The c-d conjecture

We conjecture a relation between the local dimension d of a local nearest-neighbor critical Hamiltonian in one spatial dimension and the maximum central charge, cₘₐₓ, that it can yield. Specifically, we propose that cₘₐₓ ≤ d-1, establishing a link between the short-distance lattice realization of a model and its emerging long-distance entanglement properties. This inequality can be viewed as a general form of a c-theorem establishing the reduction of effective degrees of freedom between the UV lattice and the IR conformal field theory. We support this conjecture with numerous examples.