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Coarse graining of spin net models: dynamics of intertwiners

Bianca Dittrich, Mercedes Martín-Benito, Erik Schnetter

TL;DR

This work studies continuum limits of spin foam-inspired quantum gravity by analyzing simplified spin nets with a finite group ($S_3$) using tensor network renormalization. It identifies three robust phases (disordered, $S_3$-ordered, and $\mathbb{Z}_2$-ordered) and reveals a nonstandard, triangulation-invariant fixed point outside the initial model space, along with a Barrett-Crane–analogue fixed point near a phase transition. The analysis shows that embedding maps and the intertwiner structure, governed by simplicity constraints, are the key dynamical variables controlling coarse graining; these provide guidance for extending to higher dimensions and to full spin-foam models, including quantum-group generalizations. Overall, the results demonstrate a viable coarse-graining framework for spin foams and shed light on how continuum-like behavior and diffeomorphism-inspired symmetry may emerge from discrete quantum geometries.

Abstract

Spin foams are models of quantum gravity and therefore quantum space time. A key open issue is to determine the possible continuum phases of these models. Progress on this issue has been prohibited by the complexity of the full four--dimensional models. We consider here simplified analogue models, so called spin nets, that retain the main dynamical ingredient of spin foams, the simplicity constraints. For a certain class of these spin net models we determine the phase diagram and therefore the continuum phases via a coarse graining procedure based on tensor network renormalization. This procedure will also reveal an unexpected fixed point, which turns out to define a new triangulation invariant vertex model.

Coarse graining of spin net models: dynamics of intertwiners

TL;DR

This work studies continuum limits of spin foam-inspired quantum gravity by analyzing simplified spin nets with a finite group () using tensor network renormalization. It identifies three robust phases (disordered, -ordered, and -ordered) and reveals a nonstandard, triangulation-invariant fixed point outside the initial model space, along with a Barrett-Crane–analogue fixed point near a phase transition. The analysis shows that embedding maps and the intertwiner structure, governed by simplicity constraints, are the key dynamical variables controlling coarse graining; these provide guidance for extending to higher dimensions and to full spin-foam models, including quantum-group generalizations. Overall, the results demonstrate a viable coarse-graining framework for spin foams and shed light on how continuum-like behavior and diffeomorphism-inspired symmetry may emerge from discrete quantum geometries.

Abstract

Spin foams are models of quantum gravity and therefore quantum space time. A key open issue is to determine the possible continuum phases of these models. Progress on this issue has been prohibited by the complexity of the full four--dimensional models. We consider here simplified analogue models, so called spin nets, that retain the main dynamical ingredient of spin foams, the simplicity constraints. For a certain class of these spin net models we determine the phase diagram and therefore the continuum phases via a coarse graining procedure based on tensor network renormalization. This procedure will also reveal an unexpected fixed point, which turns out to define a new triangulation invariant vertex model.

Paper Structure

This paper contains 17 sections, 55 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Spin net models as analogues of spin foams
  3. Spin foam models in holonomy representation
  4. Spin nets in holonomy representation
  5. Spin nets in the spin representation
  6. The coarse graining algorithms
  7. Contraction schemes
  8. Symmetry preserving algorithm
  9. Results for the $S_3$ spin net model
  10. The model
  11. Phase space diagram and fixed points of the renormalization flow
  12. A new fixed point
  13. Embedding maps and simplicity constraints
  14. Discussion
  15. Clebsch--Gordan coefficients and recoupling symbols
  16. ...and 2 more sections

Figures (17)

  • Figure 1: Graphical representation of the four-valent tensor $\tilde{C}_v( \{\rho_e,a_e,b_e\}_{e=1,2}, \{\rho_e^*,a_e,b_e\}_{e=3,4})$.
  • Figure 2: Rewriting of the four--valent tensor network as a three--valent tensor network, following the splitting explained in Figure \ref{['fig:split-vertices']}.
  • Figure 3: The tensor $\tilde{C}_v( \{\rho_e,a_e,b_e\}_{e=1,2}, \{\rho_e^*,a_e,b_e\}_{e=3,4})$ located at odd vertices (red circles) is regarded as a matrix $(M_1)_{A=\{\rho_e,a_e,b_e\}_{e=1,2},B=\{\rho_e^*,a_e,b_e\}_{e=3,4}}$. This matrix is split into a product of three--valent tensors, $(M_1)_{A,B}=\sum_{m=1}^\chi(S_1)_{A,m} (S_2)_{B,m}$, by performing an SVD and by truncating the sum to the $\chi$ largest singular values. For even vertices (blue squares) we proceed in the same way, now regarding the tensor as a matrix $(M_2)_{A=\{\rho_e,a_e,b_e\}_{e=2,3},B=\{\rho_e^*,a_e,b_e,\}_{e=4,1}}$, and doing the splitting $(M_2)_{A,B}=\sum_{m=1}^\chi(S_3)_{A,m} (S_4)_{B,m}$.
  • Figure 4: The contraction of four three--valent tensors yields four--valent tensors on a (rotated) square lattice.
  • Figure 5: On the left we consider a 'small' cylindrical space time built from two tensors: $\sum_{a,b}\tilde{C}(a,1,b,4)\tilde{C}(b,2,a,3)=: (M)_ {A=\{1,2\},B=\{3,4\}}$. From its SVD, $(M)_ {AB} =\sum_{i=1}^\chi V_{Ai}\lambda_i (U^\dagger)_{iB}$, we extract the embedding map $(U^\dagger)_{iB}$, which is graphically represented on the right.
  • ...and 12 more figures