(1D) quantum critical systems is of particular interest, because the many-body ground state of a system with SSD almost coincides with that of the uniform system with periodic boundary conditions (PBC). This is due to the nearly exact cancellation of the boundary scattering in the system with SSD [@GendiarKN2009; @HikiharaN2011]. In a class of 1D critical systems which are reducible to free fermions, moreover, the ground-state correspondence can be made exact. In fact, it was proved that the ground state of the spin-$1/2$ XY chain with SSD is identical to that of the uniform chain with PBC [@Katsura2011]. The same holds true for the tight-binding chains [@MaruyamaKH2011] and transverse field Ising chains at criticality [@Katsura2011B]. More generally, the ground-state correspondence between the SSD and uniform systems can be explained in terms of conformal field theory (CFT), where the Virasoro generator $L_{\pm 1} $ identified with the chiral part of the Hamiltonian of the SSD system [@chiral] annihilates the CFT vacuum [@Katsura2011B]. Another interesting aspect of the SSD is that it has a variety of applications; For instance, the nature of the quasi-localized edge excitations originating from the SSD is essential in a ‘grand-canonical’ approach for finite-size systems, where smooth magnetization 