Quantum critical phase transition between two topologically-ordered phases in the Ising toric code bilayer

Abstract

We demonstrate that two toric code layers on the square lattice coupled by an Ising interaction display two distinct phases with intrinsic topological order. The second-order quantum phase transition between the weakly-coupled ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ and the strongly-coupled ${\mathbb{Z}}_2$ topological order can be described by the condensation of bosonic quasiparticles from both sides and belongs to the 3d Ising${}^{\ast }$ universality class. This can be shown by an exact duality transformation to the transverse-field Ising model on the square lattice, which builds on the existence of an extensive number of local ${\mathbb{Z}}_2$ conserved parities. These conserved quantities correspond to the product of two adjacent star operators on different layers. Notably, we show that the low-energy effective model derived about the limit of large Ising coupling is given by an effective single-layer toric code in terms of the conserved quantities of the Ising toric code bilayer. The two topological phases are further characterized by the topological entanglement entropy which serves as a non-local order parameter.

arXiv comment6 pages, 3 figures

References in corpus (63)

• Fault-tolerant quantum computation by anyons
• Non-Abelian Anyons and Topological Quantum Computation
• Anyons in an exactly solved model and beyond
• Topological entanglement entropy
• Detecting topological order in a ground state wave function
• Quantum Spin Liquids
• A toolbox for lattice spin models with polar molecules
• Kitaev-Heisenberg Model on Honeycomb Lattice: Possible Exotic Phases in Iridium Oxides $A_2$IrO${}_3$
• $\alpha$-RuCl3: a Spin-Orbit Assisted Mott Insulator on a Honeycomb Lattice
• Relevance of the Heisenberg-Kitaev model for the honeycomb lattice iridates A_2IrO_3
• Neutron tomography of magnetic Majorana fermions in a proximate quantum spin liquid
• Local stabilizer codes in three dimensions without string logical operators
• Fracton Topological Order, Generalized Lattice Gauge Theory and Duality
• Quantum Glassiness
• A New Kind of Topological Quantum Order: A Dimensional Hierarchy of Quasiparticles Built from Stationary Excitations
• Condensate induced transitions between topologically ordered phases
• Excitations in the field-induced quantum spin liquid state of alpha-RuCl3
• Exotic topological order in fractal spin liquids
• Breakdown of a topological phase: Quantum phase transition in a loop gas model with tension
• Topological order in an exactly solvable 3D spin model
• Low-energy effective theory of the toric code model in a parallel field
• Robustness of a perturbed topological phase
• A Family of Non-Abelian Kitaev Models on a Lattice: Topological Confinement and Condensation
• Topological multicritical point in the Toric Code and 3D gauge Higgs Models
• Symmetry and Topological Order
• Autocorrelations and Thermal Fragility of Anyonic Loops in Topologically Quantum Ordered Systems
• Anyon condensation and its applications
• Entanglement and topological entropy of the toric code at finite temperature
• Topological order in a 3D toric code at finite temperature
• On thermalization in Kitaev's 2D model
• Demonstrating anyonic fractional statistics with a six-qubit quantum simulator
• Observation of Four-body Ring-exchange Interactions and Anyonic Fractional Statistics
• Machine Learning ${\mathbb{Z}}_2$ Quantum Spin Liquids with Quasi-particle Statistics
• Self-duality and bound states of the toric code model in a transverse field
• Adiabatic Preparation of Topological Order
• String and Membrane condensation on 3D lattices
• Condensation of achiral simple currents in topological lattice models: a Hamiltonian study of topological symmetry breaking
• A scheme for demonstration of fractional statistics of anyons in an exactly solvable model
• Continuous topological phase transitions between clean quantum Hall states
• Emergent Fermions and Anyons in the Kitaev Model
• Phase diagram of the toric code model in a parallel magnetic field
• Perturbative approach to an exactly solved problem: the Kitaev honeycomb model
• Anyon Condensation and Continuous Topological Phase Transitions in Non-Abelian Fractional Quantum Hall States
• Universal Signatures of Quantum Critical Points from Finite-Size Torus Spectra: A Window into the Operator Content of Higher-Dimensional Conformal Field Theories
• Broken quantum symmetry and confinement phases in planar physics
• Minimum instances of topological matter in an optical plaquette
• Bound states in two-dimensional spin systems near the Ising limit: A quantum finite-lattice study
• Numerical study of a transition between Z2 topologically ordered phases
• Superconducting Quantum Simulator for Topological Order and the Toric Code
• Anisotropic layer construction of anisotropic fracton models
• Bridging Perturbative Expansions with Tensor Networks
• Experimental simulation of anyonic fractional statistics with an NMR quantum information processor
• Experimental implementation of adiabatic passage between different topological orders
• Robustness of a topological phase: Topological color code in parallel magnetic field
• Probing topological order with Rényi entropy
• Bilayer Kitaev models: Phase diagrams and novel phases
• The breaking of quantum double symmetries by defect condensation
• Frustrated topological symmetry breaking: geometrical frustration and anyon condensation
• Persisting topological order via geometric frustration
• Quantum robustness and phase transitions of the 3D Toric Code in a field
• Local convertibility of the ground state of the perturbed Toric code
• Josephson Coupled Moore-Read States
• Spectroscopy of a topological phase

Cited by in corpus (12)

• Vestigial anyon condensation in kagome quantum spin liquids
• Competing topological orders in three dimensions: X-cube versus toric code
• Projective cluster-additive transformation for quantum lattice models
• Anyon condensation and confinement transition in a Kitaev spin liquid bilayer
• Quantum phase transitions in the $K$-layer Ising toric code
• Distinguishing Phases via Non-Markovian Dynamics of Entanglement in Topological Quantum Codes under Parallel Magnetic Field
• Gauge Field Dynamics in Multilayer Kitaev Spin Liquids
• Quantum robustness of the toric code in a parallel field on the honeycomb and triangular lattice
• Emergent Hyper-Magic Manifold in Twisted Kitaev Bilayers
• Dualities between 2+1d fusion surface models from braided fusion categories
• Mixed-State Quantum Spin Liquids and Dynamical Anyon Condensations in Kitaev Lindbladians
• Exactly solvable spin liquids in Kitaev bilayers and moiré superlattices