Fracton Phases of Matter

TL;DR

Fractons are a new class of quasiparticles with restricted mobility that emerge from higher moment conservation laws and tensor gauge theories. The paper surveys both the field-theoretic framework and concrete lattice realizations, including the X-cube and Haah's code, and develops tools such as foliated fracton order to classify 3D fracton phases. It also connects fracton physics to elasticity via fracton-elasticity duality, explores non-ergodic dynamics and gravitational/holographic behavior, and discusses experimental platforms like Majorana islands and hole-doped antiferromagnets. The work highlights open questions and outlines practical routes toward realizing and manipulating fracton phases in real materials and quantum devices, underscoring their potential for quantum information and fundamental physics insights.

Abstract

Fractons are a new type of quasiparticle which are immobile in isolation, but can often move by forming bound states. Fractons are found in a variety of physical settings, such as spin liquids and elasticity theory, and exhibit unusual phenomenology, such as gravitational physics and localization. The past several years have seen a surge of interest in these exotic particles, which have come to the forefront of modern condensed matter theory. In this review, we provide a broad treatment of fractons, ranging from pedagogical introductory material to discussions of recent advances in the field. We begin by demonstrating how the fracton phenomenon naturally arises as a consequence of higher moment conservation laws, often accompanied by the emergence of tensor gauge theories. We then provide a survey of fracton phases in spin models, along with the various tools used to characterize them, such as the foliation framework. We discuss in detail the manifestation of fracton physics in elasticity theory, as well as the connections of fractons with localization and gravitation. Finally, we provide an overview of some recently proposed platforms for fracton physics, such as Majorana islands and hole-doped antiferromagnets. We conclude with some open questions and an outlook on the field.

Figures (25)

• Figure 1: a) A single fracton cannot move freely in any direction. b) Fractons can sometimes move by forming certain bound states, such as dipoles. c) It is also possible for a fracton to move at the expense of creating new particles out of the vacuum.
• Figure 2: (a) A cube operator of the X-cube model is a product of $X$ operators of 12 spins on the edges of a cube; (b) A cross operator is a product of $Z$ operators of 4 coplanar spins touching a vertex.
• Figure 3: Visualization of particle creation operators. a) The red links correspond to a membrane geometry on the dual lattice. The product of $Z$ operators over these edges excites four fractons (the darkened cube operators at the corners); b) The product of $X$ operators over the links comprising the straight open blue string creates two lineon excitations at its endpoints (black dots).
• Figure 4: The Hamiltonian of Haah's code is a sum of two types of cube terms. Recall that there are two qubits per vertex, so $ZZ$ represents a Pauli $Z$ acting on each of the two qubits, for example. (Figure adapted from Ref. haah.)
• Figure 5: In Haah's code, fractons are created at the corners of fractal operators.