Impact of nonlinear effective interactions on group field theory quantum gravity condensates

We present the numerical analysis of effectively interacting group field theory models in the context of the group field theory quantum gravity condensate analog of the Gross-Pitaevskii equation for real Bose-Einstein condensates including combinatorially local interaction terms. Thus, we go beyond the usually considered construction for free models. More precisely, considering such interactions in a weak regime, we find solutions for which the expectation value of the number operator $N$ is finite, as in the free case. When tuning the interaction to the strongly nonlinear regime, however, we obtain solutions for which $N$ grows and eventually blows up, which is reminiscent of what one observes for real Bose-Einstein condensates, where a strong interaction regime can only be realized at high density. This behavior suggests the breakdown of the Bogoliubov ansatz for quantum gravity condensates and the need for non-Fock representations to describe the system when the condensate constituents are strongly correlated. Furthermore, we study the expectation values of certain geometric operators imported from loop quantum gravity in the free and interacting cases. In particular, computing solutions around the nontrivial minima of the interaction potentials, one finds, already in the weakly interacting case, a nonvanishing condensate population for which the spectra are dominated by the lowest nontrivial configuration of the quantum geometry. This result indicates that the condensate may indeed consist of many smallest building blocks giving rise to an effectively continuous geometry, thus suggesting the interpretation of the condensate phase to correspond to a geometric phase.

Figure 1

Probability density of the free mean field over $p$.

Figure 2

Probability density of the free mean field over $\psi$.

Figure 3

Un-normalized spectrum of the field strength with respect to the eigensolutions ${\sigma }_j(\psi )$ in arbitrary units.

Figure 4

Normalized spectrum of the volume operator with respect to the eigensolutions ${\sigma }_j(\psi )$ in arbitrary units.

Figure 5

Standard deviation of the volume operator over $j$.

Figure 6

Normalized spectrum of the area operator with respect to the eigensolutions ${\sigma }_j(\psi )$ in arbitrary units.

Figure 7

Plot of the effective potential $V_{\mathrm{eff}}[\sigma ]=\frac{\mu }{2}{\sigma }^2+\frac{\kappa }{4}{\sigma }^4$.

Figure 8

Plot of the effective potential $V_{\mathrm{eff}}[\sigma ]=\frac{\mu }{2}{\sigma }^2+\frac{\kappa }{5}{\sigma }^5$.

Figure 9

Probability density of the interacting mean field over $\psi$ for $V_{\mathrm{eff}}[\sigma ]=\frac{\mu }{2}{\sigma }^2+\frac{\kappa }{4}{\sigma }^4$.

Figure 10

Probability density of the interacting mean field over $p$ for $V_{\mathrm{eff}}[\sigma ]=\frac{\mu }{2}{\sigma }^2+\frac{\kappa }{4}{\sigma }^4$.

Figure 11

Normalized spectrum of the volume operator with respect to the interacting mean field ${\sigma }_j(\psi )$ for $\kappa =0.22$ (triangles) compared to the respective free solutions (dots).

Figure 12

Normalized spectrum of the area operator with respect to the interacting mean field ${\sigma }_j(\psi )$ for $\kappa =0.22$ (triangles) compared to the respective free solutions (dots).

Figure 13

Semilog plot of the probability density and potential for solutions $\sigma (p)$ with $\mu =-1.5$, $\kappa =0.01$, $\sigma (1)=12.2474$, and ${\sigma }^{\prime }(1)=\pm 100$ for the potential $V_{\mathrm{eff}}[\sigma ]=\frac{\mu }{2}{\sigma }^2+\frac{\kappa }{4}{\sigma }^4$. Solutions were computed by means of MATLAB’s ODE45 solver which is based on an explicit Runge-Kutta (4,5) formula. Output was generated for $10^5$ points on the interval [0, 1] while making use of highly stringent error tolerances.

Figure 14

Double-log plot of the probability density for solutions $\sigma (\psi )$ for different $\mu$, with the same $\kappa =0.01$, and the same $|{\sigma }^{\prime }(\pi /2)|$ at the respective minima $\sigma (\pi /2)$ for the potential $V_{\mathrm{eff}}[\sigma ]=\frac{\mu }{2}{\sigma }^2+\frac{\kappa }{4}{\sigma }^4$.

Figure 15

Normalized discrete spectrum of the volume operator (in arbitrary units) with respect to the interacting mean field ${\sigma }_{\mu }(\psi )$: Solutions ${\sigma }_{\mu }(\psi )$ were obtained with $\kappa =0.01$, but boundary conditions ${\sigma }^{\prime }(\pi /2)$ differ for each $\mu$ to solve around a nontrivial minimum of the respective potential $V_{\mathrm{eff}}[\sigma ]=\frac{\mu }{2}{\sigma }^2+\frac{\kappa }{4}{\sigma }^4$.

Figure 16

Normalized discrete spectrum of the volume operator (in arbitrary units) with respect to the interacting mean field ${\sigma }_{\mu }(\psi )$: Solutions ${\sigma }_{\mu }(\psi )$ were obtained for $\mu =-1.5$, different $\kappa$, and the same boundary conditions ${\sigma }^{\prime }(\pi /2)$ to solve around the respective nontrivial minima of the potential $V_{\mathrm{eff}}[\sigma ]=\frac{\mu }{2}{\sigma }^2+\frac{\kappa }{4}{\sigma }^4$.

Figure 17

Normalized discrete spectrum of the area operator (in arbitrary units) with respect to the interacting mean field ${\sigma }_{\mu }(\psi )$: Solutions ${\sigma }_{\mu }(\psi )$ were obtained with $\kappa =0.01$, but boundary conditions differ for each $\mu$ to solve around a nontrivial minimum of the respective potential $V_{\mathrm{eff}}[\sigma ]=\frac{\mu }{2}{\sigma }^2+\frac{\kappa }{4}{\sigma }^4$.

Figure 18

Semilog plot of the probability density and potential for solutions $\sigma (p)$ with $\mu =-1.5$, $\kappa =0.01$, $\sigma (1)=5.3132$, and ${\sigma }^{\prime }(1)=\pm 100$ for the potential $V_{\mathrm{eff}}[\sigma ]=\frac{\mu }{2}{\sigma }^2+\frac{\kappa }{5}{\sigma }^5$. Solutions were computed by means of MATLAB’s ODE113 procedure which is a variable order “Adams-Bashforth-Moulton predictor-corrector” solver. Output was generated for $10^5$ points on the interval [0, 1] while making use of highly stringent error tolerances.