Coarse graining flow of spin foam intertwiners

Abstract

Simplicity constraints play a crucial role in the construction of spin foam models, yet their effective behaviour on larger scales is scarcely explored. In this article we introduce intertwiner and spin net models for the quantum group $\text{SU}(2{)}_k\times \text{SU}(2{)}_k$, which implement the simplicity constraints analogous to 4D Euclidean spin foam models, namely the Barrett-Crane (BC) and the Engle-Pereira-Rovelli-Livine/Freidel-Krasnov (EPRL/FK) model. These models are numerically coarse grained via tensor network renormalization, allowing us to trace the flow of simplicity constraints to larger scales. In order to perform these simulations we have substantially adapted tensor network algorithms, which we discuss in detail. The BC and the EPRL/FK model behave very differently under coarse graining: While the unique BC intertwiner model is a fixed point and therefore constitutes a 2D topological phase, BC spin net models flow away from the initial simplicity constraints and converge to several different topological phases. Most of these phases correspond to decoupling spin foam vertices, however we find also a new phase in which this is not the case, and in which a non-trivial version of the simplicity constraints holds. The coarse graining flow of the BC spin net models indicates furthermore that the phase transitions are not of second order. The EPRL/FK model by contrast reveals a far more intricate and complex dynamics. We observe an immediate flow away from the original simplicity constraints, however, with the truncation employed here, the models generically do not converge to a fixed point. The results show that the imposition of simplicity constraints can indeed lead to interesting and complex dynamics. Thus we will need to further develop coarse graining tools to efficiently study the large scale behaviour of spin foam models, in particular for the EPRL/FK model.

33 + 8 pages, 10 figures, 2 tables; v2: 2 plots added visualizing results, references updated. Matching published version

References in corpus (69)

• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• A class of quantum many-body states that can be efficiently simulated
• The Spin Foam Approach to Quantum Gravity
• Tensor renormalization group approach to 2D classical lattice models
• LQG vertex with finite Immirzi parameter
• Relativistic spin networks and quantum gravity
• A new spinfoam vertex for quantum gravity
• ``Sum over Surfaces'' form of Loop Quantum Gravity
• Tensor Network Renormalization
• A New Spin Foam Model for 4d Gravity
• Group Field Theory: An overview
• A Lorentzian Signature Model for Quantum General Relativity
• Flipped spinfoam vertex and loop gravity
• Linking Topological Quantum Field Theory and Nonperturbative Quantum Gravity
• Asymptotic analysis of the EPRL four-simplex amplitude
• On the semiclassical limit of 4d spin foam models
• The complete LQG propagator: I. Difficulties with the Barrett-Crane vertex
• Improved and Perfect Actions in Discrete Gravity
• Spin-Foams for All Loop Quantum Gravity
• Coarse graining methods for spin net and spin foam models
• Quantum deformation of quantum gravity
• Decorated tensor network renormalization for lattice gauge theories and spin foam models
• Numerical evidence for a phase transition in 4d spin foam quantum gravity
• 4-dimensional Spin-foam Model with Quantum Lorentz Group
• Quantum deformation of two four-dimensional spin foam models
• The projector on physical states in loop quantum gravity
• (Broken) Gauge Symmetries and Constraints in Regge Calculus
• Asymptotics of 6j and 10j symbols
• From the discrete to the continuous - towards a cylindrically consistent dynamics
• Perfect discretization of reparametrization invariant path integrals
• Flowing in Group Field Theory Space: a Review
• Self-Energy of the Lorentzian EPRL-FK Spin Foam Model of Quantum Gravity
• SL(2,C) Chern-Simons Theory, a non-Planar Graph Operator, and 4D Loop Quantum Gravity with a Cosmological Constant: Semiclassical Geometry
• Asymptotics of 10j symbols
• Holonomy Spin Foam Models: Definition and Coarse Graining
• Cosmological Deformation of Lorentzian Spin Foam Models
• Self-energy and vertex radiative corrections in LQG
• On relativistic spin network vertices
• From covariant to canonical formulations of discrete gravity
• Face amplitude of spinfoam quantum gravity
• The geometry of quantum spin networks
• Path integral measure and triangulation independence in discrete gravity
• Towards a phase diagram for spin foams
• Time evolution as refining, coarse graining and entangling
• Investigation of the Spinfoam Path integral with Quantum Cuboid Intertwiners
• Operator Spin Foam Models
• Coarse graining free theories with gauge symmetries: the linearized case
• Observables in Loop Quantum Gravity with a cosmological constant
• Coarse graining of spin net models: dynamics of intertwiners
• Spin Foam Models of Riemannian Quantum Gravity
• Quantum group spin nets: refinement limit and relation to spin foams
• Discretization independence implies non-locality in 4D discrete quantum gravity
• Pachner moves in a 4d Riemannian holomorphic Spin Foam model
• Dirac's discrete hypersurface deformation algebras
• Linking covariant and canonical LQG II: Spin foam projector
• Bubble divergences and gauge symmetries in spin foams
• Spin Foam Quantization and Anomalies
• Quantum hyperbolic geometry in loop quantum gravity with cosmological constant
• Positivity of Spin Foam Amplitudes
• The EPRL intertwiners and corrected partition function
• On knottings in the physical Hilbert space of LQG as given by the EPRL model
• Feynman diagrammatic approach to spin foams
• Bulk amplitude and degree of divergence in 4d spin foams
• Topological lattice field theories from intertwiner dynamics
• q-Deformed spin foam models of quantum gravity
• On background-independent renormalization of spin foam models
• Coupled intertwiner dynamics: A toy model for coupling matter to spin foam models
• A simpler way of imposing simplicity constraints
• A look at area Regge calculus

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