A New Kind of Topological Quantum Order: A Dimensional Hierarchy of Quasiparticles Built from Stationary Excitations (1505.02576v1)

Abstract: We introduce exactly solvable models of interacting (Majorana) fermions in $d \ge 3$ spatial dimensions that realize a new kind of topological quantum order, building on a model presented in ref. [1]. These models have extensive topological ground-state degeneracy and a hierarchy of point-like, topological excitations that are only free to move within sub-manifolds of the lattice. In particular, one of our models has fundamental excitations that are completely stationary. To demonstrate these results, we introduce a powerful polynomial representation of commuting Majorana Hamiltonians. Remarkably, the physical properties of the topologically-ordered state are encoded in an algebraic variety, defined by the common zeros of a set of polynomials over a finite field. This provides a "geometric" framework for the emergence of topological order.

• The paper demonstrates a new type of topological quantum order with extensive ground-state degeneracy and immobile fracton excitations in Majorana fermion models.
• It employs a novel polynomial algebraic framework to convert Hamiltonian dynamics into an algebraic variety over a finite field, facilitating systematic analysis.
• The study establishes a dimensional hierarchy of excitations that links geometric structures to physical properties, with implications for robust quantum memory and fault-tolerant computation.

Topological Quantum Order in Majorana Fermion Models: A Novel Approach

This paper introduces a paradigm in the exploration of topological quantum order through the lens of exactly solvable Hamiltonians, specifically focusing on interacting Majorana fermions in $d \geq 3$ spatial dimensions. The central thesis is the identification of a new type of topological quantum order, characterized by a hierarchy of point-like excitations confined to specific sub-manifolds of the lattice, with the addition of ostensibly stationary fundamental excitations termed "fractons."

Key Contributions

1. Exact Solvability and Topological Degeneracy: The models proposed offer extensive topological ground-state degeneracy, showcasing fracton behavior—a phenomenon beyond the scope of traditional topological quantum field theory (TQFT). For instance, in one model in three dimensions, the fundamental excitations are strictly localized and immovable, termed as fractons.
2. Algebraic Framework: A significant methodological advancement is the use of a polynomial representation for the Hamiltonians of Majorana fermions, allowing the translation of physical characteristics into an algebraic variety defined over a finite field. This representation lends itself to systematic analysis and potentially facilitates the discovery of new models with similar exotic properties.
3. Geometric Perspective: Through a geometric framework, these models associate physical topological properties with the algebraic structure of the variety defined by common zeros of the polynomial representation, expanding the toolkit for understanding and classifying topological orders.
4. Dimensional Hierarchy of Excitations: The paper elucidates a dimensional hierarchy of excitations where particles, constrained by their freedom of movement, occupy different manifold dimensions. Specifically, in 3D models, the fundamental particles are immobile, while higher-dimensional composite particles are more mobile, resonating with the emergent concept of "fractons" with restricted mobility.

Numerical Results and Theoretical Insights

The results demonstrate that the ground-state degeneracy of these Majorana models scales extensively, which is calculated using algebraic methods by determining the dimension of a quotient ring related to stabilization maps. In particular, the degeneracy scales non-trivially with system size, reflecting the intricate interplay between topology and interaction in these models.

Implications and Future Directions

The implications of this work are both profound and multifaceted. Practically, the models may impact the design of robust quantum memory or fault-tolerant quantum computation due to the inherent stability of topological quantum states against local perturbations. Theoretically, the work suggests further exploration into non-trivial classes of particles with restricted mobility—fractons, which deviate from the traditional paradigms of quasiparticle movement and braiding statistics within a TQFT framework.

Future directions might include further refinement of the geometric-algebraic framework, exploration of potential material realizations, and expansion into higher-dimensional spaces. The theoretical construct of associating topological properties with algebraic varieties opens an avenue for unifying geometric and algebraic topology insights, potentially leading to novel quantum phases of matter.

In conclusion, this paper significantly advances the understanding of topological quantum order in fermionic systems by marrying algebraic methods with geometric intuition, challenging the existing frameworks and suggesting a roadmap for future theoretical and practical developments in the field.