Aspects of ${\mathbb{Z}}_N$ rank-2 gauge theory in $(2+1)$ dimensions: construction schemes, holonomies, and sublattice one-form symmetries

Abstract

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 ${\mathbb{Z}}_N$ rank-2 gauge theory in $(2+1)$-dimensional spacetime. Firstly, we demonstrate that $U(1)$ rank-2 gauge theory can arise from $U(1)\times U(1)$ 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 ${\mathbb{Z}}_N$ case simply by Higgsing $U(1)$ to ${\mathbb{Z}}_N$, 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 are 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.

arXiv comment26 pages, 14 figures, one append section

References in corpus (53)

• Fault-tolerant quantum computation by anyons
• Quantum Spin Liquids
• Generalized Global Symmetries
• Topological orders and Edge excitations in FQH states
• Local stabilizer codes in three dimensions without string logical operators
• Criticality, the area law, and the computational power of PEPS
• Braiding statistics approach to symmetry-protected topological phases
• Fracton Topological Order, Generalized Lattice Gauge Theory and Duality
• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• Coupling a QFT to a TQFT and Duality
• Theta, Time Reversal, and Temperature
• Quantum orders in an exact soluble model
• Quasi-adiabatic Continuation of Quantum States: The Stability of Topological Ground State Degeneracy and Emergent Gauge Invariance
• Subdimensional Particle Structure of Higher Rank U(1) Spin Liquids
• Generalized Symmetries in Condensed Matter
• Fracton Phases of Matter
• A symmetry principle for Topological Quantum Order
• Tensor-entanglement renormalization group approach to 2D quantum systems
• On 2-Group Global Symmetries and Their Anomalies
• Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d
• Quantum Field Theory of X-Cube Fracton Topological Order and Robust Degeneracy from Geometry
• Exotic Symmetries, Duality, and Fractons in 2+1-Dimensional Quantum Field Theory
• Fracton topological order via coupled layers
• Symmetry and Topological Order
• Entanglement renormalization and topological order
• Topological phases with generalized global symmetries
• Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry
• The Higgs Mechanism in Higher-Rank Symmetric $U(1)$ Gauge Theories
• Fracton topological order from Higgs and partial confinement mechanisms of rank-two gauge theory
• Emergent (anomalous) higher symmetries from topological orders and from dynamical electromagnetic field in condensed matter systems
• Global Dipole Symmetry, Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons
• Exact entanglement renormalization for string-net models
• Higher central charges and topological boundaries in 2+1-dimensional TQFTs
• The low-energy limit of some exotic lattice theories and UV/IR mixing
• Mutual Chern-Simons theory for Z_2 topological order
• Fractonic Matter in Symmetry-Enriched U(1) Gauge Theory
• Cage-Net Fracton Models
• Abelian gauge theories on the lattice: $\theta$-terms and compact gauge theory with(out) monopoles
• Symmetric Tensor Gauge Theories on Curved Spaces
• Fractons from vector gauge theory
• Fractonic Chern-Simons and BF theories
• Twisted foliated fracton phases
• Fracton phases via exotic higher-form symmetry-breaking
• Lieb-Schultz-Mattis type theorem with higher-form symmetry and the quantum dimer models
• Lattice models that realize ${\mathbb{Z}}_n$-1-symmetry protected topological states for even $n$
• Position-Dependent Excitations and UV/IR Mixing in the ${\mathbb{Z}}_N$ Rank-2 Toric Code and its Low-Energy Effective Field Theory
• Rank-2 Toric Code in Two Dimensions
• Effective Fractonic Behavior in a Two-Dimensional Exactly Solvable Spin Liquid
• Physics of Symmetry Protected Topological phases involving Higher Symmetries and their Applications
• Effective Field Theory of Dipolar Braiding Statistics in Two Dimensions
• Theory of oblique topological insulators
• Emergent higher-symmetry protected topological orders in the confined phase of $U(1)$ gauge theory
• 3+1d Boundaries with Gravitational Anomaly of 4+1d Invertible Topological Order for Branch-Independent Bosonic Systems

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