A new Hamiltonian for the Topological BF phase with spinor networks

TL;DR

This work addresses the challenge of formulating a concrete lattice description of the SU(2) BF topological phase and extracting its dynamics. It introduces a novel scalar Hamiltonian built from spinor networks and quantizes it with Schwinger bosons, yielding a fundamental $1/2$-spin recursion on the $6j$-symbol while preserving topological flatness via the constraint $g_p=\mathbb{1}$. The paper develops multiple representations of the topological equation, including spin-network, coherent-spinor, and generating-function formalisms, and shows that the ground states encode flat, Euclidean geometry through a quantum Euclidean interpretation. It further demonstrates factorization properties of physical states on cycles, connects to Wheeler–DeWitt-type equations, and generalizes the construction to arbitrary 3-valent lattices and other compact groups, highlighting the approach’s utility for topological order and background-independent quantization.

Abstract

We describe fundamental equations which define the topological ground states in the lattice realization of the SU(2) BF phase. We introduce a new scalar Hamiltonian, based on recent works in quantum gravity and topological models, which is different from the plaquette operator. Its gauge-theoretical content at the classical level is formulated in terms of spinors. The quantization is performed with Schwinger's bosonic operators on the links of the lattice. In the spin network basis, the quantum Hamiltonian yields a difference equation based on the spin 1/2. In the simplest case, it is identified as a recursion on Wigner 6j-symbols. We also study it in different coherent states representations, and compare with other equations which capture some aspects of this topological phase.

Figures (5)

• Figure 1: The tetrahedral graph we consider throughout the paper. The orientations are the same as those of the graphical representation of the 6j-symbol varshalovich-book. The three fat lines form the cycle (126) we will consider in order to explain the action of the new Hamiltonian.
• Figure 2: The 3-valent nodes represent 3jm-symbols, where legs carry the spins. A magnetic index is summed when there is a link joining two nodes. The action of $\widehat{E}_{21}$ is a grasping between $e_1$ and $e_2$, denoted by a dashed line which carries the spin 1/2. The final result is proportional to the 3jm-symbol with the spins $k_1=j_1-\frac{1}{2}, k_2=j_2+\frac{1}{2}$.
• Figure 3: A pictorial representation of \ref{['recholgen']}. The character $\chi_j$ along the closed loop acts on the left. On the right we have depicted the situation after re-coupling. A specific 6j-symbol is extracted on each node, and one has to sum over the colorings $k_1,\dotsc,k_n$. The dashed lines correspond to the dual 2d triangulation to the plaquette if we think of the latter as embedded in flat 3-space. The vertex $s$ of the 2d triangulation is then dual to the plaquette.
• Figure 4: Here we have displayed the geometric interpretation of the character operator on the plaquette as a tent move. The vertex $s$ is evolved to a new vertex $s'$, with an edge of length $j+\frac{1}{2}$, the tent pole. Between the initial and the final triangulations we have a piece of 3d triangulation. The character operator then generates the evaluation of the Ponzano-Regge amplitude on this triangulation.
• Figure 5: A graphical representation of the action of the new Hamiltonian. The basic idea is that the holonomy around a closed loop in the topological sector only depends on its homotopy type, so that we can deform the grasping on the left to that on the right, picking up this way some holonomy which must be trivial.

Theorems & Definitions (8)

• Lemma 1
• Theorem 1
• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6