In this work, we will show how the topological order of the Toric Code appears when the lattice on which it is defined discretizes a three-dimensional torus. In order to do this, we will present a pedagogical review of the traditional two-dimensional Toric Code, with an emphasis on how its quasiparticles are conceived and transported. With that, we want to make clear not only how all these same quasiparticle conception and transportation fit into this three-dimensional model, but to make it clear how topology controls the degeneracy of ground state in this new situation. September 9, 2019 0:32 WSPC/INSTRUCTION FILE mf-final-paper 28 M. F. Araujo de Resende Fig. 15. On left we use the same illustration used in Figure 12 to present a “quasiplaque” but highlighting the intersection between it, which was created due to action of σx j on the j-th edge, and the cutout of the horizontal plane that supports this edge. On right we see only the same cutout of this horizontal plane where two excitations that define a “quasiplaque” are present, the only ones that exist in this two-dimensional environment. Note that all the excitations created by a singleσx j in the two-dimensional model (which is obtained by a cut or a contraction of all planes parallel to a single plane discretizable by L2⊂L 3) are fully equivalent to the TC quasiparticles m, which are transportable at no cost to the energy. good way of thinking about this is, for example, interpreting the 3DC as a model defined in a cubic lattice that (i) can be infinite or (ii) not infinite with periodic boundary conditions in all three directions. If this is indeed the case, the only thing we need to keep in mind is that • whereas an 3DC constructed in the first lattice (which is infinite) has a unique vacuum state given by (3.4), • for the second lattice (which is not infinite but periodic), the degeneracy of the ground state is likely to be greater because this case is analogous to that of the TC, since any cubic lattice with periodic boundary conditions in all three directions can be perfectly identified, by construction, as the cubic discretization of a three-dimensional torus T3 (see Figure 16). Indeed, by remembering that the TC degeneracy is directly related to the ex- istence of non-contractile curves in T2, if we really want to understand how the ground state degeneracy of a new three-dimensional Toric Code (3TC) works, we need to understand the non-contractility that is related to T3. And certainly one of the things we could use for this purpose is the simple fact that the equivalence classes that define π1 (T3) is equal to three: after all, in the same way that happens in the TC, the non-contractility of the curves which belong to each of these classes could lead us to new vacuum states independent of (3.4). September 9, 2019 0:32 WSPC/INSTRUCTION FILE mf-final-paper 2D and 3D Toric Codes and the origin of their topological orders 29 A AB B C C Fig. 16. In the same way that a two-dimensional torus can be constructed by gluing the opposite edges of the square, it is perfectly possible to design a three-dimensional torus by an analogous gluing procedure. Despite the practical impossibility of visualizing the result of this construction, it is sufficient to take a three-dimensional cube and to glue its opposite faces. Note that, here (and here only), the letters in the figure do not refer to any of the operators mentioned in these notes: these letters serve only as indexes that highlight the faces that, “two by two”, need to be glued together. But, as we emphasize in this last statement, “could”. Because, as the action of Ox ¯γ∗ = ∏ j∈¯γ∗ σx j no longer corresponds to vacuum when done along any closed path ¯γ∗, the elements of the fundamental group π1 (T3) cannot moderate any vacuum =O x Td ⏐⏐ξ(1) 0 ⟩ , (4.1) where d = 1, 2, 3 and O x Td = ∏ j⊥Td σx j (4.2) is such that eachσx j acts on thej-th edge which is perpendicular to the discretization of a two-dimensional torus Td⊂T 3; this discretization occurs by fixing the faces that are normal to one of the three possible directions d. Of course, as well as with (2.23), these are not the only additional vacuum states related to the model: as there are four possible combinations O x T1◦O x T2 , O x T1◦O x T3 , O x T2◦O x T3 and O x T1◦O x T2◦O x T3 (4.3) which can be done by using the operators (4.2), there are also four new vacuum states which make it very clear that we are dealing with a model whose ground state is eight-fold degenerated. This eight-fold degeneracy fully agrees with the fact that the second homology group of T3 is H2 (T3) = Z⊕ Z⊕ Z . After all, since the first H1 (T3) and the second H2 (T3) homology groups of a three-dimensional torus are equal, by noting the Hurewicz Theorem shows us that H1 (T3) can be obtained through an abelianization of aπ1 (T3) [14] that is composed of eight homotopy classes, we can use this vision to make the following statement: in the same way that H1 (T3) allows us to identify the three generators that lead to the eight distinct homotopy classes defining π1 (T3), we can associate each one of the three elements of H2 (T3) with the generators that lead to the eight distinct homotopy classes that define the second homotopy group, π2 (T3). Thus, as the eight combinations of the non-contractile closed surfaces mentioned above, which index the eight vacuum states of the 3TC, correspond exactly to these eight distinct homotopy classes of π2 (T3), we can affirm that this three-dimensional Toric Code has topological order. 5. Final remarks According to all that we have just presented, what justifies the topological order in the TC is the fact that this model is defined in a two-dimensional torus T2. The fact that its vacuum state is not unique, for instance, is associated with the one- to-one relationship that exists among these vacuum states and the combinations eNote thatT2 can also be seen, for instance, as a non-contractile toroidal surface that is embedded in itself.

The toric code can be constructed as a gauge theory of finite groups on oriented two dimensional lattices. Here we construct analogous models with the gauge fields belonging to groupoids, which are categories where every morphism has an inverse. We show that a consistent system can be constructed for an arbitrary groupoid and analyze the simplest example that can be seen as the analog of the Abelian Z2 toric code. We find several exactly solvable models that have fracton-like features which include an extensive ground state degeneracy and excitations that are either immobile or have restricted mobility. Among the possibilities we study in detail the one where the ground state degeneracy scales as 2 × 2Nv , where Nv is the number of vertices in the lattice. The origin of this degeneracy can be traced to loop operators supported on both contractible and non-contractible loops. In particular, different non-contractible loops, along the same direction on a torus, result in different ground states. This is an exponential increase in the number of logical qubits that can be encoded in this code. Moreover the face excitations in this system are deconfined, free to move without an energy cost along certain directions of the lattice, whereas in certain other directions their movement incurs an energy cost. This places a restriction on the types of loop operators that contribute to the ground state degeneracy. The vertex excitations are immobile. The results are also extended to the groupoid analogs of Abelian ZN toric codes. ar X iv :2 21 2. 01 02 1v 1 [ qu an tph ] 2 D ec 2 02 2