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arXiv:2110.02658 · Physical review. B./Physical review. B

Rank-2 toric code in two dimensions

Yun-Tak Oh, Jintae Kim, Eun-Gook Moon, Jung Hoon Han

Published Jan 21, 2022• 35 citations
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Abstract

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.

DOI
10.1103/physrevb.105.045128

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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 Lx\ifmmode×\else\texttimes\fiLy{L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}Lx​\ifmmode×\else\texttimes\fiLy​ square lattice under the periodic boundary condition (PBC). There are ppp 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) DDD given by logpD=2+(1+\ensuremathδLxmodp,0)(1+\ensuremathδLymodp,0){\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})logp​D=2+(1+\ensuremathδLx​modp,0​)(1+\ensuremathδLy​modp,0​) when ppp 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 ppp lattice spacing along with certain directions. The monopole-monopole braiding phase depends on the separation of the xxx or yyy 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.
We propose two-dimensional spin models obtained by the U(1) rank-2 tensor gauge theory with scalar or vector charges on the L_x \times L_y square lattice with p degrees of freedom per orbital and three orbitals per unit cell. The resulting spin model is a stabilizer model consisting of three mutually commuting projectors which are, in turn, obtained by Higgsing the mutually commuting Gauss's law operators and the magnetic field operators in the underlying U(1) lattice gauge theory. The spin model thus obtained is exactly solvable, with the ground state degeneracy (GSD) given by log_p GSD = 2 + (1 +{\delta}_{L_x mod p},0)(1 +{\delta}_{L_y mod p},0) when p is a prime number. Two types of dipoles, pristine and emergent, are identified. Both the monopoles and the dipoles are free to move, with certain restrictions on monopoles to hop only by p lattice spacing. The monopole-monopole braiding phase depends on the separation of the x or y coordinates of the initial monopole positions, whereas the monopole-dipole braiding is anyonic and depends on the product of the monopole charge and the dipole moment. The braiding phases can be understood as the Aharonov-Bohm phases 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.
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{\mathbb{Z}}_{N}ZN​ rank-2 gauge theory in (2+1)(2+1)(2+1)-dimensional spacetime. Firstly, we demonstrate that U(1)U(1)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)U(1)U(1) rank-2 gauge theory carries over to the ZN{\mathbb{Z}}_{N}ZN​ case simply by Higgsing U(1)U(1)U(1) to ZN{\mathbb{Z}}_{N}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.
We investigate how symmetry and topological order are coupled in the (2+12+12+1)--dimensional ZN{\mathbb{Z}}_{N}ZN​ rank-2 toric code for general NNN, which is an exactly solvable point in the Higgs phase of a symmetric rank-2 U(1)\text{U}(1)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 NNN lattice sites in others, reminiscent of fracton topological order in 3+13+13+1 dimensions. We find that while there are N2{N}^{2}N2 flavors of eee charges and 2N2N2N flavors of mmm fluxes, there are not NN2+2N{N}^{{N}^{2}+2N}NN2+2N anyon types. Instead, there are N6{N}^{6}N6 anyon types, and we can use Chern-Simons theory with six U(1)\text{U}(1)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)\text{U}(1)U(1) gauge fields. Thus, the realization of translation symmetry in the U6(1){\text{U}}^{6}(1)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){N}^{3}gcd({L}_{x},N)gcd({L}_{y},N)gcd({L}_{x},{L}_{y},N)N3gcd(Lx​,N)gcd(Ly​,N)gcd(Lx​,Ly​,N), where gcdgcdgcd 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.
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 an exactly solvable toric code type of Hamiltonian in three dimensions, defined on the diamond lattice with spin-1/21/21/2 degrees of freedom at each site. The Hamiltonian is a sum of mutually commuting plaquette operators Bp{B}_{p}Bp​, all of which have eigenvalue +1 in the ground state. The excitations are ``fluxes,'' which are plaquettes with ${B}_{p}=\ensuremath{-}1$. Due to certain local kinematic constraints, fluxes form loops. The elementary flux-loop excitations are fermions, in contrast to other solvable spin-1/21/21/2 models in three dimensions, where the excitations are bosons. Furthermore, the flux loops braid nontrivially, giving rise to Abelian anyonlike statistics.