Fractons

TL;DR

This review maps the theoretical landscape of fracton phases, states of quantum matter with excitations of restricted mobility. It contrasts gapped fracton phases described by exactly solvable spin models (e.g., X-cube, Haah's code) with gapless fracton phases captured by symmetric tensor gauge theories, highlighting deep connections to elasticity theory and gravity. It also surveys dynamics and thermodynamics of fracton matter at non-zero density, including glassy dynamics and rich phase structure arising from dipole conservation and interactions. Open questions remain regarding the full taxonomy of fracton theories, simpler lattice realizations, and experimental routes to observe fracton physics.

Abstract

We review what is known about fracton phases of quantum matter. Fracton phases are characterized by excitations that exhibit restricted mobility, being either immobile under local Hamiltonian dynamics, or mobile only in certain directions. They constitute a new class of quantum state of matter, which does not wholly fit into any of the existing paradigms, but which connects to areas including glassy quantum dynamics, topological order, spin liquids, elasticity theory, quantum information theory, and gravity. We begin by discussing gapped fracton phases, which may be described using exactly solvable lattice spin models. We introduce the basic phenomena, and discuss the geometric and topological response of fracton phases. We also discuss connections to generalized gauge theories, and explain how gapped fracton phases may be obtained from more familiar theories. We then introduce the framework of tensor gauge theory, which provides a powerful complementary perspective on fracton phases. We discuss how tensor gauge theory encodes the fracton phenomenon, and how it allows us to access gapless fracton phases. We discuss the basic properties of gapless fracton phases, and their connections to elasticity theory and gravity. We also discuss what is known about the dynamics and thermodynamics of fractons at non-zero density, before concluding with a brief survey of some open problems.

Figures (4)

• Figure 1: Figures taken from Ref.prem illustrating key features of the X-cube model. (i) An illustration of the stabilizer operators that make up the model, Eq.\ref{['eq: Xcube']}, using the notation $\sigma^\ell_z = Z_\ell$, $\sigma^\ell_x = X_\ell$. (ii) Acting with a rectangular membrane of $Z_{\ell}$ operators (red lines) creates a state where the four corners of the membrane (blue cubes) have eigenvalue $-1$ under the $A$ operator. These blue cubes are the fracton excitations of the model. (iii) A fracton can be moved by a local operator (red line), but only at the cost of creating additional excitations (two more 'flipped' blue cubes).
• Figure 2: Illustration of the two stabilizer operators that make up Haah's cubic code, Eq. \ref{['eq:Haah']}. Figure taken from MaSchmitz.
• Figure 3: Illustration of a two-dimensional analog of the operators creating isolated fractons in Haah's code. Three Ising-like excitations (red dots) are created at corners of a triangle (left). To separate these excitations, triangles can be glued together, forming a fractal object (middle and right). A construction similar to this appears in Ref.castelnovo
• Figure 4: A summary of the duality mapping between fracton gauge theory and the quantum theory of elasticity in two spatial dimensions. Figure taken from Ref.leomichael, with permission.