A systematic search for conformal field theories in very small spaces

Groundstates of 1+1d conformal field theories (CFTs) satisfy a local entropic condition called the vector fixed point equation. This condition is surprisingly well satisfied by groundstates of quantum critical lattice models even at small system sizes. We perform a search in the space of states of very small systems (four qubits and four qutrits) and examine the states that satisfy this condition. By reconstructing a local Hamiltonian from each state, we are able to identify many of these solutions with known CFTs; others are gapped fixed points, or involve large relevant perturbations, and others are CFTs we have not yet identified. These ideas are also useful for identifying continuous quantum phase transitions in a given family of Hamiltonians, and for identifying the nature of the critical theory in small systems.