Interacting thermofield doubles and critical behavior in random regular graphs

We discuss numerically the nonperturbative effects in exponential random graphs which are analogue of eigenvalue instantons in matrix models. The phase structure of exponential random graphs with chemical potential for $C_4$ ${\mu }_4$ and degree preserving constraint is clarified. The first order phase transition at critical value of chemical potential for $C_4$ ${\mu }_4^{\mathrm{RRG}}$ into bipartite phase with a formation of fixed number of bipartite clusters is found for ensemble of random regular graphs (RRG). We consider the similar phase transition in mean field version of combinatorial quantum gravity based of the Ollivier graph curvature for RRG supplemented with hard-core constraint and show that a order of a phase transition at ${\mu }_4^{\mathrm{CRRG}}$ and the structure of emerging phase depend on a vertex degree $d$ in RRG. For $d=3$ the bipartite closed ribbon emerges at ${\mu }_4>{\mu }_4^{\mathrm{CRRG}}$ while for $d>3$ the ensemble of isolated or weakly interacting hypercubes supplemented with the bipartite closed ribbon gets emerged at the first order phase transition with a clear-cut hysteresis. If the additional connectedness condition is imposed the phase at ${\mu }_4>{\mu }_4^{\mathrm{CRRG}}$ gets identified as the closed chain of weakly coupled hypercubes. Since the ground state of isolated hypercube is the thermofield double we suggest that the dual holographic picture involves multiboundary wormholes. Treating RRG as a model of a Hilbert space for a interacting many-body system we discuss the patterns of the Hilbert space fragmentation at the phase transition. We also briefly comment on a possible relation of the found phase transition to the problem of holographic interpretation of a partial deconfinement transition in the gauge theories.

Figure 1

(a) The dependence of number of $C_4$ on the chemical potential ${\mu }_4$ for RRG of different sizes and $d=8$. (b) Ground state and its adjacency matrix at critical ${\mu }_4$.

Figure 2

The dependence of the spectral density of adjacency matrix on the chemical potential ${\mu }_4$ for RRG of $N=256$, $d=8$.

Figure 3

(a) The number of $C_4$ in dependence on the parameter ${\mu }_4$ in CRRG of $d=4$ and different size. (b) The portion of the nodes in the bipartite closed ribbon in dependence on the parameter ${\mu }_4$.

Figure 4

Typical subgraph configurations in the clustered phase involving hypercubes and composite bipartite strings.

Figure 5

Typical adjacency matrices in the clustered phase for RRG of $N=256$ and different values of $d$.

Figure 6

Hysteresis in dependence of the number of $C_4$ for RRG of $N=256$ and different degrees.

Figure 7

Ground state of CRRG with connectedness constraint and different degree $d$ for RRG of $N=128$.