Quantum Bose-Hubbard model with an evolving graph as a toy model for emergent spacetime

We present a toy model for interacting matter and geometry that explores quantum dynamics in a spin system as a precursor to a quantum theory of gravity. The model has no a priori geometric properties; instead, locality is inferred from the more fundamental notion of interaction between the matter degrees of freedom. The interaction terms are themselves quantum degrees of freedom so that the structure of interactions and hence the resulting local and causal structures are dynamical. The system is a Hubbard model where the graph of the interactions is a set of quantum evolving variables. We show entanglement between spatial and matter degrees of freedom. We study numerically the quantum system and analyze its entanglement dynamics. We analyze the asymptotic behavior of the classical model. Finally, we discuss analogues of trapped surfaces and gravitational attraction in this simple model.

Figure 1

Simulation of the system $H_{1/2}$ for $N=4$. The initial state is ${\rho }_1=|{\psi }_1⟩⟨{\psi }_1|$ where $|{\psi }_1⟩=|0000111111⟩$, that is, there are no particles and all edges are present. The parameters for this simulation are $U=\mu =1$, $t=k=.1$. In the figure are plotted the quantities $⟨S_i^{+}{S}_i^{-}⟩$ (red line), $\mathcal{F}(t)$ (blue line), $S(t)$ (black line), $P_{ij}(t)$, $D_i(t)/3$ (green line) as a function of time. Revivals of the expectation value of the link operator coincide with revivals in the fidelity with the initial state. The initial value of the entanglement is $S(0)=0$ because the initial state is separable. Notice that even though the fidelity is $\mathcal{F}(t)\gtrsim 0.85$, the state has a non negligible entanglement.

Figure 2

The initial state of this numerical integration is $|{\psi }_2⟩=|0000111101⟩$, the parameters are in the insulator phase, $U=\mu =1$, $t=k=0.1$. The temporal scale is in units of $\hslash$, on a range of ${10}^4$ seconds. The diagonalization of the full system has been performed by means of Householder reduction. (a) Time evolution of $⟨D_1(t)⟩$, $⟨D_4(t)⟩$. The damping of the oscillations is a sign of thermalization. (b) Expectation values $⟨P_{12}(t)⟩$, $⟨P_{24}(t)⟩$. The latter observable is thermalizing. (c) Von Neumann entropy $s_i(t)$ for the sites $i=1$, 2. We see that the entanglement dynamics is split in two different bands. The two vertices are only distinguished by the initial degree. (d) Entanglement evolution $S(t)$ and overlap with the initial state $\mathcal{F}(t)$. The damping of $\mathcal{F}(t)$ is a clear sign of thermalization. The entanglement $S(t)$ between particles and edges shows the entangling power of the system. (e) Expectation value of the particle operators at two different sites $i=1$, 4. (f) Concurrence $C(t)$ as a function of time of the particles on the site $i=2$ with the edge (2, 4)(blue). Again we notice a damping of oscillations. Instead, the concurrence between the site 1 and the link 5 (red) is identically zero.

Figure 3

The initial state of this numerical integration is $|{\psi }_2⟩=|0000111101⟩$, the parameters are in the superfluid phase, $U=\mu =1$, $t=k=0.1$. The temporal scale is in units of $\hslash$, on a range of ${10}^4$ seconds. The diagonalization of the full system has been performed by means of Householder reduction. (a) Time evolution of $⟨D_1(t)⟩$, $⟨D_4(t)⟩$. The oscillations have constant amplitude and the system does not present signs of thermalization. (b) Expectation values of the link operators $P_{14}$, $P_{24}$. There is no sign of thermalization. The two values belong to two different bands depending on the initial value of the operator. (c) Von Neumann entropy $s_i(t)$ for the sites $i=1$, 2. We see that the entanglement dynamics is split in two different bands. The two vertices are only distinguished by the initial degree and the splitting is more marked than in the insulator case. Compare the result with the higher overlap of the operators $D_{ij}(t)$. (d) Entanglement evolution $S(t)$ and overlap with the initial state $\mathcal{F}(t)$. Again the plots show no signs of thermalization. The behavior of $\mathcal{F}(t)$ implies very long recurrence times. (e) Expectation value of the particle operators at two different sites $i=1$, 4. (f) Time evolution of concurrence $C(t)$. In blue is plotted the concurrence between the vertex $i=2$ and the edge (2, 4) for the superfluid case. Unlike the insulator case, the behavior of $C(t)$ does not show any sign of thermalization. The concurrence between the site 1 and the link 5 is identically zero, as in the insulator case.

Figure 4

The graph ${\mathcal{G}}_4$. The number of vertices is 38 and 72 edges. The number of vertices of ${\mathcal{G}}_n$ is exactly the number $d_n$ of connected labeled graphs on $n$ vertices. The number $d_n$ satisfies the recurrence $n{2}^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)}=\sum _{k}k{d}_k{2}^{\left(\genfrac{}{}{0ex}{}{n-k}{2}\right)}$ [40]. The walker starts from the vertex corresponding to the graph $K_4$. Even when we add particles, the graph ${\mathcal{G}}_n$ remains the support of the dynamics. The vertices of ${\mathcal{G}}_n$ are the possible states of classical evolution. In the quantum evolution, we have a weighted superposition of vertices.

Figure 5

(a) Plot of the logarithm of the average number of particles $N_p$ as a function of the number of steps. The initial conditions were complete graphs with $N$ vertices, $K_N$, with hopping probability $P_h=1$ and interacting probability $P_i=0.1$. (b) In this plot we have the same quantity of (a), $\log (N_p(N))$, plotted as a function of the graph size $N$ at long times (when at equilibrium). (c) Plot of the equilibrium degree distributions for each graph $K_N$. The hopping and interaction probabilities for each simulation are $P_h=1$ and $P_i=0.1$ respectively. The distributions are obtained averaging over 60 simulations. At equilibrium, we obtain Poisson distributions centered on $\frac{N}{2}$, as shown in Fig. (6b), and variance $Q(N)=\frac{\sqrt{N}}{2}$, as shown in Fig. (6a). This agrees with standard results of the theory of random graphs.

Figure 6

(a) Plot of the average values of the equilibrium degree distributions of Fig. (5) (c). The red line is the fit, given $f(N)=\frac{N}{2}$. This agrees perfectly with the various $K_N$. (b) Plot of the variance of the equilibrium degree distributions of Fig. (5) (c). The red line is the fit, giving $Q(N)=\frac{\sqrt{N}}{2}$.