The contractor renormalization group (CORE) method, a new approach to solving Hamiltonian lattice systems, is introduced. The method combines contraction and variational techniques with the real-space renormalization group approach. It applies to lattice systems of infinite extent and is ideal for studying phase structure and critical phenomena. The CORE approximation is systematically improvable and can treat systems with dynamical fermions. The method is tested using the (1+1)-dimensional Ising model.
We show that artificial neural networks (ANNs) can, to high accuracy, determine the topological invariant of a disordered system given its two-dimensional real-space Hamiltonian. Furthermore, we describe a ``renormalization-group'' (RG) network, an ANN which converts a Hamiltonian on a large lattice to another on a small lattice while preserving the invariant. By iteratively applying the RG network to a ``base'' network that computes the Chern number of a small lattice of set size, we are able to process larger lattices without retraining the system. We therefore show that it is possible to compute real-space topological invariants for systems larger than those on which the network was trained. This opens the door for computation times significantly faster and more scalable than previous methods.
We develop a novel real-space renormalization group (RG) scheme which accurately estimates correlation length exponent
ν near criticality of higher-dimensional quantum Ising and Potts models in a transverse field. Our method is remarkably simple (often analytical), grouping only a few spins into a block spin so that renormalized Hamiltonian has a closed form. A previous difficulty of spatial anisotropy and unwanted terms is avoided by incorporating rotational invariance and internal
Zq symmetries of the Hamiltonian. By applying this scheme to the (2+1)-dim Ising model on a triangular lattice and solving an analytical RG equation, we obtain
ν≈0.6300. This value is within statistical errors of the current best Monte-Carlo result, 25th-order high-temperature series expansions,
ϕ4-theory estimation which considers up to seven-loop corrections and experiments performed in low-Earth orbits. We also apply the scheme to higher-dimensional Potts models for which ordinary Monte-Carlo methods are not effective due to strong hysteresis and suppression of quantum fluctuation in a weak first-order phase transition.