QFT with Tensorial and Local Degrees of Freedom: Phase Structure from Functional Renormalization

Abstract

Field theories with combinatorial non-local interactions such as tensor invariants are interesting candidates for describing a phase transition from discrete quantum-gravitational to continuum geometry. In the so-called cyclic-melonic potential approximation of a tensorial field theory on the $r$-dimensional torus it was recently shown using functional renormalization group techniques that no such phase transition to a condensate phase with a tentative continuum geometric interpretation is possible. Here, keeping the same approximation, we show how to overcome this limitation amending the theory by local degrees freedom on $\mathbb{R}^d$. We find that the effective $r-1$ dimensions of the torus part dynamically vanish along the renormalization group flow while the $d$ local dimensions persist up to small momentum scales. Consequently, for $d>2$ one can find a phase structure allowing also for phase transitions.

Figures (5)

• Figure 1: Cyclic-melonic interaction vertices diagrammatically described by bipartite $r$-colourable vertex graphs (green edges, red vertices, $r=4$ in the example) with a distinguished edge colour $c\in\{1,2,...,r\}$.
• Figure 2: Left: Critical value $\bar{\mu}_*=\bar{\mu}_*({d})$ at the Wilson-Fisher type fixed point at sextic ($n=3$) truncation as a function of the dimension ${d}$ for rank $r=1,2,3,4,5$: For any $r>0$ this function is bounded from below, $\bar{\mu}_*({d})>-1$, monotonically increasing up to a neighbourhood of $\bar{\mu}_*(4)=0$ which corresponds to the critical dimension being $d_{\textrm{crit}}=4$. With increasing $r$ the curve becomes continuously steeper. Right: Second eigenvalue $\theta_2=\theta_2(d)$ of the stability matrix for the same cases, decreasing with increasing $r=1,2,3,4,5$.
• Figure 3: Left: Comparing the flow of effective dimension for different values of $\zeta$ in the case ${d}=r=3$ (with $\bar{\kappa}=1$, $\eta_k=0$) using the asymptotic approximation Eq. (\ref{['eq:threshold function asymptotics 1']}) for the threshold function $I_1^{(3,3)}$. Right: Comparing for $\zeta=\frac{1}{2}$ the asymptotic approximation (solid curve) with the full polynomial expression Eq. (\ref{['eq:threshold function simplex']}) (dashed); this shows that details of approximation give the same qualitative results, with only minor differences in numerics.
• Figure 4: The flow of the factors $R_l$ in the case ${d}=r=3$ (with $\bar{\kappa}=1$, $\eta_k=0$) for $l=1,2,3$, comparing for $\zeta=\frac{1}{2}$ the full polynomial expression Eq. (\ref{['eq:threshold function simplex']}) (solid curves) with the asymptotic approximation Eq. (\ref{['eq:threshold function asymptotics 1']}) (dashed), and for the latter also with the case $\zeta=1$ (dotted). In any case, $R_l$ flow from one in the IR to $1/r^{l-1}$ in the UV.
• Figure 5: Left and center panel: Illustration of the flow of the dimensionful potential $U_k(\rho)$ at rank $r=3$ with $d=3$ and $\zeta=1$ in the $n=3$ truncation between $k=100$ and $k=1$ with $\bar{\kappa}=1$ in the LPA for the ensuing exemplary initial conditions at $\Lambda=100$ in the proximity of the UV non-Gaussian fixed point in this truncation: $\mu_\Lambda=-0.2041 \Lambda^2$ (left panel) and $\mu_\Lambda=-0.2153 \Lambda^2$ (center panel) each with $\lambda_{2,\Lambda}=0.0444\Lambda^{-1}$ and $\lambda_{3,\Lambda}(\Lambda)=0.0008\Lambda^{-4}$. The main qualitative result is that the first potential displays symmetry restoration towards the IR whereas for the second the global symmetry remains broken. Right panel: Flow of the modulus of $\mu_k$ in the $n=3$ truncation for: (I) the system of non-autonomous $\beta$-functions Eq. (\ref{['eq:betaCMcompact']}) with $\bar{\kappa}=1$ (dashed lines) and (II) the set of autonomous $\beta$-functions in the large-${\tilde{k}}$ limit, Sec. \ref{['sec:UV/large volume']} (continuous lines). Initial conditions are the same as in the left panel. The sign change of $\mu$ or the absence of it indicates the presence of a potential with broken symmetry or one without.