Phase Transition in Dually Weighted Colored Tensor Models

TL;DR

The paper tackles the challenge of obtaining continuum limits for higher-dimensional dynamical triangulations by introducing dually weighted colored tensor models in which the propagator assigns weights to faces (bones) of the triangulation. It derives an exact leading-order solution in $1/N$ via melonic Schwinger-Dyson equations, and shows that a DT-inspired model with a weight parameter $\beta$ undergoes a third-order phase transition with a $\gamma$-exponent that depends on $\beta$. The results provide a controlled analytic route to nontrivial continuum behavior in higher dimensions, with potential links to branched polymers and hints about nonlocal measure effects in DT. The framework also offers a platform for exploring finite-$\kappa_{D-2}$ behavior and double scaling beyond the current strong-coupling limit.

Abstract

Tensor models are a generalization of matrix models (their graphs being dual to higher-dimensional triangulations) and, in their colored version, admit a 1/N expansion and a continuum limit. We introduce a new class of colored tensor models with a modified propagator which allows us to associate weight factors to the faces of the graphs, i.e. to the bones (or hinges) of the triangulation, where curvature is concentrated. They correspond to dynamical triangulations in three and higher dimensions with generalized amplitudes. We solve analytically the leading order in 1/N of the most general model in arbitrary dimensions. We then show that a particular model, corresponding to dynamical triangulations with a non-trivial measure factor, undergoes a third-order phase transition in the continuum characterized by a jump in the susceptibility exponent.

Figures (4)

• Figure 1: The index structure of a $D=3$ vertex, in a "coarse grained" view, showing only the tensor color, and in a detailed view, showing the strands (indices should be read clockwise).
• Figure 2: The self energy $\Sigma$ at leading order in terms of the connected two-point function $G_2$. The labels $0,1,\dots D$ denote the colors of various lines.
• Figure 3: The curve $g(\beta, U)$, for $D=3$ and for $\beta=0$(a) and $\beta=0,1,2$(b), together with the critical curve $z\bigl(g_b(U),U \bigr)=1/4$.
• Figure 4: A schematic representation of the phase transition and the critical line $g(\beta,U_c(\beta))$ (in bold), intersecting the line $g(\beta_c,U)$.