  We study free fermion systems with the sine-square deformation (SSD), in
which the energy scale of local Hamiltonians is modified according to the
scaling function f(x)=sin^2[\pi(x-1/2)/L], where x is the position of the local
Hamiltonian and L is the length of the system in the x direction. It has been
revealed that when applied to one-dimensional critical systems the SSD realizes
the translationally-invariant ground state which is the same as that of the
uniform periodic system. In this paper, we propose a simple theory to explain
how the SSD maintains the translational invariance in the ground-state wave
function. In particular, for a certain one-dimensional system with SSD, it is
shown that the ground state is exactly identical with the Fermi sea of the
uniform periodic chain. We also apply the SSD to two-dimensional systems and
show that the SSD is able to suppress the boundary modulations from the open
edges extremely well, demonstrating that the SSD works in any dimensions and in
any directions.
