Topologically correct phase boundaries and transition temperatures for Ising Hamiltonians via self-consistent coarse-grained cluster-lattice models

We derive a cluster mean-field theory for an Ising Hamiltonian using a cluster-lattice Fourier transform with a cluster of size N-c and a coarse-grained (CG) lattice into cells of size N-cell. We explore forms with N-cell >= N-c, including a non-CG (NCG) version with N-cell -> infinity. For N-c = N-cell, the set of static, self-consistent equations relating cluster and CG lattice correlations is analogous to that in dynamical cluster approximation and cellular dynamical mean-field theory used in correlated electron physics. A variational N-c-site cluster grand potential based on N-c = N-cell CG lattice maintains thermodynamic consistency and improves predictions, recovering Monte Carlo and series expansion results upon finite-size scaling; notably, the N-c = 1 CG results already predict well the first-and second-order phase boundary topology and transition temperatures for frustrated lattices. The NCG version is significantly faster computationally than the CG case and more accurate at fixed Nc for ferromagnetism, which is potentially useful for cluster expansion and quantum cluster applications.