A number of exactly solvable spin models, including the Kitaev toric code in two and three dimensions and the X-cube model in three dimensions, can be related to their respective parent lattice gauge theories (LGTs) through the mathematical process of ``Higgsing.'' Field theories of the low-energy excitations of these spin models can be developed subsequently, building upon the symmetry of the parent LGTs. Recently, two of the present authors proposed a variant of the three-dimensional toric code which we now call the F3 model, whose elementary excitations consist of freeon and fluxon excitations of the three-dimensional toric code and fracton excitations of the X-cube model. In this work, we identify the parent LGT of the F3 model as the hybrid rank-1 and rank-2 U(1) LGT and develop the corresponding field theory. The resulting Lagrangian of the F3 model is that of a three-dimensional toric code with an extra term, which ties the dynamics of fractons to that of fluxons. The matter part of the effective action for the F3 model can be derived as well by carefully keeping track of the gauge invariance of the F3 model. Hydrodynamic equations of motion of the quasiparticles are derived which properly reflect their constrained dynamics. Finally, we present a tight-binding model for the quasiparticle motion in the F3 model.
A number of exactly solvable spin models, including the Kitaev toric code in two and three dimensions and the X-cube model in three dimensions, can be related to their respective parent lattice gauge theories (LGTs) through the mathematical process of ``Higgsing.'' Field theories of the low-energy excitations of these spin models can be developed subsequently, building upon the symmetry of the parent LGTs. Recently, two of the present authors proposed a variant of the three-dimensional toric code which we now call the F3 model, whose elementary excitations consist of freeon and fluxon excitations of the three-dimensional toric code and fracton excitations of the X-cube model. In this work, we identify the parent LGT of the F3 model as the hybrid rank-1 and rank-2 U(1) LGT and develop the corresponding field theory. The resulting Lagrangian of the F3 model is that of a three-dimensional toric code with an extra term, which ties the dynamics of fractons to that of fluxons. The matter part of the effective action for the F3 model can be derived as well by carefully keeping track of the gauge invariance of the F3 model. Hydrodynamic equations of motion of the quasiparticles are derived which properly reflect their constrained dynamics. Finally, we present a tight-binding model for the quasiparticle motion in the F3 model.
We study a two-dimensional spin model obtained by ``Higgsing'' the rank-2 U(1) lattice gauge theory (LGT) with scalar or vector charges on the ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$ square lattice under the periodic boundary condition (PBC). There are p degrees of freedom per orbital and three orbitals per unit cell in the spin model. The resulting spin model is a stabilizer code consisting of three mutually commuting projectors that are, in turn, obtained by Higgsing the mutually commuting Gauss's law operators and the magnetic field operators in the underlying LGT. The spin model thus obtained is exactly solvable, with the ground state degeneracy (GSD) D given by ${\text{log}}_{p}D=2+(1+{\ensuremath{\delta}}_{{L}_{x}\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.16em}{0ex}}p,0})(1+{\ensuremath{\delta}}_{{L}_{y}\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.16em}{0ex}}p,0})$ when p is a prime number. Two types of dipole excitations, pristine and emergent, are identified. Both the monopoles and the dipoles are free to move, with restrictions on monopoles to hop only by p lattice spacing along with certain directions. The monopole-monopole braiding phase depends on the separation of the x or y coordinates of the initial monopole positions, making it distinct from the ordinary anyon braiding statistics. The monopole-dipole braiding obeys the usual anyonic statistics. Despite the oddity, the monopole-monopole braiding phase can be understood as the Aharonov-Bohm phase of some emergent vector potentials.
A rank-2 toric code (R2TC) Hamiltonian in two dimensions can be constructed as a Higgsed descendant of rank-2 U(1) lattice gauge theory. As noted by the authors recently [Y.-T. Oh, J. Kim, E.-G. Moon, and J. H. Han, Phys. Rev. B 105, 045128 (2022)] the quasiparticles in that model show unusual braiding statistics that depends on the initial locations of the particles which participate in the braiding. We show that this kind of statistical phase captures the total dipole moment of quasiparticles encompassed in the braiding, in contrast to the conventional anyonic braiding seeing the total charge. An Aharonov-Bohm interpretation of such dipolar braiding statistics is made in terms of emergent, rank-1 vector potentials that are built out of the underlying rank-2 gauge fields. Pertinent field theories of the quasiparticle dynamics in the R2TC are developed, and the accompanying conservation laws derived. A dipolar BF theory of the rank-2 gauge fields is constructed and shown to correctly capture the dipolar braiding statistics, in contrast to the conventional BF theory capturing the monopolar braiding statistics of anyons in the rank-1 toric code.
