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Quantum group spin nets: refinement limit and relation to spin foams

Bianca Dittrich, Mercedes Martin-Benito, Sebastian Steinhaus

TL;DR

This work uses tensor-network renormalization to study spin nets based on the quantum group SU(2)_k as a tractable analogue of spin foams, exposing a rich fixed-point structure beyond the degenerate and BF phases. By adopting Reisenberger-inspired fixed-point intertwiners as initial data, the authors demonstrate non-tuned flows to nontrivial fixed points and reveal extended phase regions associated with each fixed point. The coarse-graining flow naturally interprets as an effective coupling between two spin-foam vertices, connecting the 2D spin-net dynamics to melonic 3D spin-foam amplitudes. The findings suggest a robust pathway to continuum-like behavior in spin-foam models and point to a range of future explorations, including phase transitions and the extension to larger structure groups.

Abstract

So far spin foam models are hardly understood beyond a few of their basic building blocks. To make progress on this question, we define analogue spin foam models, so called spin nets, for quantum groups $\text{SU}(2)_k$ and examine their effective continuum dynamics via tensor network renormalization. In the refinement limit of this coarse graining procedure, we find a vast non-trivial fixed point structure beyond the degenerate and the $BF$ phase. In comparison to previous work, we use fixed point intertwiners, inspired by Reisenberger's construction principle [1] and the recent work [2], as the initial parametrization. In this new parametrization fine tuning is not required in order to flow to these new fixed points. Encouragingly, each fixed point has an associated extended phase, which allows for the study of phase transitions in the future. Finally we also present an interpretation of spin nets in terms of melonic spin foams. The coarse graining flow of spin nets can thus be interpreted as describing the effective coupling between two spin foam vertices or space time atoms.

Quantum group spin nets: refinement limit and relation to spin foams

TL;DR

This work uses tensor-network renormalization to study spin nets based on the quantum group SU(2)_k as a tractable analogue of spin foams, exposing a rich fixed-point structure beyond the degenerate and BF phases. By adopting Reisenberger-inspired fixed-point intertwiners as initial data, the authors demonstrate non-tuned flows to nontrivial fixed points and reveal extended phase regions associated with each fixed point. The coarse-graining flow naturally interprets as an effective coupling between two spin-foam vertices, connecting the 2D spin-net dynamics to melonic 3D spin-foam amplitudes. The findings suggest a robust pathway to continuum-like behavior in spin-foam models and point to a range of future explorations, including phase transitions and the extension to larger structure groups.

Abstract

So far spin foam models are hardly understood beyond a few of their basic building blocks. To make progress on this question, we define analogue spin foam models, so called spin nets, for quantum groups and examine their effective continuum dynamics via tensor network renormalization. In the refinement limit of this coarse graining procedure, we find a vast non-trivial fixed point structure beyond the degenerate and the phase. In comparison to previous work, we use fixed point intertwiners, inspired by Reisenberger's construction principle [1] and the recent work [2], as the initial parametrization. In this new parametrization fine tuning is not required in order to flow to these new fixed points. Encouragingly, each fixed point has an associated extended phase, which allows for the study of phase transitions in the future. Finally we also present an interpretation of spin nets in terms of melonic spin foams. The coarse graining flow of spin nets can thus be interpreted as describing the effective coupling between two spin foam vertices or space time atoms.

Paper Structure

This paper contains 27 sections, 82 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Reisenberger's principle and intertwiner models
  3. Quantum group $\text{SU}(2)_k$
  4. Basic ingredients
  5. Diagrammatic Calculus
  6. Structure of quantum group spin net models
  7. Haar projector
  8. Vertex weights
  9. The coarse graining algorithm
  10. Symmetry preserving algorithm
  11. Derivation of the flow equation
  12. Geometric interpretation of the flow equation
  13. Fixed point intertwiners as initial data and coarse graining results
  14. Description of the fixed point intertwiners
  15. Only $j=0$ and $j_{max}$ excited
  16. ...and 12 more sections

Figures (12)

  • Figure 1: Vertex in the square lattice. Edges are ordered anticlockwise and have fixed orientation: edges 1 and 2 are outgoing, and edges 3 and 4 are incoming.
  • Figure 2: Rewriting of the four--valent tensor network as a three--valent tensor network.
  • Figure 3: The contraction of the three--valent tensor network yields a coarser (rotated) square lattice.
  • Figure 4: Splitting of the tensor $t(\{j\},\{m\},\{n\})$ according with the two different recoupling schemes, as expressed in equations \ref{['splitting1']}-\ref{['splitting2']}.
  • Figure 5: The new tensor $t(\{j\},\{m\},\{j'\},\{n\}, \{i\})_{1,2,3,4}$ is the result of contracting $(S_1)^{j_3,j'_3}_{m_3,n_3}(\{I\}_{\{d,c\}},i_3)$, $(S_2)^{j_1,j'_1}_{m_1,n_1}(\{I\}_{\{b,a\}},i_1)$, $(S_3)^{j_4,j'_4}_{m_4,n_4}(\{I\}_{\{a,d\}},i_4)$, and $(S_4)^{j_2,j'_2}_{m_2,n_2}(\{I\}_{\{b,c\}},i_2)$, together with the factors $(-1)^{2j_b}q^{-n_b} \; (-1)^{2j_d}q^{-n_d}$ associated to horizontal edges.
  • ...and 7 more figures