Grand canonical finite-size numerical approaches: A route to measuring bulk properties in an applied field

We exploit a prescription to observe directly the physical properties of the thermodynamic limit in a continuously applied field in one-dimensional quantum finite lattice systems. By systematically scaling down the energy of the Hamiltonian of the open system from center toward both ends, one could adopt the edge sites with a negligibly small energy scale as the grand canonical small particle bath, and equilibrium states with noninteger arbitrary conserved numbers, e. g., electron numbers or s(z), are realized in the main part of the system. This will enable the evaluation of response functions under a continuously varying external field in a small lattice without any fine-tuning or scaling of parameters while keeping the standard numerical accuracy. Demonstrations are given on quantum spin systems and on a Hubbard model by the density-matrix renormalization group.