We propose a lattice spin model on a cubic lattice that shares many of the properties of the 3D toric code and the X-cube fracton model. The model, made of Z3 degrees of freedom at the links, has the vertex, the cube, and the plaquette terms. Being a stabilizer code the ground states are exactly solved. With only the vertex and the cube terms present, we show that the ground state degeneracy (GSD) is ${3}^{{L}^{3}+3L\ensuremath{-}1}$, where L is the linear dimension of the cubic lattice. In addition to fractons, there are free vertex excitations we call the freeons. With the addition of the plaquette terms, GSD is vastly reduced to 33, with fracton, fluxon, and freeon excitations, among which only the freeons are deconfined. The model is called the AB model if only the vertex (Av) and the cube (Bc) terms are present and the ABC model if in addition the plaquette terms (Cp) are included. The AC model consisting of vertex and plaquette terms is the Z3 3D toric code. The extensive GSD of the AB model derives from the existence of both local and nonlocal logical operators that connect different ground states. The latter operators are identical to the logical operators of the Z3 X-cube model. Fracton excitations are immobile and accompanied by the creation of fluxons---plaquettes having nonzero flux. In the ABC model, such fluxon creation costs energy and ends up confining the fractons. Unlike past models of fractons, vertex excitations are free to move in any direction and pick up a nontrivial statistical phase when passing through a fluxon or a fracton cluster.
We investigate how symmetry and topological order are coupled in the (2+1)--dimensional ZN rank-2 toric code for general N, which is an exactly solvable point in the Higgs phase of a symmetric rank-2 U(1) gauge theory. The symmetry-enriched topological order present has a nontrivial realization of square-lattice translation (and rotation and reflection) symmetry, where anyons on different lattice sites have different types and belong to different superselection sectors. We call such particles ``position-dependent excitations.'' As a result, in the rank-2 toric code anyons can hop by one lattice site in some directions while only by N lattice sites in others, reminiscent of fracton topological order in 3+1 dimensions. We find that while there are N2 flavors of e charges and 2N flavors of m fluxes, there are not NN2+2N anyon types. Instead, there are N6 anyon types, and we can use Chern-Simons theory with six U(1) gauge fields to describe all of them. While the lattice translations permute anyon types, we find that such permutations cannot be expressed as transformations on the six U(1) gauge fields. Thus, the realization of translation symmetry in the U6(1) Chern-Simons theory is not known. Despite this, we find a way to calculate the translation-dependent properties of the theory. In particular, we find that the ground-state degeneracy on an ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$ torus is N3gcd(Lx,N)gcd(Ly,N)gcd(Lx,Ly,N), where gcd stands for ``greatest common divisor.'' We argue that this is a manifestation of UV/IR mixing which arises from the interplay between lattice symmetries and topological order.
Rank-2 toric code (R2TC), a prototypical archetype of the discrete rank-2 symmetric gauge theory, has properties that differ from those of the standard toric code. Specifically, it features a blending of UV and IR in its ground state, restricted mobility of its quasiparticles, and variations in the braiding statistics of its quasiparticles based on their position. In this paper, we investigate various aspects of ZN rank-2 gauge theory in (2+1)-dimensional spacetime. Firstly, we demonstrate that U(1) rank-2 gauge theory can arise from $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ rank-1 gauge theory after condensing the gauge charges in a specific way. This construction scheme of U(1) rank-2 gauge theory carries over to the ZN case simply by Higgsing U(1) to ZN, after which the resulting rank-2 gauge theory can be tuned to the R2TC. The holonomy operators of R2TC are readily identified using this scheme and are given clear physical interpretation as the pair creation/annihilation of various monopoles and dipoles. Explicit tensor network construction of the ground states of R2TC is given as two copies of the ground states of Kitaev's toric code that are ``sewn together'' according to the condensation scheme. In addition, through a similar anyon condensation protocol, we present a double semion version of rank-2 toric code whose flux excitations exhibit restricted mobility and semionic statistics. Finally, we identify the generalized discrete symmetries of the R2TC, which are much more complex than typical 1-form symmetries. They include conventional and unconventional 1-form symmetries, such as framed 1-form symmetries and what we call sublattice 1-form symmetries. Using these, we interpret the R2TC's unique properties (UV/IR mixing, position-dependent braiding, etc.) from the modern perspective of generalized spontaneous symmetry breaking and 't Hooft anomalies.
Fractons are gapped pointlike excitations in d=3 topological ordered phases whose motion is constrained. They have been discovered in several gapped models but a unifying physical mechanism for generating them is still missing. It has been noticed that in symmetric-tensor U(1) gauge theories, charges are fractons and cannot move freely due to, for example, the conservation of not only the charge but also the dipole moment. To connect these theories with fully gapped fracton models, we study Higgs and partial confinement mechanisms in rank-2 symmetric-tensor gauge theories, where charges or magnetic excitations, respectively, are condensed. Specifically, we describe two different routes from the rank-2 U(1) scalar charge theory to the X-cube fracton topological order, finding that a combination of Higgs and partial confinement mechanisms is necessary to obtain the fully gapped fracton model. On the other hand, the rank-2 Z2 scalar charge theory, which is obtained from the former theory upon condensing charge-2 matter, is equivalent to four copies of the d=3 toric code and does not support fracton excitations. We also explain how the checkerboard fracton model can be viewed as a rank-2 Z2 gauge theory with two different Gauss' law constraints on different lattice sites.