Quantum Monte Carlo simulation with a black hole

We perform quantum Monte Carlo simulations in the background of a classical black hole. The lattice discretized path integral is numerically calculated in the Schwarzschild metric and in its approximated metric. We study spontaneous symmetry breaking of a real scalar field theory. We observe inhomogeneous symmetry breaking induced by an inhomogeneous gravitational field.

Figure 1

Geometry of the simplified coordinate (6).

Figure 2

Two-point function $G(z)$ in the simplified coordinate (6). The data with $ϵ=0.1a$ are shown.

Figure 3

Condensate fraction $C(r)$ in the simplified coordinate (6).

Figure 4

Geometry of the Schwarzschild coordinate (10).

Figure 5

Two-point function $G(\phi )$ in the Schwarzschild coordinate (10). The data with $ϵ=0.1a$ and at $\theta =\pi /2$ are shown.

Figure 6

Condensate fraction $C(r)$ in the Schwarzschild coordinate (10). The data at $\theta =\pi /2$ are shown.

Figure 7

Classical solution, the tree-level results, and the full simulation result of the condensate fraction $C(r)$. The data with $ϵ=0.1a$ and at $\theta =\pi /2$ are shown.