  Quantum Hamiltonians that are fine-tuned to their so-called Rokhsar-Kivelson
(RK) points, first presented in the context of quantum dimer models, are
defined by their representations in preferred bases in which their ground state
wave functions are intimately related to the partition functions of
combinatorial problems of classical statistical physics. We show that all the
known examples of quantum Hamiltonians, when fine-tuned to their RK points,
belong to a larger class of real, symmetric, and irreducible matrices that
admit what we dub a Stochastic Matrix Form (SMF) decomposition. Matrices that
are SMF decomposable are shown to be in one-to-one correspondence with
stochastic classical systems described by a Master equation of the matrix type,
hence their name. It then follows that the equilibrium partition function of
the stochastic classical system partly controls the zero-temperature quantum
phase diagram, while the relaxation rates of the stochastic classical system
coincide with the excitation spectrum of the quantum problem. Given a generic
quantum Hamiltonian construed as an abstract operator defined on some Hilbert
space, we prove that there exists a continuous manifold of bases in which the
representation of the quantum Hamiltonian is SMF decomposable, i.e., there is a
(continuous) manifold of distinct stochastic classical systems related to the
same quantum problem. Finally, we illustrate with three examples of
Hamiltonians fine-tuned to their RK points, the triangular quantum dimer model,
the quantum eight-vertex model, and the quantum three-coloring model on the
honeycomb lattice, how they can be understood within our framework, and how
this allows for immediate generalizations, e.g., by adding non-trivial
interactions to these models.
