Matrix quantum mechanics from qubits

Abstract

We introduce a transverse field Ising model with order N2 spins interacting via a nonlocal quartic interaction. The model has an O(N, ℤ), hyperoctahedral, symmetry. We show that the large N partition function admits a saddle point in which the symmetry is enhanced to O(N). We further demonstrate that this ‘matrix saddle’ correctly computes large N observables at weak and strong coupling. The matrix saddle undergoes a continuous quantum phase transition at intermediate couplings. At the transition the matrix eigenvalue distribution becomes disconnected. The critical excitations are described by large N matrix quantum mechanics. At the critical point, the low energy excitations are waves propagating in an emergent 1 + 1 dimensional spacetime.