Foundations of Emergent Geometry: A Minimal Solvable Model and Its Phenomenology

This work presents an exhaustive and self-contained investigation of the hypothesis that spacetime geometry is an emergent phenomenon, derived from the correlation structure of an underlying quantum system. We construct a minimal model, a "spacetime atom," composed of N = 3 qubits in a Hilbert space H = (C^2)^3. The pre-geometric dynamics is governed by a pairwise interaction Heisenberg Hamiltonian, H = ∑i<j Jij σi·σj, whose coupling coefficients Jij are the fundamental parameters of the theory. A complete analytical diagonalization of the 8 × 8 Hamiltonian matrix is performed, paying special attention to the rigorous treatment of ground state degeneracy through the introduction of an infinitesimal symmetry-breaking field. From the uniquely defined ground state, |Ψ0>, bipartite correlations are quantified through Quantum Mutual Information, I(i : j). A generalized informational metric is postulated, d(i, j) = Lp[− ln(I(i : j)/Imax)]^1/p, where distance emerges as a function of correlations. We analytically demonstrate that symmetry in couplings (Jij = J) results in a discrete and flat emergent geometry (an equilateral triangle). The introduction of asymmetries (Jij ≠ Jkl) induces a non-homogeneous entanglement structure, giving rise to an intrinsically curved geometry, quantified through the angular deficit κ. The "phase space of geometries" is presented, mapping curvature κ as a function of the entanglement asymmetry parameter. Innovatively, we investigate the response of geometry to local perturbations (analogs to "matter"), defining and calculating a new observable: the geometric susceptibility tensor, χij,k = −∂dij/∂hk, which acts as an effective field equation in our model. This work not only provides an explicit and solvable demonstration of geometry emergence, but also establishes a framework for quantifying its dynamic response.