Topological Order & Fractons

In the standard physics curriculum, particles can move freely through space. A quark can propagate from one point to another; an electron can traverse a wire; even a phonon can travel through a crystal. But in the 2010s, condensed matter theorists discovered a new class of quasiparticles that violate this basic assumption: fractons: excitations that cannot move freely in isolation. This restricted mobility is not a minor curiosity; it is governed by higher-rank gauge symmetries whose mathematical structure looks suspiciously like the diffeomorphism invariance of general relativity.

This unexpected parallel between exotic condensed matter phases and gravity has opened a new frontier in theoretical physics, suggesting that gravitons themselves might be understood as fracton-like excitations.

Topological Order: Background

To understand fractons, we first need topological order. Ordinary phases of matter (solid, liquid, gas) are classified by symmetry breaking: a crystal breaks translation symmetry; a ferromagnet breaks rotation symmetry. But some quantum phases cannot be described this way. They have no local order parameter, yet they are sharply distinct from trivial phases.

Conventional Topological Order

Conventional topological order, as seen in the fractional quantum Hall effect, has distinctive features:

• Ground state degeneracy that depends on the topology of the manifold (e.g., genus of the surface).
• Anyonic excitations: particles that are neither bosons nor fermions, with fractional statistics.
• Long-range entanglement: the topological entanglement entropy $\gamma$ is nonzero: $S = \alpha L - \gamma$.
• Robustness: topological properties are immune to local perturbations, making them candidates for quantum error correction.

The key mathematical framework for conventional topological order is topological quantum field theory (TQFT). In a TQFT, the ground state degeneracy and particle statistics depend only on the topology of spacetime, not its geometry. This is elegant but also limiting, as we will see, fractons break out of this framework entirely.

Fracton Models

Haah's Cubic Code (2011)

The first fracton model, discovered by Jeongwan Haah, was remarkable for what it lacks. Haah's cubic code is a stabilizer code on a cubic lattice with the following properties:

• No string-like logical operators: Unlike the toric code, there are no string operators connecting excitations. Excitations cannot be moved by any local process.
• Fractal structure: Excitations can only be created in fractal patterns, hence "fractons."
• Sub-extensive ground state degeneracy: The ground state degeneracy grows sub-extensively with system size, unlike conventional topological codes where it is a topological invariant.
• Self-correcting potential: The energy barrier to creating errors grows with system size, suggesting Haah's code might be a self-correcting quantum memory, one of the most sought-after objects in quantum information.

The X-Cube Model

The X-cube model (Vijay, Haah, Fu, 2016) provides a more transparent example. It is defined on a cubic lattice with qubits on edges. The model supports two types of excitations:

• Fractons: Point-like excitations at cube corners that cannot move at all in isolation.
• Lineons: One-dimensional particles that can only move along lines (not in the full 3D space).
• Planons: Two-dimensional particles restricted to move within planes.

The ground state degeneracy of the X-cube model on an $L \times L \times L$ torus is $2^{6L-3}$, which depends on the linear size $L$, not just the topology. This is a hallmark of fracton order: it is sensitive to geometry, not just topology, placing it beyond TQFT.

Higher-Rank Gauge Theory

Pretko (2017) showed that fracton physics can be understood through higher-rank gauge theories. Instead of the usual vector gauge field $A_i$ of electromagnetism, fracton systems are described by a symmetric tensor gauge field $A_{ij}$.

Dipole Conservation and Restricted Mobility

The Gauss law $\partial_i \partial_j E^{ij} = \rho$ implies conservation laws beyond charge conservation. Integrating over space:

This is why fractons cannot move: their mobility is constrained by the conservation of higher multipole moments, enforced by the higher-rank gauge symmetry.

The Gravity Connection

The mathematical parallel between fracton gauge theory and linearized gravity is striking:

Both fractons and gravitons are described by symmetric tensor fields. Both have gauge invariances that restrict the mobility or localizability of their excitations. Both resist description by conventional field-theoretic frameworks. The question is: is this parallel deep or superficial?

What Could This Mean?

Fractons and Quantum Error Correction

Fracton models have deep connections to quantum error correction. Haah's code was originally discovered in a search for self-correcting quantum memories, quantum systems that passively protect quantum information through energy barriers rather than active error correction.

Beyond the Linearized Level

The most significant challenge to the fracton-gravity connection is that it operates only at the linearized level. Full general relativity is nonlinear, the metric appears in the Einstein equations in a highly nonlinear way through the Ricci tensor:

Going from the linearized Gauss law $\partial_i \partial_j E^{ij} = \rho$ to the full Einstein equations would require:

• Emergent diffeomorphism invariance: The full diffeomorphism group of GR, not just linearized gauge transformations.
• Emergent dynamics for geometry: The background geometry itself must become dynamical.
• Continuum limit: Most fracton models are lattice-based. Extracting a continuum field theory is an open problem.

None of these have been achieved. The field-theoretic treatment of fractons by Seiberg, Shao, and collaborators (2020-2021) clarified the symmetry structure of fracton theories, but the gap to full nonlinear gravity remains enormous.

Experimental Status

Unlike conventional topological phases, which have been observed in quantum Hall systems and topological insulators, fracton topological order has not yet been realized experimentally. Proposals exist for realizations in:

• Cold atom systems: Using optical lattices to engineer the required interactions.
• Spin liquids: Certain frustrated magnets may exhibit fracton-like behavior.
• Quantum simulators: Digital quantum computers could simulate fracton Hamiltonians.

Experimental verification would be a breakthrough, both for condensed matter physics and for testing the gravity connection.

Key Insights

• Fractons are quasiparticles with restricted mobility, governed by conservation of dipole moment and higher multipole moments.
• Fracton topological order goes beyond conventional TQFT: ground state degeneracy depends on geometry, not just topology.
• Higher-rank gauge theories ($A_{ij}$ instead of $A_i$) provide the field-theoretic framework for fracton physics.
• The mathematical parallel between fracton gauge theory and linearized gravity is striking: both involve symmetric tensor fields with gauge invariances that restrict mobility.
• The parallel is limited to the linearized level; reaching full nonlinear GR from fracton physics remains an open challenge.
• Fracton codes are promising candidates for self-correcting quantum memories.
• No experimental realization of fracton topological order exists yet.