The author generalizes the Jordan-Wigner Transformation to any translation-invariant fermion system, including cases where fermion parity is conserved on individual subdimensional manifolds, such as planes or fractals. The paper finds that in three dimensions, the dual spin model is an exotic gauge theory where point-like excitations have fundamentally restricted mobility called fractons, yet simultaneously are emergent fermions.
The author generalizes the Jordan-Wigner Transformation to any translation-invariant fermion system, including cases where fermion parity is conserved on individual subdimensional manifolds, such as planes or fractals. The paper finds that in three dimensions, the dual spin model is an exotic gauge theory where point-like excitations have fundamentally restricted mobility called fractons, yet simultaneously are emergent fermions.
Inspired by recent developments generalizing Jordan-Wigner dualities to higher dimensions, we develop a framework of such dualities using an algebraic formalism for translation-invariant Hamiltonians proposed by Haah. 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≥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.
We consider fermionic systems in which fermion parity is conserved within rigid subsystems, and describe an explicit procedure for gauging such subsystem fermion parity symmetries to obtain bosonic spin Hamiltonians. We show that gauging planar or fractal subsystem fermion parity symmetry in three spatial dimensions gives rise to a plethora of exactly solvable spin models exhibiting novel gapped fractonic orders characterized by emergent fermionic gauge theory. The low energy excitations of these models include fractional quasiparticles with constrained mobility and emergent fermionic statistics. We illustrate this phenomenon through a series of examples including fermionic analogs of both foliated fracton phases and fractal spin liquids. We find that the foliated analogs actually exhibit the same fractonic order as their bosonic counterparts, while this is not generally the case for fermionic fractal spin liquids.
It is well known that the noninteracting Majorana chain is dual to the one-dimensional transverse-field Ising model, either through the Jordan–Wigner transformation or by gauging fermion parity. In this correspondence, the minimal translation of the Majorana chain maps to the celebrated Kramers–Wannier (KW) duality of the spin model, with the critical point mapped to the self-dual point. In this work, we generalize this mapping to two and higher dimensions by constructing a unitary equivalence between the parity-gauged fermionic system and a spin system defined on arbitrary polyhedral decompositions of space. Imposing the flatness condition on the gauge field yields a bosonization duality between the original (ungauged) fermionic system and a gauged spin system obeying a Gauss law. The dependence of the Gauss law in the spin system on the Kasteleyn orientation (and the discrete spin structure) of the fermionic system is made explicit. Applying this bosonization to one or two copies of Majorana fermions on translationally invariant lattices, we derive higher-dimensional analogs of KW (self-) dualities in spin systems arising from fermionic minimal translations. The KW (self-) dualities are non-invertible due to projections onto eigenspaces of higher-form symmetries in the associated symmetry operators. The bosonization framework we present is intuitive, general, and systematic, encompassing other known exact bosonization methods while offering a novel approach to establish new connections between fermionic and spin systems in arbitrary dimensions.
Fractons emerge from many-body systems, featuring subdimensional particles with restricted mobility. These particles have attracted interest for their roles across disciplines, including topological quantum codes, quantum field theory, emergent gravity, and quantum information. They display unique nonequilibrium behaviors such as nonergodicity and glassy dynamics. This review offers a structured overview of fracton phenomena, especially those of gapless fracton liquids, which enable collective modes similar to gauge fluctuations in Maxwell's electromagnetic framework, yet their phenomena are distinguished by a unique conservation law that restricts the mobility of individual charges and monopoles. We delve into the theoretical basis of three-dimensional (3D) fracton liquids, exploring emergent symmetric tensor gauge theories and their properties. We also discuss the material realization of fracton liquids in Yb-based pyrochlore lattices and other synthetic quantum matter platforms.
We broaden the scope of quantum field theory by introducing a general class of discrete gauge theories that realize either topological order or fracton behavior across dimensions. We start from translation-invariant systems endowed with unconventional charge-conservation laws, which we term \textit{exponential polynomial symmetries}. Gauging these symmetries yields ZN gauge theories in 2D that exhibit topological order whose quasiparticles have constrained mobility and whose ground-state degeneracy shows ultraviolet (UV) dependence. These features are reminiscent of spatial symmetry-enriched topological order, wherein quasiparticle excitations transform nontrivially under lattice translations. We further propose a Chern-Simons variant that produces non-CSS stabilizer codes and outline a framework for exponentially symmetric subsystem SPT phases. Finally, we extend this gauging procedure to 3D, obtaining new variants of fracton topological order.
Dualities are central to our understanding of physics, in particular in the study of quantum phases of matter. In this dissertation, we explore exact lattice dualities that relate various topological phases of matter, which elucidates the interrelations between them. Moreover, searching for new dualities can even lead to the theoretical realization of new phases of matter. In the first half, we study local dualities, and how they can be used as quantum circuits to prepare Symmetry-Protected Topological (SPT) phases. We construct explicit quantum circuits that prepare fermionic SPT phases, which gives insight into the strongly interacting nature of these phases in contrast to previous free-fermion realizations. Moreover, we discuss a family of dualities, which not only generates the entanglement necessary to prepare SPT phases, but also generates anomalous symmetries which arise at transitions between such topological phases, giving new insights into topological quantum criticality. In the second half, we explore non-local dualities, exemplified by the famous Kramers-Wannier or Jordan-Wigner dualities in one dimension. we give a systematic method to generalize such non-local dualities to systems with different symmetries and to higher dimensions, which leads to explicit lattice models realizing new phases of matter. In particular, we uncover new families of fracton phases such as (i) Phases where immobile fracton excitations are emergent fermions (ii) Fracton phases that are enriched by spatial and/or time-reversal symmetries (iii) phases that "hybridizes" between conventional topological orders and fracton orders, some of which hosts non-abelian fracton excitations. Finally, we show that the two types of dualities can be related by an extra ingredient: measurement. This gives a practical method to realize non-local dualities in near-term quantum devices, paving a way to efficiently prepare a large class of topological phases of matter in experiments.