Jordan-Wigner dualities for translation-invariant Hamiltonians in any dimension: Emergent fermions in fracton topological order

Inspired by recent developments generalizing Jordan-Wigner dualities to higher dimensions [Ann. Phys. 393, 234 (2018)], we develop a framework of such dualities using an algebraic formalism for translation-invariant Hamiltonians proposed by Haah [Commun. Math. Phys. 324, 351 (2013)]. We prove that given a translation-invariant fermionic system with general $q$-body interactions, where $q$ is even, a local mapping preserving global fermion parity to a dual Pauli spin model exists and is unique up to a choice of basis. Furthermore, the dual spin model is constructive, and we present various examples of these dualities. As an application, we bosonize fermionic systems where free-fermion hopping terms are absent $(q\ge 4)$ and fermion parity is conserved on submanifolds such as higher-form, line, planar or fractal symmetry. For some cases in 3+1D, bosonizing such a system can give rise to fracton models where the emergent particles are immobile but yet can behave in certain ways like fermions. These models may be examples of new nonrelativistic 't Hooft anomalies. Furthermore, fermionic subsystem symmetries are also present in various Majorana stabilizer codes, such as the color code or the checkerboard model, and we give examples where their duals are cluster states or new fracton models distinct from their doubled CSS codes.

Figure 1

An example of an exactly solvable model with fracton excitations that are fermions, which is a twisted version of the X-Cube stabilizer code. The blue and red lines, respectively, denote Pauli $Z$ and $X$'s living on the edges of a cubic lattice. The model can be thought of as the result of gauging a fermion system which conserves fermion parity in the (001), (010), and (001) planes of a dual cubic lattice. We discuss how to obtain this model in Sec. 5f.

Figure 2

The stabilizer of the Majorana color code, written with the unit cell shown in blue. The translation vectors are denoted by $x$ and $y$.

Figure 3

(a) Fractal fermion parity symmetry acting on sites in the pattern of the Fibonacci CA. (b) ${\mathbb{Z}}_2$ fractal symmetry in the dual Pauli model. Red circles and blue squares denote Pauli $X$'s and Pauli $Z$'s, respectively. Their overlap denotes Pauli $Y$'s.

Figure 4

Constructing “twisted” stabilizer codes from known CSS codes.