illuminate the quantum error-correcting properties of gravity in AdS space. In this work we discuss a classical toy model based on fracton topological order, a class of exotic many-body systems with boundary area law of ground state degeneracy and (partially) immobile excitations. We show that such a model defined on the hyperbolic lattice satisfies some key properties of holographic correspondence. These properties include: the AdS-Rindler reconstruction is realized; the mutual information obeys the Ryu-Takayanagi formula, and a naively defined black hole's entropy scales as its horizon area. We end with an outlook of how fracton model may be used to concretely demonstrate the quantum-error correction encoding procedure of toy models for gravity.