Fracton Self-Statistics (2304.00028v2)

Abstract: Fracton order describes novel quantum phases of matter that host quasiparticles with restricted mobility, and thus lies beyond the existing paradigm of topological order. In particular, excitations that cannot move without creating multiple excitations are called fractons. Here we address a fundamental open question -- can the notion of self-exchange statistics be naturally defined for fractons, given their complete immobility as isolated excitations? Surprisingly, we demonstrate how fractons can be exchanged, and show that their self-statistics is a key part of the characterization of fracton orders. We derive general constraints satisfied by the fracton self-statistics in a large class of Abelian fracton orders. Finally, we show the existence of nontrivial fracton self-statistics in some twisted variants of the checkerboard model and Haah's code, establishing that these models are in distinct quantum phases as compared to their untwisted cousins.

• The paper introduces a systematic method to define self-exchange statistics for immobile fractons via coordinated splitting.
• It distinguishes fracton phases by identifying key invariants, including semionic statistics in twisted models like Haah’s code.
• The study provides practical calculation methods and constraints that inform potential applications in quantum memory and error correction.

An Analysis of Fracton Self-Statistics and Their Implications

The paper "Fracton Self-Statistics" by Hao Song et al. presents advanced research in the domain of fracton topological order, which postulates the existence of unique quasiparticles with constrained mobility. This work addresses a fundamental open question regarding the definition of self-exchange statistics for fractons, given their inherent immobility as isolated excitations. It provides evidence that the self-statistics of fractons are not only definable but also integral to the characterization of fracton orders.

The concept of fracton order diverges significantly from the customary framework of topological order, especially prevalent in 3D quantum systems. In contrast to conventional quasiparticles that allow free movement within their dimensional constraints, fractons are restricted; they can only move as part of a many-particle excitation or within limited one- or two-dimensional manifolds.

Key Contributions

1. Definition of Fracton Self-Statistics: The authors introduce a systematic approach to define self-exchange statistics for fractons. This involves fractons being temporarily split into coordinated parts to allow for superposition within the quantum framework, enabling an exchange.
2. Fracton Phases and Invariants: The paper identifies fracton self-statistics as a fundamental invariant crucial for distinguishing between fracton phases, including both Abelian and non-Abelian types.
3. Application to Specific Models: The researchers applied their framework to models like the checkerboard and Haah’s code to show that they host nontrivial self-statistics, particularly in twisted versions. The twisted models exhibit semionic fracton orders, contrasting with the more standard bosonic and fermionic statistics found in other topological orders.
4. Constraints and Calculation Methods: The authors also derive constraints on fracton self-statistics that mimic the algebraic relations found in lower-dimensional systems but with significant generalizations. This includes a developed method for calculating such statistics in a broad class of models.

Numerical and Theoretical Highlights

The paper includes rigorous evaluations showcasing that for certain models, such as a twisted Haah’s code, exotic self-statistics (±i for semions) can be derived. On the checkerboard lattice, explicit calculations of twists along different axes reveal the match or cancellation in fractonic order parameters.

The research predicts and confirms the existence of distinct quantum phases relying on these intricate statistics. For example, it delineates two classes of twisted checkerboard models through the presence or absence of semionic fracton statistics.

Implications and Future Directions

This research has pivotal implications for condensed matter physics and quantum information science, affecting how topological quantum devices are conceptualized and potentially realizing new paths toward fault-tolerant quantum computation. The existence of nontrivial fracton statistics gives rise to novel quantum phases that could be exploited in robust quantum memory devices and other technologies.

Additionally, the results herald a deeper exploration into the algebraic theory of fractons, motivating future endeavors to unify the unconventional characteristics of fractons with broader topological paradigms. The work suggests that these layers of statistics could inform new quantum error correction codes beyond current stabilizer formalism.

The paper reflects a sophisticated understanding and manipulation of higher-dimensional quantum systems, providing essential paths forward for constructing new physical theories that predict behaviors in complex materials and quantum technologies. The novel conceptual advances presented here are likely to stimulate considerable further discussion and exploration in the quantum physics community.