Rank-2 Toric Code in Two Dimensions

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 * 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 log_p D=2+(1+delta_{L_x mod p,0})(1+delta_{L_y mod 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.

19 pages, 9 figures

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Cited by in corpus (29)

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• 2+1d Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons
• Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections
• Effective Fractonic Behavior in a Two-Dimensional Exactly Solvable Spin Liquid
• Aspects of ${\mathbb{Z}}_N$ rank-2 gauge theory in $(2+1)$ dimensions: construction schemes, holonomies, and sublattice one-form symmetries
• Effective Field Theory of Dipolar Braiding Statistics in Two Dimensions
• Gapped Lineon and Fracton Models on Graphs
• Multipole and fracton topological order via gauging foliated SPT phases
• (SPT-)LSM theorems from projective non-invertible symmetries
• Symmetric higher rank topological phases on generic graphs
• Fractons on Graphs and Complexity
• Anyon condensation web and multipartite entanglement in 2D modulated gauge theories
• Anomaly inflow for dipole symmetry and higher form foliated field theories
• Topological order and Fractons from Gauging Exponential Symmetries
• Noninvertible operators in one, two, and three dimensions via gauging spatially modulated symmetry
• Fractal Subsystem Symmetries, 't Hooft Anomalies, and UV/IR Mixing
• Unveiling UV/IR Mixing via Symmetry Defects: A View from Topological Entanglement Entropy
• Noninvertible symmetry and topological holography for modulated SPT in one dimension
• Generalized Modulated Symmetries in ${\mathbb{Z}}_2$ Topological Ordered Phases
• Gauging Modulated Symmetries via Multiple Gauge Symmetry Operators and Adaptive Quantum Circuits
• Fractonic superfluids. III. Hybridizing higher moments
• Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
• ${\mathbb{Z}}_N$ generalizations of three-dimensional stabilizer codes
• Infinite-component $BF$ field theory: Connection of fracton order, Toeplitz braiding, and non-Hermitian amplification
• Preparing Code States via Seed-Entangler-Enriched Sequential Quantum Circuits: Application to Tetra-Digit Topological Error-Correcting Codes
• Symmetry-enriched topological order and quasifractonic behavior in ${\mathbb{Z}}_N$ stabilizer codes