Weak Finite—Size Dependence of Velocity and Strong Phase Dependence of Central Charge

We study the finite-size scaling behavior of velocity and central charge for different coupling constants and different phases in -dimensional lattice model in very short chains. Using XXZ spin 1/2 chains with 15 or fewer sites, we demonstrate the weak finite-size dependence of spinon velocity for any magnitude of coupling strength and the strong phase dependence of central charge. This behavior of velocity and central charge in different coupling constants and different phases gives a method to determine phase transitions of -dimensional models. This method is simple and efficient by utilizing only the ground state energy of very short finite-size chains. It is also general and powerful for various one-dimensional lattice models and it uncovers even the weakest Berezinski?Kosterlitz?Thouless phase transitions.